The proportions of the Great Pyramid
So many things have been written about the proportions of the Great Pyramid that it seems impossible to discover anything new about it. However, I think I've got something interesting to add to the subject.
My interest in the Great Pyramid started with a booklet that accompanied an "esoteric" magazine: not a very trustworthy source of information. It contained a few statements about the proportions of the pyramid that had been obviously written by somebody without much mathematical knowledge. If you aren't familiar with geometric terms, look at the diagram to see the meaning of the words you don't understand.

There were two statements that surprised me specially:
1) If you draw a circle of radius equal to the height of the pyramid, its length will be equal to the perimeter of the base of the pyramid. Using the formula for the length of a circle, we have:
    2 x pi x height = 4 x side ( 2piH=4S )
    In other words, the Egyptians included pi in the proportions of the pyramid.
2) The proportion between the slant height of the pyramid and the side of the base is phi/2, where phi represents the golden mean.
    slant height/side = phi/2 (Sh/S=phi/2)
    If you haven't heard before about the golden mean, it will be a good idea to jump to the section on the golden mean and the Great Pyramid.

Any mathematician would frown after reading both statements. The Egyptians could have made the pyramid adjusting its proportions to pi, or phi, but not both numbers at the same time!
Let's make a few calculations to see this. The height, the slant height and half the side make a right triangle (see the diagram). If the second statement is true, using the Pythagorean theorem we can calculate that the height of the pyramid:
    height = side x square root of phi /2 ( H=S sqr(phi)/2 )
Now, if the first statement is true at the same time, we would have:
    2piS sqr(phi)/2=4S
    pi sqr(phi)=4
In short, a mathematical relation between pi and phi, when there isn't any (at least, not so simple).

But the author of the booklet showed the measures of the Great Pyramid to prove his point, and the calculations seemed convincing. The four sides of the base aren't exactly equal, although the differences are so small that they have puzzled archeologists for many years. The average of the sides is 230.363 meters. The exact height is more difficult to calculate, because the point of the pyramid was lost. An estimation that is accepted by most archeologists is 146.595 meters. I've checked that both numbers are correct.
Now, let's suppose that the Egyptians built the pyramid using phi in their calculations. It sounds most likely, because phi is easy to obtain geometrically, and we know for sure that the Greeks used it deliberately in their temples and sculptures. We can use the formula of the second statement the other way round, to obtain the approximate value of phi they used, knowing the height of the pyramid and the length of its sides. This value for phi is 1.6198..., while the true value is 1.6180... As you can see, it's a very small error, less than two thousandths.
What happens if we suppose that they built the pyramid using pi? The formula of the first statement gives an approximate value for pi of  3.1428..., while the true value is 3.1416... Again, the error is less than two thousandths. This is even more remarkable if we know that in Egyptian texts they used this approximate formula to calculate the area of a circle, using its diameter (D):
    A=(D-D/9)^2
This is equivalent to considering pi equal to (16/9)^2=3.16..., a value that's far more inaccurate.
Perhaps this approximation of pi (3.1428...) reminds you of something. In fact, it's the exact value for one of the best rational approximations of pi, 22/7. Did the Egyptians know this value, or did the archeologists that calculated the height of the pyramid use it? It's impossible to know, but let me add an intriguing fact. If we use the false relation between pi and phi to obtain an approximate value for pi using phi, we get 3.1446..., which is a worse approximation than 3.1428... It looks like the pyramid was built trying to get as close as possible to both numbers.
 
This approximate relation between two of the most interesting irrational numbers awoke my curiosity. Is there any geometric construction that could explain it? I think there is, and I've called my discovery "the golden octogon" (see the figure to the left). You can build it from a square whose side is equal to the square root of 2phi. From the corners of the square, draw straight lines making an angle of 25.91 degrees. This angle is exactly the half of the angle that make the faces of the pyramid with the ground, or in other words, half the angle of the golden right triangle (jump to the section on the golden mean and the Great Pyramid for an explanation of what the golden right triangle is).
The golden octogon isn't regular, because its angles aren't equal, but its sides are all equal to 1. A circle of radius equal to the square root of phi, with the same center as the octogon, is surprisingly close in length and area to the golden octogon. If we draw a square of side 2, it will cut the circle in the same points as the golden octogon. Note that if this square was the base of the pyramid, the height of the pyramid would be the radius of the circle. We could say that this diagram represents the squaring of the circle according to the Great Pyramid.


The golden mean and the Great Pyramid
Phi is a number with several names, all of them quite poetic and usually referring to gold: golden mean, golden ratio, divine proportion... It was well known in antiquity, and it was called phi because the Greek sculptor Phidias used it constantly in his sculptures. Phi represents a proportion that's considered particularly harmonious. You will obtain it when you divide a line in two parts in such a way that the proportion between the greatest and the smallest part is the same as the proportion between the whole line and the greatest part. Phi appears naturally in many proportions of the human body, as it was studied by Leonardo da Vinci.
A rectangle whose sides are 1 and phi is called a golden rectangle, and it has an interesting property: if you cut a square from it, the remaining rectangle keeps the same proportions. You can keep cutting squares from the remaining rectangles, and you'll get a figure called a whirling square. You can see a drawing based on this on this page.

Phi introduces many interesting relations between the different elements of the pyramid. I've already mentioned one: the proportion between the slant height and the side of the base is phi/2 (Sh/S=phi/2). In fact, the height, the slant height and half the side make a right triangle of interesting proportions, called a golden right triangle. Its sides are proportional to 1, phi and the square root of phi. The proportion between the slant height and the height is the same as the proportion between the height and the half the side.
    slant height/height = height/(side/2) ( Sh/H=H/(S/2) )
A perpendicular line to one of the faces of the pyramid passing through the center of the base will divide the slant height according to the divine proportion. This perpendicular line allows me to talk about another interesting property, that perhaps I wouldn't mention if historians didn't tell us that the Egyptians used to build right triangles from the length of its sides. A triangle whose sides are the perpendicular, half the side of the base and the height is also a right triangle, and with the proportions of a golden right triangle. The golden right triangle is the only one with this property.
Phi appears naturally in all the regular figures with five-fold symmetry, like pentagons, decagons, pentagrams (five-pointed stars), dodecahedrons and icosahedrons. In many ancient cultures this kind of figures were considered sacred, because they represented at the same time the human being (the number five represents the hand), and perfection, because of its regularity. I'll say only a couple of relations between these figures and the pyramid, but it's easy to find many more.
Let's inscribe a circle in the base of the pyramid, inside this circle, a pentagon, and inside this pentagon, another circle. The diameter of this second circle is equal to the slant height of the pyramid.
If we build an icosahedron whose edge is equal to half the side of the base, the distance between two opposite vertices is equal to the edge of the pyramid, and the distance between any other two vertices is equal to the slant height.



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