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The Great Pyramid
...

Part 2
The Square Root Two Cubits and associated dynamics.

Be sure to review my previous post on the Square Root Two cubits,
for preceding processes and definitions.
Important introductory info there,
on the pure square root cubits:
20.61923374
20.62394778~
and how the relate to:
Ancient Square Root Two Whip

In this post I will try to blend old material in to offer some color,
to the sometimes tedious nature of defining cubits.
A couple of my oldest designs are resurrected for this post.

Image 1: is a direct follow up from the previous post on the Square Root Two cubits,
and Ancient Square Root Two.

Image 1: 
Pyramid Height Dynamics reveal ancient Egyptian calendar count dynamics,
for the Mars and Jupiter synod.

The two pure square root two cubits, 
are defined together,
by equaling -- square root 8 -- the "electron spin" tetrahedral angle tangent.

Cubit 20.61923374~ extrapolates from the 243 day Venus Rotational Day value: 243

Cubit 20.62394778~ extrapolates from the Khafre pyramid sacred seked: 5.25

Ancient Calendar Count,
Mars and Jupiter synod: 816.48 days

81648 =  100  Mars -- Jupiter synods {calendar count} =  226.8 x 360

81648 =  108 x 756 Khufu Pyramid base length = 336 x 243 Venus Rotational Day 

21st century data:
816.435 days

The two Ancient Square Root Two fractions,
are displayed,
with their respective cubit generated -- Pyramid Heights.

The modern pure square root two cubit 20.61923374~,
is also shown with it's respective cubit generated height.

All three pyramid heights are aligned,
with ancient astronomical observational and sacred geometry intent, 
to the ancient calendar count Mars - Jupiter synod -- 100 synods = 81648 days.



edit:
image below has been updated with fresher content,
specifically the cubits setting the pyramid heights are defined with relevant equations.



[Image: Oy8cDlt.jpg]





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2008-2009 image with updates.
Image 2:
The -- 2 x 2 x 2  Pyramid -- created from a 2D tetrahedral grid.
{2 units high, with a base of 2 x 2 units}

The Pentad Octagon replicating tile,
was my creation using the Tetrahedral Pentad,
representing tetrahedral geometry and the positions of the Cydonia Mounds on Mars.
{Dr. Horace Crater UTSI} -- Mounds of Cydonia 

The Tetrahedral Pentad: 
is the five sided figure in the bottom left quadrant of the Octagon, 
which circumscribes the interior five pointed star.
Each of the five points is the position of a mound in Cydonia Mars,
located just a few kilometers from The Face. 
Regardless of the mounds on Mars,
it is a tetrahedral Pentad defined by square root 2 geometry.

All the dotted line rectangles seen in the Octagon,
are -- square root two rectangles --
that will replicate the octagon to infinity -- as a replicating tile.

The Octagon is now set on it's side to present the inner parallelogram.
The interior parallelogram,
is now at the desired position for the transfer,
of the 2D octagonal grid into 3D geometry. 

The parallelogram is duplicated.
The duplicate parallelogram is now cross sected into the original one,
at 90 degrees,
through the vertical axis.

The square base of the pyramid is formed,
and shown to the right of the above duplication and cross section process.
That square base cross sects through the horizontal axis,
and the end points are connected as shown in the diagram of the base.

So actually TWO pyramids are formed -- base to base -- by the cross section process.

At the bottom of the image,
are the cross secting isosceles triangles,
formed by the:
Side Face Angles and the Corner Angles of the -- 2 x 2 x 2 pyramid. 

The Corner Angles are pure tetrahedral,
and the 70.52877937~ degree Full Apex Angle is the "electron spin angle"  {Dr. Horace Crater UTSI}

And the Side Face Angles,
produce the Khafre Pyramid Side Face Angle -- as the Full Apex Angle of the pyramid.
The progenitor of inverse phi and phi angles are found with this isosceles triangle.

Angle aa = arctangent 2,
which subdivides,
into two equal angles of -- arctan {1 / phi}.

The simple 2 x 2 x 2 pyramid has a beautiful complexity of important angles to be found within.


[Image: q20vPq3.jpg]


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2008-2009 image, hand drawn with later added color computer lines
Image 3:
This is the beginning of the replicating tile series,
with the Pentad Octagon.
The individual tiles attach and replicate to infinity, 
creating an infinite variety of polygonal forms in tetrahedral geometry.

In this image,
FOUR Octagons are assembled,
and the Pentad Hexagon Whip
 is revealed wiithin the center of the tetrahedral infrastructure.


[Image: 4UbH2dZ.jpg]



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2008-2009 image, hand drawn 
Image 4:
A cube is created with each flat face -- 2 x 2  units Whip

The 2 x 2 x 2 Pyramid is then attached to each Face of the Cube <---
The 6 pointed star is thus formed.
The electron spin angles,
and the Khafre Pyramid side face angles,
are all present as the Full Apex Angles of the attcahed pyramids. 

Pay no attention to the "acoustic Lol resonance" in the title.
That was 2008 or 2009.

[Image: 1ppzlrO.jpg]



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Grand Unification of Ancient and Modern Geometry and Mathematics

The Square Root Two Cubits:
The end game of this post is the isosceles triangle,
at the very bottom right of Image Three <---
This contains dramatic Convergence Dynamics,
in ancient planetary timeline pyramid pi cubit system code.

Image One:
The pyramid is built upon the height using the standard 280 cubits,
times:
the square root two cubit 20.61923374~ inches.

The height is then expressed in several equations that utilize square root two.
Note the equation on the image upper right:
81648 / by square root 288 <--- 
The negative tangent of ---> square root 288,
is the Full Apex Angle tangent,
of the Khafre Pyramid Corner Angles crossectional isosceles triangle.

In the pyramid in the image,
I have highlighted the Corner Angles Whip with thin blue lines.
This is because that is where this exercise goes to the end game.

A three slope pyramid,
with a very slightly rectangular base,
is presented with these three,
slope angle tangents:
4 / pi
4 / aPi
4 / {sqrt 800 / 9} = sqrt 1.62

Two parallel base lengths are 756 feet each.
These 756 foot lengths create one specific slope angle with the fixed height.
The other two parallel base lengths, have an offset center point along the base line,
to properly display the -- half base lengths, which create the other two slopes.
The vertical pyramid height line simply shifts to accommodate the two half base lengths.
The vertical pyramid height line remains totally vertical in a three slope pyramid.  

Applying the slope tangents to the square root two generated pyramid height,
the three slope positions are shown as A, B, and C.

The base lengths in inches are shown as equations with the respective cubits,
and the "ancient pi values" associated with the cubit equation.

At the bottom of the image you see the half base length as: 
378 feet = 4536 inches
                 4536 inches = 70 x aPi x cubit 20.618 18 18~

aPi = 22 / 7  ---- 1134 / 55 = 20.618 18 18~ 

[Image: 0SSFjXu.jpg]




Footnotes:
Quote:Lots of fractions, and square roots, and Pi values, etc.,  
have infinite a random or replicating series in decimal placements.

When applying cubits and pi values in calculations you have to use the fractions <---
Cubit 20.618 18 18 18~ {replicating decimal to infinity - 18 18 18},
equals:
1134 / 55 <---
If you just use -- 20.618 18 18 clicked in your hand calculator,
there may be a slight difference in the tenth decimal of your results.
When you click the fraction:
1134 / 55
into your hand calculator,
it automatically rounds off at the tenth decimal placement: 20.618 18 18 2~

Pi cubit 20.62648062~ for instance,
is actually:
{18 x 360} / Pi

Square Root Two cubit 20.61923374~ = 14.58 inches x square root two,
where 14.58 inches,
is the Egyptian remen foot in inches.

So when you see the pyramid height,
set by:
Square Root Two cubit 20.61923374~ inches -- x 280 cubit,
equals:
5773.385447~ inches  =  {8164.8 / sqrt 2}

Point being: 
if you use just the ten decimal placements 5773.385447~ inches,
in your calculations,
you may get a slight differential in the tenth decimal placement of results.
But if you use the equation for the height:
{8164.8 / sqrt 2}
then your calculations will adjust perfectly.
All the math is EXACT <---

  


-------------------------------------------------------------------------------


Image Two:

This presents the square base pyramid,
with all  four base lengths set at 756 feet.
440  x  cubit 20.618 18 18~  =  9072 inches  = 756 feet.

The square root two cubit generated height,
280  x  20.61923374~  inches,
is the same height as seen in Image One. 

I have outlined the Pyramid Corner Angles in secondary thin blue lines.
One thin blue line,
is shown as well along the base diagonal. <---
This outlines the:
cross sectional isosceles triangle <---
with designated points:
x, y, and z
Seen as well in the upper right,
with:
Full Apex Angle C <---

Obviously a second identical isosceles triangle cross sects the prior one shown,
through the opposite Corner Angles,
but this graphic design isolates only the one seen in the image.


[Image: 8Hm6vmZ.jpg]



-----


Image Three:

Next,
a series of these cross sectional isosceles triangles,
from associated cubit system generated pyramid geometries are compared,
and also defined further in ancient cubit system code.

Depending on which cubit generated geometry one uses to create a pyramid,
with the array of "ancient Pi values" and modern Pi,
with the slope tangent formula:

4 / by  Pi value = slope tangent.

The cubit generated geometries always produces Corner Angles,
with their own set of criteria.

I will display three possibilities with relative mathematics.

Pyramid with slope tangent {4 / Pi},
has a Corner Angle tangent:
-- 4 /  by  {Pi x sqrt 2} -- 
equals:
0.900316316~

Pyramid with slope tangent {4 / 3.1416},  
has a Corner Angle tangent:
-- 4 /  by  {3.1416 x sqrt 2} -- 
equals:
0.900314211~

Pyramid with slope tangent {4 / aPi},  ... aPi = 22 / 7,
has a Corner Angle tangent:
-- 4 /  by  {aPi x sqrt 2} -- 
equals:
0.899954085~ 

NOTE: how all the above Corner Angle slope tangent decimals <---
hover suspiciously and very closely to:
0.9 = 9 / 10
0.900316316~ --- Pi geometry
0.900314211~ --- 3.1416 geoemtry
0.899954085~ --- aPi --- Khufu Pyramid corner angle tangent in decimal.

It is the  Square Root 2 cubit  20.61923374~ inches <---
that creates the perfect,
cross sectional isosceles triangle of that pyramid geometry,
to ISOLATE,
the Corner Angle slope tangent : 0.9 = 9 / 10
 
Isosceles triangle on left side of the central section:
Height = 280  x  cubit 20.61923374~  inches.
Half Base Lengths:
378 feet = 4536 inches = 220 x cubit 20.618 18 18~  {1134 / 55}.

The pyramid corner angle -- angle B -- is defined two ways:
The angle tangent = 0.9 = {9 / 10}
and,
the angle tangent is defined in cubit system process:
cubit 20.61923374~
divided by:
{16.2 x sqrt 2}
equals:
0.9

ONLY the Square Root Two Cubit process,
will produce the PERFECT:
Full Apex Angle Tangent C Whip
with:
the Ancient Planetary Calendar Count Synods of Saturn and Jupiter Whip
as the angle tangent:
Ten Saturn synods with Earth, 
3780,
divided by:
399 day Jupiter synod with Earth.

On the right hand side of the central section:
I have taken the NASA data,
for the: 
Ten Saturn synods: 3780.9  days
One Jupiter synod: 398.88  days
and,
installed them as the fractional slope tangent in the Full Apex Angle.
The geometry is then back tracked Kickbut
to the:
Corner Angles,
which have the tangent defined as the:
New Cubit  20.62039981  inches <---> generated by the NASA data <---

The base corner angles seen in both isosceles triangles,
are almost exactly:  1 / 1000 degree differential.


[Image: L3QFZ58.jpg]




There will be one more post on the Square Root Two cubits,
and then on to the:
3.1416 Pi cubit system dynamics.

Guitar

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Grand Unification of Ancient and Modern Geometry and Mathematics 
The Square Root Two Cubits
20.61923374~
20.62394778~

The Mars Sidereal -- 687 Day Sidereal Progression -- In Ancient Pyramid Cubit Code
The Mars sidereal =     687 days calendar count -- 686.98 NASA

The Process:
6  x  243 Venus rotational day -- times --  square root 2  -- {tetrahedral angle tangent},
equals:
100  cubits  20.61923374~ inches.
Process:
Venus 243 -- times -- square root 8 -- {tetrahedral angle tangent},
equals:
687.3077913~  --  Mars sidereal progression position in square root two cubits.
687.3077913~
equals:
square root two cubit  20.61923374~  x  33.33333333333~

Each Pyramid Cubit - times -  33.33333333333~  =  Mars sidereal progression position.

cubit  20.625               -- times --  33.33333333333~  =  687.5
cubit 20.61923374~      - times --  33.3333333333~    =  687.3077913~               
cubit  20.618 18 18 ~   -- times --  33.33333333333~  =  687.27 27 27~       
cubit 20.61675             -- times --  33.33333333333~  =  687.225                    
cubit  20.61                -- times  --  33.33333333333~  =  687
cubit  20.606 06 06~    - times  --  33.33333333333~  =  686.86 86 86~ 

Note: 
Vedic and Saxon foot 13.2 inches   <---  {1.1  x 12 inches}
13.2 -- times:
any Mars Progression Position Value,
equals:
The corresponding Khufu Pyramid base length:

13.2 -- times -- 687.5               =  9075 inches                 =  440  x  cubit 20.625
13.2 -- times -- 687.3077913~  =  9072.462845 inches      =  440  x  cubit 20.61923374~  Whip            

13.2 -- times -- 687.27 27 27~  =  9072 inches                 =  440  x  cubit 20.618 18 18~   Royal Cubit
13.2 -- times -- 687.225          =   9071.37 inches             =  440  x  cubit 20.61675          
13.2 -- times -- 687                      9068.4  inches              =  440  x  cubit 20.61 
13.2 -- times -- 686.86 86 86~ =   9066.666666~ inches    =  440 x  cubit 20.60 60 60 60~   


And
Mars Progression Position:
687.27 27 27~
divided by -- 
540  Whip  ---  {number of degrees in the pentagon}
equals:
1.27 27 27 27~  -- the Khufu Pyramid slope tangent  =  {4 / aPi}  =  {14 / 11}.


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-----------------------------------------------------------------------------------------------------------

 IMAGE  ONE:

The Khufu Pyramid slope angle tangent:
Square Root 1.62 Whip
was generated by the square root two cubits {seen in the prior posts}.

I always test specific pyramid tangents,
in 180 degree traingle combinations to see the further results.

Triangle at the top of the image,
shows angle tangent C -- as --  sqrt. 1.62 <---
This is then combined with the:
"electron spin angle"
angle B,
with tangent -- sqrt. 8 <---

Note how angle A,
produces ancient planetary calnedar count planetary cubit system code.
The resultant tangent:
5  -- times --  the Mercury synod to Earth of 116 days,
divided by:
260 Tzolkin -- times -- sqrt. 2 <---

That fractional tangent can reduce to : 
29 -- divided by -- {13 x sqrt. 2}

------------------------------------

The 180 degree triangle in the middle of the image,
combines angle C -- arctan sqrt. 1.62 <---
with:
angle A,
showing the calendar counts of the Earth Year and Venus synod.
Arctangent {584 Venus synod  x  sqrt. 2} -- divided by -- 365 Earth year.

This angle with the Venus synod is important,
because it showed up in the Mars Hexad mounds when worked into 3D,
creating the Khafre Pyramid octahedral {see earlier posts a few months ago}.

The resultant angle B -- reveals the number 235 with sqrt. 2 <--- in the angle tangent equation.
This is significant because the Metonic Cycles,
operate exactly 235 Lunar Months
for every 19 Earth Tropical Years <----

---------------------------------------------

The 180 degree triangle at the bottom of image takes the Khafre Cake!
Angle C -- arctan sqrt. 1.62,
is combined with:
The Khafre Pyramid Full Apex Angle -- from the Corner Angles cross section. 
That is seen as angle B.

