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Journal articles on the topic 'Regular polytopes'

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Published: 10 March 2023

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1

Lalvani, Haresh. "Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes." International Journal of Space Structures 11, no. 1-2 (April 1996): 155–71. http://dx.doi.org/10.1177/026635119601-222.

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This paper presents a higher-dimensional periodic table of regular and semi-regular n-dimensional polytopes. For regular n-dimensional polytopes, designated by their Schlafli symbol {p,q,r,…u,v,w}, the table is an (n-1)-dimensional hypercubic lattice in which each polytope occupies a different vertex of the lattice. The values of p,q,r,…u,v,w also establish the corresponding n-dimensional Cartesian co-ordinates (p,q,r,…u,v,w) of their respective positions in the hypercubic lattice. The table is exhaustive and includes all known regular polytopes in Euclidean, spherical and hyperbolic spaces, in addition to others candidate polytopes which do not appear in the literature. For n-dimensional semi-regular polytopes, each vertex of this hypercubic lattice branches into analogous n-dimensional cubes, where each n-cube encompasses a family with a distinct semi-regular polytope occupying each vertex of each n-cube. The semi-regular polytopes are obtained by varying the location of a vertex within the fundamental region of the polytope. Continuous transformations within each family are a natural fallout of this variable vertex location. Extensions of this method to less regular space structures and to derivation of architectural form are in progress and provide a way to develop an integrated index for space structures. Besides the economy in computational processing of space structures, integrated indices based on unified morphologies are essential for establishing a meta-structural knowledge base for architecture.

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2

Schulte, Egon, and Asia Ivić Weiss. "Free Extensions of Chiral Polytopes." Canadian Journal of Mathematics 47, no. 3 (June 1, 1995): 641–54. http://dx.doi.org/10.4153/cjm-1995-033-7.

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AbstractAbstract polytopes are discrete geometric structures which generalize the classical notion of a convex polytope. Chiral polytopes are those abstract polytopes which have maximal symmetry by rotation, in contrast to the abstract regular polytopes which have maximal symmetry by reflection. Chirality is a fascinating phenomenon which does not occur in the classical theory. The paper proves the following general extension result for chiral polytopes. If 𝒦 is a chiral polytope with regular facets 𝓕 then among all chiral polytopes with facets 𝒦 there is a universal such polytope 𝓟, whose group is a certain amalgamated product of the groups of 𝒦 and 𝓕. Finite extensions are also discussed.

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3

CONNOR, THOMAS, DIMITRI LEEMANS, and MARK MIXER. "ABSTRACT REGULAR POLYTOPES FOR THE O'NAN GROUP." International Journal of Algebra and Computation 24, no. 01 (February 2014): 59–68. http://dx.doi.org/10.1142/s0218196714500052.

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In this paper, we consider how the O'Nan sporadic simple group acts as the automorphism group of an abstract regular polytope. In particular, we prove that there is no regular polytope of rank at least five with automorphism group isomorphic to O′N. Moreover, we classify all rank four regular polytopes having O′N as their automorphism group.

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4

Comes, Jonathan. "Regular Polytopes." Mathematics Enthusiast 1, no. 2 (October 1, 2004): 30–37. http://dx.doi.org/10.54870/1551-3440.1007.

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5

Hou, Dong-Dong, Yan-Quan Feng, and Dimitri Leemans. "Existence of regular 3-polytopes of order 2𝑛." Journal of Group Theory 22, no. 4 (July 1, 2019): 579–616. http://dx.doi.org/10.1515/jgth-2018-0155.

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AbstractIn this paper, we prove that for any positive integers {n,s,t} such that {n\geq 10}, {s,t\geq 2} and {n-1\geq s+t}, there exists a regular polytope with Schläfli type {\{2^{s},2^{t}\}} and its automorphism group is of order {2^{n}}. Furthermore, we classify regular polytopes with automorphism groups of order {2^{n}} and Schläfli types {\{4,2^{n-3}\},\{4,2^{n-4}\}} and {\{4,2^{n-5}\}}, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Period. Math. Hungar. 53 2006, 1–2, 231–255].

