Explore Platonic and Archimedean solids in 3D. Rotate, unfold to nets, and verify Euler formula V-E+F=2.
Eighteen convex solids whose vertices all look the same: the five Platonic solids, one regular polygon per face, and the thirteen Archimedean solids, which mix two or three face types. Drag any model to rotate it, unfold it into a flat net, and read off vertex, edge, and face counts with the Euler check V - E + F = 2. What surprises most people: a solid has no single net. The viewer unfolds one of many - a cube alone has 11 distinct nets - and the two snub solids come in mirror-image pairs.
A concrete input and expected output from the current implementation.
Input
Octahedron (picked from the list of five Platonic solids)
Expected output
The octahedron renders as a rotatable 3D model with the readout V = 6, E = 12, F = 8 and the Euler check 6 - 12 + 8 = 2, marked as satisfied.
Every octahedron has 6 vertices, 12 edges, and 8 triangular faces, with 4 triangles meeting at each vertex. The counts satisfy V - E + F = 2 because the octahedron is convex, which is exactly the condition the Euler check requires.
The viewer carries the five Platonic and thirteen Archimedean solids. Each solid has exact geometry - vertex positions and the regular polygons that meet there - so the displayed V, E, F counts are the true counts, from the tetrahedron's 4 vertices to the truncated icosidodecahedron's 120. You rotate the model in three dimensions; a readout shows the counts and the Euler expression.
The unfold action hinges the faces outward around shared edges until the surface lies flat in one plane, showing how the solid is stitched together. The cut pattern is chosen by the viewer, and it is not unique: cutting a different set of edges produces a different net. A cube, for example, has 11 distinct nets, and the displayed one is just one of them.
A Platonic solid uses one kind of regular face, all triangles or all squares or all pentagons, with the same arrangement at every vertex. An Archimedean solid mixes two or three regular face types, like the pentagons and hexagons of the truncated icosahedron, while every vertex still looks identical. There are 5 Platonic and 13 Archimedean solids, and the viewer lists all 18.
Because V - E + F = 2 is a theorem for every convex polyhedron, and all 18 solids here are convex, the equality holds exactly each time, for instance 120 - 180 + 62 = 2 for the truncated icosidodecahedron. The readout is a consistency check on the counts, not a filter that selects solids.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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