Abstract
A (geometric) regular polygon {p} in a euclidean space can be specified by a fine Schläfli symbol {p}, where
is a generalized fraction; here, \(0 \leq s_{1} < \cdots < s_{k} \leq \frac{1} {2}r\). This means that {p} projects onto planar polygons \(\{ \frac{r} {s_{j}}\}\) (reduced to lowest terms) in orthogonal planes, with \(\infty = \frac{1} {0}\) giving the linear apeirogon and 2 the digon (line segment). More generally, it may be possible to specify the shape or similarity class of a geometric regular polytope by means of a fine Schläfli symbol, whose data contain information about certain regular polygons occurring among its vertices in terms of generalized fractions. If so, then the fine Schläfli symbol is called rigid. This paper gives various criteria for rigidity; for instance, the classical regular polytopes are rigid. The theory is also illustrated by several examples. It is noteworthy, though, that a combinatorial description of a regular polytope – a presentation of its symmetry group – can differ considerably from its fine Schläfli symbol.
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References
Coxeter, H.S.M.: Regular skew polyhedra in 3 and 4 dimensions and their topological analogues. Proc. Lond. Math. Soc. (2) 43, 33–62 (1937). (Reprinted with amendments in Twelve Geometric Essays, Southern Illinois University Press (Carbondale, 1968), 76–105)
Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973)
Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin/New York (1980)
Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, I: Grünbaum’s new regular polyhedra and their automorphism group. Aequ. Math. 23, 252–265 (1981)
Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, II: complete enumeration. Aequ. Math. 29, 222–243 (1985)
Grünbaum, B.: Regular polyhedra – old and new. Aequ. Math. 16, 1–20 (1977)
McMullen, P.: Realizations of regular polytopes. Aequ. Math. 37, 38–56 (1989)
McMullen, P.: Realizations of regular apeirotopes. Aequ. Math. 47, 223–239 (1994)
McMullen, P.: Regular polytopes of full rank. Discret. Comput. Geom. 32, 1–35 (2004)
McMullen, P.: Four-dimensional regular polyhedra. Discret. Comput. Geom. 38, 355–387 (2007)
McMullen, P.: Regular apeirotopes of dimension and rank 4. Discret. Comput. Geom. 42, 224–260 (2009)
McMullen, P.: Regular polytopes of nearly full rank. Discret. Comput. Geom. 46, 660–703 (2011)
McMullen, P.: Regular polytopes of nearly full rank: addendum. Discret. Comput. Geom. 49, 703–705 (2013)
McMullen, P.: Geometric regular polytopes (in preparation)
McMullen, P., Schulte, E.: Regular polytopes in ordinary space. Discret. Comput. Geom. 17, 449–478 (1997)
McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and Its Applications, vol. 92. Cambridge University Press, Cambridge (2002)
Pellicer, D., Schulte, E.: Regular polygonal complexes in space, I. Trans. Am. Math. Soc. 362, 6679–6714 (2010)
Acknowledgements
This paper expands on original material contained in the mini-course The Geometry of Regular Polytopes given by me during a workshop in the Thematic Program on Discrete Geometry and Applications at the Fields Institute in October 2011. My grateful thanks go to the Institute for its support and hospitality during that visit.
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McMullen, P. (2014). Rigidity of Regular Polytopes. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_13
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