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6

Boya, Luis J., and Cristian Rivera. "On Regular Polytopes." Reports on Mathematical Physics 71, no. 2 (April 2013): 149–61. http://dx.doi.org/10.1016/s0034-4877(13)60026-9.

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7

Cuypers, Hans. "Regular quaternionic polytopes." Linear Algebra and its Applications 226-228 (September 1995): 311–29. http://dx.doi.org/10.1016/0024-3795(95)00149-l.

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8

McMullen, Peter, and Egon Schulte. "Flat regular polytopes." Annals of Combinatorics 1, no. 1 (December 1997): 261–78. http://dx.doi.org/10.1007/bf02558480.

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9

Coxeter, H. S. M. "Regular and semi-regular polytopes. II." Mathematische Zeitschrift 188, no. 4 (December 1985): 559–91. http://dx.doi.org/10.1007/bf01161657.

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10

Coxeter, H. S. M. "Regular and semi-regular polytopes. III." Mathematische Zeitschrift 200, no. 1 (March 1988): 3–45. http://dx.doi.org/10.1007/bf01161745.

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11

Conder, Marston, and Gabe Cunningham. "Tight orientably-regular polytopes." Ars Mathematica Contemporanea 8, no. 1 (May 7, 2014): 69–82. http://dx.doi.org/10.26493/1855-3974.554.e50.

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12

Coxeter, H. S. M. "Reciprocating the Regular Polytopes." Journal of the London Mathematical Society 55, no. 3 (June 1997): 549–57. http://dx.doi.org/10.1112/s0024610797004833.

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13

Allendoerfer, C. B. "Book Review: Regular polytopes." Bulletin of the American Mathematical Society 37, no. 01 (December 21, 1999): 107——107. http://dx.doi.org/10.1090/s0273-0979-99-00839-3.

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14

Hartley, Michael. "Combinatorially regular Euler polytopes." Bulletin of the Australian Mathematical Society 56, no. 1 (August 1997): 173–74. http://dx.doi.org/10.1017/s0004972700030860.

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15

McMullen, Peter. "Realizations of regular polytopes." Aequationes Mathematicae 36, no. 2-3 (June 1988): 320. http://dx.doi.org/10.1007/bf01836099.

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16

Saldanha, Nicolau C., and Carlos Tomei. "Spectra of regular polytopes." Discrete & Computational Geometry 7, no. 4 (April 1992): 403–14. http://dx.doi.org/10.1007/bf02187851.

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17

McMullen, P., and E. Schulte. "Constructions for regular polytopes." Journal of Combinatorial Theory, Series A 53, no. 1 (January 1990): 1–28. http://dx.doi.org/10.1016/0097-3165(90)90017-q.

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18

McMullen, Peter. "Locally projective regular polytopes." Journal of Combinatorial Theory, Series A 65, no. 1 (January 1994): 1–10. http://dx.doi.org/10.1016/0097-3165(94)90033-7.

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19

Cunningham, Gabe. "Mixing regular convex polytopes." Discrete Mathematics 312, no. 4 (February 2012): 763–71. http://dx.doi.org/10.1016/j.disc.2011.11.014.

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20

McMullen, Peter, and Egon Schulte. "Higher Toroidal Regular Polytopes." Advances in Mathematics 117, no. 1 (January 1996): 17–51. http://dx.doi.org/10.1006/aima.1996.0002.

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21

Brandenberg, Ren�. "Radii of Regular Polytopes." Discrete & Computational Geometry 33, no. 1 (October 20, 2004): 43–55. http://dx.doi.org/10.1007/s00454-004-1127-1.

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22

McMullen, Peter. "Realizations of regular polytopes." Aequationes Mathematicae 37, no. 1 (February 1989): 38–56. http://dx.doi.org/10.1007/bf01837943.

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23

Berestovskii, V. N., and Y. G. Nikonorov. "О конечных однородных метрических пространствах." Владикавказский математический журнал, no. 2 (June 22, 2022): 51–61. http://dx.doi.org/10.46698/h7670-4977-9928-z.

