Geometry of the Great Seal of the United
The Unfinished Pyramid
by Michael S. Schneider
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
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?
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
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
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
Of interest are the four small central Golden Rectangles, two
pairs made of a vertical Golden Rectangle above a smaller horizontal
Each pair combines to produce a "square
root of five" rectangle, well known to students of geometry
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.
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.
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)."
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.
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.
Universe Home Page
(c) 2011 Michael S. Schneider