Phi Fi Fo Fum

In my last entry I asked: how do the great pyramids of Giza have the same precise angle of slope as the location of a haunch in a barrel vault? This is an interesting question which delves into geometry and the Golden Mean, or Golden ratio.


As described in Wikipedia: “in mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, golden number, and mean of Phidias. The golden ratio is often denoted by the Greek letter phi, usually lower case (φ)” [pronounced "fee"].

The pyramids of Giza used the Golden Mean in their basic design and construction. If the edge of the base of the pyramid is taken as 2, then the height of the pyramid is the square root of phi, and the hypotenuse of this right triangle is phi.

How does the golden mean relate to a sphere? It relates to the sphere through a polyhedron known as an icosahedron, which can be traced onto a sphere. Here is an icosahedron:

These images shows three rectangular planes at right angles to each other (x,y,z) which each have the edge length proportions of the golden ratio (1:1.618...).  If the corners of these three rectangles are connected, they describe an icosahedron on a sphere.  This is a sculpture I made from clay.

Here are the three phi rectangles assembled:

Here are the rectangles being placed in the sphere:

Here is the assembled structure, showing the relationship between phi and a sphere, through the icosahedron:

So the why does the haunch of a Roman Arch (circular vault) occur at an angle which is exactly the same as the slope of a pyramid? This remains an interesting question, especially since most Egyptian ‘arches’ were not arches at all, but post and lintels. It is true that Egyptians built with arches, but they never developed the masonry arch as did the Romans.

This remains a mystery to me. If anyone has any idea, please share and let me know. It seems too precise of a coincidence that both the haunch angle in a circular arch and the phi-based geometry of the great pyramids at Giza should both manifest the specific angle of 51 degrees, 51 minutes. Furthermore, the haunch is a result of gravity acting on a body; whereas geometric constructions relying on the golden ratio appear to be independent of any force, such as gravity. This is a very curious coincidence.