Sacred geometry of the Great Pyramid: squaring the circle
In the following diagram, the triangle has the same proportions as the Great Pyramid. No measurements were made during this construction. All new points were created from existing ones by either drawing a new circle or connecting two points with a straight edge.
Relationships:
If AB = 1,
BD = BH = AB
AC = 1/2
CG = 1/2
EC = √(AC2 + AE2) = √(1/22 + 12) = √5/2
EG = √5/2 + 1/2 = (√5 + 1)/2 = φ
EF = EG = φ
AF = √(EF2 - AE2) = √(φ2 - 1) = √φ
If EA = 1, AF = Height of Great Pyramid = √φ
Slope of EF = Slope of Great Pyramid: cosA = 1/φ, 51° 50' or almost one-seventh a circle.
Graham Hancock summarizes:
. . . the sublime geometry of this building integrates π and φ as follows: its height is the radius of a circle whose circumference equals the base perimeter, and its height is the side of a square whose area equals that of each of the four sides. [1]
It is interesting to note, the perimeter of the square containing the pyramid is almost equal to the circumference of the outer circle that touches the apex. Creating a square with equal perimeter or area as a circle was a problem called "squaring the circle" worked on by geometers throughout the ages:
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. -Wikipedia
The exercise was believed to have symbolic meaning:
Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it. Nevertheless the Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, un-manifest spirit-space, while the square represents the manifest and comprehensible world. When a near-equality is drawn between the circle and square, the infinite is able to express its dimensions or qualities through the finite. -Robert Lawlor, Sacred Geometry Philosophy and Practice
The method shown above is an elegant and symbolically rich solution that isn't exact, but a close enough approximation to inspire a sense of aesthetic satisfaction.
In case you missed it, read my last post for more details about the geometry of the Great Pyramid and for a free Steemit logo built from the same "squaring the circle" architecture.
This image has been my background for 11 days now.. and I'm really starting to intuit the geometric relationships on a more perceptive, aesthetic level. Very helpful on many levels. Thanks!
Cool. You're welcome. It's amazing how doing compass and straightedge drawings and then calculating the lengths/radii really helps to understand the relationships on a deeper level. My mind has been blown multiple times while doing exercises from the book Sacred Geometry Philosophy and Practice. Actually, I think I'm going to do more posts about geometry with drawings. Ps. I thought I was already following you. am now.
Sweet, I'm going to check that book out... I think I'm long overdue for such explorations. Thanks for the reference. Yeah, i dig the geometry posts thus far, and if you find it enjoyable, then by all means... continue. :) Thanks for the follow in whatever tense. ;) Cheers!
I love mathmatics and art! Thanks for creating and posting. Upvoted/Followed.
Me too! My pleasure. Thanks for lookin. ^_^