Why Dice Have So Many Shapes: The Five Platonic Solids Explained

What Is a Platonic Solid?

A Platonic solid is a three-dimensional shape that satisfies three conditions:

1. All faces are identical regular polygons (equilateral triangles, squares, pentagons, etc.)
2. The same number of faces meet at every vertex
3. It is convex (no indentations)

There are exactly five Platonic solids. This is not a matter of discovery — it was proven by Euclid in Book XIII of his Elements around 300 BC, and the proof is essentially the same today as it was then. Only five such shapes can exist in three-dimensional Euclidean space. The five are:

These five shapes are the only geometrically fair dice possible in three dimensions — every face is equally likely to land on because every face has identical geometry.

Why Symmetry Makes a Die Fair

A die is fair (in the mathematical sense) if every face has an equal probability of landing down after a random throw. For this to hold purely from the geometry — without relying on the materials being perfectly uniform — the die must be symmetric in a specific way: every face must be physically equivalent to every other face.

For the Platonic solids, this is guaranteed by their symmetry group — the set of rotations that map the solid onto itself. For an icosahedron (D20), there are 60 distinct rotational symmetries that permute the faces. This means that no face is geometrically privileged over any other. A randomly oriented throw produces any face with probability 1/20.

Non-Platonic dice — the D10, for instance — do not have this property. The D10 is a pentagonal trapezohedron: a shape with 10 identical kite-shaped faces. While it is not a Platonic solid (because its faces are not regular polygons), it is still fair due to its trapezohedron symmetry. The D10 is a special case of a broader class of fair dice.

Plato's Cosmic Significance (360 BC)

Plato was so impressed by the five solids bearing his name that he assigned them cosmological significance in his dialogue Timaeus (360 BC). He associated each solid with one of the classical elements:

• Tetrahedron (4 faces) → Fire (sharp, piercing)
• Cube (6 faces) → Earth (stable, solid)
• Octahedron (8 faces) → Air
• Icosahedron (20 faces) → Water (fluid, many faces)
• Dodecahedron (12 faces) → The Cosmos (the shape of the universe)

These associations were not scientific claims but philosophical metaphors about the mathematical structure of reality. The assignment of the dodecahedron to "the cosmos" may have been inspired by the fact that the dodecahedron's faces are pentagons — and the regular pentagon contains the golden ratio, which the ancient Greeks regarded as aesthetically and mathematically perfect.

The D10: The Non-Platonic Standard

The standard tabletop RPG set includes a D10 (ten-sided die), which does not exist as a Platonic solid — there is no Platonic solid with 10 faces. The D10 is a pentagonal trapezohedron: five kite-shaped faces on each hemisphere, with the two halves rotated 36 degrees relative to each other so the edges interlock.

The D10's fairness comes from its trapezohedron symmetry, not Platonic symmetry. Its faces are congruent but not regular polygons. It is fair in the probabilistic sense — each face has an equal 1/10 probability — but it achieves this through a different geometric argument than the Platonic dice.

Two D10s are sometimes used together as a "percentile" combination: one shows tens (00, 10, 20, ... 90) and one shows units (1–10), giving 100 equally likely outcomes from 01 to 100 — used in percentage-based RPG systems.

Could There Be a Fair D7 or D11?

A fair die with any number of sides is theoretically possible, though the shapes get increasingly exotic. A mathematically fair D7 would need to be a heptagonal trapezohedron — a shape with 14 faces in which 7 of them are landing faces and 7 are the edges, which would require careful design. In practice, making a genuinely fair D7 is difficult because the irregular geometry makes manufacturing precision harder to achieve.

Dice with odd numbers of faces that cannot use the trapezohedron approach (D3, for example) are often approximated by using a cylinder and flattening it appropriately, or by using a longer shape where the end faces are clearly different from the side faces — technically fair but aesthetically unusual. The D3 is most commonly simulated by rolling a D6 and dividing by 2 (rounding up), which gives each value of 1, 2, 3 a probability of exactly 1/3.

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