Euclidean Geometry: Introduction to Tilings

Math 308

Fall 2023

This is a math class about tiling the plane. It is a beautiful subject, and I am excited to learn about it with you. When I asked my friend Elissa Ross what she thought was the best starting point was for this material, she suggested the book – a classic! – Tiling & Patterns by Branko Grünbaum and G. C. Shephard. This book is full of great figures and examples that will complement our lectures. Sections of it will be required reading, and my lectures will endeavour to provide additional detail on the parts of this book that the course focusses on. As a result, this course is perhaps best thought of as "a guided reading of Tiling & Patterns."

To set an overall goal, we will set out to answer the question "what is a Penrose tiling?" An example is given in the figure on the right, which is borrowed from Non-local growth of Penrose tilings by Elissa Ross.

The course is divided into units and, in particular, your grade is based entirely on the homework assignments associated with each unit.

Assignment submission and grading will make use of Canvas. Here are some essential guidelines for assignment submission:

   1. Your solutions to assignments must be submitted using canvas. In particular, do not email assignments.

   2. Your solutions must be submitted as a single file.

   3. The only accepted file type is PDF.

   4. Files should be named according to the following convention:
        YourName_YourStudentNumber_AssignmentNumber.PDF

There are free online tools available that will allow you to complete (2) and (3). Due to the volume of data being managed this term, it is really important that these 4 points be met so that we can grade your work effectively. As a result, if one (or more) of these four points are not met, your assignment will not be accepted/graded.

To view the full content of this course, including lecture summaries and assignments, turn off reader-view.

• Instructor
  Liam Watson
  liam(at)math(dot)ubc(dot)ca

• Where, when...
  The class meets Tuesdays and Thursdays from 11:00 to 12:30 in BUCH B215.

• Consultation
  Mondays (with Pamela Shah): In LSK 300B from 3:00 to 4:00
  Tuesdays (immediately after class): Outside BUCH B215 from 12:15 to 12:30
  Wednesdays (with Liam Watson): In MATH 219 from 11:30 to 12:30
  Thursdays(immediately after class): Outside BUCH B215 from 12:15 to 12:30

• Evaluation
  Your final grade will be calculated based on your submitted homework assignments. Please note that no late assignments will be accepted. Here is a detailed rubric:
  
  Assignment 0 (the test assignment) is worth 1% of your final grade; and
  Assignment N (for N=1,...,6) is worth 16.5% of your final grade.
  
  Group work is strongly encouraged, but it is an absolute requirement that you submit your own work for grading. As you are aware, copying solutions from another student, from the web or from any other source, and turning them in as your own, is a violation of the Academic Code.

• Reference
  Tiling & Patterns by Branko Grünbaum and G. C. Shephard, available at the UBC bookstore.

• On rigour
  This course expects the same level of rigour as any other 300-level mathematics course. If in doubt, provide more details in your argument. An observation: perhaps the most difficult thing about our particular course is that your intuition about why something is true comes well in advance of a mathematical argument for why something is true. I will continue to emphasize the distinction in lectures. And I am always open to discussing course material, particularly assignment questions, during office hours.

• On attendance
  In my experience, there is a very strong correlation between performance in a course and attendance/participation in lectures. It will be very difficult to complete this course without attending the lectures (just relying solely on the book, for instance). While I appreciate that there are very good reasons that someone might have to miss a lecture, but when you can come to class, I encourage you to do so. And, if you have to miss a lecture, be sure to get notes from a classmate.

              Homework assignments

Please note that no late assignments will be accepted. Assignment submission and grading will make use of Canvas. Here are some essential guidelines for assignment submission:

   1. Your solutions to assignments must be submitted using canvas. In particular, do not email assignments.

   2. Your solutions must be submitted as a single file.

   3. The only accepted file type is PDF.

   4. Files should be named according to the following convention:
        YourName_YourStudentNumber_AssignmentNumber.PDF

There are free online tools available that will allow you to complete (2) and (3). Due to the volume of data being managed this term, it is really important that these 4 points be met so that we can grade your work effectively. As a result, if one (or more) of these four points are not met, your assignment will not be accepted/graded.

• Assignment 0
  
  This is an essential test, consisting of two pages, due on Friday September 15 2020 by 11:59pm.
  
  Instructions: page 1 should consist of your name and student number together with the assertion that you have read and understood the four guidelines above, and page two should include an example of a tiling. This can come from anywhere; find one you think is interesting! Please submit your work according to the instructions above.
• Assignment 1 (linked here) is due on Friday September 29 by 11:59pm.
• Assignment 2 (linked here) is due on Saturday October 14 by 11:59pm.
• Assignment 3 (linked here) is due on Friday October 27 by 11:59pm.
• Assignment 4 (linked here) is due on Friday November 10 by 11:59pm.
• Assignment 5 (linked here) is due on Friday November 24 by 11:59pm.
• Assignment 6 (linked here) is due on Thursday December 7 by 11:59pm.

