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D. Chavey Thesis

Periodic Tilings and Tilings by Regular Polygons

Ph. D. Thesis, Darrah Chavey, UW-Madison, 1984



We assume a tiling has, under its symmetry group, v orbits of vertices; e orbits of edges; and t orbits of tiles. Inequalities are established relating these parameters, both for arbitrary tilings and for tilings by regular polygons, and we show that some of these inequalities are sharp. In the case of tilings by regular polygons, we classify those tilings with v ≤ 3, e ≤ 3, or t ≤ 2, and show that the number of tilings with some fixed number of orbits of vertices [or edges; or tiles] is finite. The edge figures that can occur in a tiling by regular polygons are classified, as are tilings which contain at most three different types of these edge figures. Progress is made towards classifying those tilings by regular polygons which contain at most two different types of vertex figures. With respect to tilings by regular polygons which contain only two types of tiles (two congruence classes of polygons), the number of possible orbits of each polygon is determined. Tilings by regular polygons in which any two congruent tiles are equivalent under the symmetries of the tiling are classified, as are tilings that satisfy a similar condition on the edges.


Results and Downloads:

The full thesis is 207 pages and 27.5 Mg (this is a scanned pdf, not a text-based pdf). The index below breaks the thesis into smaller pieces, listing the results from each independent piece, and offers a download of these individual segments. You can download the entire thesis, or click an individual chapter name to download that chapter pdf file. At the bottom of this page are a list of papers we know of which have referenced this thesis.

Full thesis: Download the entire 207 page, 27.5 Mg file at once.

Extra Material:

This includes the title page, abstract, dedication & acknowledgements, tables of contents, figures, tables, and results, and references.


Chapter 1: Definitions and Basic Properties.

§1.1 Definitions:

Basic terms and definitions, such as you would find in Grünbaum & Shephard’s “Tilings and Patterns”, but with some other terminology used in this thesis (most of which can be inferred from context elsewhere).

§1.2 Research Survey:

A necessary section, but the work of Grünbaum & Shephard makes much of this unnecessary. We do, however, present a survey of research directly relevant to the work of this thesis. Includes a discussion of what I suggest is an error in an engraved figure in Kepler’s work on tilings by polygons, “Harmonice Mundi,” 1619. This also includes the list of 10 open questions that Grünbaum & Shephard list in their preliminary version of “Tilings and Patterns” that we solve in this thesis.

§1.3 Tilings Without Singular Points:

Many “facts” which one intuitively believes are true about tilings are false unless one adds the assumption that the tiling has no singular points – points where every open neighborhood of the point intersects infinitely many tiles. Grünbaum & Shephard point out that many authors have made such mistakes, but themselves make four such false claims in the preliminary version of their book. We give counter-examples to their claims, and then give formal proofs of the truth of those claims for tilings without singular points. After this, the principle result we show is:

Theorem 1.3: If T is a tiling, then the following are equivalent:

1.     T is v-isogonal (v orbits of vertices) for some finite value of v and has no vertices of infinite valence (degree);

2.     T is e-isotoxal (e orbits of edges) for some finite value of e;

3.     T is t-isohedral (t orbits of tiles) for some finite value of t;

4.     T is periodic and has no singular points.


Chapter 2: Properties of Periodic Tilings

A (v, e, t)-tiling has v orbits of vertices under the symmetry group of the tiling, e orbits of edges, and t orbits of tilings. We ask what possible values for these parameters can occur in periodic tilings (without singular points). We establish various inequalities that relate the possible values of these parameters. Specifically, we show that for any periodic tiling:

1.     v ≤ e + 1, and for any values where v ≤ e, there exists a tiling with those parameters for v & e;

2.     t ≤ e + 1, and if (e–1)/3 ≤ t ≤ e + 1, there exists a tiling with those parameters for t & e;

3.     If v ≤ t ,  there exists a tiling with those parameters for v & t.


Chapter 3: Tilings by Regular Polygons

§3.1 Classification of the Elements of a Tiling.

We classify the local configurations around an edge that can occur in edge-to-edge tilings by regular polygons, including determining the smallest value of v for which a v-isogonal tiling exists that contains that edge configuration. This is mostly useful in the classification results of chapter 5.

§3.2 Finiteness of Tilings with k Orbits of an Element.

