Regular Polygonal Complexes of Higher Ranks in $\mathbb{E}^3$
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 6, pp. 103-110.

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The paper establishes that the rank of a regular polygonal complex in $\mathbb{E}^3$ cannot exceed $4$, and that the only regular polygonal complexes of rank $4$ in $\mathbb{E}^3$ are the eight regular $4$-apeirotopes in $\mathbb{E}^3$. The article is published in the author's wording.
Keywords: polygonal complex, abstract polytopes, regularity.
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Egon Schulte. Regular Polygonal Complexes of Higher Ranks in $\mathbb{E}^3$. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 6, pp. 103-110. http://geodesic.mathdoc.fr/item/MAIS_2013_20_6_a7/

[1] H. S. M. Coxeter, Regular Polytopes, 3rd edition, Dover | MR | Zbl

[2] H. S. M. Coxeter, “Regular skew polyhedra in 3 and 4 dimensions and their topological analogues”, Proc. London Math. Soc., 43 (1937), 33–62 ; Reprinted with amendments in Twelve Geometric Essays, Southern Illinois University Press, Carbondale, 1968, 76–105 | MR | MR

[3] L. Danzer, E. Schulte, Geom. Dedicata, 13 (1982), 295–308 | DOI | MR | Zbl

[4] A. W. M. Dress, “A combinatorial theory of Grünbaum's new regular polyhedra. I: Grünbaum's new regular polyhedra and their automorphism group”, Aequationes Math., 23 (1981), 252–265 | DOI | MR | Zbl

[5] A. W. M. Dress, “A combinatorial theory of Grünbaum's new regular polyhedra. II: Complete enumeration”, Aequationes Math., 29 (1985), 222–243 | DOI | MR | Zbl

[6] B. Grünbaum, “Regular polyhedra — old and new”, Aequat. Math., 16 (1977), 1–20 | DOI | MR | Zbl

[7] P. McMullen, “Realizations of regular polytopes”, Aequationes Math., 37 (1989), 38–56 | DOI | MR | Zbl

[8] P. McMullen, Geometric regular polytopes, monograph in preparation

[9] P. McMullen, E. Schulte, “Regular polytopes in ordinary space”, Discrete Comput. Geom., 17 (1997), 449–478 | DOI | MR | Zbl

[10] P. McMullen, E. Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, 92, Cambridge University Press, Cambridge, UK, 2002 | MR | Zbl

[11] D. Pellicer, E. Schulte, “Regular polygonal complexes in space, I”, Trans. Amer. Math. Soc., 362 (2010), 6679–6714 | DOI | MR | Zbl

[12] D. Pellicer, E. Schulte, “Regular polygonal complexes in space, II”, Trans. Amer. Math. Soc., 365 (2013), 2031–2061 | DOI | MR | Zbl

[13] E. Schulte, “Reguläre Inzidenzkomplexe, II”, Geom. Dedicata, 14 (1983), 33–56 | MR | Zbl