Six square faces, twelve edges, eight vertices. The simplest space-filling solid. Edge length: a.
A square pyramid on each of the six faces, apex at height a/2 above the centre.
At apex height a/2, adjacent pyramid triangles become coplanar — 24 triangles fuse into 12 rhombi.
The faces of the rhombic dodecahedron are uniquely defined: rhombi whose diagonals stand in ratio 1 : √2. The acute angle is arccos(1/3) ≈ 70.53°, identical to the tetrahedral bond angle of carbon and methane. This specific ratio is what allows twelve of them to close perfectly around the octahedral symmetry group.
A rhombohedron is a parallelepiped with six congruent rhombic faces, produced by stretching a cube along its [1,1,1] body diagonal. At stretch factor s=1 it is a perfect cube. As s increases the two poles move apart and the six faces elongate into diamonds. The two special cases — acute and obtuse golden rhombohedra — tile space in combination and fill the interior of an icosahedron.
Toggle ICOSA mode (◇ in the topbar) to morph from the dodecahedron to its dual, the icosahedron. This is a vertex-collapse transformation: the 12 pentagonal faces shrink toward their centroids — which become the 20 vertices of the icosahedron — while 30 rhombi and 20 triangles grow in from edges and corners to keep the surface closed at every value of T.
Every vertex is shared between adjacent face types, so the surface never tears — just continuously redistributes area among pentagons, rhombi, and triangles as the morph runs.
An interactive geometry studio built with Three.js. Morph continuously between the Rhombic Dodecahedron, the Cube, and the Rhombohedron. Load images or video onto the faces. Drag to rotate, scroll to zoom, slide to morph. Save any state as a self-contained HTML file.
Built by GenBuilt Systems · rhombohedron.fun