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Background notes

The Background Notes entitled Kaleidoscopic Tilings on Surfaces  are a continually evolving set of notes that provide a background of tiling theory and  and hyperbolic geometry. They are used on an as needed basis. The first three chapters provide a quick introduction to tiling theory, and a description of some of the main REU research problems. The remaining chapters provide  various bit of background theory.

Student authored technical reports (by year in reverse chronological order)

Summer 2003

When Abelian Groups Split: Robert C. Rhoades and Rachel L. Thomas MSTR 03-1 (

Abstract: Let S be a hyperbolic surface tiled by kaleidoscopic triangles. Let Re denote the set of fixed points by the reflection in an edge, e, of a triangle. We say that Let Re is separating if SRe has two components. Once we have a tiling, we can define a group of orientation preserving transformations, G. We develop a method for determining when a reflection is separating using the group algebra of G. Using this method we give necessary and sufficient conditions for a mirror to be separating when G is abelian. We also conjecture, that when G is simple there are no separating mirrors.

Summer 2002

Tilings of Low-Genus Surfaces by Quadrilaterals: John Gregoire and Isabel Averill MSTR 02-13 (

Abstract: In contribution to the classification of all tilings of low-genus surfaces, the kaleidoscopic and non-kaleidoscopic tilings by quadrilaterals are given up to genus 12. As part of their classification, the algebraic structure of the conformal tiling groups and the geometric structure of the tiles are specified. In addition, several infinite classes of tilings and tiling groups are presented.

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).

Summer 2001

Pigeon-Holing Monodromy Groups: Niles G. Johnson MSTR 02-07 (

Abstract: A simple tiling on a sphere can be used to construct a tiling on a d-fold branched cover of the sphere. By lifting a so-called equatorial tiling on the sphere, the lifted tiling is locally kaleidoscopic, yielding an attractive tiling on the surface. This construction is via a correspondence between loops around vertices on the sphere and paths across tiles on the cover. The branched cover and lifted tiling give rise to an associated monodromy group in the symmetric group on d symbols. This monodromy group provides a beautiful connection between the cover and its base space. Our investigation of will focus on consideration of all possible low genus branched covers for a sphere, and therefore all locally kaleidoscopic tilings of low genus surfaces. It will be carried out through the classification of their associated monodromy groups. To this end, the relationship between classifications of branched covers and classifications of monodromy groups will be stressed.

The Galois Correspondence for Branched Covering Spaces and Its Relationship to Hecke Algebras: Matt Ong, MSTR 02-08 (

Abstract: There is a very beautiful correspondence between branched covers of the Riemann sphere S2 and subgroups of the fundamental group of S2 -{branch points}, exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F) . This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois group to the case of non-Galois
covers S/S2. Specifically, we show that the isomorphism type of a Hecke algebra C[H\G/H] is completely determined by the decomposition of the induced character 1HG, and that the character of the homology representation of a Galois group generalizes to one for Hecke algebras, the decomposition of which depends on certain double cosets in the group corresponding to the Galois closure of the cover S/S2.

Applications of Graph Theory to Separability: Steve Young MSTR 02-09 (

Abstract: Let S be a surface with a kaleidoscopic triangular tiling. Let R be a reflection in one of the sides of the tiles, ie an orientation reversing isometry, of the surface. Define M to be the set of fixed points of R. We then say that the surface S separates along the reflection R if S-R has two components. This paper considers the applications of graph theoretic methods to determining whether a reflection is separating or not and compares the algorithmic efficiency of these methods to the current known methods.

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

Summer 2000

Hyperbolic Billiard Paths: Rebecca Lehman, Chad White MSTR 02-02 (

Abstract: A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by length and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard spectrum for the (2, 3, 7) tiling.

Oval Lengths for Split Metacyclic Groups: Shaun McCance and Sarah Weissman (in preparation)

Summer 1999

Lengths of Systoles on Tileable Hyperbolic Surfaces: Kevin  Woods, MSTR 00-09 (

Abstract: The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. it is determined if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, it is determined if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles Pi/2, Pi/3, and Pi/7 whose tiling groups are all PSL(2,7). These tilings can be distinguished by the lengths of their systoles.

Separability of Tilings: Nick Baeth, Jason Deblois, Lisa Powell, MSTR 00-10 (

Abstract: A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconnects the surface into two components. We consider surfaces such that the orientation preserving subgroup of the tiling group generated by the reflections is cyclic or abelian. A complete classification of those surfaces with separating reflections is obtained in the cyclic case as well as partial results for abelian, non-cyclic groups.

Summer 1998

Lengths of Geodesics on Klein’s Quartic Curve:  Ryan  Derby-Talbot, MSTR 00-03 (

Abstract: A well-known and much studied Riemann surface is Klein’s quartic curve. This surface is interesting  since it is the smallest complex curve with maximal symmetry. In addition to this high degree of symmetry, Klein’s quartic curve can be tiled by triangles, giving rise to a tiling group generated by reflections. Using the tiling group and the universal cover of the tiling group we are able to compile a list of the lengths of the short, simple, closed geodesics on this surface. In particular, we are able to determine whether the geodesic loops generated by the tiling are the systoles, i.e., the shortest closed geodesics.