The resulatnt angle A <---
has the tangent:
111  --  divided by --  {113  x  sqrt. 2}.
It is no coincidence,
that the number 113 shows up in the fraction.
It evolves from the ancient pi value : 
355 / 113 = 3.14159292


[Image: 2sLekDM.jpg]


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IMAGE  TWO:
this image expands on the previous post.
The mathematics have been presented to bring in -- square root 5 geometry math dynamics.

Two isosceles triangles are compared.
The Khufu Pyramid Corner Angles cross section -- on the left.
And,
an isosceles triangle with --> tangent 72 degrees SQUARED --> as the Full Apex Angle tangent.

tangent 72 degrees =  tangent 54 degrees  -- times -- sqrt. 5 <---- These are pentagonal angles.

tangent 72 degrees squared  =  9.472135955 
tangent 72 degrees squared  =  {4  x  phi} -- plus -- 3 <---

The resultant corner angle produces a Khufu pyramid Corner Angle tangent and CUBIT <---

This image is presented as a prelude to the combination of:
square root 2 -- and square root 5,
and the:
Grand Unification Earth Tropical Year, 
seen in the next image of unique ancient pyramid cubit system mathematics.

[Image: FeGueY4.jpg]

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IMAGE THREE -- supporting data

The Grand Unification Tropical Earth Year,
365.2430698~ days,
emerges from cubit 20.625 -- times -- 280 cubits:
Pyramid height:
5775 inches = 481.25 feet height,
With square root 5,
and square root 2, combined as a fraction, the pyramid height reveals the process.
Those two square roots are abbreviated as:
sqrt. 2  and  sqrt. 5 <---

Ten -- Grand Unification Tropical Earth Years --
3652.430698~ days,
times:
{sqrt 5 / sqrt 2}
equals
5775 inches ----- assigned as pyramid height in inches.
5775 inches = 280 x cubit 20.625

Or,
you could write it as:
10 inches -- times -- 365.2430698~ Grand Unification value,
times:
{sqrt 5 / sqrt 2
equals
5775 inches ... if you are persnickety about units in these cases.

Earth Tropical Year in days:
NASA    365.2422
Grand Unification  365.2430698~
Ancient Metonic cycle  365.2467711~

Grand Unification Tropical Earth Year
expanded value in decimal placements: 365.243069 749 44784~

Thus:
Square Root Two Cubit:
20.62394778~ = 10500  /  {360  x  sqrt.  2}  =  175  /  {6 x sqrt 2}
20.62394778~ 
also equals:
{10  x  sqrt. 5  x  365.2430698~}  --  divided by  --  396 <---
note:
396 = 19.2  x  cubit 20.625. 

Thus:
Square Root Two Cubit:
20.61923374~ = Remen foot 14.58 inches   x  sqrt 2 <---
also, 
equals:
693  x  243 Venus Rotational Day,
divided by:
{10  x  365.2430698~  x  sqrt. 5}

note: the above --  {693 x 243}  =  168399  =   cubit 20.618 18 18~  x  8167.5 cubits
cubit = {1134 / 55}

------------------------------------------


square root 5 -- divided by -- square root 2,
equals,
square root 2.5 <---
However for equation display purposes I show the fractional form.

-----------------------------------------

IMAGE  THREE: first pyramid

A two slope rectangular base pyramid is presented.
Pyramid height is seen as 5775 inches = 280 x cubit 20.625 inches.

The two angle tangents produce angles A and B <---
Tangent angle A  =  {550 / 432}  =  {4  /  by  3.14 18 18 18~}
The ancient Pi progression value:
3.14 18 18 18~  =  {1728 / 550} .

The angle A tangent --  {550 / 432},
applied to the 5775 inch height,
produces base length 9072 inches = 756 feet. 

Tangent angle B =  square root 1.62 Whip
{our original square root two cubit generated pyramid slope tangent}.

This produces the pyramid base length --->  9074.537026~ inches.
That equals:
440  x  20.62394778~ inch --  square root two cubit,

cubit 20.62394778~ = 175 -- divided by -- {6 x square root 2}.

[Image: bkPp1Eg.jpg]

IMAGE  THREE -- 2nd pyramid
A second -- two slope rectangular base pyramid -- is presented.
Pyramid height is seen as 5775 inches.
Two base lengths:
9075 inches = 440  x  20.625 cubits
9072 inches = 440  x  20.618 18 18~ 


Once you have this equation,
from the height,
you can then write the equations for the base lengths!
And using the fixed height,
5775 inches,
you can then revolve through the cubit system dynamics, 
by applying the TRUE -- Earth sidereal and tropical years, 
to the equation.

Those equations are seen to the right of the pyramid.
The series of cubits found are:
cubits:
20.625 --------------   365.2430698~  Grand Unification
20.62495089~ -----   365.2422  NASA
20.62575049~ -----   365.25636  Earth sidereal 21st century exact

Seeing how the Ancient Metonic cycle:
235 Lunar Months = 19 Earth Tropical Years
produces:
365.2467711  days for the Tropical Earth Year,
the Grand Unification Tropical Year:
365.2430698~ days,
being much more exact,
incorporates an important cubit system dynamic,
being exposed with cubit 20.625,
in the 5775 inch height.

The key, being the combinations of square root 5, and square root 2,
to accelerate the Tropical Earth Year Code,
into pyramid dimensions with cubits. 

[Image: VniJXwx.jpg]


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Vic, This is astounding evidence that an ultra-advanced geo-referencing culture existed before the ice age and passed it down to the survivors of a tumult that had to continually struggle to make such measures a mission.

[Image: naturesmostb.jpg]

This gnosis, although not universal could be re-gained if lost, merely by celestial observance and ground survey.
Held as closley gaurded secrets...when disaster strikes...you need a back-up hard drive to keep computing after a malfunction    affects the network,   therefore you teach the savages and civilize them with the rule be it any local division of a metric.

Maybe a king no true ruler.
Maybe a ruler is  true king.


Quote:Maybe a king no true ruler.
In the model, each agent (individual) was given one goal—to maximize the future possibilities available to itself.

The resultant ripple of informational shaped the movement and apparent 'behavior' of the group.
When the group reorganized into more complex formations, according to new information, it reassessed itself.
It continued gathering and exchanging information, and then reorganizing, until the goal of maximizing the space around each particle was achieved.
Maybe a ruler is  true king.
https://techxplore.com/news/2019-09-natu...gence.html




Kudos!!! LilD


also...enjoy Arrow

SEPTEMBER 6, 2019
Sum of three cubes for 42 finally solved—using real life planetary computer
by University of Bristol
[Image: 5d7282fb7c23a.jpg]
Hot on the heels of the ground-breaking 'Sum-Of-Three-Cubes' solution for the number 33, a team led by the University of Bristol and Massachusetts Institute of Technology (MIT) has solved the final piece of the famous 65-year-old maths puzzle with an answer for the most elusive number of all—42.

The original problem, set in 1954 at the University of Cambridge, looked for Solutions of the Diophantine Equation x3+y3+z3=k, with k being all the numbers from one to 100.
Beyond the easily found small solutions, the problem soon became intractable as the more interesting answers—if indeed they existed—could not possibly be calculated, so vast were the numbers required.
But slowly, over many years, each value of k was eventually solved for (or proved unsolvable), thanks to sophisticated techniques and modern computers—except the last two, the most difficult of all; 33 and 42.
Fast forward to 2019 and Professor Andrew Booker's mathematical ingenuity plus weeks on a university supercomputer finally found an answer for 33, meaning that the last number outstanding in this decades-old conundrum, the toughest nut to crack, was that firm favourite of Douglas Adams fans everywhere.
However, solving 42 was another level of complexity. Professor Booker turned to MIT maths professor Andrew Sutherland, a world record breaker with massively parallel computations, and—as if by further cosmic coincidence—secured the services of a planetary computing platform reminiscent of "Deep Thought", the giant machine which gives the answer 42 in Hitchhiker's Guide to the Galaxy.
Professors Booker and Sutherland's solution for 42 would be found by using Charity Engine; a 'worldwide computer' that harnesses idle, unused computing power from over 500,000 home PCs to create a crowd-sourced, super-green platform made entirely from otherwise wasted capacity.
The answer, which took over a million hours of calculating to prove, is as follows:
X = -80538738812075974 Y = 80435758145817515 Z = 12602123297335631
And with these almost infinitely improbable numbers, the famous Solutions of the Diophantine Equation (1954) may finally be laid to rest for every value of k from one to 100—even 42.
Professor Booker, who is based at the University of Bristol's School of Mathematics, said: "I feel relieved. In this game it's impossible to be sure that you'll find something. It's a bit like trying to predict earthquakes, in that we have only rough probabilities to go by.
"So, we might find what we're looking for with a few months of searching, or it might be that the solution isn't found for another century."




Explore further
Bristol mathematician cracks Diophantine puzzle



[b]More information:[/b] Andrew R. Booker, Cracking the problem with 33, Research in Number Theory (2019). DOI: 10.1007/s40993-019-0162-1
Provided by University of Bristol


https://phys.org/news/2019-09-sum-cubes-...-life.html
Along the vines of the Vineyard.
With a forked tongue the snake singsss...
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Grand Unification of Ancient and Modern Mathematics and Geometry
Ancient and Modern Cubits
Megalithic Yard 2.72 Feet Proven as Ancient Intent

Calendar Count
Mercury synod    116 days  {with Earth}
Mercury sidereal  88  days

An Ancient Math Lesson

The:   {1 / aPi} --  plus --  One  -- Planetary Timeline Progression.

{if you are going to follow the math with the cubits,
be sure to use the fractional equaivalents offered }

----------------------------------------------

One sees the phi progression as such:

inverse phi                                    phi
0.618033989 --- plus One = 1.618033989 -- plus one = 2.618033989
These values,
can be dimensional lengths, or they can be angle tangents.

Other important "ancient constants" can be tested in the same fashion.
The ancient pyramid cubit system constants,
were mathematically applied in a variety of math progressions.

The ancient pi progressions use this as the fundamental ancient value:
aPi  =  22 / 7
The most recognized form of ancient pi,
other than pi itself.
It is the fundamental in the Khufu Pyramid slope formula:
 
4 / aPi  =  14 / 11 = slope angle tangent

Inverse  aPi 
{1 / aPi}  =  0.318 18 18~
{7 / 22}  =  280  /  880  =  cubits in pyramid height  /  by  10 Mercury sidereal


The most amazing aspect of -- inverse aPi -- {1 / aPi}
is the:
{1 / aPi} --  plus --  One  -- planetary timeline progression. 

{1 / aPi}  plus One =  1.318 18 18 18~
equals:
116 day Mercury synod -- divided by --  88 day Mercury sidereal  <------- !

IMAGE ONE -- Top Section

{1 / aPi} --  plus --  One

The progression then adds One Unit Whip at a time,
seen as linear dimensional lengths in the image.
{think pyramid base lengths}

A profound result is found with the planet Mercury calendar count sidereal and synod,
evolving directly from the application of aPi,
in a common math progression.
One might say that could still be coincidence,
but as we proceed in the steps of:  "plus One",
the results are nothing less than spectacular in true ancient planetary timeline code,
that also details specifically,
the megalithic yard of 2.72 feet within the ancient code.

The Earth sidereal 365, and Venus synod 584 days, appear in the next step,
and the last step increment shows 360 count,
with cubit 20.625. 

A new cubit Dance2
is found within the mathematical infrastructure of:  {1 / aPi}  plus One.
This cubit is of unique outcome,
because it produces a Khufu Pyramid base length of exactly:
9071.2  inches. --------------------------------->  907120  =  3335 x 272.
440  x  20.616 36 36 36~ inches {cubit}.

The formula seen in the image shows the cubit also working with the megalithic yard.
In the fraction seen,
23 megalithic yards  =  23  x  2.72 feet  x  12 inches per foot,
{to align cubits with inches in units}

This application of  -- {1 / aPi} plus One -- in the form of a dimensional length Whip
allows the planetary timelines to express the lengths.


Bottom Section Image One
Identifying the Mars sidereal 687 days

So  Hmm2  how do we get to the Mars sidereal,
with the ancient pyramid Pi  cosmology?

Calendar Count 687 days --  x  3  =  2061 =  100 cubits 20.61 inches.

The absolute peak in ancient pyramid pi - cubit system - planetary timeline harmonic code,
is evidenced in these simple equations.
Without question this is profound evidence of intent,
in this ancient cultural sacred math and geometry.
There is no chance whatsoever that these results could be any form of coincidence.

The new math progression,
adds ---> aPi ---> to the original equation.

{1 / aPiplus  One  -->  plus  --> aPi,

{1.318 18 18 18~  +  3.142857 142857~}  =  4.461038961~
eqauls:

{4  x  687 Mars} -- divided by --  {7  x  88 Mercury} -  sidereal.


[Image: xxEtcvp.jpg]



------------------------------------------------------------------------------------------------------------------------

IMAGE TWO

Considering the above progression of "ancient constants",
it occurred to me that an even more focused test could be developed.
With the Khufu Pyramid,
the slope angle tangent:
14 /  11  = 1.27 27 27 27~ = 4  /  aPi

A progression is created:

aPi -- plus -- 14 / 11 -- plus -- 14 / 11 

In the first step of the progression,
the Megalithic Yard of 2.72 feet,
-- is proven -- to be expressed by the Khufu Pyramid cubit system,
and the aPi mathematical  infrastructure.

This eqaution shows: 
272 feet in the numerator, and 756 feet in the denominator.
100 megalithic yards -- and -- the standard Khufu Pyramid base length.
Note:
in the equation shown as equaivalent using just cubits,
the Royal Cubit 20.618 18 18~ -- {1134 / 55} -- appears in the denominator.

The cubit 20.6 06 06 06~  =  680 / 332720 / 1320  =  2720 / {64 x 20.625}
is pure ancient megalithic yard math.
It creates a Khufu Pyramid base length of:
9066.666666~ inches.
It is also tied to ancient pi value 3.1416:
314160  =  15246  x  20.6 06 06 06~
314160  =  7392  x  20.625  x  20.6 06 06 06~

This progression adds {14 / 11} again,
seen at the bottom of the image.
The resultant full length seen,
exposes the Earth year 365 days,
and the Venus synod 584 days.
Once again proving that the planetarty timelines are expressed by aPi,
and the Khufu Pyramid cubit system dynamics.


[Image: HcuOaFb.jpg]

-----------------------------------------------------------------------------------------
             

This sets the stage for the construction of a pyramid.

But this time,
instead of assigning --  {1 / aPi} --  plus -- One --  as a dimensional length,
we will assign that value,
1.318 18 18~,
as the side face slope angle tangent <---
of a pyramid with height = 2,
and the resultant base lengths return to a form of the planet Mercury timelines.

That will be in the next post. 

Guitar

...
Reply
Quote:One sees the phi progression as such:https://arxiv.org/pdf/1907.04593.pdf

Long-Standing Problem of 'Golden Ratio' and Other Irrational Numbers Solved with 'Magical Simplicity'

By Leila Sloman - Scientific American 5 days ago Strange News 
[Image: CBMtn8vnbPTjzK3eyQ2UKb-970-80.jpg]

The golden ratio is one of the most famous irrational numbers; it goes on forever and can't be expressed accurately without infinite space.
(Image: © Shutterstock)https://www.livescience.com/irrational-numbers-golden-ratio-conjecture-solved.html

The ancient pi progressions use this as the fundamental ancient value:
aPi  =  22 / 7
The most recognized form of ancient pi,
other than pi itself.

