Skip to main content Accessibility help
×

Online ordering will be unavailable from 17:00 GMT on Friday, April 25 until 17:00 GMT on Sunday, April 27 due to maintenance. We apologise for the inconvenience.

Hostname: page-component-669899f699-rg895 Total loading time: 0 Render date: 2025-04-25T19:44:36.846Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  23 March 2025

Daniel Pellicer
Affiliation:
Universidad Nacional Autónoma de México
Get access
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

A’Campo, Norbert, Ji, Lizhen, and Papadopoulos, Athanase. Actions of the absolute Galois group. In Handbook of Teichmüller Theory, Vol. VI, volume 27 of IRMA Lect. Math. Theor. Phys., pages 397435. European Mathematical Society, 2016. 165CrossRefGoogle Scholar
Araujo-Pardo, Gabriela, Hubard, Isabel, Oliveros, Deborah, and Schulte, Egon. Colorful polytopes and graphs. Israel J. Math., 195(2):647675, 2013. 306CrossRefGoogle Scholar
Arocha, Jorge L., Bracho, Javier, and Montejano, Luis. Regular projective polyhedra with planar faces. I. Aequationes Math., 59(1–2):5573, 2000. 344, 437CrossRefGoogle Scholar
Arredondo, John A., Maluendas, Camilo Ramírez, and Valdez, Ferrán. On the topology of infinite regular and chiral maps. Discrete Math., 340(6):11801186, 2017. 272CrossRefGoogle Scholar
Berman, Leah Wrenn, Mixer, Mark, Monson, Barry, Oliveros, Deborah, and Williams, Gordon. The monodromy group of the n-pyramid. Discrete Math., 320:5563, 2014. 243CrossRefGoogle Scholar
Bieberbach, Ludwig. Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann., 72(3):400412, 1912. 289CrossRefGoogle Scholar
Biggs, N. L. and White, A. T.. Permutation Groups and Combinatorial Structures, volume 33 of London Math. Soc. Lecture Note Ser. Cambridge University Press, 1979. 23CrossRefGoogle Scholar
Birkhoff, Garrett. Lattice Theory, volume 25 of Amer. Math. Soc. Colloq. Publications. American Mathematical Society, 3rd ed. 1979. 8Google Scholar
Bracho, Javier. Regular projective polyhedra with planar faces. II. Aequationes Math., 59(1–2):160176, 2000. 344, 437CrossRefGoogle Scholar
Bracho, Javier, Hubard, Isabel, and Pellicer, Daniel. A finite chiral 4-polytope in ℝ4. Discrete Comput. Geom., 52(4):799805, 2014. 303, 307, 309CrossRefGoogle Scholar
Bracho, Javier, Hubard, Isabel, and Pellicer, Daniel. Realising equivelar toroids of type {4, 4}. Discrete Comput. Geom., 55(4):934954, 2016. 233, 344, 440, 441CrossRefGoogle Scholar
Bracho, Javier, Hubard, Isabel, and Pellicer, Daniel. Chiral polyhedra in 3-dimensional geometries and from a Petrie–Coxeter construction. Discrete Comput. Geom., 66(3):10251052, 2021. 344, 388, 441CrossRefGoogle Scholar
Brahana, H. R.. Regular maps on an anchor ring. Amer. J. Math., 48(4):225240, 1926. 4, 63, 64, 91, 144CrossRefGoogle Scholar
Brahana, H. R.. Regular maps and their groups. Amer. J. Math., 49(2):268284, 1927. 64, 91CrossRefGoogle Scholar
Breda, Ana, D’Azevedo, Antonio Breda, and Nedela, Roman. Chirality group and chirality index of Coxeter chiral maps. Ars Combin., 81:147160, 2006. 194, 201, 203, 242Google Scholar
D’Azevedo, Antonio Breda, Jones, Gareth, Nedela, Roman, and Škoviera, Martin. Chirality groups of maps and hypermaps. J. Algebraic Combin., 29(3):337355, 2009. 165, 193, 194, 203, 205CrossRefGoogle Scholar
d’Azevedo, Antonio Breda, Nedela, Roman, and Širáň, Jozef. Classification of regular maps of negative prime Euler characteristic. Trans. Amer. Math. Soc., 357(10):41754190, 2005. 270CrossRefGoogle Scholar
d’Azevedo, Antonio J. Breda and Jones, Gareth A.. Double coverings and reflexive abelian hypermaps. Beiträge Algebra Geom., 41(2):371389, 2000. 257Google Scholar
Brehm, Ulrich and Kühnel, Wolfgang. Equivelar maps on the torus. European J. Combin., 29(8):18431861, 2008. 91CrossRefGoogle Scholar
Brooksbank, P. A., Ferrara, J. T., and Leemans, D.. Orthogonal groups in characteristic 2 acting on polytopes of high rank. Discrete Comput. Geom., 63(3): 656669, 2020. 340Google Scholar
Brooksbank, P. A. and Leemans, D.. Polytopes of large rank for PSL(4, 𝔽q). J. Algebra, 452:390400, 2016. 340CrossRefGoogle Scholar
Brooksbank, Peter A.. On the ranks of string C-group representations for symplectic and orthogonal groups. In Polytopes and Discrete Geometry, volume 764 of Contemp. Math., pages 3141. American Mathematical Society, 2021. 340CrossRefGoogle Scholar
Brooksbank, Peter A. and Vicinsky, Deborah A.. Three-dimensional classical groups acting on polytopes. Discrete Comput. Geom., 44(3):654659, 2010. 340CrossRefGoogle Scholar
Bujalance, Emilio, Conder, Marston D. E., and Costa, Antonio F.. Pseudo-real Riemann surfaces and chiral regular maps. Trans. Amer. Math. Soc., 362(7):33653376, 2010. 268CrossRefGoogle Scholar
Burgiel, H. and Stanton, D.. Realizations of regular abstract polyhedra of types {3, 6} and {6, 3}. Discrete Comput. Geom., 24(2-3): 241255, 2000. The Branko Grünbaum birthday issue. 437, 440CrossRefGoogle Scholar
Cameron, Peter J., Fernandes, Maria Elisa, Leemans, Dimitri, and Mixer, Mark. Highest rank of a polytope for An. Proc. Lond. Math. Soc. (3), 115(1):135176, 2017. 340CrossRefGoogle Scholar
Cayley, Professor. Desiderata and suggestions: No. 2. The theory of groups: Graphical representation. Amer. J. Math., 1(2):174176, 1878. 156CrossRefGoogle Scholar
Cohen, Arjeh M.. Coxeter groups and three related topics. In Generators and Relations in Groups and Geometries (Lucca, 1990), volume 333 of NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., pages 235278. Kluwer Academic Publishers, 1991. 112CrossRefGoogle Scholar
Colbourn, Charles J. and Weiss, Asia Ivić. A census of regular 3-polystroma arising from honeycombs. Discrete Math., 50(1):2936, 1984. 313, 318, 319, 324CrossRefGoogle Scholar
