...

The difficulty in presenting the "Square Root Two" cubits,

is that you have to explain -- ancient square root two.

So this is Part 1.

Ancient Square Root Two,

The Square Root Two Cubits In Khufu Pyramid Dynamics

The number 7 is obviously of paramount importance in Khufu Pyramid dynamics.

One subtle but revealing characteristic of the number 7,

or it's inverse {1 / 7},

is when either choice is employed as an angle tangent.

arctangent 7 = 81.86989765 degrees,

cosine 81.86989765 = square root 2 --> divided by 10.

arctangent {1 / 7} = 8.130102354 degrees

sine 8.130102354 = square root 2 --> divided by 10.

square root 2,

divided by 10 = square root 1.62 --> divided by 9.

Arctangent square root 1.62 = 1.27 27 92206~, for 51.84419346 degrees.

Arctangent {14 / 11} = 1.27 27 27 27~, for 51.84277341 degrees.

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

Ancient square root two. {99 / 70} x {140 / 99} = 2.

Every square base pyramid has a Corner Angle tangent,

that equals:

one half the base length -- times -- square root two <---

The Egyptians had a dilemma with this square root two geometry math.

Khufu did not have a handy dandy scientific calculator,

with Pi, and the square roots on demand.

They had to have a functional fraction to operate the mathematics.

Egyptian cultural math had sophisticated progressions,

and they calculated Pi and square roots to nine and ten decimal placements.

The key is "functional fraction".

Precision ancient square root two

is an expansion of ancient number 231, --- {27720 = 231 x 120}

and to cut to the chase,

the fraction reduces to:

27720 / 19601 --------- note that 27720 = 1344 x cubit 20.625

Once this fraction is accomplished,

precision fractions for tetrahedral angle tangents are a piece of cake:

tangent tetrahedral 19.47122063 degrees = 19601 / 6930 --- 6930 = 231 x 30.

accuracy: 0.999999999

Nonetheless,

the fraction 27720 / 19601 for square root 2,

is functional for pure sacred geometry,

but getting difficult to employ in written out calculations.

So the Egyptians found a splendid, dynamic, and superbly functional short cut,

to calculating with:

square root two,

by using primarily the numbers 7 and 11, in ancient square root two.

Simple fractions in the sevenths -- and -- ninety ninths {9 x 11},

formed:

Ancient square root two:

99 / 70 = 1.414 285714 285714~ = 0.45 aPi aPi = 22 / 7

140 / 99 = 1.414 14 14 14 14 14~ = 40 / {9 x aPi}

The cubit to isolate above, is the primary modern square root two cubit:

20.61923374~ inches

That cubit,

has a complimentary pure square root two cubit:

20.62394778~ inches

{18 x 360} -- divided by -- cubit 20.61923374~ = 3.142696805~ = {20 x sqrt. 2} / 9

6.5625 -- times -- cubit 20.62394778~ = 3.142696805~ = sqrt. 800 / 9

thus:

4 / Pi value = slope tangent

and,

4 / 3.142696805~ = sqrt. 1.62 = 1.27 27 92206~,

Khufu Pyramid slope tangent in pure square root two dynamics.

Arctangent {sqrt. 1.62} = 51.84419346 degrees.

cubits:

20.61923374 x 20.62394778 = 425.25 -- the square root two cubits

20.61670179 x 20.62648062 = 425.25 -- the Pi cubits

20.618 18 18~ x 20.625 = 425.25 -- primary ancient cubits

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

Image One content:

Calculating -- Pyramid Length X -- with ancient square root 2.

Square root two appears in the Khufu pyramid corner angle tangent.

Khufu Pyramid Corner Angle slope tangent = 14 -- divided by -- { 11 x sqrt. 2 },

{440 cubit base,

280 cubit height pyramid model}

square root 2 = 1.414 213562~

99 / 70 = 1.414 285714 285714~

140 / 99 = 1.414 14 14 14 14~

Either choice of "ancient square root two"

{99 / 70} or {140 / 99},

has terrible accuracy compared to modern square root two!

0.999949 accuracy,

is not good.

But the Egyptians found a splendid, dynamic, and superbly functional short cut,

to calculatuing with:

ancient square root two, 99 / 70 and 140 / 99,

which facilitated much quicker math written out,

and had an exponential jump in awesome accuracy!

The image below sets the stage for testing the "ancient square root two" method.