Quote:Many people conceptualize irrational numbers by rounding them to fractions or decimals: estimating  π as 3.14, which is equivalent to 157/50, leads to widespread celebration of Pi Day on March 14th. Yet a different approximation, 22/7, is easier to wrangle and closer to  π. This prompts the question: Is there a limit to how simple and accurate these approximations can ever get? And can we choose a fraction in any form we want?





New Proof Solves 80-Year-Old Irrational Number Problem
Mathematicians have finally proved a conjecture on approximating numbers with fractions
  • By Leila Sloman on September 16, 2019
    Golden ratio is one of the most famous irrational numbers, which run on forever and cannot be expressed accurately without infinite space. Now scientists have proved a conjecture about how to use fractions to approximate them. Credit: Getty Images
Most people rarely deal with irrational numbers—it would be, well, irrational, as they run on forever, and representing them accurately requires an infinite amount of space. But irrational constants such as  π  and √2—numbers that cannot be reduced to a simple fraction—frequently crop up in science and engineering. These unwieldy numbers have plagued mathematicians since the ancient Greeks; indeed, legend has it that Hippasus was drowned for suggesting irrationals existed. Now, though, a nearly 80-year-old quandary about how well they can be approximated has been solved.
Many people conceptualize irrational numbers by rounding them to fractions or decimals: estimating  π as 3.14, which is equivalent to 157/50, leads to widespread celebration of Pi Day on March 14th. Yet a different approximation, 22/7, is easier to wrangle and closer to  π. This prompts the question: Is there a limit to how simple and accurate these approximations can ever get? And can we choose a fraction in any form we want?
In 1941 physicist Richard Duffin and mathematician Albert Schaeffer proposed a simple rule to answer these questions. Consider a quest to approximate various irrational numbers. First, decide on how close the approximation should be for fractions of a particular denominator. (Remember, the “numerator” refers to the top of a fraction and the “denominator” the bottom. Here, all the fractions are fully simplified—so, for example, 2/4 does not count as having the denominator 4 because it simplifies to 1/2.) You might decide that simplified fractions of the form n/2 can approximate any irrational number whose true value falls within 1/10 of them—giving the approximation an “error” of 1/10. Fractions that look like n/10 are closer together on the number line than those with denominator 2, so you might limit the error in that case to only 1/100—those fractions can approximate anything within 1/100th of them.

[Image: fractions_graphic_d.png]
Credit: Amanda Montañez
Usually, larger denominators are associated with smaller errors. If this is true, and there are infinitely many denominators that one can use to approximate a number to within the corresponding error, then by increasing the denominator the approximation can be made better and better. Duffin and Schaeffer’s rule measures when this can be done based on the size of the errors.
If the chosen errors are small enough in aggregate, a randomly picked irrational number x will have only a limited number of good approximations: it might fall into the gaps between approximations with particular denominators. But if the errors are big enough, there will be infinitely many denominators that create a good approximating fraction. In this case, if the errors also shrink as the denominators get bigger, then you can choose an approximation that is as precise as you want.

UNPROVED
The upshot is that either you can approximate almost every number arbitrarily well, or almost none of them. “There’s a striking dichotomy,” says Dimitris Koukoulopoulos, a mathematician at the University of Montreal. Moreover, you can choose errors however you want, and as long as they are large enough in aggregate most numbers can be approximated infinitely many ways. This means that, by choosing some errors as zero, you can limit the approximations to specific types of fractions—for example, those with denominators that are powers of 10 only.
Although it seems logical that small errors make it harder to approximate numbers, Duffin and Schaeffer were unable to prove their conjecture—and neither was anybody else. The proof remained “a landmark open problem” in number theory, says Christoph Aistleitner, a mathematician at Graz University of Technology in Austria who has studied the problem. That is, until this summer, when Koukoulopoulos and his co-author James Maynard announced their solution in a paper posted to the preprint server arXiv.org.
The Duffin-Schaeffer conjecture “has this magical simplicity in an area of maths that’s normally exceptionally difficult and complicated,” Maynard says, a professor at the University of Oxford. He stumbled into the problem by accident—he is a number theorist, but not in the same area as most Duffin-Schaeffer experts. (He normally studies prime numbers—those that are divisible by only themselves and 1.) A University of York professor suggested Maynard tackle the Duffin-Schaeffer conjecture after he gave a talk there. “I think he had an intuition that it might be beneficial to get someone slightly outside of that immediate field,” says Maynard. That intuition turned out to be correct, although it would not bear fruit for several years. Long after that initial conversation, Maynard suggested a collaboration to Koukoulopoulos on a suspicion that his colleague had relevant expertise.

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Maynard and Koukoulopoulos knew that previous work in the field had reduced the problem to one about the prime factors of the denominators—the prime numbers that, when multiplied together, yield the denominator. Maynard suggested thinking about the problem as shading in numbers: “Imagine, on the number line, coloring in all the numbers close to fractions with denominator 100.” The Duffin-Schaeffer conjecture says if the errors are large enough and one does this for every possible denominator, almost every number will be colored in infinitely many times.
For any particular denominator, only part of the number line will be colored in. If mathematicians could show that for each denominator, sufficiently different areas were colored, they would ensure almost every number was colored. If they could also prove those sections were overlapping, they could conclude that happened many times. One way of capturing this idea of different-but-overlapping areas is to prove the regions colored by different denominators had nothing to do with one another—they were independent.
But this is not actually true, especially if two denominators share many prime factors. For example, the possible denominators 10 and 100 share factors 2 and 5—and the numbers that can be approximated by fractions of the form n/10 exhibit frustrating overlaps with those that can be approximated by fractions n/100.
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GRAPHING THE PROBLEM
[Image: xxEtcvp.jpg]Maynard and Koukoulopoulos solved this conundrum by reframing the problem in terms of networks that mathematicians call graphs—a bunch of dots, with some connected by lines (called edges). The dots in their graphs represented possible denominators that the researchers wanted to use for the approximating fraction, and two dots were connected by an edge if they had many prime factors in common. The graphs had a lot of edges precisely in cases where the allowed denominators had unwanted dependencies.
Using graphs allowed the two mathematicians to visualize the problem in a new way. “One of the biggest insights you need is to forget all the unimportant parts of the problem and to just home in on the one or two factors that make [it] very special,” says Maynard. Using graphs, he says, “not only lets you prove the result, but it’s really telling you something structural about what’s going on in the problem.” Maynard and Koukoulopoulos deduced that graphs with many edges corresponded to a particular, highly structured mathematical situation that they could analyze separately.
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The duo’s solution came as a surprise to many in the field. “The general feeling was that this was not close to being solved,” says Aistleitner. “The technique of using [graphs] is something that maybe in the future will be regarded as just as important [as]—maybe more important than—the actual Duffin-Schaeffer conjecture,” says Jeffrey Vaaler, a retired professor at the University of Texas, Austin, who proved a special case of the conjecture in 1978.
It may take other experts several months to understand the full details. “The proof now is a long and complicated proof,” says Aistleitner. “It’s not sufficient just to have one striking, brilliant idea. There are many, many parts that have to be controlled.” At 44 pages of dense, technical mathematics, even leading mathematical minds need time to wrap their heads around the paper. The community, however, seems optimistic. Says Vaaler: “It’s a beautiful paper. I think it’s correct.”



https://www.scientificamerican.com/artic...r-problem/


Arrow https://arxiv.org/pdf/1907.04593.pdf
Along the vines of the Vineyard.
With a forked tongue the snake singsss...
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...

I have to address "Magic  Herethere Simplicity" 
before I finish my prior post,
with part two of that cubit study.

Grand Unification of Ancient and Modern Geometry and Mathematics


Quote:Long-Standing Problem of 'Golden Ratio' 
and Other Irrational Numbers 
Solved with 'Magical Simplicity'

New Proof Solves 80-Year-Old Irrational Number Problem
Mathematicians have finally proved a conjecture on approximating numbers with fractions



"Magic Simplicity" is better identified as Universal Harmonic Code.
Using "primes" to isolate the ancient code in the sacred geometry,
is the basic fundamental tool,
in decoding the ancient mathematical intent of the sacred geometry.

Convergence Dynamics {in harmonic code number systems} is my label for the process.
This process was refined by ancient cultures to process the decoding of Pi,
phi, and square roots 2, 3, and 5,
into fractions.

Pi was explored by ancient cultural math systems,
and SIMPLE math progressions were designed,
to Harmonically Converge to Pi -- in fractions.
Why?
because they had to have a fraction Whip
to calculate Pi with, in volumes etc.
Khufu did not have a handy dandy scientific hand calculator with a Pi button.
He did however,
have tens of thousand of years of accumulated human cultural math science,
and the best of sacred pyramid geometry mathematicians. 

By the ways, 
if you like to learn and practice harmonic code search and review,
you only need a simple 12$ scientific calculator.
Recommend: the -- TI-30Xa ... or whatever calculator you use <---
it has to have this function in particular:
1 / X  -- the inverse of the number you are looking at.
Without that function, your search patterns and work will easily double in time taken.
Study simple replicating decimal patterns, the sevenths, elevenths, thirteenths, etc.

The Magic Simplicity in Ancient Pi and Phi Math Progressions {Convergence Dynamics}

Tens of thousands of years ago,
man tried to isolate usable and precision fractions,
for square root two, square root three, square root five,
and Pi, and Phi, in simple math progressions.
Man had one major luxury in this regard.
He had the observations of accounting for the planetary movements,
over thousands of years of observation.
By accounting for the numbers of days in planetary sidereal and synods,
the sacred geometry was born into practice.
In particular,
the Venus sidereal, and Venus synod with Earth,
created the some of the first ancient harmonic codes,
with the fibonacci.
Calendar Count:
365 day Earth year
584 day Venus synod with Earth.

365 / 584 = 0.625  =  5 / 8 ---> and right there the fibonacci progression was born.

Most readers here have seen the --> pentagonal patterns <---
produced by the Venus and Earth orbits.
It is a well known factor recognized by ancient and modern observations.
Anything truly pentagonal involves:
square root 5 and phi dynamics.

Harmonic Code and Convergence Dynamics,
will prove the Venus-Earth orbital pentagram,
with:
The Venus sidereal of 224.7 days <---> both an ancient and modern count for the sidereal. 

This functional fraction chock full of ancient code,
can express a high accuracy fraction for -- square root 5 -- with the Venus sidereal of 224.7 days.

The Convergence Dynamic processing:

224.7 Venus  x  sqrt. 5  =  502.4444 745 <---- note the long succession 4's in the decimal.

The sequence of four 4's -- in the decimal,
or any sequence of 3, 4, or 5 single digits in a decimal,
automatically tells you,
to multiply by 9, 
to arrive convergent to a whole number <---

{224.7  x  sqrt. 5} -- times --  9  =  4522.000271

The convergent result is --> 4522 = 1662.5 megalithic yards of 2.72 feet.

4522 -- divided by -- (9  x  224.7} = square root 5 --- 0.99999994 accuracy Whip
expand the code,
into a fraction with whole numbers:  --- 9 x 224.7 = 2022.3,
45220
divided by ----
20223
equals
2.236067844 -- Convergent Dynamic to sqrt 5, -- 0.999999940 accuracy.
2.236067978 -- true sqrt 5

That accuracy would equate to being off by 1.9 seconds,
on an entire Earth year.

Now that you have a new fraction for square root 5,
you can easily create a progression,
that follows fibonacci style math progression,
and find a fraction,
for PHI.

45220 + 20223
divided by
{2 x 20223}    ------------- the fraction: {65443 / 40446},
equals
PHI convergent
1.618033922 ---- 0.999999958 accuracy.

The Venus sidereal progression,
reduces the above fraction to lowest common denominator,
then runs through several fibonacci style steps,
and produces this ten decimal accuracy fraction for Phi:

75025 / 46368 = phi --- pure fibonacci
1.618033989
-----------------------------------------------------


OK, the Venus sidereal of 224.7 days works with square root 5 dynamics.
Out of nowhere it creates a fraction,
for Pi squared.

22177
divided by
2247  ----- ten Venus sidereal,
equals
Pi squared
9.869603916 -- convergent pi squared - 0.999999951 accuracy
9.869604401 -- true value

Take the square root of the convergent fraction,
and you get this Pi value:
3.141592576
This pi value exceeds the ancient fraction {355 / 113} in accuracy,
but does not quite exceed a Babylonian construct for Pi,
84823 / 27000  in accuracy.

And this brings us to the "Magic Simplicity"
of the Ancient Pi Progressions

The magic simplicity of Universal Harmonic Code,
is the fundamental missing key in human math sciences.
Military corporate industrial complex science,
did everything it could to eliminate the ancient code methods and thinking.
Religious takeover of cultures also does the same thing.
When Christianity killed off the Aztecs,
they buried the ancient math and instituted their own math.
This is why we have a BULLSHIT Gregorian calendar in our society,
with BULLSHIT months of 31 days.

Magic Simplicity,
would point to much more efficient calendar systems.
The Gregorian calendar isn't about TIME,
it is about cultural and political control of your every day life.

Here is an example of a far more efficient calendar:
We have 7 day weeks.
We have 4 week months.
We have 13 full moons per year.

7 days a week -- times -- 4 weeks per month = 28 days.

13 months x 28 days = 364 days.
with
one extra day off per year.

Modern math avoids harmonic code because it is the ancient method.

Now lets look at Pi.
A progression was created in Kerala centuries ago,
that isolated Pi.
It is the EXACT same fractional Pi value,
that the ancient Egyptians {and previous other cultures} arrived at,
with their:
Sacred Geometry and Convergence Dynamics. 
Tens of thousands of years ago,
Pi was searched for in usable fractions.
aPi = 22 / 7,
was a method of using the "sevenths" and "elevenths",
which helped create a fundamentally simple form of square root two,
and a plethora of applications to sponsor cultural math applications.
4 / aPi  = slope tangent Khufu Pyramid =  14 / 11

An ancient pi value is historically recorded as {355 / 113} = 3.14159292,
very accurate,
for common everyday volume calculations.
3.1416 
was another development in Pi -- based upon the prime 17.
There are -- hundreds of Pi values -- in the progression.
Several of them stand out in other cultural math applications. 

Note that aPi = {22 / 7}, 
and ancient pi value 3.1416 =  {3927 / 1250},
have a unique relationship <---
align them as a fraction,'and you get this:
2500 / 2499.

The ancient cultural math sciences refined the process of finding a fraction for Pi.
They took aPi  = 22 / 7 =  3.142857 142857~,
and
355 / 113 = 3.14159292~,
and created the Ultimate in Magic Simplicity,
to create a Harmonic Convergence Dynamics Progression,
into a Pi fraction,
with ten decimal accuracy.

There are several applications of the process,
and I will employ these two of the premier ancient Pi values.

Pi value: 
355 / 113
is found using: 
aPi = 22 / 7, 
and:
ancient pi value 3.14166666~  =  377 / 120.    note: 377 is fibonacci.
Like this:

377 -- minus -- 22  =  355
120 -- minus --   7  =  113

You work the numerator lines -- separately from the -- denominator lines.
Now,
just employ ancient pi values --> {355 / 113}  and  {22 / 7} <---
in:
the same form of convergence dynamics processing.

You hit -- 3.14159 0013 -- at step  27,

27  x  355  = 9585 ---> minus -- 22  = 9563
27  x  113  = 3051 ---> minus  --  7   = 3041

9563 / 3041 = 3.14159 0013

On step 239, 
the Babylonian Pi value:
84823 / 27000  is achieved,
and this exceeds pi value 355 / 113 in accuracy <---

239  x  355  =  84845 --->  minus --  22  = 84823
239  x  113  =  27007 --->  minus --    7  = 27000 

On the 294th step in the progression,
you get this Pi value,
with ten decimal accuracy:

294  x  355  = 104370 --->  minus -- 22  =  104348
294  x  113  =   33222 --->  minus --   7   =    33215

104348 / 33215  =  Pi  =  3.141592654 Whip  ten decimal accuracy.

