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Wythoffian Skeletal Polyhedra in Ordinary Space, I

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Abstract

Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff’s construction is applied to the 48 regular skeletal polyhedra (Grünbaum–Dress polyhedra) in ordinary space, new highly symmetric skeletal polyhedra arise as “truncations” of the original polyhedra. These Wythoffians are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes. The present paper describes the blueprint for the construction and treats the Wythoffians for distinguished classes of regular polyhedra. The Wythoffians for the remaining classes of regular polyhedra will be discussed in Part II, by the second author. We also examine when the construction produces uniform skeletal polyhedra.

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References

  1. Arocha, J.L., Bracho, J., Montejano, L.: Regular projective polyhedra with planar faces, part I. Aequat. Math. 59, 55–73 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bracho, J.: Regular projective polyhedra with planar faces, part II. Aequat. Math. 59, 160–176 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York (1998)

    MATH  Google Scholar 

  4. Coxeter, H.S.M.: Wythoff’s construction for uniform polytopes. Proc. Lond. Math. Soc. (2) 38, 327–339 (1935)

    Article  MATH  MathSciNet  Google Scholar 

  5. 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)

    MATH  MathSciNet  Google Scholar 

  6. Coxeter, H.S.M.: Regular and semi-regular polytopes, I. Math. Z. 46, 380–407 (1940)

    Article  MATH  MathSciNet  Google Scholar 

  7. Coxeter, H.S.M.: Regular Polytopes. Dover, New York (1973)

    MATH  Google Scholar 

  8. Coxeter, H.S.M.: Regular and semi-regular polytopes, II. Math. Z. 188, 559–591 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  9. Coxeter, H.S.M.: Regular and semi-regular polytopes, III. Math. Z. 200, 3–45 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Coxeter, H.S.M.: Regular Complex Polytopes, 2nd edn. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  11. Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, New York (1980)

    Book  MATH  Google Scholar 

  12. Coxeter, H.S.M., Longuet-Higgins, M.S., Miller, J.C.P.: Uniform polyhedra. Philos. Trans. R. Soc. Lond. Ser. A 246, 401–450 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cutler, A.M.: Regular polyhedra of index two, II. Contrib. Algebra Geom. 52, 357–387 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cutler, A.M., Schulte, E.: Regular polyhedra of index two, I. Contrib. Algebra Geom. 52, 133–161 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. del Ro Francos, M.: Chamfering operation on k-orbit maps. ARS Math. Contemp. 7, 507–524 (2014)

    MATH  MathSciNet  Google Scholar 

  16. Delgado-Friedrichs, O., Foster, M.D., O’Keefe, M., Proserpio, D.M., Treacy, M.M.J., Yaghi, O.M.: What do we know about three-periodic nets? J. Solid State Chem. 178, 2533–2554 (2005)

    Article  Google Scholar 

  17. Deza, M., Sikiric, M.D., Shpectorov, S.: Hypercube embeddings of Wythoffians. ARS Math. Contemp. 1, 99–111 (2008)

    MATH  MathSciNet  Google Scholar 

  18. Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, part I: Grünbaum’s new regular polyhedra and their automorphism group. Aequat. Math. 23, 252–265 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  19. Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, part II: complete enumeration. Aequat. Math. 29, 222–243 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  20. Farris, S.L.: Completely classifying all vertex-transitive and edge-transitive polyhedra, part I: necessary class conditions. Geom. Dedicata 26, 111–124 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  21. Grove, L.C., Benson, C.T.: Finite Reflection Groups, 2nd edn. Springer, New York (1985)

    Book  MATH  Google Scholar 

  22. Grünbaum, B.: Regular polyhedra: old and new. Aequat. Math. 16, 1–20 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  23. Grünbaum, B.: Polyhedra with hollow faces. In: Polytopes: Abstract, Convex and Computational, pp. 43–70 (1994)

  24. Grünbaum, B.: Acoptic polyhedra. In: Advances in Discrete and Computational Geometry, pp. 163–199 (1999)

  25. Grünbaum, B.: Convex Polytopes, 2nd edn. Springer, New York (2003)

    Book  MATH  Google Scholar 

  26. Grünbaum, B.: “New” uniform polyhedra. In: Discrete Geometry: In Honor of W. Kuperberg’s 60th Birthday, pp. 331–350 (2003)

  27. Grünbaum, B.: Configurations of Points and Lines. Graduate Studies in Mathematics, 103rd edn. American Mathematical Society, Providence, RI (2009)

    MATH  Google Scholar 

  28. Grünbaum, B., Johnson, N.W.: The faces of a regular-faced polyhedron. J. Lond. Math. Soc. 40, 577–586 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  29. Grünbaum, B., Shephard, G.C.: Tilings and Patterns. WH Freeman and Company, New York (1987)

