Tetrahedron · Cube · Octahedron · Dodecahedron · Icosahedron · Euler's formula V−E+F=2
Drag on the canvas to rotate manually.
A Platonic solid is a convex polyhedron whose faces are all congruent regular polygons, with the same number of faces meeting at every vertex. There are exactly five: the tetrahedron (4 triangular faces), the cube (6 square faces), the octahedron (8 triangular faces), the dodecahedron (12 pentagonal faces) and the icosahedron (20 triangular faces). This has been known since antiquity — Euclid's Elements ends by proving no other such solid can exist. Every one obeys Euler's formula, V − E + F = 2, relating vertex, edge and face counts.
The dodecahedron and icosahedron are dual polyhedra: placing a vertex at the centre of every face of one produces the other, and their vertex/face counts swap (12↔20, edges stay at 30). The cube and octahedron form another dual pair (8↔6, edges stay at 12), while the tetrahedron is self-dual. Plato associated four of the solids with the classical elements — earth (cube), air (octahedron), water (icosahedron) and fire (tetrahedron) — and the dodecahedron with the heavens, in his dialogue Timaeus (c. 360 BCE).
This simulator renders the five Platonic solids as true 3D wireframes and shaded meshes on a 2D canvas, using a hand-written rotation-and-projection pipeline (no 3D library). Each solid's exact vertex, edge and face data is built from its defining coordinates — including the icosahedron from golden-ratio coordinates and the dodecahedron as the geometric dual of the icosahedron — then rotated with standard X/Y rotation matrices and projected with a simple perspective transform.
Exact vertex coordinates for each solid (e.g. the icosahedron's twelve vertices at cyclic permutations of (0, ±1, ±φ)) are rotated in 3D, projected to 2D with perspective division, and drawn either as edges (wireframe) or as painter's-algorithm-sorted, normal-shaded polygons (solid mode).
Pick a solid from the dropdown. Auto-rotate spins it continuously at the chosen Speed, or drag on the canvas to rotate manually at any time. Toggle solid shading and wireframe edges independently, and watch the Euler's Formula panel confirm V − E + F = 2 for whichever solid is shown.
The dodecahedron (12 pentagons) and icosahedron (20 triangles) are duals of each other — swap vertices and faces and you get the other solid, while the edge count (30) stays fixed. This simulation literally constructs the dodecahedron by dualising the icosahedron's geometry.
A Platonic solid is a convex 3D shape built from one type of regular polygon, with the same number of faces meeting at every vertex. Only five such solids exist: the tetrahedron, cube, octahedron, dodecahedron and icosahedron, a fact proved geometrically over two thousand years ago.
At each vertex, the interior angles of the meeting faces must sum to less than 360 degrees, otherwise the surface cannot fold into a closed convex shape. Checking every regular polygon against every possible number of faces per vertex (3, 4 or 5, since more would already reach or exceed 360 degrees for the smallest polygon) yields exactly five valid combinations.
Euler's formula states that for any convex polyhedron, V − E + F = 2, where V, E and F are the vertex, edge and face counts. It reflects a deep topological fact: the surface of any convex polyhedron is equivalent to a sphere, and 2 is the sphere's Euler characteristic.
Rather than hardcoding its coordinates directly, the code computes it as the dual of the icosahedron: each dodecahedron vertex is the centroid of an icosahedron face, and each dodecahedron pentagon corresponds to the ring of faces around an icosahedron vertex, ordered by angle.
It fills each face with a colour whose brightness depends on the angle between the face's normal vector and a fixed light direction, and draws the faces back-to-front (painter's algorithm) using their average depth so nearer faces correctly cover farther ones.