Now,
look at the denominator:
33215
33215 = 91  x  365 Whip  --- 365 day Earth year x 91.

The Pyramid of Kukulkan has  -- four sets of -- 91 steps.
{4 x 91 = 364 = 13 x 28 day months}
and a top platform step,
to equal 365.

The ancient pi progressions are flexible and dynamic.

For pi value -- 3.1416 = 3927 / 1250 <---
11 x 355 --- 3905 ---> plus  22  =  3927
11 x 113 --- 1243 ---> plus   7  =  1250

-----------------------------------------

Magic Simplicity = Convergence Dynamics

I am light years ahead of these math scientists learning Universal Harmonic Code,
and Convergence Dynamics.
And all I needed was a 12$ scientific hand calculator.

Dr. Horace Crater UTSI,
once challenged me to use my "harmonic code" to isolate an equation,
for the tangent of the Cabibbo angle.

For me to perform high level "harmonic code",
I have to have inspiration to guide the INTUITION <----
to rule the processing direction of number strings <---
and exponentially reduce the amount of time to process a highly convergent result.
Dr. Crater always inspired me to go the extra mile.
I had this angle tangent within 6 hours to ten decimal place accuracy Whip
complete with 20.625 cubit system code,
and pure square root two.


I have several fractional equations,
that isolate the Fine Structure Constant 137.0359991 Whip


Guitar

...
Reply
Quote:The Magic Simplicity in Ancient Pi and Phi Math Progressions {Convergence Dynamics}

Tens of thousands of years ago,
man tried to isolate usable and precision fractions,
for square root two, square root three, square root five,
and Pi, and Phi, in simple math progressions.
Man had one major luxury in this regard.
He had the observations of accounting for the planetary movements,
over thousands of years of observation.

Magic Simplicity = Convergence Dynamics

I am light years ahead...


Arrow speaking of being A: Head

OCTOBER 3, 2019
Golden Ratio observed in human skulls

[Image: goldenratioo.jpg]Leonardo’s Vitruvian Man with Golden ratios highlighted. Credit: Modified by Rafael Tamargo
The Golden Ratio, described by Leonardo da Vinci and Luca Pacioli as the "Divine Proportion," is an infinite number often found in nature, art and mathematics. It's a pattern in pinecones, seashells, galaxies and hurricanes.
In a new study investigating whether skull shape follows the Golden Ratio (1.618 … ), Johns Hopkins researchers compared 100 human skulls to 70 skulls from six other animals, and found that the human skull dimensions followed the Golden Ratio. The skulls of less related species such as dogs, two kinds of monkeys, rabbits, lions and tigers, however, diverged from this ratio.
"The other mammals we surveyed actually have unique ratios that approach the Golden Ratio with increased species sophistication," says Rafael Tamargo, M.D., professor of neurosurgery at the Johns Hopkins University School of Medicine. "We believe that this finding may have important anthropological and evolutionary implications."
The researchers published their findings in the September issue of the Journal of Craniofacial Surgery.
The Golden Ratio can be calculated by taking a line and dividing it into two unequal parts, with the length of the longer part divided by the shorter length being equal to the entire length divided by the longer part. Tamargo's interest in history and anatomy led him in 2010 to publish on finding a human brain and spinal cord in the depiction of God in Michelangelo's Sistine Chapel painting.




Explore further
Michelangelo likely used mathematics when painting the creation of Adam



[b]More information:[/b] Rafael J. Tamargo et al. Mammalian Skull Dimensions and the Golden Ratio (Φ), Journal of Craniofacial Surgery (2019). DOI: 10.1097/SCS.0000000000005610
Provided by Johns Hopkins University 




https://phys.org/news/2019-10-golden-rat...kulls.html




Must Read  Arrow

Mammalian Skull Dimensions and the Golden Ratio (F) Rafael J. Tamargo, MD and Jonathan A. Pindrik, MDy Abstract: The Golden Ratio (Phi, or F ¼ 1.618...) is a potentially unifying quantity of structure and function in nature, as best observed in phyllotactic patterns in plants. For centuries, F has been identified in human anatomy, and in recent decades, F has been identified in human physiology as well. The anatomy and evolution of the human skull have been the focus of intense study. Evolving over millenia, the human skull embodies an elegant harmonization of structure and function. The authors explored the dimensions of the neurocranium by focusing on the midline calvarial perimeter between the nasion and inion (nasioiniac arc) and its partition by bregma into 2 sub-arcs. The authors studied 100 human skulls and 70 skulls of 6 other mammalian species and calculated 2 ratios: 1) the nasioiniac arc divided by the parietooccipital arc (between bregma and inion), and 2) the parietooccipital arc divided by the frontal arc (between nasion and bregma). The authors report that in humans these 2 ratios coincide (1.64 0.04 and 1.57 0.10) and approximate F. In the other 6 mammalian species, these 2 ratios were not only different, but also unique to each species. The difference between the ratios showed a trend toward convergence on F correlating with species complexity. The partition of the nasioiniac arc by bregma into 2 unequal arcs is a situation analogous to that of the geometrical division of a line into F. The authors hypothesize that the Golden Ratio (F) principle, documented in other biological systems, may be present in the architecture and evolution of the human skull.

https://insights.ovid.com/crossref?an=00...9000-00034
Along the vines of the Vineyard.
With a forked tongue the snake singsss...
Reply
Hmm2 Degeneracy may be determined by the relative deviation from the Golden Ratio in skull formation. 

[Image: Inb0ic3.jpg]
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...  

Grand Unification of Ancient and Modern Geometry and Mathematics

Still attempting to catch back up and finish the exercise on ancient math progressions.
A pyramid will eventually be supplied,
that offers the Mercury: {116 synod / 88 sidereal} as an angle tangent,
from those previous progressions. 
{See my last two posts}

To get back to Harmonic Code and Convergence Dynamics.
The ancient Pi Progressions were pure harmonic code.

The recent material posted on phi and the human body and skull,
is somewhat old hat Vitruvian Man.
Leonardo Da Vinci was high on Phi Stars 

but lean on Pi?  Dunno

Noting the ancient Pi Progressions,
I created a Pi value almost ten years ago now,
that I labeled -- Universal Harmonic Pi -- uPi <---
which evolves from square root 5 and phi mathematics.

Harmonic code in phi dynamics gets complicated and mind twisting fast.
It's magic, but not necessarily simple, until you get used to it.

Universal Harmonic Pi -- uPi <---> 3.141640787~

Phi squared x 12  =  10 uPi

{12 x Phi} -- plus -- 12  =  10 uPi

{tangent 72 degrees}squared  x  3 -- then -- Plus 3  =  10 uPi

{4 x Phi} -- plus -- 4 =  3.3333333~ x  uPi  =  {tangent 72}squared -- plus -- 1

Tangent 72 degrees is the central angle of the pentagon.

Phi squared / uPi  =  5 / 6  =  0.833333333~  = 630 / 756

Now paint all that phi,
phi squared,
and universal harmonic pi,
into the face of the Mona Lisa.
It was the tangent of 72 degrees, squared Whip
that gave her that special look.

-----------------------------------------

The Planck Constant value --  and  uPi

For many years the Planck value that was used:
6.626069573

The recently revised value:
6.626070150

https://home.cern/news/news/engineering/...definition

Quote:When analyzed by the CODATA Task Group on Fundamental Constants,
the measurements produced a final value of h of 6.62607015 × 10-34 kg·m2/s,
with an uncertainty of 10 parts per billion.
When the SI was redefined, this was set as the exact value of Planck’s constant,
which in turn defines other SI units including the kilogram.


I tried to decode Planck for 1.5 hours one evening,
and came up with this first:
5481
divided by:
{330  x  square root 2  x  square root pi}
equals:
6.62606803 --- 0.99999976  accuracy ... which is high enough accuracy to work with.
That accuracy would amount to:
7.5 seconds off, on the Earth sidereal year.
You do want to see better accuracy, but it is a very good start.

This equation has some ancient context.
Rewritten:
47.25 x 116 Mercury synod
divided by:
{16 x cubit 20.625  x  sqrt 2  x  sqrt pi}

Then by chance,
I tried the uPi value 3.141640787~,
with the exact Planck value 6.626070150,
and hit the next level of excellent high accuracy,
with a simple, straightforward, but dynamic equation.

The challenge thus is:
Can anyone find a better precision fraction for Planck?
I probably can, but I am satisfied with this result for now.

Planck Convergence Dynamic:
{1875 x uPi} -- divided by -- 8896.626070276 ---> 0.999999981 -- superb accuracy!
also
equals:
{2250 x phi squared} -- divided by -- 8896.626070276 <--- Whip


So if you don't like my Universal Harmonic Pi concept,
you cannot argue with ... phi squared ... in the above equations.

That accuracy level in my result would be:
0.6 seconds off -- the entire Earth year sidereal of 365.25636 days.

It gets better, look at the number 889 <---
889 = 7 x 127
The prime number 127 is the exclusive number in -- metric to foot conversion factor.
The fractional equation,
for metric to foot conversion factor: 3.280839895
3750 / by  {9 x 127}

Use the factor above for ---> 889 meters = 2916.666666~ feet =  35,000 inches Whip

So there it is.
The open challenge to all the harmonic code people out there.
Find an equation for the Planck constant,
that has better accuracy.
That equation cannot just be a random fraction with no meaning.
It must contain some kind of tangible mathematical substance of a universal constant,
{such as:  pi, e, sqrt 2, phi, fine structure constant, etc, etc.}

Planck:
{1875 x uPi} -- divided by -- 889 = 6.626070276  convergent value
                                                   6.626070150  current value

{2250 x phi squared} - divided by - {7 x 127} =  6.626070276 <--- Whip


[Image: brLi66G.jpg]


Guitar




-------------------------------------------------------------


In the last post I displayed a premier ancient Pi progression.
These three ancient Pi Progression fractional values,
directly produce the perfect convergent Pi fraction.

22 / 7  =  aPi = 3.142857 142857~

377 / 120 = 3.141666666~ ----------- 377 is a fibonacci number

355 / 113 = 3.14159292

The process follows through by calculating the numerator and denominator separately.
The simple process to find ancient pi value 355 / 113:

377 ---> minus --- 22 =  355
120 ---> minus ---   7 =  113

On the 294th step of the progression,
the ten decimal accuracy Pi fraction --- 104348 / 33215,
was found with the two ancient pi values,
355 / 113,
and
aPi = 22 / 7.

{294 x 355} ---> minus --- 22104348
{294 x 113} ---> minus ---   7 =    33215

104348 / 33215 = 3.141592654~

However originally,
we found the ancient pi value {355 / 113} <---
with these two ancient pi values:

377 / 120 = 3.14166666~ ----- 377 is fibonacci.
and
aPi = 22 / 7

like this:
377 -- minus -- 22  =  355
120 -- minus --  7   =  113
 
Theoretically,
we should be able to use both ancient pi progression values:
377 / 120
and
355 / 113
to produce a highly convergent ten decimal placement pi value !

Remember that on ---> step 294 <--- in the previous progression,
the pi fraction:
104348 / 33215 was achieved.

This is an amazing result.
On  ---> step 295 <--- using the ancient pi values:
377 / 120 -- and -- 355 / 113,
the EXACT same result is discovered!

{295 x 355} ---> minus --- 377  =  104348
{295 x 113} ---> minus --- 120  =    33215

104348 / 33215  =  Ancient Pi -- ten decimal placement accuracy

How about that? 
Khufu didn't have a handy dandy 21st century scientific hand calculator,
where he could plinko his fingertips right on to the pi button.
By the time that Khufu built his pyramid,
these pi progressions were likely thousands of years older.


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Grand Unification of Ancient and Modern Geometry and Mathematics

This post finishes up my prior post on "Magic Simplicity"
Prior post material at this link:
http://thehiddenmission.com/forum/showth...#pid245100

aPi = 22 / 7

IMAGE ONE

The equation:
{1 / aPi} -- plus -- One <---
equals:
116 day Mercury synod to Earth,
divided by:
88 day Mercury sideral

The fraction {116 / 88}
is installed,
as the Slope Tangent -- angle b <---

Two isosceles triangles are integrated together,
to produce interesting results.
The Khafre Pyramid Side Face Angle,
is the Apex Angle,
of the inner isosceles triangle.
Note all the angle tangents,
especially angle c <---
NOTE:
the linear dimensions supplied,
under the two isosceles triangles.
Once again,
the Earth 365 and Venus 584 are exposed,
as in the prior post images.

[Image: nMcHFzB.jpg]



IMAGE TWO:

shows a succession of linear lengths,
from the criteria of the angle b <---
in the prior image.
Once again adding ONE,
to each step,
also produces the megalithic yard again.
The progression is ended as seen,
to show how the 260 Tzolkin fits right in.

[Image: BJU703w.jpg]


IMAGE THREE:
this is the resultant actual pyramid,
with slope tangent,
angle b <---
I have supplied the:
Full Apex Angle -- slope tangents.
Full Apex Angle c <---
results from the Side Faces of the pyramid.
The megalithic yard AGAIN,
is found in the slope tangent equation!
Look at that Full Apex Angle tangent!

Full Apex Angle  d <---
results from the Corner Angles of the pyramid.
The absolutely fascinating aspect,
of this angle tangent equation,
is the presence of the prime number 127 <---
That is the prime number,
that converts -- metric -- to feet !
Conversion factor,
metric to feet:
3750 / {9 x 127}

[Image: 2bTX7Dr.jpg]


Note:
the denominator in angle d <---
127  x  square root 2 <---
use the fractional conversion factor,
to convert this value as meters,
into feet <---:
equals:
833.333333333~ feet / square root 2 Whip


Guitar


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Grand Unification of Ancient and Modern Geomtery and Mathematics

Khufu Pyramid Sacred Seked 5.5 -- as an important ancient angle tangent.

PART ONE -- sets the stage for -- Part Two {the next post after this one in a couple weeks}.

PART ONE
One of the most important equations,
that connects the ancient sacred geometry of angle tangents,
to the fundamental building block geometries of the universe,
such as the square root two - tetrahedral,
and square root 5 - phi geometry,
was this most important discovery in two steps.

Step one:
18 / 99 = 3.75 / 20.625 inches {cubit} = 2 / 11

I like to write the fraction as 18 / 99, rather than 2 / 11,
because the decimal is:
0.18 18 18 18 18~

18 / 99 -- is a very important angle tangent Whip

arctangent {18 / 99} = 10.30484647 degrees

Note:
90 degrees -- minus -- 10.30484647 degrees
equals:
79.69515353 degrees  =  arctangent 5.5  <--- 99 / 18.

99 / 18  =  5.5  =  sacred seked of the Khufu Pyramid.

Sekeds
5.666666~ = {7 / 5.666666~} =  {21 / 17} Menkaure Pyramid slope tangent.

5.5 ---- {7 / 5.5}  =  {14 / 11} = {4 / aPi}  Khufu Pyramid slope tangent.

5.25 --- {7 / 5.25} = {4 / 3}  Khafre Pyramid slope tangent. 


And this is the key focus of this Two Part Study:
Setting the Sacred Seked -- 5.5 -- of the Khufu Pyramid,
as an angle tangent,
and it's orientation in Phi geometry and Khafre Pyramid geometry infrastructures Whip
And of course the inverse angle,
with tangent = 18 / 99.
This inverse angle tangent {18 / 99},
is what directly ties into Phi geometry.

Again:
18 / 99 -- is a very important angle tangent.
arctangent {18 / 99} = 10.30484647 degrees

This angle is thus assigned as the - apex angle - of an isosceles triangle.
When this happens,
the two base angles of the isosceles triangle,
angle W Whip
seen in the image below,
reveals phi geometry angle tangents !
{go to the bottom of the image}.

