A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°.

Zonohedra with regular faces

The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated:

• Of the Archimedean solids:
  1. Truncated octahedron 4.6.6 has 6 square and 8 hexagonal faces.
  2. Great rhombicuboctahedron 4.6.8 has 12 squares, 6 hexagons, and 8 octagons.
  3. Great rhombicosidodecahedron 4.6.10 has 30 squares, 20 hexagons and 12 decagons.
• Of the infinte set of uniform prisms, 4.4.2N have 2 2N-agonal, and 2N square faces, where N=2,4...
  1. cube 4.4.4
  2. hexagonal prism 4.4.6
  3. octagonal prism 4.4.8
  4. decagonal prism 4.4.10
  5. dodecagonal prism 4.4.12
  6. ...

Zonohedra with equilateral faces

A partial common list includes:

• Of the Catalan solids: (face-uniform)
  1. Rhombic dodecahedron V3.4.3.4 has 12 identical rhombic faces.
  2. Rhombic triacontahedron V3.5.3.5 has 30 identical rhombic faces.
• Rhombohedron has 6 rhombic faces
• Rhombo-hexagonal dodecahedron has 4 hexagonal and 8 rhombic faces.
• Truncated rhombic dodecahedron has 6 square and 12 flattened hexagonal faces.
• Rhombic icosahedron has 20 rhombic faces.
• Rhombic enneacontahedron has 90 rhombic faces of two different types.

Zonohedra with nonregular faces

A partial common list includes:

• Parallelepiped has 6 parallelogrammic faces
• Cuboid has 6 rectangular faces

Polygonal face types

Some common polygonal faces with point symmetry include:

• 4 sides:
  • Equilateral: squares, rhombi
  • rectangles, parallelograms
• 6 sides:
  • Equilateral: Regular hexagons, oblate hexagons, prolate hexagons
• 8 sides:
  • Equilateral: Regular octagons, oblate octagons, prolate octagons
• ...

Properties

Although it is not generally true that any polyhedron has a dissection into any other polyhedron of the same volume, it is known that any two zonohedra of equal volumes can be dissected into each other.

Mathematically, the zonohedra can be characterised as being the Minkowski sums of line segments, and this characterisation allows the definition to be generalised to higher dimensions, giving zonotopes.

External links

• Geometry Junkyard Zonohedron page: http://www.ics.uci.edu/~eppstein/junkyard/zono.html
• http://www.georgehart.com/virtual-polyhedra/zonohedra-info.html

polyhedra | Zonohedra

zonoedro | Зоноэдр