Archimedean Solids | mysimulator.uk

Semi-regular: one step down from Platonic

A Platonic solid must be built from a single kind of regular polygon, with every vertex looking geometrically identical — that strict rule allows exactly five solids: tetrahedron, cube, octahedron, dodecahedron and icosahedron. An Archimedean solid relaxes only the face requirement: it may mix two or three different types of regular polygon around its surface, but it still demands that every vertex be surrounded by the same cyclic sequence of polygons, a property called vertex-transitivity. That single relaxed-but-still-strict rule is what defines the entire family.

Built by truncation and rectification

Many Archimedean solids are constructed directly from a Platonic solid by slicing off its corners. Cut just enough to leave a small new face at each original vertex without touching the original edges, and a tetrahedron becomes a truncated tetrahedron, a cube becomes a truncated cube, an octahedron becomes a truncated octahedron. Cut deeper, exactly down to each edge's midpoint — rectification — and the original faces shrink to half size while new faces appear where each vertex used to be: a cube rectifies into a cuboctahedron, a dodecahedron into an icosidodecahedron.

truncation depth t = 0     →  original Platonic solid, untouched
truncation depth t = 1/3   →  Archimedean truncated form (e.g. truncated cube)
truncation depth t = 1/2   →  rectified solid (cut to the edge midpoints)

The full list: 13 solids

The complete family numbers exactly 13 convex Archimedean solids (a count that treats the two chiral snub forms as one solid each). This simulation focuses on five of them, generated live from truncation and rectification: the truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron and icosidodecahedron — each rotatable, with its vertex, edge and face counts displayed alongside a running Euler check.

live demo · Archimedean solids generated by truncating Platonic solids● LIVE

Euler's formula as a live check

Every convex polyhedron obeys Euler's formula, V - E + F = 2, relating its vertex, edge and face counts. Computing all three for a generated solid and checking that they balance is a fast, reliable sanity check that a truncation or rectification was built correctly. A truncated cube, for instance, has 24 vertices, 36 edges and 14 faces (6 octagons plus 8 triangles): 24 - 36 + 14 = 2, exactly as required.

Why they matter beyond aesthetics

Archimedean solids show up well outside pure geometry. The truncated icosahedron is the classic soccer-ball pattern, and the same vertex arrangement describes the carbon cage of the buckminsterfullerene molecule, C60. Viral capsids frequently adopt icosahedral and related semi-regular symmetries to enclose genetic material efficiently, and geodesic domes and other architectural structures borrow the same family of shapes for their combination of structural efficiency and near-spherical coverage.

Frequently asked questions

What makes a solid Archimedean rather than Platonic?

A Platonic solid must use only one kind of regular polygon for every face, with every vertex identical. An Archimedean solid relaxes that face restriction — it can mix two or three different types of regular polygon — but still requires every vertex to be surrounded by the same cyclic sequence of polygons, a property called vertex-transitivity. That relaxed but still-strict symmetry rule is exactly what produces the family of 13 semi-regular solids.

Why are there exactly 13 Archimedean solids?

The count of 13 comes from an exhaustive geometric classification: only a limited set of vertex configurations (cyclic sequences of regular polygons meeting at a point) can actually close up into a convex, vertex-transitive solid without leaving gaps or overlaps. Archimedes is credited with the original enumeration in antiquity, and it was independently reconstructed and confirmed in the Renaissance and later by rigorous modern proof — no fourteenth one exists.

How does Euler's formula check whether a polyhedron is valid?

Euler's formula, V - E + F = 2, holds for every convex polyhedron, relating its vertex count V, edge count E and face count F. Computing all three for a generated solid and checking that the equation balances is a fast, reliable sanity check that a truncation or rectification was constructed correctly — if the count comes out to anything other than 2, something in the geometry generation went wrong.

Try it live

Everything above runs in your browser — open Archimedean Solids and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Archimedean Solids simulation

Archimedean Solids: Truncating the Platonic Solids