This old image sets the complete stage and for Khafre pyramid geometry,
and it's close connections,
into phi geometry infrastructures, with angle W <---
 
Note as well the equation below the isosceles triangle in the bottom right corner of the image.
80 -- times -- any cubit,
divided by:
angle tangent:
{18 / 99},
equals:
the corresponding base length in inches, as 440 cubits.

80 x 20.618 18 18 18~    ---      {cubit = 1134 / 55},
divided by:
{18 / 99}
equals:
9072 inches = 440 cubits.

These equation below of angle tangents,
is the "magic simplicity" of Khafre Pyramid angle.
26.56505118 degrees:
arctangent {1 / 2}  +  arctangent {1 / 2}  =  arctangent {4 / 3} Khafre Pyramid

arctangent {1 / 2}  +  arctangent {1 / 2}  +  arctangent {1 / 2} = arctangent 5.5 Whip

arctangent {4 / 3} Khafre Pyramid  +  arctangent {18 / 99} = arctangent 2

arctangent {1 / phi}  +  arctangent {1 / phi} = arctangent 2

Review IMAGE ONE,
to understand standard 3-4-5 triangle Khafre Pyramid dynamics.
Note as well,
in the actual pyramid design itself,
the Corner Angle dimensional length -- square root 34 <---
automatically introduces,
the megalithic yard of 2.72 feet <---  34 = {12.5 x 2.72}.

[Image: 0Uk34w3.jpg]






IMAGE TWO
Trying to find a way to best display the specific angle combinations,
I came up with this:
360 degree circle design.
Look straight to the 12 o'clock position of the circle.
You see these numbers and letters:
m = angle m --- {1 / 2} is the angle tangent. -- arctangent {1 / 2} = 26.56505118 degrees.

h = angle h --- {18 / 99} is the angle tangent. -- arctangent {18 / 99} = 10.30484647 degrees.

A = angle A --- Phi is the angle tangent.

a = angle a --- {1 / phi} is the angle tangent.

These angles all refer to the interior angles surrounding the central point.
This design sets the stage for further angles to subdivide what is seen now.
But for now,
the design system needs to be kept simple for the viewer to understand in stages.

This setting shows the inter related nature of these angles.
The goal:
is to reveal the angle with tangent 5.5 -- sacred seked of the Khufu pyramid.
It is found in the bottom half of the circle,
after the angles therein,
are subdivided into their important components,
presented in PART TWO <--- 
a second post of this material in a couple of weeks or so.
It is impossible to present both designs properly in one post.

Of course,
the surpise feature is angle j Whip
The -- angle j -- tangent,
aligns 360 and 260 count systems respectively in the numerator and denominator.
angle j = arctangent {180 / 260 Tzolkin}  -- or {360 / 520}.

This is the ancient code form of the angle tangent,
and important to be expressed that way.
The tangent also equals:
{360 x 3600} -- divided by -- 1872000 Mayan Long count.

The reduced form is:
{180 / 260} = {9 / 13}

IMAGE TWO

[Image: WacrS0d.jpg]



This is a preview of PART TWO:
Scan through the angles in the circle.
Note the angle i <---
with tangent = {1 / 3}.
This angle has two extremely important building block angles,
in this complexity of interactive angles.
8.130102354         +  10.30484647                18.43494882 degrees
arctangent {1 / 7}   +  arctangent {18 / 99}  =  arctangent {1 / 3}.

This is important because the angle -->  arctan {1 / 7} = 8.130102354 degrees,
has the sine:
square root 2 -- divided by -- 10, 
and those square root two dynamics augment the angle infrastructures.

last but not least:
arctangent {1 /3}  +  arctangent {1 / 3}  = K* = inverse slope Khafre Pyramid -- arctan {3 / 4}.



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Grand Unification of Ancient and Modern Geometry and Mathematics

Khufu Pyramid sacred seked 5.5 -- as an angle tangent.
Review my last post and image for complimentary data involved <---
In the last post image:
You see different angle combinations that equal the Khafre pyramid slope in degrees.

arctan{1 / 2} -- plus -- arctan{1 / 2} = Khafre Pyramid side face in degrees.
26.56505118              26.56505118  = 53.13010235  degrees

arctan{1 / 3} -- plus -- arctan{180 / 260} = Khafre Pyramid
18.43494882               34.69515353          = 53.13010235 degrees

The angle tangent {180 / 260} encodes pure ancient calendar count system dynamics.
It aligns the 360 count into the numerator,
and the 260 Tzolkin count into the denominator.
Angle tangent {180 / 260} = 360 / by {2 x 260 Tzolkin}.

The reduced form of the fraction 180 / 260  =  9 / 13.
The 9 and the 13 are as Mayan and MesoAmerican in count as can be hoped for.
9 x 13 = 117 -- an important number in the Dresden Codex.
Their own cultural geometry no doubt took into account this angle tangent,
as an important angle in the 3-4-5 triangle generated Khafre Pyramid geometry.
The ancient and modern code unifies within the sine and cosine as well {next post}.

This angle tangent {180 / 260},
can be subdivided as well.

arctan{1 / 2} -- plus -- arctan{1 / 7} = arctan{180 / 260}
26.56505118              8.130102354  = 34.69515353

Thus the interconnected angle tangent combination follows as such:
arctan{1 / 2} -- plus -- arctan{1 / 3} -- plus -- arctan{1 / 7}  =  Khafre pyramid
26.56505118              18.43494882               8.130102354  =  53.13010235

To then proceed along the same routes of approach,
the simple fundamental pathway to arctangent 5.5 <---
goes like this:

arctan{1 / 2} -- plus -- arctan{1 / 2} = Khafre pyramid -- plus --> arctan{1 / 2} = arctangent 5.5
26.56505118                26.56505118  = 53.13010235                    26.56505118     79.69515353

artan{1 / 3} -- plus -- arctan{1 / 2}  = arctan 1 -- plus --> arctan{180 / 260}  = arctangent 5.5
18.43494882             26.56505118    = 45                         34.69515353               79.69515353

--------------------------------------------------------------------------------------------------


My last image introduced the most basic fundamentals,
in simplistic angle tangent connectivities,
attached to Khafre Pyramid geometry.
The next step,
is to integrate ---> golden rectangle phi geometry into the Kahfre pyramid geometry constructs,
while also exposing how the Khufu Pyramid sacred seked 5.5,
functions as an angle tangent within those geometry infrastructures.

This is a bridge post between my last post,
and the summation of it all in my last post on this subject.
What was "magic simplicity" in previous posts,
suddenly can become quite complex,
when phi dynamics are introduced.

I felt that this geometry display was needed to help the viewer,
understand how specific angles with tangents like:
{1 / 3}, and {1 / 7}, or {11 / 2} = 5.5 {Khufu Pyramid seked},
integrate with phi geometry angle tangents,
to account for the Khafre Pyramid geometry.
Khafre Pyramid side face slope tangent = 4 / 3 <---
Khafre Pyramid inverse slope tangent = 3 / 4 <---
Remember,
that angle tangent 5.5 = {11 / 2} = {99 / 18},
the Khufu pyramid seked,
has the inverse angle tangent:
{18 / 99} = 2 / 11.

If you are doing the math seen in the images,
it is ALL exact Whip
but you have to perform the math processing correctly.
Scientific hand calculators,
round off at the tenth decimal placement,
so you have to use proper process to get exact results.

In other words,
if you use the golden rectangle -- angle tangent = Phi,
you cannot just calculator click the numbers:
1.618033989 = Phi.
You have to do it like this,
for your calculator to properly calculate the decimal placements:
click:
Square root 5 <---
add ONE. ------ your result will be: 3.236067977,
Now,
divide by Two,
equals,
true Phi placed into your calculator properly for exact results <---

You cannot just type in:
1.618033989
into your calculator,
you have to perform the process then continue the equations.
So using that exact process,
to obtain inverse phi <---
you simply click:
1 / x
function on your scientific calculator.
If your calculator does not have the option {1 / x},
you cannot perform geometry tangent work,
that has square roots and so forth.

----------------------------------------------------

IMAGE

Top section of the image.
Where you see the angle letter designation:
c a b = arctangent Phi,
that means:
angle c + angle a + angle c = the angle with tangent Phi.

This design is created to show how the angle with tangent 5.5,
integrates with these angles tangents:
Phi  and  {1 / phi},
Khafre Pyramid slope tangent {4 / 3},
and especially,
simple building block angles,
with tangents:
{1 / 3} and {1 / 7}.

Angle c,
in the image,
is subdivided into angles d and e.
Note that the building block angles,
d and e,
are functions of square root 5 and phi geometry tangents.

Look at the angle a b b a = arctangent 5.5 Whip
The premier angle featured in this study,
with the tangent 5.5 -- Khufu pyramid seked 5.5,
is constructed in a primary combination of these two angles:
arctangent {1 / 7
and 
arctangent {1 / phi}.

Now look to the central section of the image.
The exact same geometry is offered,
with an addition of the -- inverse phi tangent-- angle a,
on each side of the original design in the top section of the image.

This creates a perfect half circle of 180 degrees.

Now,
look to the -- Khafre pyramid -- inverse slope angle:
angle a e <---
Here the combination of two phi related geometry angles,
come together to form the Khafre inverse slope.
angle a + angle e = K* = inverse slope Khafre Pyramid = arctangent {3 / 4}.

K* = 36.8698765 degrees.

But here is where the "magic simplicity" comes back into play:
K* = arctangent{3 / 4} = angle a + angle e <---
K* = arctangent{3 / 4} = angle c + angle c <----

angle c = arctangent {1 / 3} = 18.43494882 degrees

thus:
arctangent{1 / 3}  +  arctangent{1 / 3} = K* = arctangent {3 / 4}.

Last but not least,
by adding the inverse phi tangent angles to the original design,
angle a c a = arctangent 7
emerges in the angle infrastructures.

Note angle: b + b -- featured as arctangent {2625 / 9000}  =  2625 /  by {25 x 360}.
You see the extrapolation at the top of the image,
using the number: 2625 <---

fibonacci ancient phi progression starts like this;
5 / 8 = 365 / 584
5 / 8 = 0.625

0.625 --->  + 1 = 1.625 ---> + 1 = 2.625 
-------------------------------------------------------

The bottom section of the image goes through related data,
to aid the viewer in understanding,
both the simplicity and complexity of these building block angles.

In particular,
angle b + b = arctangent {21 / 72},
is featured.
I have supplied the extrapolations of the angle tangent fraction,
to point to the ancient code within. 
Most dynamically,
the square root two cubit 20.62394778,
functions within this tangent infrastructure.
But look to last equation presented there.
in that sequence of angle tangent equations.
This is the ancient code within.
angle b + angle b <----
equals:
arctan{1 / 7}  +  arctan{1 / 7} = arctan {21 / 72}  --- {21 / 72}  =  {105 / 360}.

To access the dynamic ancient angle tangent code,
you set the -- denominator -- to pure 360 count.

angle bb = arctan{105 / 360} = arctan{21 / 72}.

105 = 16  x  6.5625 Whip

Thus the tangent of angle bb,
equals:
16  x  6.5625 -- divided by -- 360.

The ancient code key is in the number 6.5625:
aPi = 22 / 7

6.5625  x  aPi = cubit 20.625

6.5625 x 3.1416 = cubit 20.61675
6.5625 x 3.14 18 18 18~ = cubit 20.618 18 18~

6.5625 x  pi = pi cubit 20.61670179,
thus to access,
the mirror or complimentary pi cubit,
you use the formula:
{18 / 360} -- divided by -- Pi value = cubit.

{18 x 360} -- divided by -- Pi = pi cubit 20.62648062

Multiply the two pi cubits together:
20.61670179~  x  20.626489062~  =  425.25
and
the corresponding ancient cubits:
20.618 18 18 18~  x  20.625 = 425.25 = 162 x 2.625 <---



Much more to follow,
with magnificent ancient angle tangent code to be revealed.
Check out the sine and cosine of arctangent 5.5,
or 79.69515353 degrees,
in the meantime.


[Image: goYhRwF.jpg]



Guitar

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Grand Unification of Ancient and Modern Geometry and Mathematics

Khufu Pyramid sacred seked 5.5 as an angle tangent.
There will be important followup posts to this one.

The featured angle from the start of this exercise was:
arctangent{18 / 99}            {review last two posts}
10.30484647 degrees

This angle is the complimentary angle,
to the sacred seked assigned angle:
arctangent 5.5
79.69515353 degrees

arctangent{18 / 99}  plus  arctangent 5.5  =  90 degrees.
10.30484647          plus    79.69515353    =  90 degrees.

sine 10.30484647  =  cosine 79.69515353
cosine 10.30484647 =  sine  79.69515353

It is in the sine and cosine of these complimentary angles,
that superb ancient planetary timeline code exists.
but that is jumping ahead,
let's look at image one.

----------------------------

IMAGE ONE

The original 360 degree circle of angles is restructured,
and sub divided with additional important angles,
from within this ancient geometry infrastructure.

In the circle are --- letters -- and numbers.

The letters designate the specific angle.
The numbers are specific angle tangents of the predominant angles.
Example:
see the letters in the angles -- m -- that is angle m Whip
With letter m for angle m,
there is the fraction 1 / 2 -- that is the tangent of angle m.

angle i -- tangent = 1 / 3 -- arctangent {1 / 3} = 18.43494882 degrees.

angle K* {inverse slope Khafre pyramid}
tangent = 3 / 4 -- arctangent{3 / 4} = 36.86989765 degrees.

The rest of the angles marked with letters are defined in the data supplied.

These are the most fundamental simple building block angles,
{and their inverse complimentary angles}
evolving into the Khafre pyramid geometry infrastructures:
arctan{1 / 2}
arctan{1 / 3}
arctan{1 / 7}
arctan{18 / 99}  --- 18 / 99 = 2 / 11
and
arctan{180 / 260}  =  360 count / {2 x 260 Tzolkin}.

Within these Khafre Pyramid geometry infrastructures,
the phi and inverse phi geometry is intertwined.
If you look closely at the angle combinations offered,
then you should see how the phi and square root 5 related geometries,
work together with the above simpler fractional angle tangents.

Of special note is the introduction of angle f <----.
arctangent{1 / 7} = 8.130102354 degrees
This angle,
has a very important angle sine:
sine 8.130102354 degrees,
equals:
square root 2 -- divided by -- 10  <---
also equals:
square root 1.62 -- divided by -- 9 <---
and,
square root 1.62 -- is the Khufu Pyramid slope angle tangent,
in pure square root 2 dynamics,
with specific cubits 20.61923374 and 20.62394778.

In the Khufu Pyramid,
slope tangents:
{14 / 11} = 1.27 27 27 27~ ---> arctan {14 / 11}  = 51.84277341
sqrt 1.62 = 1.27 27 92206 ----> arctan {sqrt 1.62} = 51.84419346

Look it over,
try to follow the angle combinations in the data,
with the angles within the circle.
Note angles {j + j} -- for the next image.
This image sets the stage for IMAGE TWO.

[Image: NLckgW1.jpg ]


IMAGE TWO:

The bottom 180 degrees of the circle are highlighted,
with very important angle subdivisions.

One highly exposing method of angle tangent combinations,
uses angle f <---> like this:
arctan{1 / 7}
plus
arctan{1 / 7}  =  angle n <---
the tangent of angle n --- seen in the image,
is written to expose the ancient pyramid math coding,
with:
2625 in the numerator,
and
25 x 360 -- in the denominator {ancient 60 or 360 count system}.
Look,
to the bottom left in the image and the equation with 2625,
in the numerator.
The equation exposes the Khufu pyramid slope tangent with cubit 20.625.
You can install -- any cubit,
and get a corresponding pyramid slope tangent.

