What is a dodecahedron?

Worked example. A dodecahedron with 2 cm edges has a surface area of 3√(25 + 10√5) × 2² ≈ 82.58 cm² and a volume of ((15 + 7√5) ÷ 4) × 2³ ≈ 61.30 cm³.

The golden ratio is everywhere in it. The diagonal of each pentagon face is φ ≈ 1.618 times its edge, and the 20 corners can be written using nothing but ±1, φ and 1/φ. That is why the golden ratio keeps turning up in its formulas as √5.

Make one

The net of a dodecahedron

A dodecahedron has 43,380 different nets, the same number as its dual, the icosahedron. Press New net above as many times as you like: you will almost never see the same one twice.

You always cut 19 of the 30 edges. Eleven edges stay as folds holding the 12 pentagons together, and 19 are cut: one less than the 20 vertices.

To build it: press Print this net, cut along the solid lines, fold along the dashed lines, put glue on the grey tabs and close it up one face at a time. Albrecht Dürer printed some of the first nets like this in his 1525 book on measuring with compass and ruler. All five Platonic solids, with every net, are in Platonic Solids 3D.

Name and history

Plato’s fifth shape and the Roman mystery

The name is Greek: dōdeka, twelve, and hedra, a seat or face.

Plato never actually names it. After giving fire, air, water and earth their solids, the Timaeus says only that “there was yet a fifth combination which God used in the delineation of the universe.” Readers have taken that to be the dodecahedron ever since, but the word itself appears only in translators’ introductions.

Roman dodecahedra. Around 130 small hollow copper-alloy dodecahedra are known. Live Science: “Each corner of a Roman dodecahedron is topped with a small sphere, and each pentagonal face has a hole of varying diameter.” They come only from the north-western Roman provinces: Britain, Gaul and Germania, today’s England, Wales, France, Belgium, the Netherlands and Germany. Live Science puts them all in the late 2nd to late 4th centuries, from 4 to 10 cm tall. “No writing has been found on any dodecahedron,” and of “the 50 or more theories,” the researcher Guggenberger wrote, “most can now be ruled out.” None has been proven.

The newest one was dug up in June 2023 at Norton Disney in Lincolnshire, by volunteers with no metal detector. It is about 8 cm tall, weighs 245 g, and is made of 75% copper, 7% tin and 18% lead. It was buried with 4th century pottery, and the group that found it thinks it was used “for ritual and religious purposes.”

An older one in stone was found at Monte Loffa in northern Italy and is believed to date from before 500 BCE, carved in soapstone with figures on its faces.

Out in the world

Where you see a dodecahedron

Dice. A fair die needs identical faces, so twelve-sided dice are regular dodecahedra.

Crystals that almost make it. Pyrite grows a twelve-sided crystal called the pyritohedron. It looks like a dodecahedron, but MathWorld calls it “an irregular dodecahedron composed of identical irregular pentagons”. Nature cannot grow the true one in a crystal.

The shape of the universe? In 2003 a team led by Jean-Pierre Luminet proposed in Nature that space might be finite and wrap around like the Poincaré dodecahedral space, to explain a pattern in the cosmic microwave background. It was a suggested model, not a proven one, but it put Plato’s fifth shape back into cosmology.

Kepler’s planets. In 1596 Johannes Kepler placed a dodecahedron between the orbits of Mars and Earth in his nested-solids model of the solar system.

The dodecahedron is one of exactly five Platonic solids, and Euclid proved around 300 BCE that there can never be a sixth.