Collins, José and Montero, Antonio. Equivelar toroids with few flag-orbits. Discrete Comput. Geom., 65(2):305330, 2021. 85, 289CrossRefGoogle Scholar
Conder, Marston. Regular maps with small parameters. J. Austral. Math. Soc. Ser. A, 57(1):103112, 1994. 263CrossRefGoogle Scholar
Conder, Marston. The smallest regular polytopes of given rank. Adv. Math., 236:92110, 2013. 66, 277, 288CrossRefGoogle Scholar
Conder, Marston. Chiral polytopes with up to 4000 flags. www.math.auckland.ac.nz/~conder/ChiralPolytopesWithUpTo4000Flags-ByOrder.txt, 2014. 150, 164, 270, 273, 287Google Scholar
Conder, Marston. Regular polytopes with up to 4000 flags. www.math.auckland.ac.nz/~conder/RegularPolytopesWithUpTo4000Flags-ByOrder.txt, 2014. 150, 164, 270, 273, 287Google Scholar
Conder, Marston and Cunningham, Gabe. Tight orientably-regular polytopes. Ars Math. Contemp., 8(1):6881, 2015. 280, 331Google Scholar
Conder, Marston and Dobcsányi, Peter. Determination of all regular maps of small genus. J. Combin. Theory Ser. B, 81(2):224242, 2001. 269CrossRefGoogle Scholar
Conder, Marston, Hubard, Isabel, O’Reilly-Regueiro, Eugenia, and Pellicer, Daniel. Construction of chiral 4-polytopes with alternating or symmetric automorphism group. J. Algebraic Combin., 42(1):225244, 2015. 75, 328, 339CrossRefGoogle Scholar
Conder, Marston, Hubard, Isabel, and Pisanski, Tomaž. Constructions for chiral polytopes. J. Lond. Math. Soc. (2), 77(1):115129, 2008. 65, 66, 236, 238CrossRefGoogle Scholar
Conder, Marston, Potočnik, Primož, and Širáň, Jozef. Regular maps with almost Sylow-cyclic automorphism groups, and classification of regular maps with Euler characteristic −p2. J. Algebra, 324(10):26202635, 2010. 270CrossRefGoogle Scholar
Conder, Marston D. E.. Generators for alternating and symmetric groups. J. London Math. Soc. (2), 22(1):7586, 1980. 157CrossRefGoogle Scholar
Conder, Marston D. E.. Regular maps and hypermaps of Euler characteristic –1 to –200. J. Combin. Theory Ser. B, 99(2):455459, 2009. 51, 59, 64, 115, 162, 163, 164, 174, 178, 236, 269, 270CrossRefGoogle Scholar
Conder, Marston D. E., Hucíková, Veronika, Nedela, Roman, and Širáň, Jozef. Chiral maps of given hyperbolic type. Bull. Lond. Math. Soc., 48(1):3852, 2016. 64, 268, 269, 273CrossRefGoogle Scholar
Conder, Marston D. E., Širáň, Jozef, and Tucker, Thomas W.. The genera, reflexibility and simplicity of regular maps. J. Eur. Math. Soc. (JEMS), 12(2):343364, 2010. 64, 270, 271CrossRefGoogle Scholar
Conder, Marston D. E. and Zhang, Wei-Juan. Abelian covers of chiral polytopes. J. Algebra, 478:437457, 2017. 66CrossRefGoogle Scholar
Conder, Marston D. E. and Zhang, Wei-Juan. The smallest chiral 6-polytopes. Bull. Lond. Math. Soc., 49(4):549560, 2017. 66, 287, 288CrossRefGoogle Scholar
Conway, John, Burgiel, Heidi, and Goodman-Strauss, Chaim. The Symmetries of Things. A K Peters, 2008. 290, 431, 455Google Scholar
Cori, Robert and Machì, Antonio. Maps, hypermaps and their automorphisms: a survey. I, II, III. Exposition. Math., 10(5):403427, 429447, 449467, 1992. 194Google Scholar
Coxeter, H. S. M.. Discrete groups generated by reflections. Ann. of Math. (2), 35(3):588621, 1934. 4, 113, 147, 265, 266CrossRefGoogle Scholar
Coxeter., H. S. M. Configurations and maps. Rep. Math. Colloquium (2), 8:1838, 1949. 4, 51, 63, 64, 144, 259Google Scholar
Coxeter, H. S. M.. Self-dual configurations and regular graphs. Bull. Amer. Math. Soc., 56:413455, 1950. 263CrossRefGoogle Scholar
Coxeter, H. S. M.. Regular honeycombs in hyperbolic space. In Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, pages 155169. Erven P. Noordhoff N. V., 1956. 265, 290, 291, 293, 301, 344Google Scholar
Coxeter, H. S. M.. Regular compound tessellations of the hyperbolic plane. Proc. Roy. Soc. London Ser. A, 278:147167, 1964. 266Google Scholar
Coxeter, H. S. M.. Introduction to Geometry. John Wiley & Sons, Inc., 2nd ed., 1969. 64, 178Google Scholar
Coxeter, H. S. M.. Twisted Honeycombs. American Mathematical Society, 1970. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 4. 65, 193, 290, 291, 292, 293, 300, 301, 335Google Scholar
Coxeter, H. S. M.. Regular Polytopes. Dover Publications, Inc., 3rd ed., 1973. 11, 13, 51, 80, 81, 99, 112, 113, 213, 219, 228, 229, 230, 289, 290, 295Google Scholar
Coxeter, H. S. M.. Ten toroids and fifty-seven hemidodecahedra. Geom. Dedicata, 13(1):8799, 1982. 292, 312, 313CrossRefGoogle Scholar
Coxeter, H. S. M.. A symmetrical arrangement of eleven hemi-icosahedra. In Convexity and Graph Theory (Jerusalem, 1981), volume 87 of North-Holland Math. Stud., pages 103114. North-Holland, 1984. 312Google Scholar
Coxeter, H. S. M.. Regular Complex Polytopes. Cambridge University Press, 2nd ed., 1991. 351Google Scholar
Coxeter, H. S. M. and Moser, W. O. J.. Generators and Relations for Discrete Groups, 2nd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 14. Springer-Verlag, 1965. 51, 64, 79, 91Google Scholar
Coxeter, H. S. M. and Weiss, Asia Ivić. Twisted honeycombs {3, 5, 3}t and their groups. Geom. Dedicata, 17(2):169179, 1984. 65, 299, 300CrossRefGoogle Scholar
Coxeter, Harold Scott MacDonald. Regular skew polyhedra in three and four dimensions, and their topological analogues. Proc. London Math. Soc., 43:3362, 1937. 2, 220, 342, 367, 437Google Scholar
Cunningham, Gabe. Constructing self-dual chiral polytopes. European J. Combin., 33(6):13131323, 2012. 255, 256CrossRefGoogle Scholar
Cunningham, Gabe. Internal and external invariance of abstract polytopes. 2012. PhD thesis, Northeastern University. 207, 256Google Scholar
Cunningham, Gabe. Mixing chiral polytopes. J. Algebraic Combin., 36(2):263277, 2012. 243, 254, 255, 261CrossRefGoogle Scholar
