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 tent...so 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:

Johannes Kepler: Sleepwalker-Founder of Modern Science

While I have been focused here lately on incompetence in modern climate science (and in modern science generally), it needs to be emphasized that scholars throughout history have been thoroughly misguided by the ancient knowledge that was handed down among all the peoples of the world as the "sacred truth", first about the "gods" of earliest myth, and later about "God" Himself. (And it was a rare scholar, philosopher, or theologian, who could both start and end with the humble position of admitting that he/she didn't know what "God" really entailed, or why the most ancient peoples worshipped whole families of "gods", with one god understood as being above all the others). The men who founded modern science were steeped in the old "knowledge", and they had to learn how to focus upon hard physical evidence, and master strict mathematics and logic, as their proper guides, over the "sacred wisdom" passed down from times immemorial, in order to show the way for true science. Dogma had to be set aside, but its ages-old attraction was compelling, so they reached out for hard evidence fitfully and far from consistently, like sleepwalkers in an enchanted-like dream-state (the author Arthur Koestler wrote a beautiful book, "The Sleepwalkers", about these earliest scientists, many years ago now).

I received a comment (from one "garlo") here, in which Kepler's more misguided mystical musings, masquerading as science in his mind, were brought up. I realized a proper response required that I make a distinction between my scientific discovery and verification of a real Great Design of the world, and the mystical musings of Kepler and earlier (and even modern) religiously-minded scholars. I hasten to add, the Design IS of fundamental religious, or more correctly spiritual, importance; but the primary truth of it, that above all needs to be grasped, is that it is a hard, objective, scientific fact. The following is that fuller response to "garlo", and to anyone of like mind.

The verifiable dodecahedral arrangement in the layout of landmasses on the Earth is just one aspect, among many, of the Great Design I found--by the way, I am capitalizing it as a real, unified, and all-encompassing design (not a mere mass of "conjectures", or a formless pile of unconnected and odd observations, as all past researchers have produced, throughout history). The Great Design was responsible for ALL the so-called "ancient mysteries", the latter being just the religious and ritually precise human expressions of the intellectual awe with which the smallest knowledge fragments from the design were passed down among men--as literally the "sacred truth" about the origin of the world and mankind. (But in the present tattered intellectual atmosphere, particularly among academics and those in authority in science, just getting anyone to seriously and professionally consider ANY of the overwhelming mass of evidence, for the Great Design I uncovered, takes on the dimensions of a life's work.)

Kepler's belief, that the Earth was somehow made according to the dodecahedron, was nothing new, it goes back to ancient times; indeed, it qualifies as one of the "ancient mysteries", and Kepler picked up on the regular solids as such, and used them, not so much dispassionately, as a scientist, but as a mystical expression of the will of God (he noted their ancient provenance himself, writing, "More than 2,000 years ago Pythagoreans said that these figures [the regular solids] were the figures of the world..."). Kepler's attempt to match them to the planetary orbits was really just his own mathematical elucidation of the original Pythagorean teaching, which every mathematician knew of as the "harmony of the spheres". Kepler called them the regular solids (not platonic solids, because he knew they predated Plato), as we do today.

But as I wrote in my book, "The End of the Mystery":

"I do not want to take away from those who were already assured of a dodecahedral design of the Earth, including Socrates; but I think I am the first person to reveal its objective foundation, and thus prove it to science, and more importantly, to prove it an integral part of a comprehensive design."

In other words, as I said earlier, the dodecahedral pattern in the layout of the continents is just one aspect of the Great Design of the Earth and solar system, by the "gods" who were our real forebears.

As just one of uncounted many checks I made as part of my research, I naturally checked Kepler's attempt to match the planetary orbits with the regular solids, and found the fit was not that good, certainly not good enough to warrant his claim that the planetary orbits were a God-designed matching of the relative sizes of the regular solids--particularly since the Great Design I had already found is much more precise than any such planetary fitting. And the real meaning behind the ancient association of Earth with the dodecahedron is not to be found in the sizes of the planetary orbits--it is because (as I have explained in my book, "The End of the Mystery") the Earth is the planet of life in the solar system, and the mathematics of the dodecahedron is tied to the life on Earth, on every level from the smallest physical expression (DNA) to the level of metaphorical spiritual insights in the Pythagorean and other ancient mystery traditions.

While Kepler's mystical matching of the regular solids with the planetary orbits was misguided, I did get one important piece of ancient knowledge from him, and later in fine detail in Euclid (Proposition 17, Book XIII): The dodecahedron was anciently, and most simply, known as being built up from the cube, very simply, using the famous "golden section", another key Pythagorean tradition, that goes back to the Great Design itself as a matter of course. I discussed Euclid's lucid construction of the dodecahedron, for the benefit of lay readers, in an appendix in "The End of the Mystery".

Kepler's approach to the "harmony of the spheres" was to consider the sizes of the two spheres associated with a regular solid (like the dodecahedron), one inscribed within and the other circumscribing the solid. For each of the five regular solids, Kepler gave the ratio of the radii of the inscribed (Ri) and circumscribing (Ro) spheres. In modern form, those ratios are:

cube: Ri/Ro = .57735
tetrahedron: .33333
dodecahedron: .79465
icosahedron: same as for dodecahedron
octahedron: same as for cube

You can yourself compare these ratios with the relative sizes of the orbits of the 6 innermost planets (with the mean Earth-Sun distance as the standard unit),

Mercury .387, or .4
Venus .723, or .7
Earth 1.000, or 1
Mars 1.524, or 1.5
Jupiter 5.202, or 5.2

as many have no doubt done both before and after Kepler, and in vain. The Great Design is otherwise.

More empirical-minded scientists later crafted the Titius-Bode "law" for the successive planetary orbits:

mean orbital radius = (n + 4)/10, with n = 0, 3, 6, 12, 24, 48, etc.,

with Earth at the standard distance (1.0, for n = 6), and the asteroid belt, between Mars and Jupiter, being in the right position to represent a former, now-shattered planet (an hypothesis no one has ever proved), but an extra planet, Neptune, where it shouldn't be, if the Titius-Bode formula was actually a law (so it is not a law--any more than current climate models are physical laws, as the incompetent consensus supposes).