Six square faces, twelve edges, eight vertices. The simplest space-filling solid. Edge length: a.
ii.
Raise a pyramid on each face.
A square pyramid on each of the six faces, apex at height a/2 above the centre.
iii.
Triangles fuse into rhombi.
At apex height a/2, adjacent pyramid triangles become coplanar — 24 triangles fuse into 12 rhombi.
02 · DEFINITION
A polyhedron of twelve congruent rhombi.
Faces
12
All identical rhombi — face-transitive.
Edges
24
Every edge has identical length and dihedral angle.
Vertices
14
8 of order 3 (acute) + 6 of order 4 (obtuse).
Symmetry
Oh, m3̄m
Full octahedral symmetry — order 48.
Diagonal ratio
1 : √2
Short to long rhombus diagonal.
Acute angle
≈ 70.53°
arccos(1/3) — tetrahedral bond angle of carbon.
Volume
2·a³
Exactly twice the cube's volume.
Class
Catalan · Parallelohedron
One of 13 Catalan solids; one of 5 space-fillers.
03 · GEOMETRY OF THE FACE
Each face is a specific rhombus, not just any.
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.
04 · THE RHOMBOHEDRON
A cube elongated along its body diagonal.
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.
05 · ABOUT
RHOMBOHEDRON.FUN
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.