04-08-2018, 02:06 PM

...

It is difficult to draw 3D representations,

of hexagonal or pentagonal pyramids.

No matter what you do, the true perspective is a bit skewed.

Essentially, these image designs presented are to facilitate --> Apex Angle <-- testing,

just to determine,

what the angle tangents are,

in the new pyramid designs --- full apex angles.

If you study Apex Angles in ancient Egytian pyramids,

you had better get a handle on modern geometry constructs as well.

Ever wonder why ...

you never see ... hexagonal or pentagonal pyramids on Earth?

Or giant tetrahedrons?

Ancient cultures only needed to express their cultural math science -- with a square base pyramid,

and,

it was easier to construct as a square base pyramid.

In a square base pyramid ... such as in ancient Egypt,

there are --- cross secting -- isosceles triangles -- that define: -- the full Apex Angles <---

These isosceles triangles,

cross sect ---> the Side Face Angles of the pyramid,

and

the Corner Angles of the pyramid.

A square base pyramid has -- FOUR equal corner angles -- and -- FOUR equal side face angles.

A hexagonal pyramid:

has SIX equal corner angles -- and SIX equal side face angles <---

A PENTAGONAL PYRAMID ... however;

cannot have,

a true cross secting --- isosceles triangle <---> like a square or hexagonal base pyramid has.

In the case of a pentagonal pyramid,

the corner angles go up to the APEX,

and that's it,

on the other side of the apex,

{or reflective side --- as in a square or hexagonal base}

on the other side of the apex,

of the pentagonal Corner Angle --- is a Side Face Angle <---

so you result in an oddball dual geometry ... as a cross secting apex angle ...

ie,

there really is no viable isosceles triangulated Apex Angle --- like in a square or hexagonal base.

That probably confused the issue.

But the process here is to ascend from -- square base pyramid,

to:

a true 2D hexagonal base <---> with 120 degree angles -- that sits on a flat surface.

Almost all the stacked molecular latticing seen in physics is displayed with:

Hexagonal based ... pyramidal forms.

It is the molecular latticing and stacking,

that becomes important in modern technological production.

So this exercise addresses -- hexagonal pyramids -- that would interlink on a flat surface,

and then be designed,

to comply to some sort of molecular latticing and stacking schemes.

The simplest way to confront the hexagonal pyramid then,

is with the fundamental building blocks of geometry,

by combining:

the hexagonal base <---> which has -- square root three geometry <---

with heights <---

that create --- tetrahedral square root two geometry,

or heights <---

that create pentagonal square root five geometry

-- or specifically ---> phi or inverse phi geometry <---.

Then, the study test is to determine,

the new hexagonal pyramids -- Apex Angle -- tangents.

When you see the word --- arctan --- in the images,

that means ---> arctangent <---

ie

take the defined equation -- seen with each "arctan" -- click 2nd -- then click tangent,

-- and you have your angle --

or,

the equation next to the word --arctan -- IS THE TANGENT -- of the angle in question.

The FIRST hexagonal pyramid --

installs the Phi angle tangent ---> into the Side Face Angles <---

So golden rectangle -- phi angles -- cross sect the hexagonal pyramid,

through the Side Face Angles.

The SECOND hexagonal pyramid --

installs the Phi angle tangent -- into the Corner Angles <---

So golden rectangle -- phi angles -- cross sect the hexagonal pyramid,

through the Corner Angles.

NOTE: the ugly duckling angle tangents -- for the Apex Angles

in both pyramids

{angle Q -- first pyramid ... angle R -- second pyramid}

Those are really tough angle tangents to ferret out.

When it comes to phi oriented angle tangents -- in the apex angles --

even I cannot ferret the equations out in all the cases,

because it gets so complex.

These Apex Angle tangents,

have components of:

square root three -- AND -- sqrt 5 based phi geometry functions.

{sqrt 12 = 2 sqrt3}

and

2Phi = {sqrt5 -- plus -- One}.

The Apex Angles are a ... hybrid angle ... of sorts,

that have to incorporate both geometry functions.

Pyramid One --- The Phi angle is installed into the Side Face Angles B.

I was unable to draw in the angle B on the front Side Face.

and

Pyramid Two -- the Phi angle is installed into the Corner Angles A :

...