NOTE:

Length X ... in the image, is the base diagonal to base midpoint.

What is length X ?

We will use "ancient square root two" to acquire a Length X,

and compare the accuracy to modern square root 2 results.

There are two -- Side Face Angle --> slope tangents in this pyramid:

{4 / Pi} and {4 / aPi} -- where aPi = 22 / 7.

NOTE:

the Height,

is fixed by using one of the Pi cubits -- 20.62648062~ inches --- times 280 cubits.

cubit 20.62648062~ = {4 / Pi} x 16.2

A rectangular pyramid base,

creates the Two Slope Pyramid,

generating two Side Face slope angles automatically.

see image math:

Length X -- using modern square root 2:

534.68032 feet

Length X -- using ancient square root 2:

534.6803044 feet

A difference of 0.0000156 feet with superb ---> 0.999999971 accuracy.

When just comparing ancient sqrt. 2,

to modern sqrt. 2, in accuracy,

ancient sqrt. 2 displayed very low accuracy.

However, the ancient square root two math operation for Length X -- dramatically increased accuracy,

by a factor of 10,000 !

In terms of the Earth Lunar Month 29.53059 days,

0.999999971 accuracy

yields a Lunar Month of 29.53058914 days ---> off by 0.074 seconds <---

That is extreme accuracy.

The early Egyptians perhaps at Saqqara,

looked at their result of 534.6803044 feet and probably said:

534.68 feet,

close enough for pyramid length X measurement.

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

A second pyramid is offered with two different slope tangents.

More of a pure ancient pyramid here.

The same process with ancient square root 2 is applied.

Spectacular accuracy is accomplished with the ancient process.

A height of 481.25 feet is used for simpler calculation effort, and display.

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

A more sophisticated geometry approach is presented from old material,

constructing an isosceles triangle with square root 2 dynamics.

The angle tangents are decoded into cubit system dynamics.

All three cubit formulas for -- Full Apex Angle B -- are equal.

This was specifically done to show how the modern square root 2 cubits intermixed,

in the mathematics.

...

The difficulty in presenting the "Square Root Two" cubits,

is that you have to explain -- ancient square root two.

So this is Part 1.

Ancient Square Root Two,

The Square Root Two Cubits In Khufu Pyramid Dynamics

The number 7 is obviously of paramount importance in Khufu Pyramid dynamics.

One subtle but revealing characteristic of the number 7,

or it's inverse {1 / 7},

is when either choice is employed as an angle tangent.

arctangent 7 = 81.86989765 degrees,

cosine 81.86989765 = square root 2 --> divided by 10.

arctangent {1 / 7} = 8.130102354 degrees

sine 8.130102354 = square root 2 --> divided by 10.

square root 2,

divided by 10 = square root 1.62 --> divided by 9.

Arctangent square root 1.62 = 1.27 27 92206~, for 51.84419346 degrees.

Arctangent {14 / 11} = 1.27 27 27 27~, for 51.84277341 degrees.

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

Ancient square root two. {99 / 70} x {140 / 99} = 2.

Every square base pyramid has a Corner Angle tangent,

that equals:

one half the base length -- times -- square root two <---

The Egyptians had a dilemma with this square root two geometry math.

Khufu did not have a handy dandy scientific calculator,

with Pi, and the square roots on demand.

They had to have a functional fraction to operate the mathematics.

Egyptian cultural math had sophisticated progressions,

and they calculated Pi and square roots to nine and ten decimal placements.

The key is "functional fraction".

Precision ancient square root two

is an expansion of ancient number 231, --- {27720 = 231 x 120}

and to cut to the chase,

the fraction reduces to:

27720 / 19601 --------- note that 27720 = 1344 x cubit 20.625

Once this fraction is accomplished,

precision fractions for tetrahedral angle tangents are a piece of cake:

tangent tetrahedral 19.47122063 degrees = 19601 / 6930 --- 6930 = 231 x 30.

accuracy: 0.999999999

Nonetheless,

the fraction 27720 / 19601 for square root 2,

is functional for pure sacred geometry,

but getting difficult to employ in written out calculations.

So the Egyptians found a splendid, dynamic, and superbly functional short cut,

to calculating with:

square root two,

by using primarily the numbers 7 and 11, in ancient square root two.

