YOUNGS'S DISK


20cm × 20cm × 1cm
3D printed PLA and PETG
Bridges 2025 Art Exhibition

BLURB

"In 1963, J. W. T. Youngs discovered a symmetric, combinatorial embedding of a 'neighborly' map with 12 countries, where each country borders every other country, in the orientable surface of genus 6. In a proper coloring of such a map, every country must be given a unique color. Using a partially automated process, I found a geometric realization of this map that preserves the entire dihedral symmetry. I then 3D printed each color individually and assembled them into the final model."


ADDITIONAL COMMENTS
Using the method of current graphs, J. W. T. Youngs (see [7]) made an attempt at finding genus embeddings of the complete graphs \(K_{12s}\). He was only successful at solving the \(s = 1\) case, which was previously known to Heffter [1]:


Unlike Heffter's solution, Youngs's construction resulted in a combinatorial embedding with nontrivial symmetries: the theory of current graphs already naturally produces embeddings with cyclic symmetry, and Youngs's current graph has a further reflectional symmetry (though note that this is not a sufficient condition).

Ultimately, "Case 0" was first solved in a different manner by Terry, Welch, and Youngs [6] using nonabelian current graphs. Later, a complete solution using Youngs's ideas was announced by Pengelly and Jungerman [3], though they only gave details for \(s \equiv 0 \pmod{8}\), and much later, Korzhik [2] and Sun [4] filled in the remaining cases.

The above model was generated using the method developed in a paper I will be presenting at the same conference [5]. In short, I used a computer program to find a set of six non-separating cycles that respect the original symmetries of Youngs's combinatorial solution, and manually drew the rest.



REFERENCES
  1. L. Heffter, Über das problem der Nachbargebiete, Mathematische Annalen, 38(4):477-508, 1891.
  2. V. P. Korzhik, Exponentially many nonisomorphic orientable triangular embeddings of \(K_{12s}\), Discrete Mathematics, 308(7) 1046-1071, 2008.
  3. D. J. Pengelley, M. Jungerman, Index four orientable embeddings and case zero of the Heawood conjecture. Journal of Combinatorial Theory, Series B, 26(2) 131-144, 1979.
  4. T. Sun, A simple construction for orientable triangular embeddings of the complete graphs on \(12s\) vertices, Discrete Mathematics 342(4) 1147-1151, 2019.
  5. T. Sun, Visualizing and 3D Printing Colorful Maps on Surfaces, to appear inBridges Conference Proceedings, 2025.
  6. C. M. Terry, L. R. Welch, J. W. T. Youngs, The genus of \(K_{12s}\), Journal of Combinatorial Theory 2(1) 43-60, 1967.
  7. J. W. T. Youngs, The Heawood Map Coloring Conjecture, Graph Theory and Theoretical Physics, Academic Press, 313-354, 1967.