There are five Platonic solids: the tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron. They’re a unique group of three-dimensional shapes that have identical polygons on each face and the same number of polygons meeting at each corner. These same-surface solids aren’t new to the mathematical party — they’re called “Platonic” because Athenian philosopher Plato famously associated them with elements of nature — but they still remain a source a mathematical mystery. Now, a new dodecahedron discovery is shaking up the five solids framework with pathway permutation.

Solid Gold

So, what sets a Platonic solid apart from other shapes? It’s all about repetition. Each Platonic solid has the same polygon on every face and the same number of polygons touching at each corner (which is called a vertex), and the internal angles of these meeting points add up to less than 360 degrees.

For an easy example, consider the cube. It has six sides, each of which is a square, with three squares meeting at each corner. Each internal angle is 90 degrees, meaning that the total at each vertex is 270 degrees. Three of the other five Platonic solids — the tetrahedron, octahedron and icosahedron — have triangles on each face, while the dodecahedron has pentagons.

Here’s a quick breakdown:

• Tetrahedron — four triangle faces, three triangles meet at each corner with combined angles of 180 degrees
• Hexahedron (cube) — six square faces, three squares meet at each corner with combined angles of 270 degrees
• Octahedron — eight triangle faces, four triangles meet at each corner with combined angles of 240 degrees
• Dodecahedron — 12 pentagon faces, three pentagons meet at each corner with combined angles of 324 degrees
• Icosahedron — 20 triangle faces, five triangles meet at each corner with combined angles of 300 degrees

Five for Fighting

Why five? Why not ten, twenty or a thousand Platonic solids?

The answer is simple, if not immediately obvious: internal angles. Consider a hexagon, which has internal angles of 120 degrees at each corner. To create a three-dimensional polygon, at least three faces need to touch, meaning the internal angle of identical hexagon vertices is 360 degrees (3×120) and the shape goes flat. The same problem happens if we add more corners to the existing line of Platonic solids — while three pentagons meeting give a combined angle of 324 degrees at 108 degrees each, adding one more corner bumps this up to 432 degrees and makes creating a cohesive polygon impossible.

While it’s possible to solve this angle problem by changing the shape of some sides to bring internal totals under 360 degrees, the solid is no longer uniform. Despite mathematical efforts to prove otherwise over two millennia, there are only five shapes that satisfy all three conditions: identical faces, uniform corner connections and combined vertex angles less than 360 degrees.

Found in Translation

Despite the ample time to study the five Platonic solids, questions still remain. One of the most popular centers on corners and pathways: Is it possible to stand at the corner of any Platonic solid, walk in a straight line, and arrive back at your starting point without passing through any other corners along the way? Recent research found that no such corner-to-corner or “geodesic” paths exist across tetrahedrons, octahedrons, icosahedrons or cubes. But what about pentagon-based dodecahedrons?

As it turns out, there are an infinite number of ways to start at a corner, walk across a dodecahedron and end up back where you started, all without crossing your own path. Research published in Experimental Mathematics, three mathematicians — Jayadev Athreya, David Aulicino and Patrick Hooper — found “exactly 31 equivalence classes of closed saddle connections” that can be used as the basis for non-overlapping routes. They relied on the use of computer algorithms to create what are known as “translation surfaces.”

These surfaces start as Platonic solids that have been pulled apart at the corners to lie flat. Each iteration of this lay-flat surface is called a “net” and offers a staring point for our potential Platonic path. When this path reaches the edge of the net, another net is added, rotated by some multiple of 36 degrees to represent the next circuit around the dodecahedron. Lay out 10 nets and you’ve covered all 360 degrees, meaning the 11th net is effectively the same as the first and can be joined to its corresponding counterpart on the first net to produce a complete translation surface.

Using advanced computer modeling, the team was able to calculate all possible paths from all possible corners and demonstrate that there are, in fact, a host of non-crossing, verified dodecahedron routes that lead away from and back to the same vertex.

Shaping the Future

While the mathematical and statistical modeling that underpins investigation into the five Platonic shapes may not capture public interest like the ongoing evolution of more well-known theories of quantum mechanics and relativity, research makes it clear that even well-known and seemingly well-understood frameworks still contain secrets that could help shape our understanding of the physical world.