Geometry of the Great Seal of the United States:
The Unfinished Pyramid

by Michael S. Schneider

The Great Seal of the United States has an interesting history and may be the only seal of any nation to have two sides, known as the Pyramid and the Eagle. Look at any dollar bill to see them both. A great deal has been written about the geometry of each half of the seal, which they invite by being composed in circles. And even more has been speculated about what may be the intended symbolism of the Seal with its intriguing phrases, objects, details, numbers and proportions. Most curiously, what nation's Seal puts the architecture of another country -- an Egyptian pyramid no less, the granddaddy of all speculation -- on it's own seal? But I'm not going to speculate much on the usual symbolism here, nor will I use the circle as the basis of its geometric exploration. Instead, we'll only consider the face of the pyramid, looking straight at it. Does it contain any proportions of interest to a geometer?

I began by drawing a rectangle around the extremes of the face of the pyramid, beginning with a line across the base and up to it's tip (Figure 1).
I then divided the rectangle into its three natural parts with a line across the top of the unfinished pyramid and one above it, across the base of the triangle containing the "Eye of Providence".
If there is any interesting geometry to this rectangle these three rectangles would have to relate to each other by a consistent proportion. What could it be?

Figure 1

A little investigation reveals that this whole rectangle is composed solely of Golden Rectangles in four related sizes. (Figures 2 and 3)

Figure 2 Figure 3

The first idea to notice is that these eight Golden Rectangle form a left-right rmirror symmetry, and that each half "spirals" around by decreasing sizes.
Next, we notice that the pyramid's solid stone base (of 72 stones in 13 layers) is perfectly contained within the two largest Golden Rectangles.
The upper layers are formed by six Golden Rectangles, each mirror pair made of three.

As if a miniature of the pyramid's base, the triangle containing the eye is framed in two Golden Rectangles as well but it's base doesn't reach the corners (more on this below).

Below the vertical Golden Rectangles framing the Eye are two small, horizontal Golden Rectangles.
These define the height of the gap between the base of the pyramid and triangle of the Eye.
It's as if the two large Golden Rectangles of the base rose up, shrunk and spread apart to reveal the four Golden Rectangles of the Eye and gap between them.

This sort of design using only Golden Rectangles (and squares) is routinely found in the proportions of ancient Egyptian art, crafts and architecture, sacred design in general.
Of interest are the four small central Golden Rectangles, two pairs made of a vertical Golden Rectangle above a smaller horizontal one.
Each pair combines to produce a "square root of five" rectangle, well known to students of geometry and Dynamic Symmetry art theory.

The mathematics of the "square root of five" rectangle shows that it relates, by definition, to the Golden Ratio. (Figure 4)
Curiously, the square root of five equals the Golden Ratio plus it's own inverse.

Figure 4

In most every ancient culture, the appearance of Golden Rectangle symmetry indicated something sacred.
To the ancient Egyptians, as seen in the proportions of their art (Figures 5 and 6), especially those scenes relating to Osiris, the small horizontal Golden Rectangle represented Osiris in his coffin, dead, while the larger vertical Golden Rectangle adjoining it symbolized Osiris alive. Together, as the square root of five rectangle, they represented the resurrection of Osiris.

Figure 5 Figure 6


As for the divine triangle containing the right eye of Deity, it seems to be proportioned within a square across it's base.
Connecting the square's lower corners to the midpoint at it's top perfectly frames the triangle. (Figure 7)
Symbolically this simply says "the divine (triangle) beomes manifest on earth (square)."

Figure 7


This geometry also relates to the Golden Ratio through the square root of five, because if the length of each side of the square is 2,
then the length of each diagonal equals the square root of five (Figure 8) and the Golden Ratio is defined as half of the sum of one plus the square root of five.
So the proportions of the divine triangle relate to the dimensions of the pyramid's base as well as the gap between them.

Figure 8


The construction of a square root of five rectangle is simple. (Figure 9)
It emerges from an arc made by swinging the semi-diagonals of a square, the same ones that frame the Eye.
In fact, the geometry of the triangle containing the Eye is the beginning of the construction of the square root of five rectangle.
This construction also leads to the five-pointed star (Figure 10), a symbol closely associated with the United States through it's pentagonal iconography.

Figure 9 Figure 10


Of course, I have no way of knowing whether the Seal was intentionally designed with this or any geometry,
or intentionally expresses the symbolism of sacred regeneration, but it's interesting to consider.
Perhaps it will inspire us to "complete the unfinished pyramid" within ourselves and society
for a divine life on earth.


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(c) 2011 Michael S. Schneider