Quest for Tilings on Riemann Surfaces of Genus Six and Seven: Robert M. Dirks and Maria T. Sloughter,  MSTR 00-08  (

Abstract: The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, $\theta$, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7.

Summer 1997

Symmetry and Tiling Groups for Genus 4 and 5:  C. Ryan Vinroot, MSTR 98-02,  (

Abstract: All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of non-tiling symmetry groups is presented, as well as a class of odd order non-abelian tiling groups.

Quadrilaterals Subdivided by Triangles in the Hyperbolic Plane: Dawn M. Haney and Lori T. McKeough, MSTR 98-04  (

Abstract: In this paper, we consider triangle-quadrilateral pairs in the hyperbolic plane which “kaleidoscopically” tile the plane simultaneously. These tilings are called divisible tilings }or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condition, in which a single triangle occurs at each vertex of the quadrilateral. All possible such divisible tilings are catalogued as well as determining the minimal genus surface which the divisible tiling exists. The tiling groups of these surfaces are also determined.

Tilings which Split at a Mirror: Jim Belk, MSTR 99-01, (

Abstract: Let r be a reflection on a surface S. The mirror of r is the fixed point subset Sr ={x in S : r(x)=x} which is a disjoint union of circles . We say that S splits at the mirror of  r if  S-Sr is disconnected. We further assume that the reflection is a symmetry of a tiling of S by triangles. In this paper we investigate a number of conditions on the tiling that guarantee that  S splits at a mirror.

Summer 1996

Ovals Intersections in Tilings on Surfaces: Dennis A. Schmidt, MSTR 97-03  (

Abstract: A tiling is a covering by polygons, without gaps or overlapping, of a compact, orientable surface. We are particularly interested in tilings by triangles that generate a large symmetry group of the surface. An oval of the tiling is a simple, closed curve that is a union of edges of the tiling. We investigate the number of points of intersection of two ovals. We have found that the number of intersections is bounded when the subgroup of orientation preserving symmetries is abelian. However, there is no upper bound on the number of intersections in the non-abelian case.

Published papers and technical reports (latest first)

Triangular Surface Tiling Groups for Low Genus: S. Allen Broughton, Robert M. Dirks, Maria T. Sloughter, C. Ryan Vinroot MSTR 01-01 ( preprint technical report,

Abstract: Consider a surface, S, with a kaleidoscopic tiling by non-obtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G*, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups G* are also classified, up to isometric equivalence. The computer algebra system Magma is used extensively.

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)

Abstract: All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of symmetry groups that are not tiling groups is presented, as well as a class of odd order non-abelian tiling groups.

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)

Abstract: We consider triangle-quadrilateral pairs in the hyperbolic plane which “kaleidoscopically” tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1, 2, and 3 parameter families as well as a finite number of exceptional cases

Related papers and technical reports

Counting Ovals on a Symmetric Riemann Surface: S. Allen Broughton, MSTR 97-04  (

Abstract: Let S be a compact Riemann surface without boundary. A symmetry,  t,  of S is an anti-conformal, involutary automorphism. The fixed point set of  is a disjoint union of circles, each of which is called an oval of t. A method is presented for counting the ovals of a symmetry when S admits a large group G of automorphisms, normalized by t. The method involves only calculations in G, based on the geometric description of S/G, and the knowledge of the action of t on G.  As an illustrative example, the family of generic, symmetric surfaces of genus three with conformal automorphism group S4  is constructed as several 1-parameter families.

Constructing Kaleidoscopic Tiling Polygons in the Hyperbolic Plane, S. Allen Broughton, American Math. Monthly, October, 2000 ( 

Tiling the hyperbolic plane by iterated reflection in the sides of a kaleidoscopic polygon can be employed to make many beautiful and artistic patterns in the plane. These constructions may be greatly assisted by computer methods, provided we can construct the sides and vertices of these polygons, and hence the reflections in the sides of these polygons. We show how simple analytic geometry may be used to construct kaleidoscopic triangles and quadrilaterals. This work was motivated by work with undergraduates Dawn Haney, Lori McKeough, and Brandi Smith in the Rose-Hulman NSF-REU tilings project site. Indeed, the methods of the paper have been used for visualization in a classification project of all divisible tilings of the hyperbolic plane. A description of the project is available at the same site. More importantly, figures depicting the divisible tilings in the classification — using the methods of this paper.

Splitting tiled surfaces with abelian conformal tiling group, S. Allen Broughton, MSTR 99-03  (

Abstract: Let r be a reflection on a closed Riemann S i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sr denote the fixed point subset of r, which is also called the mirror of r.  If S-Sr has two components then r is called separating and that S splits at the mirror Sr.  Otherwise r is called non-separating. We assume that the system of mirrors, Sr, as r varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines a subgroup G, lying in Aut*(S),  consisting of conformal automorphisms of S.  We give a simple criterion, derived from the geometry of the tiling, for determining whether the reflection is separating by means of equations in the rational group algebra of G. Examples for abelian G, where the computations are especially simple, are presented.