“Kepler-Poinsot Solid”的概念、定义、习题解题方法及翻译-数学辞典

单词

Kepler-Poinsot Solid

释义

Kepler-Poinsot Solid

The Kepler-Poinsot solids are the four regular Concave Polyhedra with intersecting facialplanes. They are composed of regular Concave Polygons and were unknown to the ancients.Kepler discovered two and described them about 1619. These two were subsequently rediscovered by Poinsot, who also discoveredthe other two, in 1809. As shown by Cauchy, they are stellated forms of the Dodecahedron andIcosahedron.

The Kepler-Poinsot solids, illustrated above, are known as the GreatDodecahedron, Great Icosahedron, Great Stellated Dodecahedron, and Small Stellated Dodecahedron. Cauchy (1813) proved that these four exhaust all possibilities for regular star polyhedra (Ball and Coxeter 1987).

A table listing these solids, their Duals, and Compounds is givenbelow.

Polyhedron	Dual Polyhedron	Compound
Great Dodecahedron	Small Stellated Dodecahedron	Great Dodecahedron-Small Stellated Dodecahedron Compound
Great Icosahedron	Great Stellated Dodecahedron	Great Icosahedron-Great Stellated Dodecahedron Compound
Great Stellated Dodecahedron	Great Icosahedron	Great Icosahedron-Great Stellated Dodecahedron Compound
Small Stellated Dodecahedron	Great Dodecahedron	Great Dodecahedron-Small Stellated Dodecahedron Compound

The polyhedra and fail to satisfy the Polyhedral Formula

where

is the number of faces,

the number of edges, and

the number of faces, despite the fact that theformula holds for all ordinary polyhedra (Ball and Coxeter 1987). This unexpected result led none less than Schläfli(1860) to conclude that they could not exist.

In 4-D, there are 10 Kepler-Poinsot solids, and in -D with , there are none. In 4-D, nine of the solids havethe same Vertices as , and the tenth has the same as . TheirSchläfli Symbols are , , ,, , , , ,, and .

Coxeter et al. (1954) have investigated star ``Archimedean'' polyhedra.

See also Archimedean Solid, Deltahedron, Johnson Solid, Platonic Solid, PolyhedronCompound, Uniform Polyhedron

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 144-146, 1987.

Cauchy, A. L. ``Recherches sur les polyèdres.'' J. de l'École Polytechnique 9, 68-86, 1813.

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. ``Uniform Polyhedra.'' Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.

Pappas, T. ``The Kepler-Poinsot Solids.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989.

Quaisser, E. ``Regular Star-Polyhedra.'' Ch. 5 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 56-62, 1986.

Schläfli. Quart. J. Math. 3, 66-67, 1860.

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