              Course Units

• Unit 1: Foundations
  Reading: Introduction, Sections 1.1 and 1.2
• Lecture 1 (September 12)
  What is a tiling?
  We argued that the difference between (a+b)² and a²+b² could be interpreted geometrically, and in doing so we proved the Pythagorean theorem. This followed from decomposing a square of area (a+b)² in two apparently different ways, leading us to think about the definition of a tiling that will be used through the course. And, based on this definition, we proved that the two tilings proposed were indeed different.
  
  Lecture 2 (September 14)
  Contraints on tiles
  The tilings we first think of are relatively simple, given the tilings encountered in nature. Towards considering tilings at large, we need to have some restriction on what constitutes a valid tile. Grünbaum and Shephard give a constraint that tiles should be topological disks, and we worked though what that means.
  
  Lecture 3 (September 19)
  Equivalence of tilings
  Grünbaum and Shephard call two tilings congruent if one can be taken to the other by a congruence of the plane, and they call two tilings equal (and regard them as the same tiling) if one can be taken to the other by a similarity of the plane. A planar similarity is a congruence composed with some positive scaling factor λ, which may be 1 (that is, a congruence is a similarity, but a similarity is not a congruence). Before turning to the linear algebra needed to make this precise, we saw that scaling the hexagonal lattice by λ=2 gives rise to a self-similarity of the honeycomb tiling. As an aside, I noted that self-similarities of this form are in one-to-one correspondence with non-negative integer solutions (x,y) to the equation x²+xy+y²=n for various choice of natural number n. This is not an obvious fact, but if you would like to read more about it is discussed in Section 2.2 of the book Sphere packings, lattices, and groups.
  
  Lecture 4 (September 21)
  Affine transformations of the plane
  One issue with linear transformations is that they fix the origin. To deal with this, we introduced affine transformations. These are functions g for which differences may be measured by a linear transformation f, in the sense that g(X)-g(Y)=f(X-Y). As a result, we saw that a translation in the plane gives rise to an equality of tilings, as desired.
  
  Lecture 5 (September 26)
  Orthogonal Matrices
  To conclude the discussion linking material from Math 221 to tilings, today we consiered congruent tilings related by an affine transformation. This, in turn, is equivalent to asking that the linear part of the affine transformation be representable by an orthogonal matrix, and we checked that an affine transformation with orthogonal linear part did indeed preserve angles. I want to point out that, while we are not carrying out every detail of the proof in this class, you now have all of the tools in place needed to give a complete classification of planar congruences.
• Assignment 1 (linked here) is due Friday September 29 by 11:59pm.

• Unit 2: Symmetries
  Reading: Sections 1.2 and 1.3
• Lecture 6 (September 28)
  Monohedral tilings
  We looked at tilings that can be constructed from a single prototile, and say that a given prototile can admit many tilings – we used the material from earlier lectures to check. Some of Escher's works are generated by planar monohedral tilings. Then we thought about the symmetry group of a tile, before giving a definition of a symmetry of a tiling: This is a congruence of the plane taking tiles to tiles. A tiling is symmetric if it has a non-trivial symmetry; and a tiling of the plane is periodic if the symmetry group of the tiling contains at least two distinct translations. We saw that periodic implies symmetric, but the converse is not true; Figure 1.3.6 in Grünbaum and Shephard shoes some examples of this. For people interested more in group theory: I encourage you to consider taking Math 322.
  
  Lecture 7 (October 3)
  Isohedral tilings
  An isohedral tiling is one where every tile can be taken to another by a symmetry of the tiling. Isohedral tilings are monohedral, but the converse is definitely not the case. We ended with a question: Is every periodic monohedral tiling isohedral?
  
  Lecture 8 (October 5)
  Isohedral versus monohedral
  Today we discussed an answer to the question from last time, and I suggested a "proof" that every periodic monohedral tiling is in fact isohedral. The flaw in this argument leads to the idea of transitivity classes. In a monohedral tiling all tiles are congruent; in an isohedral tiling this congruence can be realized by an element of the tiling's symmetry group. The latter is a stronger condition, as we saw: Just because you can move one tile to another by a congruence of the plane does not mean that this congrucence will be part of the symmetry group of the tiling. This is where the notion of a k-isohedral tiling comes from, and we saw that every periodic monohedral tiling is k-isohedral for some k.
• Assignment 2 (linked here) is due Saturday October 14 by 11:59pm.