Cunningham, Gabe. Minimal equivelar polytopes. Ars Math. Contemp., 7(2):299315, 2014. 277, 278, 279, 280CrossRefGoogle Scholar
Cunningham, Gabe. Non-flat regular polytopes and restrictions on chiral polytopes. Electron. J. Combin., 24(3):Paper No. 3.59, 14, 2017. 66, 70, 285, 288CrossRefGoogle Scholar
Cunningham, Gabe. Tight chiral polyhedra. Combinatorica, 38(1):115142, 2018. 283, 284, 439CrossRefGoogle Scholar
Cunningham, Gabe. Chiral polytopes whose smallest regular cover is a polytope. J. Combin. Theory Ser. A, 204:Paper No. 105839, 2024. 240, 243CrossRefGoogle Scholar
Cunningham, Gabe, Río-Francos, Mar ía Del, Hubard, Isabel, and Toledo, Micael. Symmetry type graphs of polytopes and maniplexes. Ann. Comb., 19(2):243268, 2015. 54CrossRefGoogle Scholar
Cunningham, Gabe and Pellicer, Daniel. Chiral extensions of chiral polytopes. Discrete Math., 330:5160, 2014. 76, 329, 330, 331CrossRefGoogle Scholar
Cunningham, Gabe and Pellicer, Daniel. Classification of tight regular polyhedra. J. Algebraic Combin., 43(3):665691, 2016. 280, 284CrossRefGoogle Scholar
Cunningham, Gabe and Pellicer, Daniel. Open problems on k-orbit polytopes. Discrete Math., 341(6):16451661, 2018. 207, 441, 448CrossRefGoogle Scholar
Cunningham, Gabe and Pellicer, Daniel. Tight chiral polytopes. J. Algebraic Combin., 54(3):837878, 2021. 284, 285CrossRefGoogle Scholar
Cunningham, Gabe, Pellicer, Daniel, and Williams, Gordon. Stratified operations on maniplexes. Algebr. Comb., 5(2):267287, 2022. 208Google Scholar
Danzer, L.. Regular incidence-complexes and dimensionally unbounded sequences of such. I. In Convexity and Graph Theory (Jerusalem, 1981), volume 87 of North-Holland Math. Stud., pages 115127. North-Holland, 1984. 273, 326, 327Google Scholar
Danzer, L. and Schulte, E.. Reguläre Inzidenzkomplexe. I. Geom. Dedicata, 13(3):295308, 1982. 3CrossRefGoogle Scholar
Davis, Michael W.. Regular convex cell complexes. In Geometry and Topology (Athens, Ga., 1985), volume 105 of Lecture Notes in Pure and Appl. Math., pages 5388. Dekker, 1987. 80Google Scholar
D’Azevedo, Antonio Breda, Jones, Gareth A., and Schulte, Egon. Constructions of chiral polytopes of small rank. Canad. J. Math., 63(6):12541283, 2011. 66, 194, 201, 205, 243, 255CrossRefGoogle Scholar
Diestel, Reinhard. Ends and tangles. Abh. Math. Semin. Univ. Hambg., 87(2):223244, 2017. 272CrossRefGoogle Scholar
do Carmo, Manfredo P.. Differential Geometry of Curves and Surfaces. Dover Publications, Inc., 2016. Revised and updated second edition. 80Google Scholar
Doro, Stephen and , Stephen E. Wilson. Rotary maps of type {6, 6}4. Quart. J. Math. Oxford Ser. (2), 31(124):403414, 1980. 62CrossRefGoogle Scholar
Douglas, Ian, Hubard, Isabel, Pellicer, Daniel, and Wilson, Steve. The twist operator on maniplexes. In Discrete Geometry and Symmetry, volume 234 of Springer Proc. Math. Stat., pages 127145. Springer, 2018. 66, 327Google Scholar
Dress, Andreas W. M.. A combinatorial theory of Grünbaum’s new regular polyhedra. I. Grünbaum’s new regular polyhedra and their automorphism group. Aequationes Math., 23(2–3):252265, 1981. 3, 342, 344, 437CrossRefGoogle Scholar
Dress, Andreas W. M.. A combinatorial theory of Grünbaum’s new regular polyhedra. II. Complete enumeration. Aequationes Math., 29(2–3):222243, 1985. 3, 366, 367, 390, 437, 443CrossRefGoogle Scholar
Euclid, . Euclid’s Elements. Green Lion Press, 2002. All thirteen books complete in one volume, the Heath, Thomas L. translation, edited by Densmore, Dana. 2Google Scholar
Fernandes, Maria Elisa and Leemans, Dimitri. String C-group representations of alternating groups. Ars Math. Contemp., 17(1):291310, 2019. 340CrossRefGoogle Scholar
Fernandes, Maria Elisa, Leemans, Dimitri, and Mixer, Mark. All alternating groups An with n ≥ 12 have polytopes of rank . SIAM J. Discrete Math., 26(2):482498, 2012.CrossRefGoogle Scholar
Fernandes, Maria Elisa, Leemans, Dimitri, and Mixer, Mark. Polytopes of high rank for the alternating groups. J. Combin. Theory Ser. A, 119(1):4256, 2012. 340CrossRefGoogle Scholar
Maria Elisa Fernandes, Leemans, Dimitri, and Mixer, Mark. An extension of the classification of high rank regular polytopes. Trans. Amer. Math. Soc., 370(12):88338857, 2018. 340Google Scholar
Gallier, Jean and Xu, Dianna. A Guide to the Classification Theorem for Compact Surfaces, volume 9 of Geometry and Computing. Springer, 2013. 264CrossRefGoogle Scholar
Garbe, Dietmar. A generalization of the regular maps of type {4, 4}b, c and {3, 6}b, c. Canad. Math. Bull., 12:293297, 1969. 263CrossRefGoogle Scholar
Garbe, Dietmar. Über die regulären Zerlegungen geschlossener orientierbarer Flächen. J. Reine Angew. Math., 237:3955, 1969. 64, 269Google Scholar
Garbe, Dietmar. A remark on nonsymmetric compact Riemann surfaces. Arch. Math. (Basel), 30(4):435437, 1978. 274CrossRefGoogle Scholar
Gardiner, A., Nedela, R., Širáň, J., and Škoviera, M.. Characterisation of graphs which underlie regular maps on closed surfaces. J. London Math. Soc. (2), 59(1):100108, 1999. 51CrossRefGoogle Scholar
Garner, C. W. L.. Coordinates for vertices of regular honeycombs in hyperbolic space. Proc. Roy. Soc. London Ser. A, 293:94107, 1966. 290Google Scholar
Garner, Cyril W. L.. Regular skew polyhedra in hyperbolic three-space. Canad. J. Math., 19:11791186, 1967. 344CrossRefGoogle Scholar
Garner, Cyril Wilbur Luther. Polyhedra and honetcombs in hyperbolic space. ProQuest LLC, 1964. PhD thesis, University of Toronto (Canada). 290Google Scholar
Garza-Vargas, Jorge and Hubard, Isabel. Polytopality of maniplexes. Discrete Math., 341(7):20682079, 2018. 29, 31, 32, 35CrossRefGoogle Scholar