Now look to the far right in the image for this equation:
angles:
f  +  f  +  f  =  arctangent {365 / 805} =  arctan {584 / 1288}
Suddenly,
the ancient Earth year 365 and Venus synod 584 cycle numbers are seen.
But what does the denominator:
805
have to do with numerator 365?

This test shows the ancient code:
805 + 365 = 1170 = 4.5 x 260 Tzolkin

805 - 365  =  440 -- number of cubits in the Khufu Pyramid base length.

Hmm2

Now look at the tangent of angles: {j + j} --->   seen in IMAGE ONE.
arctangent {1170 / 440} = angles {j + j}.

And the grand finale using angle f Whip
angles:
f + f + f + f  ---> has the best angle tangent written in the ancient code.
You can see the angle tangent there in the image.
the numerator contains:
the Khufu Pyramid slope tangent -- times -- the mile in feet 5280.
the denominator:
3875 megalithic yards of 2.72 feet.

Two new angles,
are also found within this infrastructure of angles:

angle s = arctan {272 / 496}  --- 272 - megalithic yard.
and
angle {s + i} = arctan {41 / 38}

Very important angle sub divisions.
These two angles will get explained further,
along with the Earth year angle tangent {365 / 805},
in the next post.


[Image: 0IoTa6N.jpg]



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Grand Unification of Ancient and Modern Geometry and Mathematics

Sacred seked 5.5 and Khafre Pyramid geometry

Khafre Pyramid geometry connects phi related geometry,
within the side face angle in 3-4-5 triangle geometry.

arctangent {4 / 3} = K = 53.13010235 degrees.

As shown in the previous diagrams,
a specific genre of angles all intertwine uniquely.
The keynote angle that initiated the study was:
arctan {18 / 99} = 10.30484647 degrees = angle A

90 degrees - minus - angle A = 79.69515353 =  arctangent 5.5 = arctan {99 / 18}

18 / 99  =  2 / 11  =  1 / 5.5  =  3.75 / 20.625 cubit

Essentially both angles,
arctan {18 / 99}  and  arctan 5.5,
qualify as the ancient sacred seked.
Though diffferent angles in degrees, they are inserparable.
The sines,
and cosines,
of the two angles, simply switch in placement.
sine 10.30484647  =  cosine 79.69515353, and vice versa.

A combination of simple fractional angles,
provides a platform of building block angles,
in both the ancient and modern perspective.
The Khafre Pyramid side face angle,
K = 53.13010235 degrees,
employs the building block angles:
{unraveling the mix}

arctan {1 / 7} - plus - 45 degrees = K = Khafre Pyramid side face angle

arctan {18 / 99} - plus - arctan {180 /260} =  45 degrees.

arctan {18 / 99} = 10.30484647 degrees
equals:
arctan{ 1 / {5Phi + 3} - plus - arctan{ 1 / {5Phi + 3}
        5.152423234                    5.152423234  =  10.30484647 degrees

arctangent Phi - minus - arctan{ 1 / {5Phi + 3}  = K = Khafre Pyramid
58.28252559                            5.152423234         = K = 53.13010235 degrees

The Image of Pyramids:
employs two of these building block angles,
arctan {18 / 99} and arctan {1 / 7}
to perform a pyramid geometry test,
to generate the dimensional lengths as observational targets.
The dimensional lengths,
offer specific prime numbers that are generated into those lengths,
and those are the target observations.
Those prime numbers generate into the later geometry angle tangents,
seen in the prior diagrams.

Top pyramid of Image -- dimensional lengths
A single slope pyramid.
arctangent {18 / 99} -- side face angle.
Length Y displays the phi related component.
Length W displays a square root component -- sqrt 242 = {11 x sqrt 2}.
Length Z,
generates the prime -- 41.
This prime evolves,
and manifests in the angle tangent {41 / 38} {see prior post diagram}.

The highly important ancient,
and extremely short term planetary time line cycle,
is revealed by the prime 41,
as seen in the Jupiter and Earth cycle.

Second Pyramid in Image
A two slope pyramid:
created using both angles:
arctan{18 / 99}
and
arctan{1 / 7}
Length Y displays the phi related component.
Length X displays the square root two component.
Length Z,
generates the prime 107,
which in turn produces the Venus sidereal x 10 = 2247 = 21 x 107.

Length W suddenly shows up in the Khufu Pyramid:


Third Pyramid in Image.
Khufu Pyramid arctangent {14 / 11}.
This is displayed to show that:
Length W <---
from each of the last two pyramids,
is generated by the Khufu Pyramid.
Length Z,
generates the prime 73,
the fundamental prime,
in the calendar count 365 day Earth year, and 584 day Venus synod.


Bottom Pyramid in Image
This two slope pyramid is generated from:
arctan {18 / 99} - and - arctan {1 / 2}.
You can see the equation using the two angles,
which results in the -- inverse slope of the Khafre Pyramid = K*

Length X generates the square root 5 geometry component:
sqrt 20  =  2 x sqrt 5  =  {sqrt 5 + 5} / Phi <---

Length Y generates the phi component as well.
Length Z,
generates the prime 47 ---> 3 x 47 = 141.
The spectacular cycle of the Earth Lunar month,
with the Saturn sidereal is revealed with the prime 47.
47000 Lunar months = 129 Saturn sidereal.
The prime 47,
is also the fundamental prime in the ancient Metonic cycles.

And the icing on the Khufu Cake !
Length W,
generates the prime Kabbalah 137 Whip



[Image: s5UBFFU.jpg]



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Grand Unification of Ancient and Modern Geometry and Mathematics

Masterpiece Ancient Angle Tangent Code


The current geometry investigation began with the isosceles triangle,
seen below in:
IMAGE ONE.
A unique perspective on this triangle geometry is presented.

The tangent of angle A = {18 / 99} = {2 / 11} = {1 / 5.5} = {3.75 / 20.625 cubit},
angle A = 10.30484647 degrees,
and occupies the full apex angle of the isosceles triangle in image one.

The two identical base angles,
angles B,
have a fascinating angle tangent:

{5 Phi + 3} = Phi --> to the fifth power,
are equivalent formulas for the angle B tangent.

Think about the beauty of this triangle.
The two equal base angles both have the unique angle tangents:
Phi--to the fifth power Whip
and those two angles,
as base angles in the isosceles triangle,
generate the ancient angle of particular focus,
arctangent {18 / 99} = 10.30484647 degrees,
as the Full Apex Angle!

The angle attributed to the sacred seked 5.5,
arctangent 5.5 = arctangent {99 / 18} = 79.69515353 degrees,
is essentially,
the complimentary angle of arctangent {18 / 99}.

90 degrees -- minus -- arctangent {18 / 99}  =  arctangent {99 / 18}
                               10.30484647 degrees       79.69515353 degrees

90 degrees -- minus -- arctangent {1 / 5.5}   =   arctangent {5.5}
                               10.30484647 degrees        79.69515353 degrees

The sines and cosines,
of the two angles are identical,
only difference being that they switch positions.
The sine of 10.30484647 degrees,
is the cosine of 79.69515353 degrees. 

------------------------------------

So what do we know?
The isosceles triangle base angles clearly display components of Phi.
That is square root five geometry and mathematics.
The square root five and phi geometry math present in the base angles tangents,
is transferred to the sine and cosine,
of the fractional angle tangent:
arctangent {18 / 99} = 10.30484647 degrees <---

The masterpiece ancient code pointed to,
is exposed using square root five to generate that code from the sine and cosine.
The results seen,
with the ancient process unveiled,
uses the numerators of the sine and cosine ---> as angle tangents.
This then reveals the planetary calendar count timelines,
in the process shown.
BOTH:
the Venus synod = 584 days,
and the Mercury synod of 116 days,
are exposed from the original ancient angle geometry code.

It is statistically impossible Whip
for that result displayed to happen by sheer mathematical random chance.
No rabbit was pulled out of the math magicians hat,
and there is no rabbit hole full of chance possibilities.
Two highly important calendar count planetary time lines,
that are distinctly generated by the ancient math and pyramid geometry,
come to presence in the angles, sines and cosines.

These results,
repeat themselves in grand fashion,
in the next set of equations in the bottom section of the image,
by using Phi -- as a catalyst to generate the ancient code,
from the sines and cosines.
Take special note of the cosine equation,
which features the Khufu pyramid slope tangent <----

This is a major reason why,
arctangent {18 / 99} and arctangent 5.5,
have such intensely connected presence and ancient relevance,
in the cumulative Khafre Pyramid and phi geometries, here to fore presented.


[Image: 7kXhhUB.jpg]




----------------------------------------------------------------------------------

IMAGE TWO

How magnificent is that first isosceles triangle in the image?
The original triangle -- in the above image one,
with the Full Apex Angle A -- arctangent {18 / 99},
is placed perfectly within:
the cross section isosceles triangle of the Khafre Pyramid side faces,
displayed with the two base angles --> K = arctangent {4 / 3} = 53.13010235 degrees.

Thus a dual isosceles triangle diagram is seen.
This dual isosceles triangle alignment,
was created to isolate ---> the two equal angles B ---> which beautifully display as:
arctangent {1 / Phi} Whip

Think about it.
That is a beautiful, masterpiece ancient Khafre Pyramid geometry, fully exposed.
And you saw it here first.

From there,
the next two isosceles triangles are shown as a bit of a progression.
That idea is to show how the two unusual new angles,
and their inverses <---
arctangent {272 / 496} ... arctan {496 / 272} --- 272 megalithic yard.
and
arctangent {38 / 41}  ... arctan {41 / 38},
emerge in the Khafre Pyramnid geometry,
by subdividing <---
the inverse phi angle B -- as seen subdivided.
These angles are re-introduced from earlier,'
and more specifically defined in other words.

In the far bottom right of the image,
I extrapolate the content and code,
in the angle AC tangent, with the megalithic yard number 272.
The angle -- arctangent {272 / 496} = 28.73979529 degrees.
It was quite the surprise to see the number 496,
generate the ancient pi progression value seen with the process.
Sure,
the fraction {272 / 496} reduces to {17 / 31}.
the ancient code is built upon the multiples of specific prime numbers,
generated by specific geometries <---

Note that -- 17 squared -- plus -- 31 squared  = 1250 <---
now,
take the square root of 1250 <---
equals:
100 -- times -- the tangent of tetrahedral 19.47122063 degrees :holy cow:

The numerators and the denominators of the fractional tangents,
of those important angles in the overall Khafre pyramid geometry infrastructure,
all perform like that.

The angle CAC seen in the just prior isosceles triangle,
arctangent {41 / 38} = 47.17474411 degrees,
incredibly shows a known Khufu Pyramid cubit and height shown.

Note that -- 41 squared -- plus -- 38 squared =  3125.
now,
take the square root of 3125 <---
equals:
10 x {Phi to fifth power} -- minus -- 55 <--- Whip

On that note ...
in prior post images {two posts ago},
we see the angle tangent {805 / 365} emerge in the subdivided geometry.
The 365 correlation being the Earth year = {5 x 73}
{the prime 73 is also generated,
into the corner angle dimensional length}.

The reduced fraction is {161 / 73}.
Once you perform the operation above,
you will see the decimal:
176.7766953
but then,
you have to know how to make it work.
A hop, skip, and jumping jack Guitar flash away!



[Image: G0gWFs4.jpg]




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Following through with the series of recent posts,
and the exposure of the important angle combinations displayed.
{see the images and text in the last few posts}

This exercise started with the angle:
arctangent {18 / 99} = 10.30484647 degrees.
Another extremely important angle emerges in the diagrams:
arctangent {180 / 260} = 34.69515353 degrees.

You will now see the most important and dynamic test ever applied,
to Khufu Pyramid geometry,
in this post with a very special: Tetrahedral Geometry Test Whip

The process is easy to see, and understand the applications.
Nobody in the last 3000 years has accomplished this,
and produced such definitive and unparalleled results.

This is a simple and fundamental angle tangent application,
into the Khufu Pyramid -- Corner Angles <---

Tetrahedral Geometry Testing
Within The Khufu Pyramid Corner Angles Cross Section:
Isosceles Triangle -- pyramid positional points -- D P D --

Every square base pyramid has this - side face - slope formula:
The pyramid angle slope tangent, 
equals:
height -- divided by - one half the full base length.

The Khufu Pyramid slope tangent,
results from the ancient formula:
280 cubit height,
440 cubit base.
This produces the slope tangent:
{14 / 11} = {4 / aPi} --- where aPi = 22 / 7.
That is the inescapable math <---

THE PYRAMID has a height ---> 14 units <--- {or it can be 280 cubits}
and the HALF base length --> 11 units <-- {or that can be 220 cubits}
thus
the Corner Angle,
has height 14 units,
and HALF base length -->  {11 x sqrt 2} units <--

Khufu Pyramid Corner Angle
tangent = 14 / {11 x sqrt 2}  = square root 242 =  dimensional length W 


[Image: h3Gk0tA.jpg]



Grand Unification of Ancient and Modern Mathematics and Geometry

Sacred Seked 5.5 as an angle tangent in Khafre Pyramid Dynamics

Tetrahedral Testing in the Khufu Pyramid Geometry



IMAGE ONE

Starting with angle:
arctangent {18 / 99},
a procession of "building block angles",
are generated within the Khafre Pyramid realm of geometry,
that contain both fractional,
and phi geometry angle tangents which collectively compliment each other.
{see previous posts and images}

At the top of the first image below,
a set of specific angle tangent equations are displayed,
which add up to the Khafre Pyramid side face angle in degrees.

Of distinct note as well,
in the semi-circle of angle tangents displaying the angle combinations,
are the two side by side angles:
arctangent {180 / 260} = 34.695215353 degrees.

angle tangent
180 / 260  = 9 / 13  =  360 / {2 x 260}

The angle tangent fraction is written as:
180 / 260 Tzolkin <---
to indicate the ancient intent of interpretation of that angle.
Mayan cultural calendar math components of pyramid geometry:
260 and 360, are seen in the angle tangent.
Mayans no doubt,
like any other advanced cultural math science builders of pyramids,
understood the 3-4-5 triangle geometry of the Khafre Pyramid,
and the associated synergistic angles displayed in the image.

Ancient cultural geometry observed the exact same "building block angles",
that align to produce the Khafre Pyramid geometry and mathematics.
The Olmec and Mayans calendar count dynamics,
evolve directly from this aspect of the angle geometry:
arctangent {9 / 13} =
arctan {180 / 260 Tzolkin} = arctan {360 / {2 x 260}.

The numerator in {180 / 260}:
is one half of the Mayan civic calendar Earth year of 360 days <---
Mayans {and Olmec},
aligned their calendars in this fashion:
The Mayan Civic Earth Year was defined as 360 days -- plus the 5 Uayeb days.
Mayan:
360 = 18 x 20 --- Egyptians and Sumerians used: {12 x 30} = 360.
180 = 9 x 20 -- half year -->                                {6 x 30} = 180

260 Tzolkin spiritual calendar = 13 x 20.

Mayan Long Count 1872000 <---
equals:
5200 x 360 = 7200 x 260 Tzolkin <---

Mayan Calendar Round 18980 =
52 x 365 Earth year = 32.5 x 584 Venus synod.

The incredible ancient code Whip
found in angle:
arctangent {180 / 260}  = 34.695215353 degrees,
resides within:
the sine and cosine -- via -- square root 5 Whip

The numerator of the sine and cosine,
directly signify two very important angle tangents.
Both of which emerge directly from Khufu Pyramid geometry dynamics.

IMAGE ONE
arctangent {180 / 260 Tzolkin}
within Khafre Pyramid geometry infrastructures.

Geometry design has been truncated from a recent previous image.

[Image: pf4Eqxs.jpg]


---------------------------------------------------------------------------


The distinct angle,
arctangent {180 / 260} -- AND -- it's inverse angle:
arctan {260 / 180}.
are clearly defined and generated within the Khafre Pyramid realm,
of important "building block angles".
You see the equation of angle tangents that lead to the Khafre Pyramid.
In the semi-circle of angle tangents seen in the above image,
arctangent 5.5 Whip
is interwoven in there as well.

