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Hyperbolic Space Forms with Crystallographic Applications and Visualizations

  • Emil MolnárEmail author
  • Jenő Szirmai
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 809)

Abstract

In this paper we study a family of compact 3-dimensional manifolds, i.e. space forms - more popularly, finite worlds - that are derived from famous Euclidean and non-Euclidean polyhedral tilings by the unified method of face identification, i.e. logical gluings. All these seem to have application in modern crystallography, as fullerenes and nanotubes!?!.

References

  1. 1.
    Cavichioli, A., Molnár, E., Spaggiari, F., Szirmai, J.: Some tetrahedron manifolds with Sol geometry. J Geomet. 105(3), 601–614 (2014).  https://doi.org/10.1007/s00022-014-0222-6MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cavicchioli, A., Telloni, A.I.: On football manifolds of E. Molnár. Acta Math. Hungar. 124(4), 321–332 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hahn, Th. (ed.): International Tables for Crystallography, vol. A, 5th edn. Kluwer, Dordrecht (2002)Google Scholar
  4. 4.
    Kellerhals, R.: On the volume of hyperbolic polyhedra. Math. Ann. 245, 541–569 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Molnár, E.: Projective metrics and hyperbolic volume. Annales Univ. Sci. Budapest Sect. Math. 32, 127–157 (1989)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Molnár, E.: Space forms and fundamental polyhedra. In: Proceedings of the Conference on Differential Geometry and Its Applications, Nové Mésto na Moravé, Czechoslovakia 1983. Part 1. Differential Geometry, pp. 91–103 (1984)Google Scholar
  7. 7.
    Molnár, E.: Two hyperbolic football manifolds. In: Proceedings of International Conference on Differential Geometry and Its Applications, Dubrovnik Yugoslavia, pp. 217–241 (1988)Google Scholar
  8. 8.
    Molnár, E.: Combinatorial construction of tilings by barycentric simplex orbits (D symbols) and their realizations in Euclidean and other homogeneous spaces. Acta Cryst. A61, 541–552 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Molnár, E.: The projective interpretation of the eight 3-dimensional homogeneous geometries. Beitr. Alg. Geom. (Contr. Alg. Geom.) 38/2, 261–288 (1997)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Molnár, E.: Polyhedron complexes with simply transitive group actions and their realizations. Acta Math. Hung. 59(1–2), 175–216 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Molnár, E.: On non-Euclidean crystallography, some football manifolds. Struct. Chem. 23/4, 1057–1069 (2012)CrossRefGoogle Scholar
  12. 12.
    Molnár, E., Szirmai, J.: Symmetries in the 8 homogeneous 3-geometries. Symmetry Cult. Sci. 21(1—-3), 87–117 (2010)zbMATHGoogle Scholar
  13. 13.
    Molnár, E., Szirmai, J.: Classification of Sol lattices. Geom. Dedicata 161(1), 251–275 (2012).  https://doi.org/10.1007/s10711-012-9705-5MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Molnár, E., Szirmai, J.: Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups. Publications de l’Institut Mathmatique, (2017), (to appear), arXiv: 161204541v1
  15. 15.
    Molnár, E., Szirmai, J.: On hyperbolic cobweb manifolds. Stud. Univ. Zilina. Math. Ser. 28, 43–52 (2016)zbMATHGoogle Scholar
  16. 16.
    Molnár, E., Szirmai, J.: Infinite series of compact hyperbolic manifolds, as possible crystal structures. Submitted manuscript, (2018), arXiv:1711.09799
  17. 17.
    Molnár, E., Prok, I., Szirmai, J.: Classification of tile-transitive 3-simplex tilings and their realizations in homogeneous spaces. In: Prékopa, A., Molnár, E. (eds.). Non-Euclidean Geometries, János Bolyai Memorial Volume, Mathematics and Its Applications, Springer, (2006), vol. 581, 321–363Google Scholar
  18. 18.
    Molnár, E., Prok, I., Szirmai, J.: The Euclidean visualization and projective modelling the 8 Thurston geometries. Stud. Univ. Zilina. Math. Ser. 27(1), 35–62 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Molnár, E., Szirmai, J., Vesnin, A.: Packings by translation balls in \({\widetilde{{\mathbf{S}}{\mathbf{L}}_2{\mathbf{R}}}}\). J. Geometry 105(2), 287–306 (2014).  https://doi.org/10.1007/s00022-013-0207-xMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Molnár, E., Szirmai, J., Vesnin, A.: Geodesic and translation ball packings generated by prismatic tessellations of the universal cover of \( SL _2 R \). Results. Math. 71, 623–642 (2017).  https://doi.org/10.1007/s00025-016-0542-yMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Prok, I.: Data structures and procedures for a polyhedron algorithm. Periodica Polytechnica Ser. Mech. Eng. 36(3–4), 299–316 (1992)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Prok, I.: Classification of dodecahedral space forms. Beitr. Alg. Geom. (Contr. Alg. Geom.) 38/2, 497–515 (1998)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Prok, I.: On Maximal Homogeneous 3-GeometriesA Polyhedron Algorithm for Space Tilings. Universe 4/3, 49 (2018).  https://doi.org/10.3390/universe4030049CrossRefGoogle Scholar
  24. 24.
    Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15, 401–487 (1983)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Szirmai, J.: The optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic \(d\)-space. Beitr. Alg. Geom. (Contr. Alg. Geom.) 48/1, 35–47 (2007)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Szirmai, J.: The densest geodesic ball packing by a type of Nil lattices. Beitr. Alg. Geom. (Contr. Alg. Geom.) 48/2, 383–397 (2007)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Szirmai, J.: The densest translation ball packing by fundamental lattices in Sol space. Beitr. Alg. Geom. 51/2, 353–373 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Szirmai, J.: Geodesic ball packing in \(\mathbf{S}^2 \times \mathbf{R}\) space for generalized Coxeter space groups. Beitr. Alg. Geom. (Contr. Alg. Geom.) 52, 413–430 (2011).  https://doi.org/10.1007/s13366-011-0023-0MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Szirmai, J.: A candidate to the densest packing with equal balls in the Thurston geometries. Beitr. Algebra Geom. 55(2), 441–452 (2014).  https://doi.org/10.1007/s13366-013-0158-2MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vinberg, E.B. (ed.): Geometry II. Spaces of Constant Curvature. Spriger Verlag Berlin-Heidelberg, New York-London-Paris-Tokyo-Hong Kong-Barcelona-Budapest (1993)Google Scholar
  31. 31.
    Weeks, J.R.: Real-time animation in hyperbolic, spherical, and product geometries. In: Prékopa, A., Molnár, E. (eds.). Non-Euclidean Geometries, János Bolyai Memorial Volume, Mathematics and Its Applications, vol. 581, pp. 287–305. Springer (2006)Google Scholar
  32. 32.
    Wolf, J.A.: Spaces of Constant Curvature. McGraw-Hill, New York (1967). (Russian translation: Izd. “Nauka” Moscow, 1982)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics, Department of GeometryBudapest University of Technology and EconomicsBudapestHungary

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