Girondo, Ernesto and González-Diez, Gabino. Introduction to Compact Riemann Surfaces and Dessins d’Enfants, volume 79 of London Math. Soc. Student Texts. Cambridge University Press, 2012. 266Google Scholar
Gradolato, Monique and Zimmermann, Bruno. Finite quotients of hyperbolic tetrahedral groups. In Proceedings of the Eleventh International Conference of Topology (Trieste, 1993), volume 25, pages 211222, 1993. 302Google Scholar
Graver, Jack E. and Watkins, Mark E.. Locally finite, planar, edge-transitive graphs. Mem. Amer. Math. Soc., 126(601):vi+75, 1997. 54Google Scholar
Grothendieck, Alexandre. Esquisse d’un programme. In Geometric Galois Actions, 1, volume 242 of London Math. Soc. Lecture Note Ser., pages 548. Cambridge University Press, 1997. With an English translation on pp. 243283. 165Google Scholar
Grove, L. C. and Benson, C. T.. Finite Reflection Groups, volume 99 of Graduate Texts in Mathematics. Springer-Verlag, 2nd ed., 1985. 290, 351, 449CrossRefGoogle Scholar
Grünbaum, Branko. Regular polyhedra – old and new. Aequationes Math., 16(1–2):120, 1977. 2, 27, 220, 342, 344, 355, 367, 390, 419, 437, 445CrossRefGoogle Scholar
Grünbaum, Branko. Regularity of graphs, complexes and designs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 191197. CNRS, 1978. 312, 313, 314Google Scholar
Grünbaum, Branko. Uniform tilings of 3-space. Geombinatorics, 4(2):4956, 1994. 289, 452Google Scholar
Grünbaum, Branko. Convex Polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag, 2nd ed., 2003. Prepared and with a preface by Kaibel, Volker, Klee, Victor, and Ziegler, Günter M.. 27, 346CrossRefGoogle Scholar
Grünbaum, Branko and Shephard, G. C.. Edge-transitive planar graphs. J. Graph Theory, 11(2):141155, 1987. 54CrossRefGoogle Scholar
Grünbaum, Branko and Sheppard, Geoffrey Colin. Tilings and Patterns. Freeman, 1987. 80Google Scholar
Hantzsche, W. and Wendt, H.. Dreidimensionale euklidische Raumformen. Math. Ann., 110(1):593611, 1935. 290CrossRefGoogle Scholar
Hartley, Michael. All polytopes are quotients, and isomorphic polytopes are quotients by conjugate subgroups. Discrete Comput. Geom. 21, 2:289298, 1999. 164, 180, 183CrossRefGoogle Scholar
Hartley, Michael, Pellicer, Daniel, and Williams, Gordon. Minimal covers of the prisms and antiprisms. Discrete Math., 20:30463058, 2012. 207, 243CrossRefGoogle Scholar
Hartley, Michael I.. More on quotient polytopes. Aequationes Math., 57(1):108120, 1999. 164, 183CrossRefGoogle Scholar
Hartley, Michael I.. Quotients of some finite universal locally projective polytopes. Discrete Comput. Geom., 29(3):435443, 2003. 314CrossRefGoogle Scholar
Hartley, Michael I.. Locally projective polytopes of type {4, 3, ..., 3, p}. J. Algebra, 290(2):322336, 2005. 314, 328CrossRefGoogle Scholar
Hartley, Michael I.. An atlas of small regular abstract polytopes. Period. Math. Hungar., 53(1–2):149156, 2006. 51, 59, 115, 162, 163, 171, 173, 287CrossRefGoogle Scholar
Hartley, Michael I.. An exploration of locally projective polytopes. Combinatorica, 28(3):299314, 2008. 314CrossRefGoogle Scholar
Hartley, Michael I., Hubard, Isabel, and Leemans, Dimitri. Two atlases of abstract chiral polytopes for small groups. Ars Math. Contemp., 5(2):371382, 2012. 64, 236CrossRefGoogle Scholar
Hartley, Michael I. and Leemans, Dimitri. Quotients of a universal locally projective polytope of type {5,3,5}. Math. Z., 247(4):663674, 2004. 314CrossRefGoogle Scholar
Hartley, Michael I. and Leemans, Dimitri. Errata: “Quotients of a universal locally projective polytope of type {5,3,5}” [Math. Z. 247 (2004), no. 4, 663–674; mr2077414]. Math. Z., 253(2):433434, 2006.CrossRefGoogle Scholar
Hartley, Michael I. and Leemans, Dimitri. On locally spherical polytopes of type {5,3,5}. Discrete Math., 309(1):247254, 2009. 302, 314CrossRefGoogle Scholar
Hartley, Michael I., McMullen, Peter, and Schulte, Egon. Symmetric Tessellations on Euclidean Space-Forms. Canad. J. Math., 51:12301239, 1999. Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday. 4, 80, 290Google Scholar
Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2002. 24Google Scholar
Heffter, L.. Ueber metacyklische Gruppen und Nachbarconfigurationen. Math. Ann., 50(2–3):261268, 1898. 63, 144CrossRefGoogle Scholar
Henle, Michael. A Combinatorial Introduction to Topology. A Series of Books in Mathematical Sciences. W. H. Freeman and Co., 1979. 264Google Scholar
Hess, Edmund. Über die regulären polytope ho¨herer art. Sitz. Ges. Naturwiss Marburg, pages 3157, 1885. 2Google Scholar
Hiller, Howard. Geometry of Coxeter Groups, volume 54 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), 1982. 112Google Scholar
Hua, Loo Keng. Introduction to Number Theory. Springer-Verlag, 1982. Translated from the Chinese by Shiu, Peter. 456Google Scholar
Hubard, Isabel. Two-orbit polyhedra from groups. European J. Combin., 31(3):943960, 2010. 54, 103, 105, 112CrossRefGoogle Scholar
Hubard, Isabel and Leemans, Dimitri. Chiral polytopes and Suzuki simple groups. In Rigidity and Symmetry, volume 70 of Fields Institute Communications, pages 155175. Springer, 2014. 340CrossRefGoogle Scholar
Hubard, Isabel, Mixer, Mark, Pellicer, Daniel, and Weiss, Asia Ivić. Cubic tessellations of the didicosm. Adv. Geom., 14(2):299318, 2014. 290Google Scholar
Hubard, Isabel, Mixer, Mark, Pellicer, Daniel, and Weiss, Asia Ivić. Cubic tessellations of the helicosms. Discrete Comput. Geom., 54(3):686704, 2015. 290CrossRefGoogle Scholar
Hubard, Isabel and Mochán, Elías. All polytopes are coset geometries: Characterizing automorphism groups of k-orbit abstract polytopes. European J. Combin., 113:Paper No. 103746, 2023. 29, 106CrossRefGoogle Scholar
