Saturday, November 13, 2021

Constructing the Dodecahedron Easily from the Cube, as Ancient Geometers Knew

"The dodecahedron was, to these ancient mathematicians, the most mysterious of the solids; it was by far the most difficult to construct, the accurate drawing of the regular pentagon necessitating a rather elaborate application of Pythagoras' great theorem. Hence the conclusion, as Plato put it, that 'this (the regular dodecahedron) the Deity employed in tracing the plan of the Universe.' (H. Stanley Redgrove, in Bygone Beliefs.)"

from "The Life and Philosophy of Pythagoras", in "The Secret Teachings of All Ages" by Manly P. Hall

Yet Johannes Kepler (in "The Epitome of Copernican Astronomy", Book IV, c. 1621) understood that the dodecahedron could be constructed from the cube, by the addition of what he called a "prism" -- and which I call a "roof", or rooflike structure -- to each of the cube's 6 sides.

I was so taken with Kepler's brief mention of those "prisms" -- which I thought suggested their construction might be done rather easily after all -- that I supposed he must have gotten it from long before; and the most obvious place he could have gotten it was from Euclid, the great expositor of geometry at the beginning of modern mathematical thought.

And so it was (!), when I looked into Euclid's "Elements" (c. 300 BC). (By the way, the history of my discovery of the Great Design of the "gods" is replete with the quick discovery of such connections with the ancient world, and answers to ancient mysteries throughout history, which I have likened to finding one gold nugget after another, scattered everywhere, in a previously unknown valley of gold, of worldwide extent.)

I honored that feeling of discovery, in this case via Kepler and Euclid, by including a simplified version of Euclid's demonstration (cf. Proposition 17, Book XIII), of precisely constructing the dodecahedron from the cube, in an appendix to my book, "The End of the Mystery". It goes as follows, quickly:

Begin with a cube, and make its edges 2 units long. Then on the upper square face of the cube, draw two lines dividing the face into four equal squares of 1 unit side length. On one of these lines, on both the left and right halves (each of 1 unit length), find the golden section (φ) of the unit lines, and construct lines of length φ (= .6180339887... = .618...) as upright poles. ***

These are the ridge poles of the roof structure we are making on this face of the cube. Then add the ridge line between their tops, and add four lines from each ridge pole to the corners of the cube's face (like pegging down a the "roof" is also a "tent", in construction)

Now it is just a matter of repeating this simple process for each face of the cube, to make a full dodecahedron out of the cube.

***To find the golden section of a line is also elementary, Watson: