Tilings, Hyperbolic Geometry and Computational Group Theory
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Overview
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 180^{o}/l,
180^{o}/m and 180^{o}/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 180^{o}/2 +180^{o}/3
+180^{o}/5 = 186^{o} .
Now
if the surface has genus higher than 2, say as in the picture 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:
180^{o}/l + 180^{o}/m + 180^{o}/n <
180^{o}.
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 page). Now it is more than a coincidence that there are
no pictures of hyperbolic surface in 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 to it will be one of the goals of the program
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.
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. This site includes
- a list of contributors,
- project descriptions and progress reports,
- technical reports, publications, background notes, and related papers
- a growing archive of images
- archives of Maple, Magma and Matlab Scripts for computation and creation
of images, and
- links to useful sites.
Project descriptions
Since 1996, twenty-seven 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:
- Problem Description: Determine all tilings on surfaces of
low genus.
- Contributors: Ryan Vinroot, Robert Dirks, Maria Sloughter,
John Gregoire, and Isabel Averill
- Progress: Classification of triangular and quadrilaterla tilings
in genus 2-13 is complete
- Reports and Papers
- Symmetry and Tiling Groups for Genus 4 and 5: C. Ryan
Vinroot, MSTR 98-02 (TRvinroot.pdf)
- Symmetry and Tiling Groups for Genus 4 and 5: C. Ryan Vinroot,
Rose-Hulman Institute of Technology Undergraduate Mathematics Journal,
1#1 (2000). (link
to journal, link
to paper)
- Quest for Tilings on Riemann Surfaces of Genus Six and Seven:
Robert Dirks and Maria Sloughter MSTR 00-08 (TRdirkssloughter.pdf)
- Triangular Tiling Surface Tiling Groups for Low Genus: C.
Ryan Vinroot, Robert Dirks, Maria Sloughter and S. Allen Broughton
(Tech Report MSTR 01-01 : tileclass.pdf,
summary tables: tileclasstables.pdf)
- Tilings of Low-Genus Surfaces by Quadrilaterals: John Gregoire
and Isabel Averill MSTR 02-13 (TRaverillgregoire.pdf)
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 (TRhaneymckeough.pdf)
- 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) (divquadplane.pdf).
- 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 Galiois 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
- The Galois Correspondence for Branched Covering Spaces and
Its Relationship to Hecke Algebras: Matt Ong, MSTR 02-08 (TRong.pdf)
- Pigeon-Holing Monodromy Groups: Niles G. Johnson MSTR 02-07
(TRjohnson.pdf).
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 (TRschmidt.pdf)
- Oval Lengths for Split Metacyclic Groups: Shaun McCance and
Sarah Weissman (in prepration)
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
- Lengths of geodesics on Klein's quartic curve: Ryan Derby-Talbot
MSTR 00-03 (TRderbytalbot.pdf)
- Lengths of systoles on tileable hyperbolic surfaces: Kevin
Woods MSTR 00-09 (TRwoods.pdf)
- Hyperbolic Billiard Paths, Rebecca Lehman, Chad White MSTR
02-02 (TRlehmanwhite.pdf)
Separability of Symmetries
- Problem Description: Determine when the mirror of a symmetry
separates the plane.
- Contributors: Jim Belk, Nick Baeth, Jason Deblois, Lisa Powell,
Robert Rhoades, Rachel Thomas, Steve Young.
- Progress: Some initial results. Many examples calculated,
complete results on cyclic groups and abelian groups .
- Reports and papers
- Tilings which split at a mirror: Jim Belk MSTR 99-02 (TRbelk.pdf)
- Separability of tilings: Nick Baeth, Jason Deblois, Lisa
Powell MSTR 00-10 (TRbaethdebloispowell.pdf)
- Applications of Graph Theory to Separability, Steve Young
MSTR 02-09 (TRyoung.pdf).
- When Abelian Groups Split, Rovert Rhoades and Rachel Thomas
MSTR 03-01 (TRrhoadesthomas.pdf).
Tilings and Cwatsets
- Problem Description: Determine when the mirror of a symmetry
separates the plane.
- 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.
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Send questions and comments to: allen.broughton@rose-hulman.edu
Last modified December 26, 2002 |