Hubard, Isabel, Mochán, Elías, and Montero, Antonio. Voltage operations on maniplexes, polytopes and maps. Combinatorica, 43(2):385420, 2023. 208CrossRefGoogle Scholar
Hubard, Isabel, Orbanić, Alen, and Weiss, Asia Ivić. Monodromy groups and selfinvariance. Canad. J. Math., 61(6):13001324, 2009. 209CrossRefGoogle Scholar
Hubard, Isabel, Orbanić, Alen, Pellicer, Daniel, and Weiss, Asia Ivić. Symmetries of equivelar 4-toroids. Discrete Comput. Geom., 48(4):11101136, 2012. 85, 290Google Scholar
Hubard, Isabel, Schulte, Egon, and Weiss, Asia Ivić. Petrie–Coxeter maps revisited. Beiträge Algebra Geom., 47(2):329343, 2006. 225, 226Google Scholar
Hubard, Isabel and Weiss, Asia Ivić. Self-duality of chiral polytopes. J. Combin. Theory Ser. A, 111(1):128136, 2005. 79CrossRefGoogle Scholar
Hubard, Isabel A.. From geometry to groups and back: The study of highly symmetric polytopes. ProQuest LLC, 2007. PhD thesis, York University (Canada). 54Google Scholar
Humphreys, James E.. Reflection Groups and Coxeter Groups, volume 29 of Camb. Studies Adv. Math. Cambridge University Press, 1990. 112, 113, 118, 147, 425CrossRefGoogle Scholar
Ireland, Kenneth and Rosen, Michael. A Classical Introduction to Modern Number Theory, volume 84 of Graduate Texts in Mathematics. Springer-Verlag, 2nd ed., 1990. 271, 457CrossRefGoogle Scholar
James, Lynne D. and , Gareth A. Jones. Regular orientable imbeddings of complete graphs. J. Combin. Theory Ser. B, 39(3):353367, 1985. 263CrossRefGoogle Scholar
Jendroľ, Stanislav. A noninvolutory self-duality. Discrete Math., 74(3):325326, 1989. 46Google Scholar
Johnson, Norman W.. Geometries and Transformations. Cambridge University Press, 2018. 155CrossRefGoogle Scholar
Jones, G. A. and Thornton, J. S.. Operations on maps, and outer automorphisms. J. Combin. Theory Ser. B, 35(2):93103, 1983. 208CrossRefGoogle Scholar
Jones, Gareth, Nedela, Roman, and Škoviera, Martin. Complete bipartite graphs with a unique regular embedding. J. Combin. Theory Ser. B, 98(2):241248, 2008. 63CrossRefGoogle Scholar
Jones, Gareth A.. Chiral covers of hypermaps. Ars Math. Contemp., 8(2):425431, 2015. 268, 269, 273CrossRefGoogle Scholar
Jones, Gareth A. and Long, Cormac D.. Epimorphic images of the [5,3,5] Coxeter group. Math. Z., 275(1–2):167183, 2013. 302CrossRefGoogle Scholar
Jones, Gareth A. and Singerman, David. Theory of maps on orientable surfaces. Proc. London Math. Soc. (3), 37(2):273307, 1978. 23, 263, 266CrossRefGoogle Scholar
Jones, Gareth A. and Wolfart, Jürgen. Dessins d’Enfants on Riemann Surfaces. Springer Monographs in Mathematics. Springer, 2016. 165, 267CrossRefGoogle Scholar
Kato, Mitsuyoshi. On combinatorial space forms. Sci. Papers College Gen. Ed. Univ. Tokyo, 30(2):107146, 1980. 80Google Scholar
Kepler, Johannes. The Harmony of the World, volume 209 of Memoirs Amer. Philos. Soc. American Philosophical Society, 1997. Translated from the Latin and with an introduction and notes by Aiton, E. J., Duncan, A. M., and Field, J. V., with a preface by Duncan and Field. 2, 7Google Scholar
Koike, Hiroki, Pellicer, Daniel, Raggi, Miguel, and Wilson, Steve. Flag bicolorings, pseudo-orientations, and double covers of maps. Electron. J. Combin., 24(1):Paper No. 1.3, 23, 2017. 63, 257, 258CrossRefGoogle Scholar
Leemans, Dimitri. Almost simple groups of Suzuki type acting on polytopes. Proc. Amer. Math. Soc., 134(12):36493651, 2006. 340CrossRefGoogle Scholar
Leemans, Dimitri. String C-group representations of almost simple groups: a survey. In Polytopes and Discrete Geometry, volume 764 of Contemp. Math., pages 157178. American Mathematical Society, 2021. 340CrossRefGoogle Scholar
Leemans, Dimitri and Liebeck, Martin W.. Chiral polyhedra and finite simple groups. Bull. Lond. Math. Soc., 49(4):581592, 2017. 205CrossRefGoogle Scholar
Leemans, Dimitri, Moerenhout, Jérémie, and O’Reilly-Regueiro, Eugenia. Projective linear groups as automorphism groups of chiral polytopes. J. Geom., 108(2):675702, 2017. 205CrossRefGoogle Scholar
Leemans, Dimitri and Schulte, Egon. Groups of type L2(q) acting on polytopes. Adv. Geom., 7(4):529539, 2007. 340CrossRefGoogle Scholar
Leemans, Dimitri and Schulte, Egon. Polytopes with groups of type PGL2(q). Ars Math. Contemp., 2(2):163171, 2009. 340CrossRefGoogle Scholar
Leemans, Dimitri, Schulte, Egon, and Maldeghem, Hendrik Van. Groups of Ree type in characteristic 3 acting on polytopes. Ars Math. Contemp., 14(2):209226, 2018. 340CrossRefGoogle Scholar
Leemans, Dimitri and Vandenschrick, Adrien. On chiral polytopes having a group PSL(3,q) as automorphism group. J. Lond. Math. Soc. (2), 106(1):85111, 2022. 205CrossRefGoogle Scholar
Leemans, Dimitri and Vauthier, Laurence. An atlas of abstract regular polytopes for small groups. Aequationes Math., 72(3):313320, 2006. 163CrossRefGoogle Scholar
Matteo, Nicholas. Combinatorially two-orbit convex polytopes. Discrete Comput. Geom., 55(3):662680, 2016. 80, 289CrossRefGoogle Scholar
Matteo, Nicholas. Two-orbit convex polytopes and tilings. Discrete Comput. Geom., 55(2):296313, 2016. 289CrossRefGoogle Scholar
McMullen, P.. Combinatorially regular polytopes. Mathematika, 14:142150, 1967. 80CrossRefGoogle Scholar
McMullen, P. and Schulte, E.. Self-dual regular 4-polytopes and their Petrie–Coxeter-polyhedra. Results Math., 12(3–4):366375, 1987. 220, 222, 224CrossRefGoogle Scholar
McMullen, P. and Schulte, E.. Regular polytopes in ordinary space. Discrete Comput. Geom., 17(4):449478, 1997. Dedicated to Wills, Jörg M.. 344, 366, 378, 381, 387, 414, 419, 437, 450CrossRefGoogle Scholar
McMullen, Peter. On the combinatorial structure of convex polytopes. 1968. PhD thesis, University of Birmingham (UK). 4, 63, 67, 80, 289Google Scholar
