What is an icosahedron?

Worked example. An icosahedron with 3 cm edges has a surface area of 5√3 × 3² = 45√3 ≈ 77.94 cm² and a volume of (5 ÷ 12)(3 + √5) × 3³ ≈ 58.91 cm³.

The golden ratio is hidden in it. Its 12 corners sit on three golden rectangles, each φ ≈ 1.618 times as long as it is wide, crossing at right angles. The midradius, from the centre to the middle of an edge, is exactly half of φ times the edge. More on that number on the golden ratio page.

Make one

The net of a icosahedron

An icosahedron has 43,380 different nets, the same as the dodecahedron, its dual. The classic one is a zigzag strip of 10 triangles with 5 flaps above and 5 below.

You always cut 11 of the 30 edges. Nineteen edges stay as folds and 11 are cut: one less than its 12 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

Water, Theaetetus and the dual

The name is Greek: eikosi, twenty, and hedra, a seat or face.

Plato made it water. Of the three triangle solids it is the biggest and the steadiest, so the Timaeus gives it to the heaviest of the moving elements: “to water we assign that one of the remaining forms which is the least moveable,” and “the greatest to water.”

Theaetetus first studied it. MacTutor: “Theaetetus was the first to study the octahedron and the icosahedron and it is believed that Book XIII of Euclid’s Elements is based on his work.”

It is the dodecahedron’s dual. Put a dot in the middle of each of the icosahedron’s 20 faces and join them: you get a dodecahedron with 20 corners. Both have 30 edges.

Out in the world

Where you see a icosahedron

Viruses. Many viruses wrap their genes in a protein shell with icosahedral symmetry. The simplest kind is built from “60 asymmetric units made of 1 protein” (ViralZone). Donald Caspar gave the first evidence for this symmetry in 1956.

Domes. Buckminster Fuller’s 1954 patent for his geodesic domes says “the grids are formed on the faces of a spherical icosahedron.” He made them famous, but MathWorld notes the first geodesic dome was built in 1922 on the Zeiss works in Jena, Germany.

The d20. Twenty-sided dice are far older than tabletop games. The Metropolitan Museum of Art holds one from Egypt, carved in serpentinite with Greek letters on its faces, dated about the 2nd century BCE to the 4th century CE.

Plankton. In 1904 Ernst Haeckel opened Art Forms in Nature with a tiny sea creature he named Circogonia icosahedra.

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