Note this unique angle combination,
using the -- inverse angle -- to arctan {180 / 260}:

arctan {260 / 180} -- plus -- arctan 5.5    =  135 degrees
       55.30484647    -- plus -- 79.69515353 =  135 degrees

And 135 degrees is the corner angle of the octagon <---

The -- Tzolkin -- 260,
is not just a Meso American calendar count construct.
The 260 COUNT system far predates the Mayans.
260 uses the prime number 13,
distinctly generated in a variety Khufu and Khafre Pyramid geometry constructs.
I have shown this many times in previous posts.


NOW  to prove the statement above,
that the 260 count also is evidenced in ancient Khufu pyramid dynamics <---
look at the isosceles triangle <---
in IMAGE THREE. 

IMAGE THREE
displays a Dual Overlay Isosceles Triangle,
with one isosceles triangle,
perfectly positioned within the:
Khufu Pyramid Corner Angles <---

Tetrahedral Geometry Testing Whip
Within The Khufu Pyramid Corner Angles -- Cross Sectional Isosceles Triangle
The angle corresponds to points D P D


IMAGE THREE
Tetrahedral Geometry Testing
Dual Overlay Isosceles Triangles in Khufu Pyramid Corner Angles A

What do you see in image 3?

The test <---> is performed by positioning TWO:
tetrahedral Hi 19.47122063 degree angles Whip
into the:
Full Apex Angle -- of the corner angles cross sectional isosceles triangle.

The test question:
is to determine the geometry <-- of the isosceles triangle within <--
that is produced,
by the test positioning of the most important tetrahedral 19.47122063 degree angles.
Note the dimensional lengths,
of the base line !
Note angle D !
The masterpiece ancient codes,
are initially found in the sine and cosine of the angle:
arctangent {180 / 260} <----


[Image: Dt4v8Rq.jpg]
-------------------------------------------------------------------------------------------------------------------------

.
The resultant isosceles triangle created by the test,
and seen within the Khufu Pyramid Corner Angles,
has two base angles B,
of:
square root 3.38  = {1.3 x sqrt 2} =  5.2 x {tangent 19.47122063 degrees} --- 338 = 260 Tzolkin x 1.3 

The Full Apex Angle D,
clearly displays the Tzolkin 260 in the angle tangent numerator.

angle D = arctangent {260 Tzolkin -- divided by -- {119 x sqrt 2}.
NOTE:
This fractional angle tangent's whole numbers, 260 and 119,
cannot be reduced any further <---
By applying the test angles of tetrahedral 19.47122063 degrees,
the interior isosceles triangle created,
within the Khufu Pyramid corner angles isosceles, 
produces a Full Apex Angle -- tangent <---
that clearly and solely <---
identifies,
the 260 Tzolkin calendar count value -- in the tangent numerator!

Right there, within PURE ANCIENT Khufu Pyramid geometry, 
tested with the most important tetrahedral angle Smoke 

119 <--- is seen in the denominator.
119 = {7 x 17}  =  ancient Pi value 31416 - divided by - 264.
Which means:
you can legitimately reconstruct further in ancient code,
the angle D tangent to:

264 x 260 Tzolkin
divided by
31416  x  sqrt 2

31416 = 17 x 1848.
All prime number 17 values,
like 31416,
the megalithic yard 2.72 feet, {272 = 16 x 17}
will function in defining the angle tangent denominator.
31416 = 264 x 119 = 168 x 187 = 115.5 x 272 = 88 x 357,
and
314160 = 15232 x 20.625 cubit,
gives us:
the icing on the Khufu Cake: 
My Grand Unification Tropical Earth Year Dance2 {see earlier posts this year} 
365.2430697 days 
equals:
{28 x 20.625 x sqrt 2} -- divided by -- sqrt 5, 
can be written,
into the equation with the megalithic yard:

264 x 260 x sqrt 5                               33 x 260 x sqrt 5
divided by                    --- reduces to ---      divided by
272  x  365.2430697                           34 x 365.2430698  
equals:
tangent angle D <--- 

The base line of the isosceles triangles overlay,
displays lengths that reveal the 365 day Earth year in the calculations.

The sine and cosine of angle D,
also reveal functions of the 365 day Earth Year or the 584 day Venus synod.

The ancient codes are revealed in the mathematical evidence presented.
The ancient Khufu Pyramid geometry,
was tested with:
PURE tetrahedral angles Whip
and the results are there to prove that the 260 count system,
was clearly understood,
by ancient cultures that revered sacred geometry,
and it is not exclusive to just the Mayans and the Olmec.

---------------------------------------------------------



The sine of the  angle:
arctangent {180 / 260} = 34.69515353 degrees,
displays this equation:

sine 34.69515353 degrees,
equals:
square root 1.62
divided by
square root 5

If you go back a couple of months to my - square root two cubits - review,
you will see that in the Khufu Pyramid,
square root 1.62 = 1.27 27 92206
is:
the slope tangent of the Khufu pyramid Side Faces,
in PURE,
square root two dynamics.
NOTE:
slope tangent:
14 / 11  =  1.27 27 27 27 27~
slope tangent:
sqrt 1.62  =  1.27 27 92206 <----

The best way to define this angle -- in a -- Tetrahedral Geometry Test Whip
is seen in the triangle below,
showing three angles that add up 180 degrees.
{all triangles have 180 degrees}.

The Khufu Pyramid slope tangent in pure square root two dynamics:
square root 1.62
is seen as angle C.
The tetrahedral test angle -- is the famous "electron spin angle" -- {Horace Crater UTSI}
with angle B tangent:
square root 8 

Angle A -- results with the incredible angle tangent:
5 x 116 Mercury synod
divided by:
260 Tzolkin  x  sqrt 2 

The angle has been written to express the ancient codes. 

[Image: R4fMKfb.jpg]


Merry Christmas
Happy New Year

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Grand Unification of Ancient and Modern Geometry and Mathematics

Khufu Pyramid Seked  5.5  as an angle tangent

The Khafre Pyramid Circle of Geometry

This is a wrap up post for the series of images and texts over the last month.
We started with attributing the angle with tangent 5.5 <---
as a representation of the Khufu pyramid seked 5.5 <---
The most important development -- was in the last post:
the Khufu Pyramid Corner Angles Cross Section
Dual Overlay Isosceles Triangle.
The interior isosceles triangle proves that by itself,
the number 260 <--- {Tzolkin},
was an important count system number within Khufu Pyramid dynamics.
This process and method will be built upon in the next post <---

But for now,
I need to post a couple of images for reference to the series of images,
which identified SPECIFIC angles within the Khafre Pyramid dynamics,
all related and interconnected with Phi geometry dynamics.

A series of angles and angle tangents,
all in fractional form,
were presented in the circles of geometry.
They all have a distinct common thread of connection.
I will explain this with the definitive process.

arctan 5.5 --- and it's inverse -- arctan {18 / 99} ----- {5.5 = 99 / 18}

Process:
18 squared
plus
99 squared
equals
10125 ---> = 16.2  x  625
What do you see there?

Take ANY cubit  Whip
--- divide by 16.2 -- 
and you have a researched Khufu Pyramid slope tangent:
Cubit 20.618 18 18 18~  {756 foot base length}
divided by:
16.2
equals
{14 / 11} --- Khufu Pyramid slope tangent <--- =  {4 / aPi}

Pi cubit 20.62648062
divided by
16.2
equals
{4 / Pi} -- Khufu Pyramid slope tangent -- in Pi dynamics.

review again:
10125 ---> = 16.2  x  625  <--- the 625 multiple!

fibonacci 0.625 = {5 / 8} = 365 Earth  / 584 Venus synod
625 = {584000 / 365}

So this is the simple connection:
ALL the angle tangents,
with whole numbers as numerators and denominators in fractions,
end up as a multiple of 5 <--- 25, 625 and 1250 etc etc,
all the way to:
390625 = 625 squared {see angle--->  f + f + f + f }

SO,
if you take the ----> square root <---- of every angle tangent's {numerator sq. + denominator sq.}
you will always get:
a result with a multiple of:
--> sqrt 2, --> sqrt 5, --> sqrt 2 x sqrt 5, --> sqrt 5 / sqrt 2, ---> or 5 {in the form of 625, 1250 etc}
Now review:

Process:
18 squared
plus
99 squared
eqauls
10125 ---> = 16.2  x  625

NOW -- take the:  square root of 10125 <----

sqrt 10125 = {225 / sqrt 5}  =  225 / {2Phi - 1}


OK,
the angle tangent fraction {99 / 18}  =  {11 / 2}  =  5.5 <---
repeat the process with the reduced fraction:

11 squared,
plus
2 squared,
equals:
125 <---
Now take the square root of 125,
equals:
11.18033989  = {10Phi minus 5} = {25 / sqrt 5}
equals:
11.18033989  = {tetrahedral tangent 19.47122063} x {square root 1000}
note:
sqrt 1000 = {10 x sqrt 2  x  sqrt 5}

SO,
if you look at Khafre Pyramid tangent {4 / 3},
4 squared,
plus
3 squared,
equals:
25
sqrt 25 = 5 <--- {3-4-5 triangle geometry}

It is the --->  3-4-5  triangle geometry of the Khafre pyramid,
that connects all the angle tangents I have revealed together,
and produces this process above.

IMAGE ONE
The half circle of Khafre pyramid geometry is at the top of the image,
for angle identification,
and thus displays the process more clearly with all the math presented in the image.
Without going into pages of proofs,
you can figure it out,
if you take the time to slowly go through the image math <---

IMAGE ONE

[Image: LKCxLrg.jpg]



The second image here offers a series of triangles.
Triangles ALL have 180 degrees -- as a combination of the three angles.
In this selection,
I combine a series of the angles from the Khafre Pyramid Circle of Geometry,
to show a different perspective on how they combine and integrate together.

This wraps up the Khafre Pyramid Circle of Geometry,
and the Khufu pyramid seked 5.5 as an angle tangent.

NEXT -- I will take the Khufu Pyramid Corner Angles Cross Section <---
where I used tetrahedral 19.47122063 degrees as the test angle <---
and take that into spectacular ancient geometry connections.

[Image: SGf9qzP.jpg]


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Grand Unification of Ancient and Modern Geometry and Mathematics

Magnificent Ancient Khafre and Khufu Pyramid
Angle Tangent Code Process Revealed in Tetrahedral Angle Testing


This post is the fun part of all this ancient geometry.
It takes less explanation I suppose,
and introduces novel process for defining ancient geometry code.

Two posts back,
the Cross Section of the Khufu Pyramid Corner Angles was displayed,
with a very special process:
Two tetrahedral angles of 19.47122063 degrees,
were inserted into the cross section as a geometry test,
as seen just below in the image.
This tetrahedral test performed on the Corner Angles Cross Section,
unveiled the amazing geometry found in the:
Dual Overlay Isosceles Triangle design and process.

The most important Khufu Pyramid geometry code exposed in current history,
exposes the angle D <---
with the awesome angle tangent:
260 {Tzolkin}
divided by:
119 x square root two. -- where 119 = 43.75 x 2.72 {megalithic yard}.

Here again is that masterpiece geometry.

[Image: 5jQgD1a.jpg]



So the fun part of all this is very simple!
What you do now,
when inspecting ancient geometries,
is to expand such testing into other ancient pyramid geometry.

Let's repeat the above process right into the Khafre Pyramid Whip


Khafre Pyramid Cross Section tested with tetrahedral angles.

The Khafre Pyramid Corner Angles Cross Section,
produces it's own unique isosceles triangle.
Since the Khafre Pyramid standard Side Face,
has an angle tangent:
{4 / 3} -- from the 3-4-5 triangle geometry,
the Base Corner Angle,
thus,
has this angle tangent:
4 -- divided by -- {3 x square root 2} <---
also equals:
{4 / sqrt 18} = ---> the square root of fraction {8 / 9}
{cosine tetrahedral 19.47122063 degrees}

Now we have our Khafre Pyramid Corner Angles Cross section,
as seen in the image below.
The PROCESS,
dictated that we insert the exact same:
tetrahedral 19.47122063 degree angles,
into the Khafre Pyramid Corner Angles Cross Section,
as they were inserted into the Khufu Pyramid Corner Angles Cross Section.

The purpose:
is to determine the secondary isosceles triangle geometry,
created by the insertion of the tetrahedral angles.

Below is the image that shows the process and results!
The results are discussed below the image.

[Image: fjSKes5.jpg]




The two tetrahedral angles d,
of 19.47122063 degrees each,
are seen -- on each side -- of the Apex Angle C <---
of the newly created isosceles triangle,
within the Khafre Pyramid Corner Angles Cross Section.

You are looking at a "dual overlay" of isosceles triangles.
The interior isosceles triangle,
gives us the test results:
the angles B b*,  and the Full Apex Angle C <---

Now look at angle B.
The premier mathematical definition of the angle tangent,
displays this angle B tangent as:
{square root  3.78125} <---
The ancient code is completely obvious here.
Look at the number 3.78125 <---> 378.125 ---> and think -- possible Khufu Pyramid base lengths.
220 cubits,
of 20.625 inches,
equals 4537.5 inches = 378.125 feet.
This is not a coincidence.
The cubit 20.625,
is arranged into the angle B tangent by those mathematics.

Now,
look at the resultant -- Full Apex Angle C Whip
Look at that awesome angle tangent!
The calendar count Mercury sidereal -- 88 days ... {NASA 87.969 days}
AND
fibonacci 89 Whip
appear in the denominator and numerator respectively.

This is highly important,
as you look at the equation ---> to the above right of angle C <----
Restructuring the angle C tangent equation,
we see a direct connection,
between the Khufu pyramid and Khafre pyramid -- Side Face angles,
like this:
The Khafre Pyramid Side Face angle in degrees,
minus:
the Khufu Pyramid Side Face angle in degrees,
equals:
1.287328942 degrees --> and that angle tangent = {2 / 89} <---

Therefore, in the interior isosceles triangle -- the Apex Angle C <---
can be defined by:
the angle tangent --> {2 / 89} -- of the specific angle <---
{found by subtracting,
the two pyramid angles in degrees},

TIMES:
{Mercury sidereal 88 days -- divided by square root two} Whip

Like this:
{2 / 89 fibonacci} ------ the angle tangent for 1.287328942 degrees,
times:
{88 Mercury sidereal / sqrt 2}
equals Apex Angle C -- tangent.

This is profound ancient pyramid geometry dynamics.
Note that angle b* Whip
has this tangent written in cubit code:
15
divided by
{cubit 20.625  x  sqrt 2}.

I am going to leave it up to you to study the designs.
But after you look at the image above,
we take this ALL,
back into:
The Mars Hexad Mounds -- tetrahedral geometry  <----

NOTE: if your calculator does not offer the EXACT same results on these calculated presentations,
due to ten decimal placement capacity,
ALL equations presented are thoroughly researched and correctly defined.
there is no percentage of error.
The equation where the Khufu and Khafre Pyramid angles are subtracted, for instance,
has the exact and precision result seen.

NOW,
Look back in the above image ---> at angle  b*.
Angle b* ---> tangent,
equals:
15
divided by:
{cubit 20.625  x  sqrt 2}.

NOW, 
look at the Mars Hexad image below,
the tetrahedral Hexad -- 
the 6 mounds on Mars at Cydonia,
{Dr. Horace Crater -- University Tennessee Space Institute}
evidenced,
as the tetrahedral hexad in the diagram.

The blue line placed between mound P and mound D,
unveils the Khafre Pyramid Corner Angle  y
within the tetrahedral square root rectangle.