McMullen, Peter. Realizations of regular polytopes. Aequationes Math., 37(1):3856, 1989. 343, 366, 437CrossRefGoogle Scholar
McMullen, Peter. Locally projective regular polytopes. J. Combin. Theory Ser. A, 65(1):110, 1994. 314CrossRefGoogle Scholar
McMullen, Peter. Realizations of regular apeirotopes. Aequationes Math., 47(2–3):223239, 1994. 366, 437CrossRefGoogle Scholar
McMullen, Peter. Regular polytopes of full rank. Discrete Comput. Geom., 32(1):135, 2004. 3, 388, 389CrossRefGoogle Scholar
McMullen, Peter. Four-dimensional regular polyhedra. Discrete Comput. Geom., 38(2):355387, 2007.CrossRefGoogle Scholar
McMullen, Peter. Regular apeirotopes of dimension and rank 4. Discrete Comput. Geom., 42(2):224260, 2009. 3CrossRefGoogle Scholar
McMullen, Peter. Realizations of regular polytopes, III. Aequationes Math., 82(1–2):3563, 2011. 366CrossRefGoogle Scholar
McMullen, Peter. Regular polytopes of nearly full rank. Discrete Comput. Geom., 46(4):660703, 2011. 3, 210, 441CrossRefGoogle Scholar
McMullen, Peter. Regular polytopes of nearly full rank: Addendum. Discrete Comput. Geom., 49(3):703705, 2013. 3CrossRefGoogle Scholar
McMullen, Peter. Realizations of regular polytopes, IV. Aequationes Math., 87(1–2):130, 2014. 366, 437CrossRefGoogle Scholar
McMullen, Peter. Geometric Regular Polytopes, volume 172 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2020. vii, 366, 437CrossRefGoogle Scholar
McMullen, Peter. Quasi-regular polytopes of full rank. Discrete Comput. Geom., 66(2):475509, 2021. 61CrossRefGoogle Scholar
McMullen, Peter. Realizations of the 120-cell. In Polytopes and Discrete Geometry, volume 764 of Contemp. Math., pages 193219. American Mathematical Society, 2021. 437CrossRefGoogle Scholar
McMullen, Peter and Monson, Barry. Realizations of regular polytopes. II. Aequationes Math., 65(1–2):102112, 2003. 366, 437CrossRefGoogle Scholar
McMullen, Peter and Schulte, Egon. Hermitian forms and locally toroidal regular polytopes. Adv. Math., 82(1):88125, 1990. 314CrossRefGoogle Scholar
McMullen, Peter and Schulte, Egon. Locally toroidal regular polytopes of rank 4. Comment. Math. Helv., 67(1):77118, 1992.CrossRefGoogle Scholar
McMullen, Peter and Schulte, Egon. Higher toroidal regular polytopes. Adv. Math., 117(1):1751, 1996. 79, 100CrossRefGoogle Scholar
McMullen, Peter and Schulte, Egon. Twisted groups and locally toroidal regular polytopes. Trans. Amer. Math. Soc., 348(4):13731410, 1996. 314CrossRefGoogle Scholar
McMullen, Peter and Schulte, Egon. Flat regular polytopes. Ann. Comb., 1(3):261278, 1997. 274CrossRefGoogle Scholar
McMullen, Peter and Schulte, Egon. Abstract Regular Polytopes. Cambridge University Press, 2002. vii, 2, 3, 6, 20, 47, 95, 100, 111, 115, 120, 121, 208, 213, 225, 227, 247, 250, 290, 302, 312, 313, 314, 324, 327, 366, 415CrossRefGoogle Scholar
Mixer, Mark, Pellicer, Daniel, and Williams, Gordon. Minimal covers of the Archimedean tilings, part II. Electron. J. Combin., 20(2):Paper 20, 19, 2013. 243CrossRefGoogle Scholar
Mochán, Elías. Abstract polytopes from their symmetry type graphs. 2021. PhD thesis, Universidad Nacional Autónoma de México (Mexico). 58Google Scholar
Monson, B., Pellicer, Daniel, and Williams, Gordon. Mixing and monodromy of abstract polytopes. Trans. Amer. Math. Soc., 366(5):26512681, 2014. 20, 117, 118, 164, 237, 240, 251, 256CrossRefGoogle Scholar
Monson, B. and Schulte, Egon. Reflection groups and polytopes over finite fields. II. Adv. in Appl. Math., 38(3):327356, 2007. 314CrossRefGoogle Scholar
Monson, B. and Schulte, Egon. Reflection groups and polytopes over finite fields. III. Adv. in Appl. Math., 41(1):7694, 2008. 314CrossRefGoogle Scholar
Monson, B. and Schulte, Egon. Modular reduction in abstract polytopes. Canad. Math. Bull., 52(3):435450, 2009. 65CrossRefGoogle Scholar
Monson, B. and Schulte, Egon. Locally toroidal polytopes and modular linear groups. Discrete Math., 310(12):17591771, 2010. 314CrossRefGoogle Scholar
Monson, B. and Schulte, Egon. Semiregular polytopes and amalgamated C-groups. Adv. Math., 229(5):27672791, 2012. 61CrossRefGoogle Scholar
Monson, B. and Schulte, Egon. Finite polytopes have finite regular covers. J. Algebraic Combin., 40(1):7582, 2014. 257CrossRefGoogle Scholar
Monson, B. and Weiss, A. I.. Realizations of regular toroidal maps of type {4,4}. Discrete Comput. Geom., 24:453465, 2000. The Branko Grünbaum birthday issue. 437, 440CrossRefGoogle Scholar
Monson, B. and Ivić Weiss, A.. Eisentein [Eisenstein] integers and related C-groups. Geom. Dedicata, 66(1):99117, 1997. 65, 314CrossRefGoogle Scholar
Monson, B. and Weiss, A. Ivić. Realizations of regular toroidal maps. Canad. J. Math., 51:12401257, 1999. Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday. 437, 440CrossRefGoogle Scholar
Monson, B. and Weiss, Asia Ivić. Polytopes related to the Picard group. Linear Algebra Appl., 218:185204, 1995. 65CrossRefGoogle Scholar
Monson, Barry. On Roli’s cube. Art Discrete Appl. Math., 5(3):Paper No. 3.10, 17, 2022. 307, 309CrossRefGoogle Scholar
Monson, Barry, Pellicer, Daniel, and Williams, Gordon. The tomotope. Ars Math. Contemp., 5(2):355370, 2012. 237CrossRefGoogle Scholar
Montero, Antonio. Chiral extensions of toroids. 2019. PhD thesis, Universidad Nacional Autónoma de México (Mexico). 338, 339Google Scholar
Montero, Antonio. On the Schläfli symbol of chiral extensions of polytopes. Discrete Math., 344(11):Paper No. 112507, 16, 2021. 155, 330, 331CrossRefGoogle Scholar
Nedela, Roman and Škoviera, Martin. Regular maps on surfaces with large planar width. European J. Combin., 22(2):243261, 2001. 165CrossRefGoogle Scholar
