Platonic Solids and High Genus Covers of Lattice Surfaces

Jayadev S. Athreya, David Aulicino, and W. Patrick Hooper, with an appendix by Anja Randecker


Trajectories on the Dodecahedron

Press the button to start the animation.
DodecahedronAnimation

Student Activities: Make your own dodecahedra! Click the figure or link below for instructions.

Click the figure below to download the complete pdf of representatives for all closed saddle connections. The pdf contains nets of the dodecahedron such that each net contains a representative closed saddle connection corresponding to the coset listed by its number in Appendix C of the second paper. These can be printed on paper, cut out, and taped together to get the models of the dodecahedra in the animation above.

Dodecahedron
A Shortest Closed Saddle Connection on the Dodecahedron

Papers

  1. J. S. Athreya and D. Aulicino, A Trajectory from a Vertex to Itself on the Dodecahedron. arXiv:1802.00811, (Accepted. To appear in Amer. Math. Monthly). Preprint: 1-2 (2018).
  2. J. S. Athreya, D. Aulicino, and W. P. Hooper, with an appendix by A. Randecker, Platonic Solids and High Genus Covers of Lattice Surfaces. arXiv:1811.04131. Preprint: 1-51 (2018).

Sage Notebooks and Auxiliary Files

For instructions on how to download and install Sage, please visit SageMath.

After installing Sage, two separate packages are needed to run the code in the auxiliary files. The first is the surface_dynamics package, and the second package is flatsurf.

All .ipynb files are intended to be read in Jupyter.

Download all of the files below in a single zip file (Updated: November 9, 2018)

  1. code_from_the_article.ipynb, (html Version): This goes through all of the primary code occuring in the main text of the second paper. (Updated: November 9, 2018)
  2. Figures.ipynb, (html Version): This contains all of the code necessary to generate the figures from the second paper. (Updated: November 9, 2018)
  3. wordlist: This contains all of the coset representatives reduced under the unfolding-symmetry relation needed in Section 9.5 of the second paper. (Updated: November 9, 2018)
  4. r, t: These are the monodromy permutations in S2106 for the branched covering of the Teichmueller curve of the unfolded dodecahedron to the Teichmueller curve of the double pentagon. (Updated: November 9, 2018)
  5. 3D_Figures.ipynb: This generates the code needed to run the 3D images of the dodecahedra on this webpage from the second paper. (Updated: November 9, 2018)

The following versions of the software and packages listed above have been used in the production of these files.

Acknowledgements

J. S. A. was partially supported by NSF CAREER grant DMS 1559860.
D. A. was partially supported by NSF DMS - 1738381, DMS - 1600360 and PSC-CUNY grants 60571-00 48 and 61639-00 49.
W. P. H. was partially supported by NSF DMS 1500965 and PSC-CUNY grant 60708-00 48.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or PSC-CUNY.

The authors are grateful to Peter Coe for help with the student activities and to Derrick Koo for help with the website design.

Updated: November 9, 2018