“Quasiregular Polyhedron”的概念、定义、习题解题方法及翻译-数学辞典

单词

Quasiregular Polyhedron

释义

Quasiregular Polyhedron

A quasiregular polyhedron is the solid region interior to two Dual regular polyhedra withSchläfli Symbols and . Quasiregular polyhedra are denoted using aSchläfli Symbol of the form , with

(1)

Quasiregular polyhedra have two kinds of regular faces with each entirely surrounded by faces of the other kind, equalsides, and equal dihedral angles. They must satisfy the Diophantine inequality

(2)

But

, so

must be 2. This means that the possible quasiregular polyhedra have symbols

, and

. Now

(3)

is the Octahedron, which is a regular Platonic Solid and not considered quasiregular. This leaves only two convexquasiregular polyhedra: the Cuboctahedron

and the Icosidodecahedron

If nonconvex polyhedra are allowed, then additional quasiregular polyhedra are the Great Dodecahedron and the Great Icosidodecahedron (Hart).

For faces to be equatorial ,

(4)

The Edges of quasiregular polyhedra form a system of Great Circles: theOctahedron forms three Squares, the Cuboctahedron four Hexagons, and theIcosidodecahedron six Decagons. The Vertex Figures of quasiregularpolyhedra are Rhombuses (Hart). The Edges are also all equivalent, a propertyshared only with the completely regular Platonic Solids.

See also Cuboctahedron, Great Dodecahedron, Great Icosidodecahedron, Icosidodecahedron,Platonic Solid

References

Coxeter, H. S. M. ``Quasi-Regular Polyhedra.'' §2-3 in Regular Polytopes, 3rd ed. New York: Dover, pp. 17-20, 1973.

Hart, G. W. ``Quasi-Regular Polyhedra.'' http://www.li.net/~george/virtual-polyhedra/quasi-regular-info.html.

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