Nostrand, B. and Schulte, E.. Chiral polytopes from hyperbolic honeycombs. Discrete Comput. Geom., 13(1):1739, 1995. 144, 324CrossRefGoogle Scholar
Nostrand, Barbara. Chiral honeycombs. ProQuest LLC, 1993. PhD thesis, Northeastern University. 302, 324Google Scholar
Nostrand, Barbara. Ring extensions and chiral polytopes. In Proceedings of the Twenty-fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1994), volume 102, pages 147153, 1994. 65, 302Google Scholar
Nostrand, Barbara, Schulte, Egon, and Weiss, Asia Ivić. Constructions of chiral polytopes. In Proceedings of the Twenty-fourth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1993), volume 97, pages 165170, 1993. 65, 324Google Scholar
Nowacki, Werner. Die euklidischen, dreidimensionalen, geschlossenen und offenen Raumformen. Comment. Math. Helv., 7(1):8193, 1934. 290CrossRefGoogle Scholar
O’Keeffe, M. and Andersson, S.. Rod packings and crystal chemistry. Acta Cryst. A, 33(6):914923, 1977. 455CrossRefGoogle Scholar
Orbanić, Alen. F -actions and parallel-product decomposition of reflexible maps. J. Algebraic Combin., 26(4):507527, 2007. 165CrossRefGoogle Scholar
Orbanić, Alen, Pellicer, Daniel, and Weiss, Asia Ivić. Map operations and k-orbit maps. J. Combin. Theory Ser. A, 117(4):411429, 2010. 59, 185CrossRefGoogle Scholar
Paoluzzi, Luisa. PSL(2,q) quotients of some hyperbolic tetrahedral and Coxeter groups. Comm. Algebra, 26(3):759778, 1998. 302CrossRefGoogle Scholar
Pasechnik, Dmitrii V.. Locally toroidal polytopes of rank 6 and sporadic groups. Adv. Math., 312:459472, 2017. 314, 324CrossRefGoogle Scholar
Pellicer, Daniel. Extensions of regular polytopes with preassigned Schläfli symbol. J. Combin. Theory Ser. A, 116(2):303313, 2009. 327CrossRefGoogle Scholar
Pellicer, Daniel. A construction of higher rank chiral polytopes. Discrete Math., 310(6–7):12221237, 2010. 66, 76, 243, 325, 336, 338, 339CrossRefGoogle Scholar
Pellicer, Daniel. Extensions of dually bipartite regular polytopes. Discrete Math., 310(12):17021707, 2010. 327, 330CrossRefGoogle Scholar
Pellicer, Daniel. Developments and open problems on chiral polytopes. Ars Math. Contemp., 5(2):333354, 2012. 5, 64, 65, 66, 193, 291, 303, 311, 324, 341, 366, 441, 459CrossRefGoogle Scholar
Pellicer, Daniel. The higher dimensional hemicuboctahedron. In Symmetries in Graphs, Maps, and Polytopes, volume 159 of Springer Proc. Math. Stat., pages 263271. Springer, 2016. 61CrossRefGoogle Scholar
Pellicer, Daniel. Regular and chiral polyhedra in Euclidean nets. Symmetry, 8(11):Art. 115, 14, 2016. 437CrossRefGoogle Scholar
Pellicer, Daniel. A chiral 4-polytope in R3. Ars Math. Contemp., 12(2):315327, 2017. 398CrossRefGoogle Scholar
Pellicer, Daniel. Chiral 4-polytopes in ordinary space. Beitr. Algebra Geom., 58(4):655677, 2017. 403, 437, 438, 439CrossRefGoogle Scholar
Pellicer, Daniel. Cleaved abstract polytopes. Combinatorica, 38(3):709737, 2018. 226CrossRefGoogle Scholar
Pellicer, Daniel. A chiral 5-polytope of full rank. Discrete Math., 344(6):Paper No. 112370, 10, 2021. 441, 447, 448CrossRefGoogle Scholar
Pellicer, Daniel. Chiral polytopes of full rank exist only in ranks 4 and 5. Beitr. Algebra Geom., 62(3):651665, 2021. 442CrossRefGoogle Scholar
Pellicer, Daniel and Weiss, Asia Ivić. Combinatorial structure of Schulte’s chiral polyhedra. Discrete Comput. Geom., 44(1):167194, 2010. 206, 269, 412, 430, 434, 435CrossRefGoogle Scholar
Pellicer, Daniel, Potočnik, Primož, and Toledo, Micael. An existence result on two-orbit maniplexes. J. Combin. Theory Ser. A, 166:226253, 2019. 58CrossRefGoogle Scholar
Pellicer, Daniel and Weiss, Asia Ivić. Generalized CPR-graphs and applications. Contrib. Discrete Math., 5(2):76105, 2010. 76, 157, 193, 205, 206, 269, 328Google Scholar
Pellicer, Daniel and Williams, Gordon. Minimal covers of the Archimedean tilings, part 1. Electron. J. Combin., 19(3):paper 6, 37, 2012. 243CrossRefGoogle Scholar
Poinsot, Louis. Mémoire sur les polygones et les polyèdres. J. École Polytech., 10:1648, 1810. 2Google Scholar
Potočnik, Primož. Primož Potočnik’s home page. https://users.fmf.uni-lj.si/potocnik/work.htm. Accessed: 2023-07-13. 64Google Scholar
, John G. Ratcliffe. Foundations of Hyperbolic Manifolds, volume 149 of Graduate Texts in Mathematics. Springer, 3rd ed., 2019. 79, 80, 290, 431Google Scholar
Raymond, Frank. The end point compactification of manifolds. Pacific J. Math., 10:947963, 1960. 272CrossRefGoogle Scholar
Richards, Ian. On the classification of noncompact surfaces. Trans. Amer. Math. Soc., 106:259269, 1963. 272CrossRefGoogle Scholar
Schläfli, Ludwig. Theorie der vielfachen Kontinuität. George & Company, 1901. 2CrossRefGoogle Scholar
Schreier, Otto. Die Untergruppen der freien Gruppen. Abh. Math. Sem. Univ. Hamburg, 5(1):161183, 1927. 157CrossRefGoogle Scholar
Schulte, Egon. On arranging regular incidence-complexes as faces of higherdimensional ones. European J. Combin., 4(4):375384, 1983. 327CrossRefGoogle Scholar
Schulte, Egon. Extensions of regular complexes. In Finite Geometries (Winnipeg, Man., 1984), volume 103 of Lecture Notes in Pure and Appl. Math., pages 289305. Dekker, 1985.Google Scholar
Schulte, Egon. Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J. Combin. Theory Ser. A, 40(2):305330, 1985. 327CrossRefGoogle Scholar
Schulte, Egon. Amalgamation of regular incidence-polytopes. Proc. London Math. Soc. (3), 56(2):303328, 1988. 310, 313CrossRefGoogle Scholar
Schulte, Egon. On a class of abstract polytopes constructed from binary codes. Discrete Math., 84(3):295301, 1990. 327CrossRefGoogle Scholar
Schulte, Egon. Classification of locally toroidal regular polytopes. In Polytopes: Abstract, Convex and Computational (Scarborough, ON, 1993), volume 440 of NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., pages 125154. Kluwer Academic Publishers, 1994. 314CrossRefGoogle Scholar
