Abstract
An isohedron is a 3-dimensional polyhedron all faces of which are equivalent under symmetries of the polyhedron. Many well known polyhedra are isohedra; among them are the Platonic solids, the polars of Archimedean polyhedra, and a variety of polyhedra important in crystallography. Less well known are isohedra with nonconvex faces. We establish that such polyhedra must be starshaped and hence of genus 0, that their faces must be star-shaped pentagons with one concave vertex, and that they are combinatorially equivalent to either the pentagonal dodecahedron, or to the polar of the snub cube or snub dodecahedron.
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References
H.-G. Bigalke, Die flächenäquivalenten Pentagon-Dodekaeder.Didaktik der Mathematik 3 (1986), 204–221.
M. Brückner,Vielecke und Vielflache. Teubner, Leipzig 1900.
H. S. M.Coxeter, P. DuVal, H. T.Flather and J. F.Petrie,The Fifty-Nine Icosahedra. University of Toronto Press, 1938; 2nd ed. Springer, New York 1982.
H. S. M. Coxeter andW. O. J. Moser,Generators and Relations for Discrete Groups. 4th ed. Springer, Berlin 1980.
H. M. Cundy andA. P. Rollett,Mathematical Models. 2nd ed., Clarendon Press, Oxford 1961.
P. J. Federico,Descartes on Polyhedra. Springer, New York 1982.
D. Gale,Boomerang problem: Quickie 91-3.Mathematical Intelligencer 13(1991), 33. Solution,ibid. 14(1992), 69–70.
B. Grünbaum andG. C. Shephard, Spherical tilings with transitivity properties. In “The Geometric Vein — The Coxeter Festschrift”, C. Davis et al, eds. Springer, New York 1982, pp. 65–98.
B. Grünbaum andG. C. Shephard, Polyhedra with transitivity properties.C. R. Math. Rep. Acad. Sci. Canada 6(1984), 61–66.
B. Grünbaum andG. C. Shephard,Tilings and Patterns. Freeman, New York 1987.
B. Grünbaum andG. C. Shephard, Duality of polyhedra. In“Shaping Space: A Polyhedral Approach”, Proc. “Shaping Space” Conference, Smith College, April 1984, M. Senechal and G. Fleck, eds. Birkhäuser, Boston 1988, pp. 205–211.
Z. Har'El, Uniform solutions for uniform polyhedra.GeometriaeDedicata 47(1993), 57–110.
L. Lines,Solid Geometry. Dover, New York 1965.
J. Ounsted, An unfamiliar dodecahedron.Mathematics Teaching 83(1978), 46–47.
A. J. Schwenk, Boomerangs cannot tile convex polygons. Solution of Problem 1244,Math. Magazine 60(1987), 182.
B. M. Stewart,Adventures Among the Toroids. The author, Okemos, MI, 1980.
L. Szilassi, Regular toroids.Structural Topology, 13(1986), 69–80.
M. J.Wenninger,Polyhedron Models. Cambridge University Press, 1971.
M. J.Wenninger,Dual Models. Cambridge University Press, 1983.
J. M. Wills, Semi-Platonic manifolds. In“Convexity aud its Applications”, P. M. Gruber and J. M. Wills, eds. Birkhäuser, Basel 1983, pp. 413–421.
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Supported in part by grants from the USA National Science Foundation.
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Grünbaum, B., Shephard, G.C. Isohedra with nonconvex faces. J Geom 63, 76–96 (1998). https://doi.org/10.1007/BF01221240
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DOI: https://doi.org/10.1007/BF01221240