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This survey is devoted to recently obtained results on finite homogeneousmetric spaces. The main subject of discussion is the classification of regular and semiregular polytopes in Euclidean spacesby whether or not their vertex sets have the normal homogeneity property or the Clifford - Wolf homogeneity property.Every finite homogeneous metric subspace of an Euclidean space represents the vertex set of a compact convex polytope with the isometry group that is transitive on the set of vertices, moreover, all these vertices lie on some sphere. Consequently, the study of such subsets is closely related to the theory of convex polytopes in Euclidean spaces. The normal generalized homogeneity and the Clifford - Wolf homogeneity describe more stronger properties than the homogeneity. Therefore, it is natural to first check the presence of these properties for the vertex sets of regular and semiregular polytopes. In addition to the classification results, the paper contains a description of the main tools for the study of the relevant objects.

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24

Schulte, Egon. "Amalgamation of Regular Incidence-Polytopes." Proceedings of the London Mathematical Society s3-56, no. 2 (March 1988): 303–28. http://dx.doi.org/10.1112/plms/s3-56.2.303.

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25

Adams, Joshua, Peter Zvengrowski, and Philip Laird. "Vertex embeddings of regular polytopes." Expositiones Mathematicae 21, no. 4 (2003): 339–53. http://dx.doi.org/10.1016/s0723-0869(03)80037-3.

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26

Saldanha, Nicolau C., and Carlos Tomei. "Spectra of semi-regular polytopes." Boletim da Sociedade Brasileira de Matem�tica 29, no. 1 (March 1998): 25–51. http://dx.doi.org/10.1007/bf01245867.

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27

McMullen, Peter. "Regular Polytopes of Full Rank." Discrete and Computational Geometry 32, no. 1 (May 1, 2004): 1–35. http://dx.doi.org/10.1007/s00454-004-0848-5.

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28

Effenberger, Felix, and Wolfgang Kühnel. "Hamiltonian Submanifolds of Regular Polytopes." Discrete & Computational Geometry 43, no. 2 (March 31, 2009): 242–62. http://dx.doi.org/10.1007/s00454-009-9151-9.

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29

Cantwell, Kristal. "All regular polytopes are Ramsey." Journal of Combinatorial Theory, Series A 114, no. 3 (April 2007): 555–62. http://dx.doi.org/10.1016/j.jcta.2006.08.001.

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30

McMullen, P., and E. Schulte. "Regular Polytopes in Ordinary Space." Discrete & Computational Geometry 17, no. 4 (June 1997): 449–78. http://dx.doi.org/10.1007/pl00009304.

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31

Pellicer, Daniel. "CPR graphs and regular polytopes." European Journal of Combinatorics 29, no. 1 (January 2008): 59–71. http://dx.doi.org/10.1016/j.ejc.2007.01.001.

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32

McMullen, Peter. "Lattices compatible with regular polytopes." European Journal of Combinatorics 29, no. 8 (November 2008): 1925–32. http://dx.doi.org/10.1016/j.ejc.2008.01.005.

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33

McMullen, Peter. "Realizations of regular polytopes, III." Aequationes mathematicae 82, no. 1-2 (February 3, 2011): 35–63. http://dx.doi.org/10.1007/s00010-010-0063-9.

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34

McMullen, Peter. "Realizations of regular polytopes, IV." Aequationes mathematicae 87, no. 1-2 (February 26, 2013): 1–30. http://dx.doi.org/10.1007/s00010-013-0187-9.

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35

McMullen, P., and B. Monson. "Realizations of regular polytopes, II." aequationes mathematicae 65, no. 1 (February 2003): 102–12. http://dx.doi.org/10.1007/s000100300007.

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36

B�r�czky, Jr., K., and M. Henk. "Random projections of regular polytopes." Archiv der Mathematik 73, no. 6 (December 1, 1999): 465–73. http://dx.doi.org/10.1007/s000130050424.

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37

Zhang, Wei-Juan. "Some simplifications of the intersection condition of chiral form for polytopes." Journal of Algebra and Its Applications 18, no. 11 (August 19, 2019): 1950203. http://dx.doi.org/10.1142/s0219498819502037.