Schulte, Egon. Chiral polyhedra in ordinary space. I. Discrete Comput. Geom., 32(1):5599, 2004. 4, 206, 268, 362, 388, 420, 424, 430, 431, 432, 434, 435, 436, 437CrossRefGoogle Scholar
Schulte, Egon. Chiral polyhedra in ordinary space. II. Discrete Comput. Geom., 34(2):181229, 2005. 4, 362, 388, 394, 400, 405, 411, 412, 430, 437CrossRefGoogle Scholar
Schulte, Egon and Weiss, Asia Ivić. Chiral polytopes. In Applied Geometry and Discrete Mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 493516. American Mathematical Society, 1991. 1, 62, 65, 67, 111, 122, 317, 318, 321, 325Google Scholar
Schulte, Egon and Weiss, Asia Ivić. Chirality and projective linear groups. Discrete Math., 131(1–3):221261, 1994. 65, 77, 144, 154, 155, 193, 314, 324, 458CrossRefGoogle Scholar
Schulte, Egon and Weiss, Asia Ivić. Free extensions of chiral polytopes. Canad. J. Math., 47(3):641654, 1995. 65, 206, 329CrossRefGoogle Scholar
Schulte, Egon and Weiss, Asia Ivić. Problems on polytopes, their groups, and realizations. Period. Math. Hungar., 53(1–2):231255, 2006. 5, 311, 324, 366, 459CrossRefGoogle Scholar
Servatius, Brigitte and Servatius, Herman. Self-dual maps on the sphere. Discrete Math., 134(1–3):139150, 1994. 23CrossRefGoogle Scholar
Sherk, F. A.. The regular maps on a surface of genus three. Canad. J. Math., 11:452480, 1959. 263CrossRefGoogle Scholar
Sherk, F. A.. A family of regular maps of type {6,6}. Canad. Math. Bull., 5:1320, 1962. 64, 144, 259, 268, 270, 271CrossRefGoogle Scholar
So˘ıfer, G. A.. Affine crystallographic groups. In Algebra and Analysis (Irkutsk, 1989), volume 163 of Amer. Math. Soc. Transl. Ser. 2, pages 165170. American Mathematical Society, 1995. 431Google Scholar
Spiga, Pablo. On the rank of Suzuki polytopes: an answer to Hubard and Leemans. Discrete Math., 344(5):Paper No. 112331, 5, 2021. 340CrossRefGoogle Scholar
Tits, J.. Sur les analogues algébriques des groupes semi-simples complexes. In Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, pages 261289. Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, 1957. 112Google Scholar
Tutte, William T.. What is a map? In New Directions in the Theory of Graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), pages 309325. Academic Press, 1973. 23Google Scholar
Vince, Andrew. Combinatorial maps. J. Combin. Theory Ser. B, 34(1):121, 1983. 22, 164CrossRefGoogle Scholar
Vince, Andrew. Regular combinatorial maps. J. Combin. Theory Ser. B, 35(3):256277, 1983. 64, 267, 268, 269, 273CrossRefGoogle Scholar
Širáň, Jozef. Regular maps on a given surface: a survey. In Topics in Discrete Mathematics, volume 26 of Algorithms Combin., pages 591609. Springer, 2006. 263CrossRefGoogle Scholar
Širáň, Jozef, Tucker, Thomas W., and Watkins, Mark E.. Realizing finite edgetransitive orientable maps. J. Graph Theory, 37(1):134, 2001. 54CrossRefGoogle Scholar
Weber, C. and Seifert, H.. Die beiden Dodekaederräume. Math. Z., 37(1): 237253, 1933. 65, 291CrossRefGoogle Scholar
Weiss, Asia Ivić. On trivalent graphs embedded in twisted honeycombs. In Combinatorics ’81 (Rome, 1981), volume 78 of North-Holland Math. Stud., pages 781787. North-Holland, 1983. 299Google Scholar
Weiss, Asia Ivić. Twisted honeycombs {3, 5, 3}t. C. R. Math. Rep. Acad. Sci. Canada, 5(5):211215, 1983. 65, 300Google Scholar
Weiss, Asia Ivić. An infinite graph of girth 12. Trans. Amer. Math. Soc., 283(2):575588, 1984. 314CrossRefGoogle Scholar
Weiss, Asia Ivić and Lučić, Zoran. Regular polyhedra in hyperbolic three-space. Mitt. Math. Sem. Giessen, (165):237252, 1984. 344Google Scholar
Wilson, Stephen E.. Non-orientable regular maps. Ars Combin., 5:213218, 1978. 263Google Scholar
Wilson, Stephen E.. Riemann surfaces over regular maps. Canad. J. Math., 30(4):763782, 1978. 263CrossRefGoogle Scholar
Wilson, Stephen E.. Operators over regular maps. Pacific J. Math., 81(2):559568, 1979. 23, 209CrossRefGoogle Scholar
Wilson, Stephen E.. Parallel products in groups and maps. J. Algebra, 167(3):539546, 1994. 243, 244, 246CrossRefGoogle Scholar
Wilson, Steve. The smallest nontoroidal chiral maps. J. Graph Theory, 2(4):315318, 1978. 274CrossRefGoogle Scholar
Wilson, Steve. Uniform maps on the Klein bottle. J. Geom. Graph., 10(2):161171, 2006. 290Google Scholar
Wilson, Steve. Maniplexes: Part 1: Maps, polytopes, symmetry and operators. Symmetry, 4(2):265275, 2012. 28, 29, 165CrossRefGoogle Scholar
Wolf, Joseph A.. Spaces of Constant Curvature. AMS Chelsea Publishing, 6th ed., 2011. 290CrossRefGoogle Scholar
Zhang, Wei-Juan. Some simplifications of the intersection condition of chiral form for polytopes. J. Algebra Appl., 18(11):1950203, 15, 2019. 66CrossRefGoogle Scholar
Zhang, Wei-Juan. More on chiral polytopes of type {4,4, ...,4} with solvable automorphism groups. J. Group Theory, 26(6):12311265, 2023. 66Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Daniel Pellicer, Universidad Nacional Autónoma de México
  • Book: Abstract Chiral Polytopes
  • Online publication: 23 March 2025
  • Chapter DOI: https://doi.org/10.1017/9781108695046.011
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Daniel Pellicer, Universidad Nacional Autónoma de México
  • Book: Abstract Chiral Polytopes
  • Online publication: 23 March 2025
  • Chapter DOI: https://doi.org/10.1017/9781108695046.011
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Daniel Pellicer, Universidad Nacional Autónoma de México
  • Book: Abstract Chiral Polytopes
  • Online publication: 23 March 2025
  • Chapter DOI: https://doi.org/10.1017/9781108695046.011
Available formats
×