    MATH  Google Scholar 

  30. Har’El, Z.: Uniform solution for uniform polyhedra. Geom. Dedicata 47, 57–110 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. Johnson, N.W.: Convex solids with regular faces. Can. J. Math. 18, 169–200 (1966)

    Article  MATH  Google Scholar 

  32. Johnson, N.W.: Uniform polytopes (In preparation)

  33. Martini, H.: Hierarchical classification of euclidean polytopes with symmetry properties. In: Polytopes: Abstract, Convex and Computational, pp. 71–96 (1994)

  34. McMullen, P.: Realizations of regular polytopes. Aequat. Math. 37, 38–56 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  35. McMullen, P., Monson, B.: Realizations of regular polytopes, II. Aequat. Math. 65(1), 102–112 (2003)

    MATH  MathSciNet  Google Scholar 

  36. McMullen, P.: Regular polytopes of full rank. Discrete Comput. Geom. 32, 1–35 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. McMullen, P.: Four-dimensional regular polyhedra. Discrete Comput. Geom. 38, 355–387 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. McMullen, P.: Geometric regular polytopes (In preparation)

  39. McMullen, P., Schulte, E.: Regular polytopes in ordinary space. Discrete Comput. Geom. 17, 449–478 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  40. McMullen, P., Schulte, E.: Abstract Regular Polytopes. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  41. Monson, B., Pellicer, D., Williams, G.: Mixing and monodromy of abstract polytopes. Trans. AMS 366, 2651–2681 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  42. O’Keeffe, M.: Three-periodic nets and tilings: regular and related infinite polyhedra. Acta Crystallogr. A 64, 425–429 (2008)

    Article  MathSciNet  Google Scholar 

  43. O’Keeffe, M., Hyde, B.G.: Crystal Structures, No. 1: Patterns and Symmetry. Mineralogical Society of America, Chantilly (1996)

    MATH  Google Scholar 

  44. Pellicer, D., Schulte, E.: Regular polygonal complexes in space, I. Trans. Am. Math. Soc. 362, 6679–6714 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  45. Pellicer, D., Schulte, E.: Regular polygonal complexes in space, II. Trans. Am. Math. Soc. 365, 2031–2061 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  46. Pellicer, D., Weiss, A.I.: Combinatorial structure of Schulte’s chiral polyhedra. Discrete Comput. Geom. 44, 167–194 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  47. Pisanski, T., Randic, M.: Bridges between geometry and graph theory. Geom. Work, MAA Notes 53, 174–194 (2000)

    MathSciNet  Google Scholar 

  48. Pisanski, T., Žitnik, A.: Representing graphs and maps. In: Topics in Topological Graph Theory. Encyclopedia of Mathematics and Its Applications, vol. 128, pp. 151–180 (2009)

  49. Schulte, E.: Chiral polyhedra in ordinary space, I. Discrete Comput. Geom. 32, 55–99 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  50. Schulte, E.: Chiral polyhedra in ordinary space, II. Discrete Comput. Geom. 34, 181–229 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  51. Schulte, E.: Polyhedra, complexes, nets and symmetry. Acta Crystallogr. Sect. A 70(3), 203–216 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  52. Seifert, H., Threlfall, W.: A Textbook of Topology. Academic Press, New York (1980)

    MATH  Google Scholar 

  53. Senechal, M.: Shaping Space: Exploring Polyhedra in Nature, Art and the Geometrical Imagination. Springer, New York (2013)

    Book  MATH  Google Scholar 

  54. Skilling, J.: The complete set of uniform polyhedra. Philos. Trans. R. Soc. Lond. Ser. A 278, 111–135 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  55. Skilling, J.: Uniform compounds of uniform polyhedra. Math. Proc. Camb. Philos. Soc. 79, 447–457 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  56. Wells, A.F.: Three-Dimensional Nets and Polyhedra. Wiley, New York (1977)

    Google Scholar 

  57. Williams, A.: Wythoffian skeletal polyhedra. PhD Thesis, Northeastern University (2015)

  58. Williams, A.: Wythoffian skeletal polyhedra in ordinary space, II (In preparation)

  59. Wolfram Research Inc.: Mathematica 10.0. Wolfram Research, Inc., Champaign. http://www.wolfram.com (2010)

  60. Wythoff, W.A.: A Relation Between the Polytopes of the C600-family, vol. 20. Koninklijke Akademie van Wetenschappen te Amsterdam, Amsterdam (1918)

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Acknowledgments

We are grateful to the anonymous referees for their careful reading of our original manuscript and their helpful suggestions that have improved our paper. Supported by NSA-Grant H98230-14-1-0124.

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  1. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Egon Schulte & Abigail Williams

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  1. Egon Schulte
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Correspondence to Abigail Williams.

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Schulte, E., Williams, A. Wythoffian Skeletal Polyhedra in Ordinary Space, I. Discrete Comput Geom 56, 657–692 (2016). https://doi.org/10.1007/s00454-016-9814-2

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