Grünbaum & Shephard asked whether for a fixed value of v, could there be an infinite number of different v-isogonal tilings by regular polygons? What about e-isotoxal tilings or t-isohedral tilings? We show this is impossible by giving a bound on the number of such tilings. Unfortunately, this is a very loose (inaccurate) bound.

§3.3 Bounds on the Number of Orbits of an Element

We specialize the results of chapter 2 to the case of tilings by regular polygons, which allows us to give stronger bounds. Specifically:

    Theorem 3.6: Let T be an edge-to-edge (v, e, t)-tiling by regular polygons. Then:

            1.         v ≤ e                and      t ≤ e + 1;

            2.         e ≤ 5v              and      t ≤ 4v + 1;

            3.         e ≤ 6t – 2         and      v ≤ 6t – 2.

As a corollary, we can combine this result with Krötenheerdt’s classification of 2-isogonal tilings to classify the isotoxal and 2-isotoxal tilings (an open question from Grünbaum & Shephard). We also show a variety of ranges of these parameters for which an edge-to-edge tiling by regular polygons can always be found with those parameters.


Chapter 4: Local Regularity in Tilings by Regular Polygons

A tiling is referred to as k-gonal (respectively k-toxal, k-hedral) if there are k combinatorially different configurations of the tiling elements around a vertex (resp. edge, tile). This does not assume that there is any symmetry of the tiling that maps one configuration to an equivalent one; indeed the tiling may not have any symmetries at all. In general, even for tilings by regular polygons, it isn’t possible to satisfactorially classify the k-gonal, k-toxal, or k-hedral tilings if k > 1. We do, however, proof some partial results, specifically:

1.     We classify all k-toxal tilings (by regular polygons) with k ≤ 3.

2.     We give partial results on classifying 2-hedral tilings.

3.     For 2-gonal tilings, there are 16 possible combinations of pairs of vertex figures that can give rise to a 2-gonal tiling. We classify the 2-gonal tilings that exist for 10 of these 16 possible pairs.


Chapter 5: Global Regularity in Tilings by Regular Polygons

Here we analyze tilings whose symmetry groups result in a small number of orbits of either vertices, edges, or tiles. We also classify “homogenous” tilings. A “vertex-homogeneous” tiling is one where for any pair of combinatorially equivalent vertex figures there will always be a symmetry of the tiling that maps one such vertex to the other, and similarly for edge-homogeneous and tile-homogeneous.

§5.1: Vertex Regularity

We classify all vertex-homogenous tilings, and all 3-isogonal tilings by regular polygons. The vertex-homogeneous tilings were previously classified by Krötenheerdt; we provide

§5.2: Tile Regularity

We classify all tile-homogenous tilings, and all 2-isohedral tilings by regular polygons.

§5.3: Edge Regularity

We classify all 3-isotoxal tilings by regular polygons. We also classify all “strongly edge-homogenous” tilings by regular polygons. A tiling is strongly edge-homogenous if whenever two edges each form the intersection of the same type of polygons (e.g. two edges which both meet a hexagon and a triangle) there is a symmetry of the tiling which maps one edge to the other.


Chapter 6: Summary and Suggestions for Further Research

The traditional final chapter of any thesis. The summary includes my analysis of what the most important results and techniques were, and which results were just “interesting” or intended to fill certain gaps. It includes 20 suggestions for additional research, only a few of which have been answered (by other authors).



This work has been cited by the following papers (that I know of):

1.     Aslaksen, H., 2006, “In search of Demiregular Tilings,” Bridges London, Mathematical Connections in Art, Music and Science, Reza Sarhangi and John Sharp (eds.), Tarquin Publications, 533-536.

2.     Deza, M., M. Shtogrin, 2002, “Mosaics and their isometric embeddings,” Izvestia Math., 66(3), 443-462.

3.     Deza, M., M. Shtogrin, 2002, “Isometric embeddings of mosaics into cubic lattices,” Discrete Math., 244(1-3), 43-53.

4.     Deza, M., V Grishukhin, M. Shtogrin, 2004, Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices: Polytopes in Hypercubes & Zn, World Scientific Publishing Co., 188 pp., ISBN 978-1860944215.

5.     Roth, Johannes, 2008, “Restricted square-triangle tilings,” Z. Kristallogr. 223, 761-764.

6.     Sloane, Neal, 2003, “The On-Line Encyclopedia of Integer Sequences” (Sequence A068599).