Home▸Geometry▸Archimedean Solids

🔶 Archimedean Solids

Interactive Archimedean solids explorer. Rotate the truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron and icosidodecahedron, generated live by truncating Platonic solids, with V/E/F stats and Euler's formula.

Geometry3DModerate60 FPS

🔶 What It Demonstrates

The Archimedean solids are 13 convex, semi-regular polyhedra: every face is a regular polygon, but (unlike the Platonic solids) two or more different polygon types meet at each vertex — and every vertex is surrounded identically (they are vertex-transitive). This simulator generates five of them — the truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron and icosidodecahedron — geometrically, by slicing the corners off the corresponding Platonic solid (tetrahedron, cube or octahedron) with a plane at each vertex, a process called vertex truncation.

Note on accuracy: the truncation ratio used here is a fixed, simplified value rather than the exact Archimedean ratio needed for perfectly regular faces at every cut. The resulting shapes are topologically identical to the true Archimedean solids (same V, E, F counts, same face-adjacency pattern) and visually very close, but the octagons/hexagons are only approximately regular. This is an intentional simplification for an interactive geometry demo, not a precision CAD model.

How to Use

Pick a solid from the Solid dropdown.
Toggle Auto-rotate and adjust its Speed, or drag the canvas to rotate manually.
Switch between Solid shaded faces and Wireframe edges.
Check the Statistics panel: V, E and F always satisfy Euler's formula V − E + F = 2, since truncation keeps the solid convex.

Did You Know?

A cuboctahedron is what you get by truncating a cube exactly at each edge's midpoint (a "rectification") — it has the same vertex arrangement whether you start from a cube or an octahedron, because the two are dual solids. The truncated octahedron is the only Archimedean solid that tiles 3D space by itself with no gaps, a shape used to model the tightest possible packing of foam bubbles (the Kelvin conjecture). Buckminster Fuller's geodesic domes and the classic football's black-and-white panel pattern both borrow from Archimedean-style semi-regular tilings of a sphere.

About this simulation

This simulator builds five Archimedean solids live, by slicing the corners off a Platonic solid with a generic vertex-truncation algorithm written for this page — no hardcoded coordinate tables. Each cut vertex is replaced by a small polygon, and each original face shrinks accordingly, producing the same combinatorial structure (vertex/edge/face counts) as the true Archimedean solids.

🔬 What it shows

Vertex truncation applied to the tetrahedron, cube and octahedron. At a 50% truncation ratio, opposite cuts meet at edge midpoints, producing the "rectified" cuboctahedron and icosidodecahedron; at a smaller ratio, distinct corner-cut polygons and shrunken original faces produce the truncated forms.

🎮 How to use

Choose a solid from the dropdown. Enable Auto-rotate and set its Speed, or drag to rotate manually. Toggle Solid shaded faces and Wireframe edges. The Statistics panel reports V, E, F live and confirms Euler's formula V − E + F = 2 for every solid, since all remain convex.

💡 Did you know?

The truncated octahedron is the only Archimedean solid that tiles 3D space on its own with no gaps — a fact relevant to the geometry of foam and cellular packing (the Kelvin conjecture on minimal-surface-area space-filling cells).

Frequently asked questions

What makes a solid "Archimedean"?

An Archimedean solid is convex, has only regular polygons as faces (of two or more different types), and looks identical from every vertex — the same cyclic sequence of polygons surrounds each corner. There are exactly 13 such solids, plus two infinite families of prisms and antiprisms which are usually classified separately.

How does vertex truncation work?

Truncation slices off each corner of a polyhedron with a small plane, replacing the vertex with a new polygon whose number of sides equals the number of edges meeting there. Each original face survives but with its corners cut off, effectively doubling its side count unless the cut reaches all the way to the edge midpoints.

Why aren't the shapes on this page perfectly regular?

The exact Archimedean solids require a specific truncation ratio for every face to be a perfectly regular polygon. This simulator uses one fixed, simplified ratio for clarity of the construction process, so faces are close to regular but not mathematically exact — the topology (V, E, F and adjacency) is still fully correct.

What is the difference between cuboctahedron and icosidodecahedron?

Both are "rectified" solids made by truncating exactly to each edge's midpoint. The cuboctahedron rectifies the cube (or equivalently the octahedron, its dual), producing 8 triangles and 6 squares. The icosidodecahedron rectifies the icosahedron (or dodecahedron), producing 20 triangles and 12 pentagons.

Do Archimedean solids satisfy Euler's formula?

Yes. Every Archimedean solid is convex, so V − E + F = 2 always holds, exactly as it does for the simpler Platonic solids. This simulator recomputes V, E and F from the generated mesh on every solid change and displays the live check.

⚙ Under the hood

Canvas 2DArchimedean SolidsVertex Truncation3D ProjectionSemi-Regular

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install