Platonic Solids and High Genus Covers of Lattice Surfaces

References

• Athreya, J. S., Aulicino, D. (2019). A trajectory from a vertex to itself on the dodecahedron. The Amer. Math. Monthly. 126(2): 161–162. doi:10.1080/00029890.2019.1538475
   Web of Science ®Google Scholar
• Atiyah, M., Sutcliffe, P. (2003). Polyhedra in physics, chemistry and geometry. Milan J. Math. 71: 33–58. MR 2120915
   Google Scholar
• Aulicino, D. (2019). A new approach to the automorphism group of a Platonic surface. Rocky Mountain J. Math. Advance publication.
   Web of Science ®Google Scholar
• Bainbridge, M., Chen, D., Gendron, Q., Grushevsky, S, Möller, M. (2019). Strata of k-differentials. Algebr. Geom. 6(2): 196–233.
   Web of Science ®Google Scholar
• Bobenko, A. I., and Springborn, B. A. (2007). A discrete Laplace–Beltrami operator for simplicial surfaces. Discrete Comput. Geom. 38(4): 740–756. doi:10.1007/s00454-007-9006-1
   Web of Science ®Google Scholar
• Davis, D., Dods, V., Traub, C, Yang, J. (2017). Geodesics on the regular tetrahedron and the cube. Discrete Math. 340(1): 3183–3196.
   Web of Science ®Google Scholar
• Delecroix, V., Fougeron, C, Lelièvre, S. (2018). The surface_dynamics package for SageMath [Computer Software. Available at: https://pypi.org/project/surface-dynamics/
   Google Scholar
• Delecroix, V., Hooper, W. P. Flat surfaces in SageMath [Computer Software]. Available at: https://github.com/videlec/sage-flatsurf
   Google Scholar
• Davis, D., Lelièvre, S. (2018). Periodic paths on the pentagon, double pentagon and golden L. arXiv:1810.11310.
   Google Scholar
• Fuchs, D. B, Fuchs, E. (2007). Closed geodesics on regular polyhedra. Moscow Mathematical Journal. 7(2): 265–279. doi:10.17323/1609-4514-2007-7-2-265-279
   Web of Science ®Google Scholar
• Finster, M. (2014). Veech groups and translation coverings. Karlsruhe: KIT Scientific Publishing.
   Google Scholar
• Fox, R. H., Kershner, R. B. (1936). Concerning the transitive properties of geodesics on a rational polyhedron. Duke Math. J. 2: 147–150. (English). doi:10.1215/S0012-7094-36-00213-2
   Google Scholar
• Farb, B., Margalit, D. (2011). A primer on mapping class groups (PMS-49). Princeton, NJ: Princeton University Press.
   Google Scholar
• Freidinger, M. (2008). Stabilisatorgruppen in Aut(Fz) und Veechgruppen von überlagerungen. Diploma thesis. Universität Karlsruhe (TH),
   Google Scholar
• Fuchs, D. (2014). Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra. Geom. Dedicata. 170: 319–333.
   Web of Science ®Google Scholar
• Fuchs, D. (2016). Geodesics on regular polyhedra with endpoints at the vertices. Arnold Math. J. 2(2): 201–211.
   Google Scholar
• Galperin, G. (2003). Convex polyhedra without simple closed geodesics. Regul. Chaotic Dyn. 8(1): 45–58.
   Web of Science ®Google Scholar
• Gardner, M. (2008). The five Platonic solids. In: Origami, Eleusis, and the Soma cube: Martin Gardner’s mathematical diversions Cambridge: Cambridge University Press, 1–10.
   Google Scholar
• Gutkin, E., Judge, C. (2000). Affine mappings of translation surfaces: geometry and arithmetic. Duke Math. J. 103(2): 191–213. doi:10.1215/S0012-7094-00-10321-3
   Web of Science ®Google Scholar
• Göttingen collection of mathematical models and instruments. Available at: http://modellsammlung.uni-goettingen.de/.
   Google Scholar
• Hooper, W. P. (2013). Grid graphs and lattice surfaces. Int. Math. Res. Notices. 2013(12): 2657–2698. doi:10.1093/imrn/rns124
   Web of Science ®Google Scholar
• Hubert, P., Schmidt, T. A. (2001). Invariants of translation surfaces.. Ann. Inst. Fourier (Grenoble). 51(2): 461–495.