It is difficult to draw 3D representations,

of hexagonal or pentagonal pyramids.

No matter what you do, the true perspective is a bit skewed.

Essentially, these image designs presented are to facilitate --> Apex Angle <-- testing,

just to determine,

what the angle tangents are,

in the new pyramid designs --- full apex angles.

If you study Apex Angles in ancient Egytian pyramids,

you had better get a handle on modern geometry constructs as well.

Ever wonder why ...

you never see ... hexagonal or pentagonal pyramids on Earth?

Or giant tetrahedrons?

Ancient cultures only needed to express their cultural math science -- with a square base pyramid,

and,

it was easier to construct as a square base pyramid.

In a square base pyramid ... such as in ancient Egypt,

there are --- cross secting -- isosceles triangles -- that define: -- the full Apex Angles <---

These isosceles triangles,

cross sect ---> the Side Face Angles of the pyramid,

and

the Corner Angles of the pyramid.

A square base pyramid has -- FOUR equal corner angles -- and -- FOUR equal side face angles.

A hexagonal pyramid:

has SIX equal corner angles -- and SIX equal side face angles <---

A PENTAGONAL PYRAMID ... however;

cannot have,

a true cross secting --- isosceles triangle <---> like a square or hexagonal base pyramid has.

In the case of a pentagonal pyramid,

the corner angles go up to the APEX,

and that's it,

on the other side of the apex,

{or reflective side --- as in a square or hexagonal base}

on the other side of the apex,

of the pentagonal Corner Angle --- is a Side Face Angle <---

so you result in an oddball dual geometry ... as a cross secting apex angle ...

ie,

there really is no viable isosceles triangulated Apex Angle --- like in a square or hexagonal base.

That probably confused the issue.

But the process here is to ascend from -- square base pyramid,

to:

a true 2D hexagonal base <---> with 120 degree angles -- that sits on a flat surface.

Almost all the stacked molecular latticing seen in physics is displayed with:

Hexagonal based ... pyramidal forms.

It is the molecular latticing and stacking,

that becomes important in modern technological production.

So this exercise addresses -- hexagonal pyramids -- that would interlink on a flat surface,

and then be designed,

to comply to some sort of molecular latticing and stacking schemes.

The simplest way to confront the hexagonal pyramid then,

is with the fundamental building blocks of geometry,

by combining:

the hexagonal base <---> which has -- square root three geometry <---

with heights <---

that create --- tetrahedral square root two geometry,

or heights <---

that create pentagonal square root five geometry

-- or specifically ---> phi or inverse phi geometry <---.

Then, the study test is to determine,

the new hexagonal pyramids -- Apex Angle -- tangents.

When you see the word --- arctan --- in the images,

that means ---> arctangent <---

ie

take the defined equation -- seen with each "arctan" -- click 2nd -- then click tangent,

-- and you have your angle --

or,

the equation next to the word --arctan -- IS THE TANGENT -- of the angle in question.

The FIRST hexagonal pyramid --

installs the Phi angle tangent ---> into the Side Face Angles <---

So golden rectangle -- phi angles -- cross sect the hexagonal pyramid,

through the Side Face Angles.

The SECOND hexagonal pyramid --

installs the Phi angle tangent -- into the Corner Angles <---

So golden rectangle -- phi angles -- cross sect the hexagonal pyramid,

through the Corner Angles.

NOTE: the ugly duckling angle tangents -- for the Apex Angles

in both pyramids

{angle Q -- first pyramid ... angle R -- second pyramid}

Those are really tough angle tangents to ferret out.

When it comes to phi oriented angle tangents -- in the apex angles --

even I cannot ferret the equations out in all the cases,

because it gets so complex.

These Apex Angle tangents,

have components of:

square root three -- AND -- sqrt 5 based phi geometry functions.

{sqrt 12 = 2 sqrt3}

and

2Phi = {sqrt5 -- plus -- One}.

The Apex Angles are a ... hybrid angle ... of sorts,

that have to incorporate both geometry functions.

Pyramid One --- The Phi angle is installed into the Side Face Angles B.

I was unable to draw in the angle B on the front Side Face.

and

Pyramid Two -- the Phi angle is installed into the Corner Angles A :

...