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The Tiling Book
Communicated by Notices Associate Editor Emily Olson
The Tiling Book
An Introduction to the Mathematical Theory of Tilings
By Colin Adams. American Mathematical Society, 2022, 298 pp. https://bookstore.ams.org/mbk-142
Colin Adams’s book, The Tiling Book, is the most colorful math book I have ever seen. First of all, the cover attracts our eyes as it is covered with bright red, orange, and yellow lightning bolt shapes. Sooner or later I noticed it is a tiling of the plane!
Once I opened the book, I saw 310 figures and photos listed. Many of them were illustrated by Adams using Adobe Illustrator. Some of them are photos taken by Adams, including his private collection of vintage Kimberly Clark toilet paper, as seen in Figure 1. On the toilet paper a Penrose rhombic tiling is creased. The episode of the toilet paper (p.191) is interesting. Unfortunately, production of the toilet paper was stopped after Sir Roger Penrose sued the company.
Toilet paper with Penrose tiling.
Perhaps for a typical mathematician a tiling means the so-called regular tilings (as seen in Figure 2) and the usual subway tile. The book contains all kinds of exotic tilings beyond my imagination. I was fascinated by Casey Mann and his students’ counterintuitive tiling.
Tilings with regular polygons.
This book is fun to read. Readers do not need to be tile otaku or kitchen-bathroom designers. The book is accessible for general people who love math. As a mathematician studying low-dimensional topology including knots and braids, I had never studied tilings before.
While its target audience may be undergraduate and graduate students, I expect a motivated and interested high school student could read this book. Chapter 0 (12 pages with 8 figures) provides the background that is needed for the entire book and is a friendly introduction to point-set topology and group theory. I read this book seminar-style with five undergraduate students at the University of Iowa and one high school student from Iowa City. Some of them in fact had no previous experience in the material in Chapter 0, but we all enjoyed the book.
The book contains numerous projects suitable for undergraduates to do either independently or in a group. By reading proofs of theorems in the book, students will be excited to see how concepts recently learned in analysis or abstract algebra courses are actually applied.
For instance, in Section 2.6, the Bolzano–Weierstrauss theorem taught in an introductory analysis course, is used to prove the extension theorem. Each section also contains open problems for researchers and graduate students. I can imagine (partially) solving some of those would make an excellent PhD thesis. I want to emphasize that only small advanced training is required to understand the problems, as the background and motivation of the problems are included together with appropriate references.
The Appendix of the book can be useful for children and their teachers. In the Appendix, Adams teaches us how to create our own tilings. Now we have an idea how the bright tiling of the book cover was created. Inspired by his method, I recently taught 3rd and 4th graders a 40-minute origami tile-making course.
Adams uses colors all over the book not only for illustrations but also in narratives. It is a clever idea that he highlights definitions in green, lemmas and theorems in orange, open problems/questions in blue, and projects in red. Glancing at the orange highlights, readers can see that there are only one or two theorems per section, which enables them to focus on what is most important in the section. In addition, almost every definition and theorem comes with concrete examples and figures that I found very helpful.
1. Chapter 1
Chapter 1 starts with a detailed introduction of isometries of a plane and moves into symmetries. If a tiling admits translational symmetries all of which are parallel then its symmetry group is called a Frieze group. A periodic tiling is a tiling whose symmetry group possesses at least two nonparallel translational isometries. These symmetry groups are called the wallpaper groups. Adams presents the complete list of 7 Frieze groups and 17 wallpaper groups along with illustrations of tilings whose symmetry groups are those groups. See Figure 3 for examples.
Illustrations of one of the Frieze groups (left) and one of the wallpaper groups (right).
A convenient feature of Adams’s illustrations is that symmetry data are embedded in the tiling pictures. For example, a line of reflection is a solid red line and the center of a 3-fold rotation is a hollow triangle. In addition to the International Union of Crystallography notation, John Conway’s orbifold notation is provided for each of the 24 symmetry group types. The orbifold notation records the translation/rotation/reflection/glide reflection symmetry by a finite word in , and positive integers. For example, in Figure 3 is the orbifold notation for the symmetry group of the tiling. Conway came up a way to assign a certain rational number to each letter in the word and proved an amazing theorem: The orbifold notation symbols add up to exactly 2 if and only if the orbifold notation corresponds to an actual symmetry group of a tiling. In fact for each of the 24 tilings, you can enjoy staring at the image, identifying its orbifold notation and verifying that the sum is exactly 2.
