Polyhedron?

The duals of the semi-regular polyhedra are face-uniform. These are, correspondingly:

• the bipyramids
• the trapezohedra
• 11 of the Catalan solids

Other polyhedra with regular faces

See main entries: Johnson solid and Deltahedron

Norman Johnson sought which non-uniform polyhedra had regular faces. In 1966, he published a list of 92 convex solids, the Johnson solids, and gave them their names and numbers. He did not prove there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:

• 3 regular convex polyhedra (3 of the Platonic solids)
  • the tetrahedron
  • the octahedron
  • the icosahedron
• 5 non-uniform convex polyhedra (5 of the Johnson solids)
  • the triangular dipyramid
  • the pentagonal dipyramid
  • the snub disphenoid
  • the triaugmented triangular prism
  • the gyroelongated square dipyramid

With regard to polyhedra whose faces are all squares: if coplanar faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (a,b,c), with integers a,b,c of which at most one is even.

There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron.

There exists no polyhedron whose faces are all regular polygons with six or more sides.

There exist an infinite number of non-uniform non-convex polyhedra.

General polyhedron

More recently mathematics has defined a polyhedron as a set in real affine (or Euclidean) space of any dimensional n that has flat sides. It could be defined as the union of a finite number of convex polyhedra, where a convex polyhedron is any set that is the intersection of a finite number of half-spaces. It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron.

All classical polyhedra are general polyhedra, and in addition there are examples like

• A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: { ( x, y ) : x ≥ 0, y ≥ 0 }. Its sides are the two positive axes.
• An octant in Euclidean 3-space, { ( x, y, z ) : x ≥ 0, y ≥ 0, z ≥ 0 }
• A prism of infinite extent. For instance a doubly-infinite square prism in 3-space, consisting of a square in the xy-plane swept along the z-axis: { ( x, y, z ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }
• Each cell in a Voronoi tessellation is a convex polyhedron. In the Voronoi tessellation of a set S, the cell A corresponding to a point c∈S is bounded (hence a classical polyhedron) when c lies in the interior of the convex hull of S, and otherwise (when c lies on the boundary of the convex hull of S) A is unbounded.

Topological polyhedron

A topological polyhedron is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way that needs better description.

Relation with graphs

Any polyhedron gives rise to a graph, called skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra:

• The Archimedean solids give rise to regular graphs: 7 Archimedean solids are degree 3, 4 solids are degree 4, and the remaining 2 are chiral pairs of degree 5.

• The octahedron gives rise to a strongly regular graph, because adjacent vertices have always two common neighbors, and non-adjacent vertices always four.

• Only the tetrahedron gives rise to a complete graph (K₄).

• Due to Steinitz theorem convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.

See also

• Antiprism
• Archimedean solid
• Bipyramid
• Defect
• Deltahedron
• Deltohedron
• M. C. Escher
• Johnson solid
• Kepler-Poinsot solid
• Platonic solid
• Polyhedral compound
• Polyhedron models
• Prism
• Semiregular polyhedra
• Spidron
• Trapezohedron
• Zonohedron
• Overview of many polyhedra, with images

External links

• Polyhedra Index Page
• Stella: Polyhedron Navigator
• The Uniform Polyhedra
• Virtual Reality Polyhedra - The Encyclopedia of Polyhedra
• Paper Models of Polyhedra Many links
• Paper Models of Uniform (and other) Polyhedra
• Interactive 3D polyhedra in Java

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