   Web of Science ®Google Scholar
• Hubert, P, Schmidt, T. A. (2004). Infinitely generated Veech groups. Duke Math. J. 123(1): 49–70. doi:10.1215/S0012-7094-04-12312-8
   Web of Science ®Google Scholar
• Indermitte, C., Liebling, T. M., Troyanov, M., Clémençon, H. (2001). Voronoi diagrams on piecewise flat surfaces and an application to biological growth. Theoret. Comput. Sci. 263(1/2): 263–274. doi:10.1016/S0304-3975(00)00248-6
   Web of Science ®Google Scholar
• Kenyon, R., Smillie, J. (2000). Billiards on rational-angled triangles. Comment. Math. Helv. 75(1): 65–108.
   Web of Science ®Google Scholar
• Lelièvre, S., Monteil, T., Weiss, B. (2016). Everything is illuminated. Geom. Topol. 20(3): 1737–1762.
   Web of Science ®Google Scholar
• Masur, H. (1986). Closed trajectories for quadratic differentials with an application to billiards. Duke Math. J. 53(2): 307–314.
   Web of Science ®Google Scholar
• McMullen, C. T. (2003). Teichmüller geodesics of infinite complexity. Acta Math. 191(2): 191–223. doi:10.1007/BF02392964
   Web of Science ®Google Scholar
• McMullen, C. T. (2006). Prym varieties and Teichmüller curves. Duke Math. J. 133(3): 569–590. doi:10.1215/S0012-7094-06-13335-5
   Web of Science ®Google Scholar
• Möller, M. (2006). Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve. Invent. Math. 165(3): 633–649. doi:10.1007/s00222-006-0510-3
   Web of Science ®Google Scholar
• Mirzakhani, M., Wright, A. (2018). Full-rank affine invariant submanifolds. Duke Math. J. 167(1): 1–40.
   Web of Science ®Google Scholar
• Petrunin, A. (2009). Puzzles in geometry that I know and love. arXiv:0906.0290.
   Google Scholar
• Puchta, J.-C. (2001). On triangular billiards. Comment. Math. Helv. 76(3): 501–505.
   Web of Science ®Google Scholar
• Rodenberg, C. (1906). Geodätlsche Linien auf Polyederflächen (Einige Bemerkungen zur gleichnnamigen Arbeit des Herrn Paul Stäckels in den” Rendiconti”, Bd. XXII). Supplemento ai Rendiconti del Circolo matematico di Palermo I. (5): 60–61. (German). Available at: https://books.google.com/books?id=H21OAAAAMAAJ&pg=PA60&lpg=PA60
   Google Scholar
• Rodenberg, C. (1907). Geodätische Linien auf Polyederflächen. Rend. Circ. Mat. Palermo. 23: 107–125. German).
   Google Scholar
• Rosen, D. (1954). A class of continued fractions associated with certain properly discontinuous groups. Duke Math. J. 21: 549–563. doi:10.1215/S0012-7094-54-02154-7
   Web of Science ®Google Scholar
• Schmithüsen, G. (2004). An algorithm for finding the Veech group of an origami. Exper. Math. 13: 459–472. doi:10.1080/10586458.2004.10504555
   Web of Science ®Google Scholar
• Stäckel, P. (1906). Geodätische Linien auf Polyederflächen. Rend. Circ. Matem. Palermo. 22: 141–151. (German). doi:10.1007/BF03018610
   Google Scholar
• Steinitz, E. Zentralblatt review of “Geodätische linien auf polyederflächen.” Available at: https://zbmath.org/?q=an%3A38.0518.03.
   Google Scholar
• The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.9) (2019). Available at: http://www.sagemath.org.
   Google Scholar
• Veech, W. A. (1989). Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97: 553–583. (3)MR 1005006 (91h:58083a) doi:10.1007/BF01388890
   Web of Science ®Google Scholar
• Veech, W. A. (1992). The billiard in a regular polygon. Geom. Funct. Anal. 2(3): 341–379.
   Google Scholar
• Renteln, M. (2002). Die Mathematiker an der Technischen Hochschule Karlsruhe: (1825–1945). Erscheinungsort nicht ermittelbar: Verlag nicht ermittelbar.
   Google Scholar
• Zemlyakov, A. N, Katok, A. B. (1975). Topological transitivity of billiards in polygons. Math. Notes Acad Sci. USSR. 18(2): 760–764. doi:10.1007/BF01818045
   Web of Science ®Google Scholar