Introduction to tilings | Site map | Project descriptions | NSF support

Introduction to tilings

Icosahedral tiling of sphere and (3,3,3) tiling of torus

A tiling of a Riemann surface is a covering by polygons, without gaps or overlaps, of a two-dimensional surface. The two tilings pictured above are the icosahedral (2,3,5)-tiling of the sphere and the (3,3,3)-tiling of the torus. (An (l,m,n)triangle has degree angles of 180o/l, 180o/m and 180o/n.) If the torus tiling is cut up and flattened out the tiling can be replicated to completely cover the plane with a regular pattern of equilateral triangles, i.e., (3,3,3)-triangles. Thus the tiling of the torus exhibits the characteristics of ordinary Euclidean geometry. Obviously, the icosahedral tiling exhibits spherical characteristics, indeed the angle sum of any of its triangles is

180o/2 +180o/3 +180o/5 = 186o .

Now if the surface has genus 2 or higher, say as in the surface at the left, then the geometry will be hyperbolic. What that means is that if we cut apart the surface, flatten it and try to join together replicates to tile a plane, we will end up with a tiling of the hyperbolic plane. Thus the edges of the tiling will follow the the curved lines of hyperbolic geometry and the angle sum of the geometry will satisfy:

180o/l + 180o/m + 180o/n  < 180o.

Here is an example of an unwrapped tiled surface giving a tiling of the hyperbolic plane by (4,3,3) triangles. For more examples go to the images and archives page. Now it is more than a coincidence that there are no pictures of tilings of hyperbolic surfaces on this web site. That is because it is quite difficult to draw them, and since it is impossible to have a geometrically true 3D realization of these surfaces (the same is true of the torus). Of course it would be wonderful to obtain reasonable renditions of many such surfaces. For the time being we need to content ourselves with the unwrapped versions of the tilings as tilings of the hyperbolic plane and some recipe for abstractly constructing such a surface and understanding its geometry. The icosahedral tiling yields the answer. The 120 triangles in the icosahedral tilings are all congruent to each other by means of a rotation or reflection of the sphere that preserves the tiling. The same is true of higher genus surfaces with highly symmetric tilings, i.e., there will be a “tiling group” of the surface that will move any tile congruently onto another. The groups can be almost any finite group, and therefore we use the methods of computational group theory, especially using computer algebra systems like Magma (Maple and Matlab are used for more geometric aspects). That now explains all the terms in the title.

Site map

The Rose-Hulman Tilings, Hyperbolic Geometry and Computational Group Theory website serves as a resource to the REU participants and others who have contributed, and as a dissemination site for their work, available to anyone who is interested. The site pages include

Project descriptions

Since 1996, thirty-one students have worked on tilings research. The following nine main projects have emerged, and there has been some progress on each one.

The contributing students and their technical reports are listed in the details below. (Note that MSTR refers to our Mathematical Sciences Technical Report series. Links to the pdf versions of these papers are supplied.

Low genus classification of tilings:

Divisible tilings

  • Problem Description: Determine all tilings by quadrilaterals which may be tiled by triangles
  • Contributors: Dawn M. Haney, Lori T. McKeough, Brandy M. Smith
  • Progress: Classification in the hyperbolic plane completed. Classification on surfaces under consideration
  • Reports and papers

    • Quadrilaterals Subdivided by Triangles in the Hyperbolic Plane: Dawn M. Haney and Lori T. McKeough, MSTR 98-04 (https://scholar.rose-hulman.edu/math_mstr/111)
    • Triangle Tilings of Quadrilaterals in the Hyperbolic Plane, Brandy M. Mayfield (Smith), (in preparation)
    • Divisible Tilings in the Hyperbolic Plane: Dawn M. Haney, Lori T. McKeough, Brandy M. Smith, Allen Broughton, MSTR 99-04 (preprint for paper submitted to New York Journal of Mathematics) (https://scholar.rose-hulman.edu/math_mstr/60/)
    • Divisible Tilings in the Hyperbolic Plane: Dawn M. Haney, Lori T. McKeough, Brandy M. Smith, Allen Broughton, New York Journal of Mathematics 6 (2000), 237-283. (link to journal, link to paper)

Fundamental domains for tilings

  • Problem Description: For a tiling on a surface, determine an “interesting”, fundamental domain , consisting of tiles.
  • Contributors: Yvonne Lai
  • Progress: Some algorithms constructed.
  • Reports and papers

    • Towards Finding Fundamental Domains for Hurwitz Groups: Yvonne Lai (in preparation).

Moduli of quadrilateral tilings

  • Problem Description: Investigate the moduli of tilings by quadrilateral and Galois, 4-point branched coverings of the sphere.
  • Contributors: Michael A. Burr and Katrhyn M. Zuhr
  • Progress: Discovery of and progress on vanishing cycles. Significant progess on classification of moduli space of 4-point branched coverings of the sphere.
  • Reports and papers
    • Descripton of the Limiting Surfaces of Hyperbolic Surfaces Tiled by Quadrilaterals: Michael A. Burr and Katrhyn M. Zuhr (in preparation).
    • Constructing the Moduli Space of Riemann Surfaces with a G -(k,l,m,n) Action: Katrhyn M. Zuhr (in preparation).

Non-Galois covers and Hecke algebras

  • Problem Description: Classification of low-degree non-Galois covers and their monodromy groups. Study of the allied Hecke algebra, a generalization of the Galois group.
  • Contributors: Niles G. Johnson and Matthew Ong.
  • Progress: Classification of all mondromy groups of low degree. Introductory study of the Hecke algebra of a surface.
  • Reports and papers

Oval intersection problems

  • Problem Description: Determine the patterns of intersections of ovals on tiled surfaces, relating them to the tiling group
  • Contributors: Dennis Schmidt, Shaun McCance, Sarah Weissman
  • Progress: Initial results and theorems. Some infinite families of examples. Some rather compete results on ovals for abelian and metacyclic groups
  • Reports and papers
  • Ovals Intersections in Tilings on Surfaces: Dennis A. Schmidt MSTR 97-03 (https://scholar.rose-hulman.edu/math_mstr/69/)
  • Oval Lengths for Split Metacyclic Groups: Shaun McCance and Sarah Weissman (in preparation)

Length spectrum of a surface and hyperbolic billiards

  • Problem Description: Determine the lengths of geodesics on a surface with symmetry relating them to the tiling group and geometric properties of the surface, in particular, hyperbolic billiards.
  • Contributors: Ryan Derby-Talbot, Kevin Woods, Rebecca Lehman, Chad White
  • Progress: Initial results for Klein’s quartic curve, additional results for genus 14 Hurwitz surfaces with PSL(2,13) automorphism group. Billiards of 50 bounces or less have been completely classified for the (2,3,7) tiling.
  • Reports and papers

Separability of symmetries

Tilings and cwatsets

  • Problem Description: Determine the relationship between tilings and Cwatsets (another Rose-hulman REU project).
  • Contributors: Patrick Swickard and Reva Schweitzer
  • Progress: Some initial results. Many examples calculated.

NSF support and disclaimer

This project is based upon work supported by the National Science Foundation under Grant No’s. 9619714 and 0097804. Per NSF grant policy please note the following: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.