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To determine if a poset of type [Formula: see text] is a directly regular or chiral polytope, it is necessary to test whether or not its rotation group (as a quotient of the orientation-preserving subgroup of the Coxeter group [Formula: see text]) satisfies the so-called intersection condition of chiral form. However, due to the fact that many cases need to be checked, this process is often very tedious and takes much time. In this paper, under certain circumstances, we give some simplifications for checking the intersection condition, which leads to certain constructions for directly regular or chiral polytopes.

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38

Monson, B., and Egon Schulte. "Modular Reduction in Abstract Polytopes." Canadian Mathematical Bulletin 52, no. 3 (September 1, 2009): 435–50. http://dx.doi.org/10.4153/cmb-2009-047-7.

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AbstractThe paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in ℤ[τ] (with τ the golden ratio), to construct new regular 4-polytopes of hyperbolic types ﹛3, 5, 3﹜ and ﹛5, 3, 5﹜ with automorphism groups given by finite orthogonal groups.

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39

Kabluchko, Zakhar, and Hauke Seidel. "Convex cones spanned by regular polytopes." Advances in Geometry 22, no. 2 (April 1, 2022): 245–67. http://dx.doi.org/10.1515/advgeom-2021-0041.

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Abstract We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace.

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40

Katunin, Andrzej. "Fractals based on regular convex polytopes." Scientific Research of the Institute of Mathematics and Computer Science 11, no. 2 (June 2012): 53–62. http://dx.doi.org/10.17512/jamcm.2012.2.06.

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41

McMullen, Peter. "Quasi-Regular Polytopes of Full Rank." Discrete & Computational Geometry 66, no. 2 (July 6, 2021): 475–509. http://dx.doi.org/10.1007/s00454-021-00304-5.

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42

Montagard, Pierre-Louis, and Nicolas Ressayre. "Regular lattice polytopes and root systems." Bulletin of the London Mathematical Society 41, no. 2 (February 24, 2009): 227–41. http://dx.doi.org/10.1112/blms/bdn120.

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43

McMullen, Peter, and Egon Schulte. "Locally unitary groups and regular polytopes." Advances in Applied Mathematics 29, no. 1 (July 2002): 1–45. http://dx.doi.org/10.1016/s0196-8858(02)00001-5.

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44

Filliman, P. "The largest projections of regular polytopes." Israel Journal of Mathematics 64, no. 2 (June 1988): 207–28. http://dx.doi.org/10.1007/bf02787224.

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45

McMullen, P., and E. Schulte. "Regular polytopes from twisted Coxeter groups." Mathematische Zeitschrift 201, no. 2 (June 1989): 209–26. http://dx.doi.org/10.1007/bf01160678.

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46

Proskurnikov, A. V., and Yu R. Romanovskii. "Regular triangulations of non-convex polytopes." Russian Mathematical Surveys 57, no. 4 (August 31, 2002): 817–18. http://dx.doi.org/10.1070/rm2002v057n04abeh000546.

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47

Akopyan, Arseniy, and Roman Karasev. "Inscribing a regular octahedron into polytopes." Discrete Mathematics 313, no. 1 (January 2013): 122–28. http://dx.doi.org/10.1016/j.disc.2012.09.004.

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48

Brehm, Ulrich, Wolfgang Kühnel, and Egon Schulte. "Manifold structures on abstract regular polytopes." Aequationes Mathematicae 49, no. 1 (February 1995): 12–35. http://dx.doi.org/10.1007/bf01827926.

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49

Pellicer, Daniel. "Extensions of dually bipartite regular polytopes." Discrete Mathematics 310, no. 12 (June 2010): 1702–7. http://dx.doi.org/10.1016/j.disc.2009.11.023.

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50

McMullen, Peter. "Regular Polytopes of Nearly Full Rank." Discrete & Computational Geometry 46, no. 4 (March 9, 2011): 660–703. http://dx.doi.org/10.1007/s00454-011-9335-y.

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