2. Chapter 2
In Chapter 2, Adams shows various kinds of tilings by polygons. An edge-to-edge tiling by regular polygons is called a uniform tiling if all vertices have the same type. Using elementary computations, Adams shows that there are 11 such uniform tilings, which included the three regular tilings; see Figure 4. The numerical notation under each illustration describes the vertex type. For example, (4.6.12) means a 4-gon, a 6-gon and a 12-gon are meeting at a vertex with this cyclic order. The fact was originally published by Johannes Kepler in 1619.
The 11 uniform tilings, where the two instances of are equivalent via reflection.
The second kind is called Laves tilings, as seen in Figure 5; a monohedral tiling by a polygon (no need to be regular type) is called a Laves tiling if at every vertex angles between adjacent edges are equal ( for some ). Adams shows us there are 11 Laves tilings, which also includes the three regular tilings. Adams explains why it is not a coincidence that the number 11 shows up in both uniform tilings and Laves tilings.
The 11 Laves tilings, where the two instances of are equivalent via reflection.
After describing random tilings, a general study of periodic tilings is given. The periodicity theorem (2.14) states that if a set of polygonal prototiles (called a protoset) admits an edge-to-edge tiling with a translational symmetry then the same protoset admits a periodic tiling. This statement connects the Frieze groups and the wallpaper groups as follows. A polygonal protoset that generates a tiling with Frieze group symmetry must also generate a tiling with wallpaper group symmetry.
The chapter ends with discussion of the problem: Which convex polygons tile? The answer is given: Any convex polygon with seven or more sides cannot tile the plane. In the hexagon case, Karl Reinhardt showed that there are exactly three families of hexagons that can tile the plane, as we see in Figure 6. As for pentagons, there are exactly 15 families that can tile the plane. This result involves at least ten people including Karl Reinhardt, Richard Kerschner, Martin Garner, Richard James, Marjorie Rice, Rolf Stein, Casey Mann, Jennifer McLoud-Mann, David Von Derau, and Michäel Rao. As a corollary, every monohedral tiling by a convex polygon is a periodic tiling.
The three families of hexagons that can tile the plane.
3. Chapter 3
In Chapter 3, aperiodic protosets are studied. By definition, every tiling generated by an aperiodic protoset is non-translational.
The substitution method is a method to create non-translational tilings. Adams provides the chair reptile example (see Figure 7) and the pinwheel tiling example. It’s interesting that these examples somehow give me the wrong impression that they have translational symmetry. The Robinson aperiodic protoset, the Penrose aperiodic protosets, and the Taylor-Socolar hexagonal tile generate non-translational tilings.
The chair reptile example.
I was attracted to the two aperiodic protosets by Penrose discussed in Section 3.4. As seen in Figure 8, the first protoset consists of a kite and a dart where and . The second protoset consists of two rhombi. This is the protoset generating the pattern of the toilet paper mentioned earlier. It was a fine surprise to learn that these kite, dart, and two rhombi shapes can be decomposed into two types of triangles. Adams applies the substitution method to these triangles and proves aperiodicity of the two Penrose protosets. The fact also plays an essential role in the proof.
The kite and dart protoset and the two rhombi protoset.
4. Chapter 4
In Chapter 4, Adams discusses tilings of the sphere , the hyperbolic plane (Poincaré disk) , the Euclidean 3-space , and more general spaces. This chapter starts with an introduction to non-Euclidean geometries for beginners. Basic comparisons of the three geometries , , and are presented with detailed studies of triangles and computations of distances. Adams also explains isometries carefully. Classification of isometries of is nicely contrasted with that of . There are five types: rotation, translation, reflection, glide reflection, and parabolic isometry. The fifth one is unique because it does not have analogy from Euclidean or spherical geometries.
