Abstract
Even in our decade there is still an extensive search for analogues of the Platonic solids. In a recent paper Schulte and Wills [13] discussed properties of Dyck's regular map of genus 3 and gave polyhedral realizations for it allowing self-intersections. This paper disproves their conjecture in showing that there is a geometric polyhedral realization (without self-intersections) of Dyck's regular map {3, 8}6 already in Euclidean 3-space. We describe the shape of this new regular polyhedron.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Antonin,Ein Algorithmusansatz für Realisierungsfragen im E d getestet an kombinatorischen 3-Sphären, Staatsexamensarbeit, Bochum, 1982.
J. Bokowski and K. Garms, Altshuler's sphereM 10425 is not polytopal,European J. Combin., to appear.
J. Bokowski and I. Shemer, Neighborly 6-polytopes with 10 vertices,Israel J. Math. 58 (1987), 103–124.
J. Bokowski and B. Sturmfels, On the coordinatization of oriented matroids,Discrete Comput. Geom. 1 (1986), 293–306.
J. Bokowski, G. Ewald, and P. Kleinschmidt, On combinatorial and affine automorphisms of polytopes,Israel J. Math. 47 (1984), 123–130.
W. Dyck, Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemannscher Flächen,Math. Ann. 17 (1880), 473–508.
W. Dyck, Notiz über eine reguläre Riemannsche Fläche vom Geschlecht 3 und die zugehörige Normalkuve 4. Ordnung,Math. Ann. 17 (1880), 510–516.
W. Dyck, Gruppentheoretische Studien,Math. Ann. 20 (1882), 1–45.
D. Garbe, Über die regulären Zerlegungen geschlossener Flächen,J. Reine Angew. Math. 237 (1969) 39–55.
P. McMullen, C. Schulz, and J. M. Wills, Equivelar polyhedral manifolds inE 3,Israel J. Math. 41 (1982), 331–346.
P. McMullen, E. Schulte, and J. M. Wills, Infinite series of combinatorially regular polyhedra in three-space, in preparation.
E. Schulte and J. M. Wills, A polyhedral realization of Felix Klein's map {3, 7}8 on a Riemannian manifold of genus 3,J. London Math. Soc. 32 (1985), 539–547.
E. Schulte and J. M. Wills, Geometric realizations for Dyck's regular map on a surface of genus 3,Discrete Comput. Geom. 1 (1986), 141–153.
E. Schulte and J. M. Wills, On Coxeter's regular skew polyhedra,Discrete Math. 60 (1986), 253–262.
E. Schulte and J. M. Wills,Combinatorially Regular Polyhedra in Three-Space, Preprint, Siegen, Vol.179, 1986.
F. A. Sherk, The regular maps on a surface of genus 3,Canad. J. Math. 11 (1959), 452–480.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bokowski, J. A geometric realization without self-intersections does exist for Dyck's regular map. Discrete Comput Geom 4, 583–589 (1989). https://doi.org/10.1007/BF02187748
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02187748