Now look to:
angle b -- plus-- angle b1 <--- in the hexad diagram.
RIGHT THERE,
is the angle  b* <---
in the previous image of the Khafre Pyramid Dual Overlay Isoscles tetrahedral test angles!

90 degrees -- minus -- angle b* = angle B {prior image} <---> tangent = sqrt 3.78125 <---

That is where the geometries all tie together.


[Image: 2naSm2u.jpg]



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Grand Unification of Ancient and Modern Geometry and Mathematics

The Tzolkin 260 and 360 Calendar Count Pyramid
A masterpiece in ancient planetary timeline code.

Khafre and Khufu Pyramid Corner Angles Cross Sectional Isosceles Triangles
with dual overlay geometries.

I had a nearly complete essay for this post written in Notepad,
and the computer ... either malfunctioned,
or I was in the midst of a hack as my mouse wasn't mine anymore.
I had to decide to pull the plug fast,
with nothing saved,
because I had no control left on the machine.
I am swamped with things to do,
so here are the images,
and hope to be back in a day or two with the essay rewritten with more detail.

IMAGE ONE
The Tzolkin 260 and 360 Calendar Count Pyramid,
using the geometry from the earlier posted "wheel or circle of angles"
from the Khafre Pyramid infrastructures.
Pyramid height -- 13 or 260  units
half base length - 9 or 180

My favorite angle from the selection of angles,
in the "wheel or circle of Khafre pyramid inter related angles":
arctan {9 / 13} = 180 / 260 Tzolkin <---
or it's inverse:
arctan {13 / 9= 260 Tzolkin / 180 <---> this angle is chosen to build the pyramid.
thus:
the corner angle:
arctangent {13 / {9 x sqrt 2} 

[Image: y0VCxhR.jpg]




IMAGE TWO
Khafre and Khufu Pyramid Corner Angles Cross Sectional Isosceles Triangles
-- with dual overlay geometries --

Top section:
the Tzolkin 260 and 360 count Pyramid -- Corner Angles Cross Section,
is inserted into the Khufu Pyramid Corner Angles Cross Section.

A masterpiece of planetary timeline equations is produced in the angle tangents.
Saturn sidereal 10759 days        {7 x 1537} -- aligns as a seventh  -- NASA 10759.2
Mercury sidereal 116 days
260 Tzolkin
360 calendar count
AND
the number of Mars Sols in a Martian year:
668.57142857~ days Whip
aligns
as an ancient Egyptian "seventh"

668.57142857~ = 32.5  x  cubit 20.57142857~ --- Hancock root cubit Whip
NASA data rounded:
668.595 days


Bottom section:
I suddenly realized Hmm2
that the actual Aztec calendar --- displayed from my old study,
has the:
13 Dots -- in the box -- above the 8 pointed star.

I had always used that fraction as an angle as well.
arctan {13 / 8}
arctan 1.625 ----> from the ancient phi - fibonacci progression first step.

Thus:
create a pyramid the exact same way:
height 13,
half base length 8,
and
the corner angle becomes:
arctan {13 / {8 x sqrt2}

This above arctangent geometry is inserted into the:
Khufu Pyramid Corner Angles Cross Section.

The fantastic surprise,
Kaballah  137 Whip
shows up in angle C  Dance2

You can clearly see the results.


[Image: gv85nWh.jpg]

Guitar

-------------------------------------

Note as well,
when creating the pyramid side face angle  -- {rather than focusing on the corner angles},
pyramid side face:
{height 13 -- half base length 8}
arctangent
{13 / 8} = 58.39249775 degrees

we take that angle --  58.39249775 degrees,
and subtract:
the Khafre Pyramid side face angle -- arctan {4 / 3} = 53.13010235 degrees
the
result:
5.2623954  degrees <----
which has EXACTLY,
this tangent:
399 --- Jupiter synod to Earth -- calendar count in days
divided by:
4332 Jupiter sidereal calendar count in days

arctangent {399 / 4332}  =  5.2623954  degrees <----

This aligns the two timelines under the prime number 19,
399 = 19 x 21 .... 4332 = 19 x 228

I will come back and clean this up with more details in a few days Whip



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Grand Unification of Ancient and Modern Geometry and Mathematics

Khafre Pyramid base length defined in spectacular angle tangent

This pyramid concept is one of my favorites,
due to the unusual results found in the angle tangent,
of the 2D cross sectional plane through the corner angles.

For this result,
we have to look back at the Khafre Pyramid base lengths.
Just like in the Khufu pyramid,
there are 4 uneven base lengths as measured by Petrie.

If you go back many posts to the Khafre pyramid work in this thread,
you will see several choices of base lengths,
which comply to ancient math and geometry code.

The premier base length,
aPi  x  224.7 Venus sidereal = 706.2 feet -- where aPi = {22 / 7}.

Petrie's south side:
8477 inches  -- most closely aligns with:
411 x cubit 20.625 = 8476.875 inches.

The number of cubits -- 411 -- equals 3 x 137 Kabbalah.

In this exercise,
we note Petrie's East side,
8475.2 inches.
My proposal back in the original Khafre pyramid work here several months ago,
is the:
706.25 feet <---
8475 inch base length. A simple direct easy measurement.
Why?
because this base length of:
706.25 feet  ---> 70625 --> equals:  625  x  113 <--- we have our 625 multiple, {25 squared}
noted in the recent Khafre Pyramid work,
and
113 --- from the all important Pi Progression Constant {355 / 113}.

I consider the above three base lengths to be the best choices available,
as Khafre pyramid base lengths.

So the old determinations of these base lengths,
prioritized the 8475 inch -- 706.25 foot base length as very important.


IMAGE ONE

The Tzolkin 260 and Saturn synod 378 pyramid construct.

Two of the most important ancient calendar count timelines:
260 Tzolkin spiritual calendar
378 day calendar count Saturn synod -- is also the --> Half base length of the Khufu Pyramid <---

So to construct this particular pyramid,
Height is set at:
378 feet
half base length:
260 feet.
Thus,
the Corner Angle of the pyramid,
has the angle tangent:
378 -- divided by -- {260  x  sqrt 2}

Below the first pyramid,
the cross section of the Corner Angles is seen as an isosceles triangle.
We want to determine:
what the Full Apex Angle of that isosceles cross section details as the angle tangent <---

Now look at the results.
The Apex Angle tangent:
260 x 378
divided by:
{1921  x  sqrt. 2}

The spectacular ancient code is found in the number 1921 Dance2

1921  = megalithic yard 2.72 feet -- times -- exactly 706.25 Whip

You cannot ask for a better result of ancient code evidence,
that basically confirms,
one of the Khafre Pyramid base lengths as: -- 706.25 feet --

IMAGE ONE
[Image: 7fnqKBt.jpg]



IMAGE TWO

It is statistically impossible for the above results to be a random chance result.
Pyramid dimensional lengths were constructed from:
ancient calendar count timelimes:
260 Tzolkin
378 Saturn synod
and the Apex Angle of the cross sectional isosceles,
spits out and angle tangent,
with the megalithic yard -- 2.72 feet -- and the exact 706.25 feet --
of a previously proposed and well defined ancient base length.

OK,
but perhaps what we want,
is to see the ACTUAL Khufu Pyramid base length -- 756 feet <---
using our two aforementioned,
calendar count timelines.

To do this --> we simply set the Height = {260  x  sqrt 2} <-- Whip
and
the Half base length as -- 378 feet <--

we now have a standard Khufu Pyramid base length of 756 feet <---
or 9072 inches = 440 x cubit 20.618 18 18~

I created this pyramid to test the cross sectional apex angle tangent.

The difference in this pyramid from the last pyramid,
is that our cross sectional isosceles triangle,
is aligned through:
The Side Face Angles <--> as opposed to the Corner Angles of the last pyramid.
Thus,
we view the resultant Apex angle tangent.
the tangents are exactly the same tangent -- in both pyramids,
except that,
the first pyramid tangent --> has a positive tangent <---
and the second pyramid ---> has a negative tangent <---

So the angle tangent math,
is exactly identical,
{aside from positive and negative denotation}
but the actual apex angles are slightly different.

Apex Angle -- Pyramid One -- cross section via Corner Angles -- 88.416605 degrees <---

Apex Angle -- Pyramid Two -- cross section via Side Face Angles {180 minus 88.416605} degrees <---

Now that we have Pyramid Two designed,
we can actually define,
the Height:
{260 x sqrt. 2} feet
equals
{3120 x sqrt. 2} inches
and,
assign the cubit count to that height,
with:
the two well known -- square root two cubits --
as defined from all the previous long standing square root two cubit dynamics:
cubit 20.61923374~
cubit 20.62394778~
are the two premier identified square root two cubits from the Khufu Pyramid.
{See earlier work a couple months ago}.

so in IMAGE TWO
I show two equations that define the pyramid height in number of cubits.
Note how the 360 calendar count,'
and the Venus Rotational day 243 <---> appear with the respective cubits in the math.


For the record,
the Side Face angle of this pyramid -- arctan {260sqrt.2  /  378} = 44.2083025 degrees
and the base Corner Angle:
arctan {260 x sqrt. 2} -- divided by -- {378 x sqrt. 2} = 34.52138688 degrees.

If you would like to know,
what the Full Apex Angle of the --> Corner Angles cross section of this second pyramid is:

negative 
arctangent:
{260  x  378  x  2}
divided by:
{649  x  116 Mercury synod} <---  for 110.9572262 degrees <---

IMAGE TWO

[Image: HBaOnf9.jpg]





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Grand Unification of Ancient and Modern Geometry and Mathematics
Masterpiece Ancient Khufu Pyramid Angle Tangent Code
Khufu Pyramid Sacred Seked 5.5 as an angle tangent

In recent work from just previous posts,
I proposed the concept of the Khufu Pyramid Sacred Seked 5.5, ... {7 / 5.5 = 1.27 27 27 27~}
as an angle tangent in it's own right.
arctangent 5.5 = arctangent {11 / 2} = arctangent {99 / 18}  = 79.69515353 degrees.

The inverse angle of the above:
{90 - minus - 79.69515353 degrees}  =  10.30484647 degrees  =  arctangent {18 / 99} <---

So just for fun,
here is the Sacred Seked 5.5 pyramid,
where arctangent 5.5,
is set as the:
Side Face of the pyramid,
complete with the two Full Apex Angles defined

[Image: MkCeZHT.jpg]




The last several posts I made previously were predicated and then developed upon,
a specific angle tangent,
which is constructed from elements of phi geometry.
That angle tangent:
{18 / 99}  =  {2 / 11}  =  3.75 inches - divided by - cubit 20.625 inches.
I use the numbers:
{18 / 99}
because the actual decimalized tangent value is:
0.18 18 18 18 18~ 

The isosceles triangle that defines this process,
is shown below.
Two base angles of the isosceles triangle,
have the angle:
84.84757677 degrees  =  arctangent {5 Phi + 3},
thus,
the top of the isosceles triangle has the Full Apex Angle <---
with angle:
10.30484647 degrees  =  arctangent {18 / 99} <---

here is that isosceles triangle:

[Image: VqVjEU9.jpg]



Likewise,
if you take the angle:
10.30484647 degrees  = arctangent {18 / 99},
and 
bisect it evenly in half,
you get two angles,
of exactly:
5.152423234 degrees  =  arctangent {1 / 5Phi}.

This is a very difficult pyramid to construct or draw with this angle:
by using it as the Side Face Angle of the pyramid.
This creates a very shallow or low height pyramid,
height: 18 units,
half base length: 99 units.

So this is the pyramid now with these dimensions.
Note the lettered positions of the angles.

The image directly after this one,
will define the process,
of reconstructing this pyramid into another pyramid,
then extracting the specific isosceles triangle from that,
and inserting it into the Khufu Pyramid Corner Angles Cross Section,
as a dual overlay of isosceles triangles,
for the inevitable test procedure.

Here is the pyramid using arctangent {18 / 99},
as the side face of the pyramid.
Note now, where arctangent 5.5 is located as angle B
[Image: 10oqFkY.jpg]


Now that we have the above pyramid,
in the image below,
the triangles which define the Side Face Angle,
and the Corner Angle,
of the above shallow pyramid are shown in the top half of the image <---

The Side Face triangle at the very top,
displays the bisecting line,
which shows the Phi component of those two equal bisecting angles.
Now,
look to the Corner Angle triangle with the double arrow pointing to it.
These are the critical angles or specific triangle,
from which we will construct ---> the new pyramid from.

In the bottom half of the image,
the shallow pyramid Corner Angles <---> are rotated 90 degrees, as seen.

Two -- now upright -- pyramid Corner Angle triangles,
are then assembled together to form:
a larger isosceles triangle <---
which then becomes:
one of the ---> cross sectional isosceles triangles of the New Pyramid <---

The Full Apex Angle tangent of angle CC -- is defined with an equation.

THE MAJOR ALTERATION,
of the geometry positioning to be aware of:
The new isosceles triangle that you see,
at the bottom of this image,
assembled from the previous pyramid triangles,
{that were turned 90 degrees to be very upright, and then conjoined},
will become:
The New Pyramid's ---> Cross Sectional Isosceles ---> THROUGH the Side Face Angles Whip 
rather than being assigned into the Corner Angles Cross Section,
of the new pyramid.

[Image: bLviIVB.jpg]


This above aligned geometry allows the - new pyramid - to express:
arctangent 5.5,
through the corner angles Whip
due to the height of the pyramid being assigned the {99 x sqrt2} height value.

The cross section of the pyramid that you want to focus on,
the cross sectional isosceles triangle,
will bisect the pyramid through the ---> Side Face Angles <---
The dimension of the bisecting isosceles triangle:
height:  99 x sqrt 2
base: 2 x 18

So in this pyramid,
we have preserved the important angle:  arctangent 5.5,
and the novel geometry developed to focus on,
bisects the Side Face Angles.

[Image: 2hYopTc.jpg]



And the grand finale in the actual TEST:

The selected Isosceles Triangle Cross Section, {The side face angles cross section}
is now inserted into the:
Khufu Pyramid Corner Angles Full Cross Sectional Isosceles Triangle.

This creates:
the two angles F Whip
and these are the angles we want to inspect the resultant angle tangent of.

The angle tangent of angle F,
is written to express the Venus sidereal of 224.7 days.
That is how the ancient code is interpreted and extracted.
It is NO COINCIDENCE,
that the Venus sidereal appears in this angle F geometry.
The dimensional constructs of the new pyramid generate the prime number 107,
into the angle F tangent,
and thus the Venus sidereal of 224.7 days, is the ancient angle tangent code.
2247 = 21 x 107


[Image: 1ph2fHF.jpg]


Note length X <---

Masterpiece Ancient Khufu Pyramid Angle Tangent Code


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Perhaps a little break from all the ancient angle tangent code and numbers.

It is somewhat difficult to maintain the motivation, inspiration and desire,
to produce, create, and improvise new material,
in a revived mix with less recent and older material,
during the coronavirus apocalypse.
 
So a quick dash to the past,
and my tetrahedral Pentad Tiles from the very first pdf,
showing the original Pentad Octagon,
set as a tile replication,
of four tetrahedral pentad octagons.
The Pentad of course,
being the original 5 mound configuration on Mars at Cydonia {H Crater amd S McDaniel -- UTSI}.

When my four tetrahedral octagon tiles are assembled together,
the tetrahedral Pentad Hexagon,
evolves in the center of the framework.
best viewed at 125-150%

[Image: nDlRPvk.jpg]


As you look for various polygonal shapes possible by the replicating grids,
there are numerous possibilities,
and then I noticed a somewhat angular -- heart shaped construct -- within the quadra-octagonal grid.

So just a simple flashback to the past,
with a fresh heart for the future.
Heart outline in dark blue.

[Image: dsOonub.jpg]




Two yellow outlined hearts cross secting each other from upper and lower directions,
form the central eight sided figure within:

[Image: UMQlxkv.jpg]


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