The five Platonic solids are the only convex polyhedra whose faces are congruent regular polygons with the same number meeting at each vertex. They were classified by Plato in the Timaeus (~360 BC): tetrahedron (4 equilateral triangles), cube (6 squares), octahedron (8 triangles), dodecahedron (12 pentagons), and icosahedron (20 triangles). That exactly five exist follows from a dihedral-angle argument: the interior angles of the faces meeting at a vertex must sum to less than 360°, which limits the possibilities.
The 13 Archimedean solids are semi-regular convex polyhedra — every face is a regular polygon but faces may come in two or more types, and every vertex is surrounded identically. They were first enumerated by Archimedes (lost work) and rediscovered by Kepler in 1619. The truncated icosahedron — 12 pentagons and 20 hexagons — is the shape of a standard football and of the C60 buckminsterfullerene molecule.
Euler's formula V − E + F = 2 holds for every convex polyhedron (and any polyhedron topologically equivalent to a sphere). First proved by Euler in 1758, it expresses the Euler characteristic of the sphere. Every solid in this simulation satisfies it — check the stats panel to confirm.
Frequently Asked Questions
The five Platonic solids are the tetrahedron (4 equilateral-triangle faces, V=4, E=6, F=4), cube or hexahedron (6 square faces, V=8, E=12, F=6), octahedron (8 equilateral-triangle faces, V=6, E=12, F=8), dodecahedron (12 regular-pentagon faces, V=20, E=30, F=12), and icosahedron (20 equilateral-triangle faces, V=12, E=30, F=20). Each solid has identical regular polygon faces and the same vertex configuration throughout. Plato associated them with fire, earth, air, ether/cosmos, and water respectively.
Euler's formula V − E + F = 2 relates the number of vertices (V), edges (E), and faces (F) of any convex polyhedron. For a cube: V=8, E=12, F=6, so 8−12+6=2. For an icosahedron: V=12, E=30, F=20, so 12−30+20=2. The formula was first proved by Leonhard Euler in 1758 and follows from the fact that every convex polyhedron is topologically equivalent to a sphere, which has Euler characteristic 2.
There are exactly 13 Archimedean solids. They are convex polyhedra with two or more types of regular polygon faces and every vertex surrounded identically (vertex-transitive). The 13 are: truncated tetrahedron, cuboctahedron, truncated cube, truncated octahedron, rhombicuboctahedron, truncated cuboctahedron, snub cube, icosidodecahedron, truncated dodecahedron, truncated icosahedron (football shape), rhombicosidodecahedron, truncated icosidodecahedron, and snub dodecahedron. All 13 satisfy Euler's formula.
Platonic solids have only one type of regular polygon face (e.g., all equilateral triangles or all squares). Archimedean solids have two or more types of regular polygon face (e.g., squares and triangles) but every vertex is still surrounded identically — they are vertex-transitive. Both families are convex and highly symmetric. Archimedean solids can often be constructed from Platonic solids by truncation (cutting corners), rectification, or snubbing.
Every convex polyhedron is topologically equivalent to a sphere, which has Euler characteristic χ = 2. One classic proof: remove one face and flatten the polyhedron onto a plane. The result is a planar graph. Then repeatedly remove edges that border two regions (V−E+F unchanged) or "trim" leaf edges (V−E+F unchanged) until a single vertex remains (V=1, E=0, F=1, so 1−0+1=2). Therefore the original V−E+F must also be 2.
The dual polyhedron is formed by placing a new vertex at the centre of each face, then connecting new vertices that share an original face-edge. The tetrahedron is self-dual (its dual is another tetrahedron). The cube and octahedron are duals of each other — cube has V=8, F=6 while octahedron has V=6, F=8; the counts swap. Likewise the dodecahedron (V=20, F=12) and icosahedron (V=12, F=20) are duals. The duals of Archimedean solids are the 13 Catalan solids.
The traditional 32-panel football (soccer ball) follows the shape of a truncated icosahedron: begin with an icosahedron and truncate each of its 12 vertices, producing a regular pentagon at each cut. The result has 12 pentagons and 20 hexagons (F=32), V=60, E=90, and 60−90+32=2. The same geometry describes the C60 buckminsterfullerene carbon molecule, dubbed a "buckyball" after Buckminster Fuller.
The five Platonic solids belong to three symmetry groups: the tetrahedron has full tetrahedral symmetry T_d (order 24); the cube and octahedron share full octahedral symmetry O_h (order 48); and the dodecahedron and icosahedron share full icosahedral symmetry I_h (order 120). Archimedean solids inherit the same three groups depending on which Platonic solid they derive from by truncation or rectification.
Yes — for polyhedra that are not topologically equivalent to a sphere. A polyhedron shaped like a torus (genus g=1) satisfies V − E + F = 0. In general V − E + F = 2 − 2g, where g is the genus (number of handles). The formula also fails for non-orientable surfaces (like a Klein bottle) and for self-intersecting polyhedra. The 18 regular star polyhedra (Kepler-Poinsot solids) also violate the simple formula.
Select any solid from the dropdown menu and rotate it with click-drag (or touch-drag on mobile). Read V, E, F from the stats panel and confirm V − E + F = 2. Switch to wireframe mode to see the edge skeleton clearly without face colours. Use flat shading to highlight individual faces. Try all five Platonic solids, then explore all 13 Archimedean solids — notice how each Archimedean solid relates to a Platonic solid by truncation, snubbing or rectification.