The Fifth Platonic Solid

As most of you know, my husband Allan lives squarely in the round world of footbag (Hacky Sack©). He's a four-time world champion, he's traveled the world demonstrating the sport, and he also designs and stitches footbags. When I say he “designs” them, it might be more accurate to say that he's discovered and decoded many, many spherical shapes involving as many as 452 faces.

What's amazing is that he was able to then mentally construct a 452-faced sphere, cut those 452 panels out of Ultrasuede material, and stitch them all together! In order to close a footbag this complicated, you must work simultaneously with five threaded needles. His is an amazing talent.

But Allan really has designed many visually stunning variations on a footbag.

When I met him eight years ago, I immediately noticed that Allan had an incredible ability to visualize and conceptualize sacred geometries in three dimensions. He does similar work with wood and in constructing the truncated icosahedron lamp below, he consulted a math professor to find the exact angles necessary for fitting the dowels between the blocks of wood. Precision, here, is key; if the angles are off, the result won't be a sphere.

Together, Allan and I have produced a stitching course for a 12-panel footbag, a shape geometers call a dodecahedron. This was mind expanding for me because, as the writer of the course, I had to learn his stitching method. I now fully understand the structure of a dodecahedron and how it goes together, which is, sort of... mentally liberating! It definitely helps me in meditation.

A dodecahedron has 12 faces: do (2) and deca (10). Each face is an identical pentagon. Forming the faces with pliable fabric allows this geometrical shape to pull simultaneously in all directions, forming a sphere.

Below is an excerpted page from the stitching manual, where we briefly describe the shape as the fifth Platonic solid. (A Platonic solid is defined as a shape whose vertices are inscribed within a sphere.) In Timaeus, Plato described this shape as the one, “...the god used for arranging the constellations on the whole heaven.”

Timaeus is also where Plato describes the creation of the world soul by the demiurge, and how the world soul is structured using the same ratios as a musical scale. This scale doesn't use Pythagorean intervals, however. Instead, its intervals are tempered by arithmetic and harmonic means. (I discuss this in more detail in this video:)

If I look at a dodecahedron or a 12-panel footbag harmonically, it represents the note of G as the 12th harmonic. This note is the mese of the harmonic chromatic C scale between Harmonics 16 and 32.

The mese is the center of the ancient Greek Lesser Perfect System, as shown below:

Another way to say this is that G (scaled up to Harmonic 24, as it's valued in the scale between Harmonics 16 and 32) is the perfect 5th off the tonic of C (Harmonic 16). The ratio of the perfect 5th is 3 : 2, and that's also the ratio of G : C (24 : 16 = 3 : 2).

Looking at the 12-panel or dodecahedron as reflecting the concept of the central mese echoes what Allan said about the shape from his own perspective, at the top of the manual's first page: “If I had to pick the most basic geometrical sphere, one that relates and scales to most other spherical geometries, it would be the 12 panel or 12-faced sphere. It’s the biggest brick in the foundation of the three-dimensional sphere.” I feel the same way about the perfect 5th and its mese position.

When asked this question: How does the dodecahedron relate to the ancient Greek musical mese?, Google AI gives the following response:

If your curiosity is piqued regarding how the 12 faces of a dodecahedron all fit together and would like to deep dive this geometrical mese for yourself, please visit Allan's website to learn more about the 12-panel Stitching Course: https://www.hanedanefootbags.com/12-panel-get-started