Recall (Lemma 1.12) that every isometry of is a composition of at most three reflections about lines. Similarly, every isometry of is a composition of at most four reflections about planes. A new type of isometry is called a screw motion, which is a composition of four reflections.
4.1. Hyperbolic tilings
Adams shows us five differences between tilings of and tilings of . For example, for integers , there is a regular tiling of by regular -gons such that tiles meet at a vertex if and only if . This was a nice surprise to me. A picture of tiling by -gons three of which meet at each vertex ( and , so ) can be seen in Figure 9, which looks like the surface of a sink full of bubbles. This chapter also contains the more familiar example of the tiling by -gons meeting four to a vertex ( and ), which is the universal covering space of a genus 2 hyperbolic surface.
A tiling of by -gons.
Two recent research results that highlight differences between Euclidean tilings and hyperbolic tilings are also introduced here. One is on prototiles for strongly aperiodic tilings and the other is on the realization of arbitrary high Heesch number.
4.2. Tiling of
Tilings of Euclidean 3-space has application to the study of crystals and quasicrystals. A simple way to construct a tiling of from one of is by thickening.
A tiling of that is non-face-to-face.
The cubical tiling is the only regular tiling of . A simple warm-up exercise is left to readers to show that the regular tetrahedron or the regular octahedron cannot tile . In Figure 10, Adams shows a non-face-to-face tiling of by a nonconvex prototile that looks like a precious rock in a science museum. The 3D illustration made by Adams is painted with 4 colors and helps us imagine the complicated structure.
The section ends with the Schmitt-Conway-Danzer aperiodic prototile of that doesn’t admit any -action. This shows a clear contrast between and . There is no such prototile found for .
4.3. Knotted tilings
So far, every prototile is topologically a disk or a ball. We can get rid of this restriction and consider a prototile topologically to be a genus handlebody. In this case, tiles possibly link together. Adams shows us six examples. He starts with a genus 1 prototile with no linking, as in Figure 11. The most impressive example that blew me away was a trefoil knot in the prototile, as seen in Figure 12.
A genus prototile with no linking.
The trefoil knot in the prototile.
4.4. 3-manifolds
The last section of the book provides a bridge to the study of covering spaces, which is an important topic taught in algebraic topology courses, from the viewpoint of tilings.
Starting with dimension 2, Adams shows us how to obtain infinitely many distinct tori from parallelogram tilings of the plane . We say that covers a torus. Depending on the corner angles and the edge length ratio of the parallelogram, the resulting torus has different metrics. A rectangular tiling also can yield a Klein bottle.
Increasing the dimension from 2 to 3, Adams discusses compact orientable 3-manifolds that result from tilings of Euclidean -space . Six compact -manifolds are explicitly defined by identifying faces of cubes and hexagonal prisms. Adams says: “Surprisingly, there are a total of only six such -manifolds, including the -torus. If the universe if compact and orientable …and Euclidean, then our universe must be one of these six possibilities.” At the very end we see compact orientable -manifolds including the Poincaré manifold and the Seifert-Weber manifold that come from tilings of the 3-sphere and the hyperbolic 3-space , respectively.
5. Conclusion
In his book, Adams provides many fantastic hiking paths suitable for a wide range of readers. The paths are all walkable, and there is no need to bring ropes or carabiners. The hiking paths contain many scenic points where we can take rest and see interesting tilings. Occasionally along the path, the dimension leaps from 2 to 3. A hiker might notice that the metric system changes from Euclidean to hyperbolic. The paths are open-ended with the possibility that a hiker constructs a brand new path to a new scenic point.
Keiko Kawamuro is a professor of mathematics at the University of Iowa. Her email address is keiko-kawamuro@uiowa.edu.
Article DOI: 10.1090/noti2771
Credits
Figures 1–7 and Figure 8 (left) are courtesy of Colin Adams.
Figure 8 (right) is from Branko Grunbaum and G.C. Shephard, Tilings and Patterns, Second edition, Dover Publications, Inc., 2016. Reprinted with permission.
Figure 9 is by Tomruen. Licensed under CC-BY-SA 3.0.
Figure 10 was created by John Petrucci.
Figures 11 and 12 are courtesy of Colin Adams.
Author photo is courtesy of Kazumi Noguchi.