Visualization of Regular Maps

A puzzle:
  • Take six squares, cut out from an elastic fabric;
  • Sew them together, to get a closed surface;
  • Stuff tightly, with polyester fibre;
  • Now, what shape do we get?

That's not difficult. But now try the same with 24 octagons, what do we get then? The resulting shape is an example of a regular map. This is a concept from surface topology: A graph embedded in a closed surface in 3D, such that all faces, vertices, and edges are topologically the same. Or, in other words, a tiling of a closed surface such that the symmetry is maximal. For instance, all faces have the same number of sides, and at all vertices the same number of edges meet.

During my sabattical in 2008/2009 I have worked on this puzzle. For a quick overview of the results, see the video:

Visualization of regular maps, .wmv, 56.4 MB or

For more details on regular maps and how to generate space models, see the article:

Jarke J. van Wijk, Symmetric Tiling of Closed Surfaces: Visualization of Regular Maps. ACM Transactions on Graphics vol. 28, no. 3, Article 49, 12 pages, August 2009. Proceedings ACM SIGGRAPH'09.


6 squares give a cube, blown up to a sphere

24 octagons give a shape like this.