Simple fractions in the sevenths -- and -- ninety ninths {9 x 11},

formed:

Ancient square root two:

99 / 70 = 1.414 285714 285714~ = 0.45 aPi aPi = 22 / 7

140 / 99 = 1.414 14 14 14 14 14~ = 40 / {9 x aPi}

Quote:The square root two -- trio of cubits:

Egyptian remen foot:

14.58 inches

14.58 inches x {140 / 99} = cubit 20.618 18 18 18 ~ = {1134 / 55}

14.58 inches x modern square root two = cubit 20.61923374 ~ = {16.2 x square root 1.62}

14.58 inches x {99 / 70} = cubit 20.620 285714 285714~ = {81 squared x aPi} / 1000

The cubit to isolate above, is the primary modern square root two cubit:

20.61923374~ inches

That cubit,

has a complimentary pure square root two cubit:

20.62394778~ inches

{18 x 360} -- divided by -- cubit 20.61923374~ = 3.142696805~ = {20 x sqrt. 2} / 9

6.5625 -- times -- cubit 20.62394778~ = 3.142696805~ = sqrt. 800 / 9

thus:

4 / Pi value = slope tangent

and,

4 / 3.142696805~ = sqrt. 1.62 = 1.27 27 92206~,

Khufu Pyramid slope tangent in pure square root two dynamics.

Arctangent {sqrt. 1.62} = 51.84419346 degrees.

cubits:

20.61923374 x 20.62394778 = 425.25 -- the square root two cubits

20.61670179 x 20.62648062 = 425.25 -- the Pi cubits

20.618 18 18~ x 20.625 = 425.25 -- primary ancient cubits

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

Image One content:

Calculating -- Pyramid Length X -- with ancient square root 2.

Square root two appears in the Khufu pyramid corner angle tangent.

Khufu Pyramid Corner Angle slope tangent = 14 -- divided by -- { 11 x sqrt. 2 },

{440 cubit base,

280 cubit height pyramid model}

square root 2 = 1.414 213562~

99 / 70 = 1.414 285714 285714~

140 / 99 = 1.414 14 14 14 14~

Either choice of "ancient square root two"

{99 / 70} or {140 / 99},

has terrible accuracy compared to modern square root two!

0.999949 accuracy,

is not good.

But the Egyptians found a splendid, dynamic, and superbly functional short cut,

to calculatuing with:

ancient square root two, 99 / 70 and 140 / 99,

which facilitated much quicker math written out,

and had an exponential jump in awesome accuracy!

The image below sets the stage for testing the "ancient square root two" method.

NOTE:

Length X ... in the image, is the base diagonal to base midpoint.

What is length X ?

We will use "ancient square root two" to acquire a Length X,

and compare the accuracy to modern square root 2 results.

There are two -- Side Face Angle --> slope tangents in this pyramid:

{4 / Pi} and {4 / aPi} -- where aPi = 22 / 7.

NOTE:

the Height,

is fixed by using one of the Pi cubits -- 20.62648062~ inches --- times 280 cubits.

cubit 20.62648062~ = {4 / Pi} x 16.2

A rectangular pyramid base,

creates the Two Slope Pyramid,

generating two Side Face slope angles automatically.

see image math:

Length X -- using modern square root 2:

534.68032 feet

Length X -- using ancient square root 2:

534.6803044 feet

A difference of 0.0000156 feet with superb ---> 0.999999971 accuracy.

When just comparing ancient sqrt. 2,

to modern sqrt. 2, in accuracy,

ancient sqrt. 2 displayed very low accuracy.

However, the ancient square root two math operation for Length X -- dramatically increased accuracy,

by a factor of 10,000 !

In terms of the Earth Lunar Month 29.53059 days,

0.999999971 accuracy

yields a Lunar Month of 29.53058914 days ---> off by 0.074 seconds <---

That is extreme accuracy.

The early Egyptians perhaps at Saqqara,

looked at their result of 534.6803044 feet and probably said:

534.68 feet,

close enough for pyramid length X measurement.

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

A second pyramid is offered with two different slope tangents.

More of a pure ancient pyramid here.

The same process with ancient square root 2 is applied.

Spectacular accuracy is accomplished with the ancient process.

A height of 481.25 feet is used for simpler calculation effort, and display.

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

A more sophisticated geometry approach is presented from old material,

constructing an isosceles triangle with square root 2 dynamics.

The angle tangents are decoded into cubit system dynamics.

All three cubit formulas for -- Full Apex Angle B -- are equal.

This was specifically done to show how the modern square root 2 cubits intermixed,

in the mathematics.

...