• Unit 3: Tilings with few tiles
  Reading: Sections 2.1,2.2,2.3,2.4,2.5
• Lecture 9 (October 10)
  Regular polygons meeting at a vertex
  Today we considered constraints imposed at vertices in order for regular polygons to arange around a fixed point without gaps or overlaps. This gives a constraint on which configurations of regular polygons have any hope of tiling the entire plane. We saw, in the monohedral setting, how equilateral triangles, squares, and regular hexagons fit into this framework. We also saw more interesting configurations, like the triple (3,7,42). For more on 42 showing up all over the place, see this link; to see some pictures of how this configuration fails to give a tiling look at this. Finally, we ended with a problem: Find a systematic means of enumerating all triples of integers (a,b,c) giving rise to so-called vertex species. Our first observation was that, ordering so that a is the smallest, the a-gon contributing to this triple has at most 6 sides.
  
  Lecture 10 (October 17)
  Counting triples
  Grünbaum and Shephard enumerate all possible tuples of regular polygons that might arrise as vertices in a tiling; they call these species. Today we looked at how this enumeration goes for triples. For example, we found that (4,5,20), (4,6,12) and (4,8,8) were all possible. Two of these three possibilities actually arise in (edge-to-edge) planar tilings by regular polygons. Here's a great picture for the triple (4,6,12).
  
  Lecture 11 (October 19)
  Uniform tilings: guest lecture by Pamela Shah
  
  Lecture 12 (October 24)
  2-uniform tilings: guest lecture by Pamela Shah
• Assignment 3 (linked here) is due Friday October 27 by 11:59pm.

• Unit 4: Tilings by polygons
  Reading: Chapter 9
• Lecture 13 (October 26)
  Symmetry and deformation
  We looked at how to deform a tiling continuously in such a way that a symmetry is broken and hence a new tiling is obtained. We started by creating new polygonal tilings from old, and placed these into 1-parameter families of tilings (for some real parameter between 0 and 1/2). This involved many choices, and by relaxing these we saw that a much greater diversity of variation was possible, reminiscent of our discussion of "topological disks" when restricting tiles. Ultimately, all of this looked closely at 3 of the 81 distinct types of isohedral tilings listed by Grünbaum and Shephard in Section 6.2; compare their Figure 6.2.4.
  
  Lecture 14 (October 31)
  Tiling types
  Our goal is to understand what the possible types of isohedral proper tilings by triangles can arise. We started by deforming the tiling (3⁶) in various ways.
  
  Lecture 15 (November 2)
  Homothety equivalence
  A homothety is a homeomorphism of the plane that preserves the symmetry groups of two tilings. This is an equivalence relations on tilings that is weaker than equality (discussed earlier in the course). The tiling types that we are in the process of understanding are homothety classes.
• Assignment 4 (linked here) is due Friday November 10 by 11:59pm.

• The remainder of the course focuses on constructing tilings that are not periodic and, in particular, the search for tile-sets that are aperiodic (so that no tiling realized by the set is periodic). This is the context for Penrose tilings and, amazingly, research published this year establishes the existence of a single aperiodic tile. You can read about the discovery here.

• Unit 5: Aperiodic sets
  Reading: Chapter 10 (mainly sections 10.1, 10.2, and 10.3)
• Lecture 16 (November 7)
  Similarity tilings
  We gave a definition of tiling composition, which made precise a process that has come up in various settings earlier in the course, namely, we created new tiles by considering patches of a given tiling. This gives rise to the idea of similarity tilings, which are tilings that compose to themselves. In settings where such a similarity is unique, we proved that the tiling cannot have a translation in its symmetry group. This is a strong statement, because it provides a means of constructing non-periodic tilings.
  
  Lecture 17 (November 9)
  Robinson aperiodic prototiles
  Starting from unit square tiles, we constructed a set of 6 prototiles, and checked that these could be fit together into 3 by 3 patches when appropriate choices are made. Then, by passing to larger and larger 2ⁿ-1 by 2ⁿ-1 patches, we made some observations about the coarse behaviour of these patches. Crucially, we observed that, even as these patches grow, they always contain a central cornerless cross. With this observation in hand, it becomes possible to prove that no tiling of the plane by Robinson tiles is periodic.
  
  Lecture 18 (November 16)
  Pentaplexity
  Penrose's article Pentaplexity, which first appeared in Eureka in 1978, gives a first-hand description of the discovery of an aperiodic tile-set. We stepped through this construction in order to deduce the six tiles appearing in Section 10.3 of Grünbaum and Shephard. This is also the starting point for the discovery of a smaller aperiodic tile set, which is the focus of the final unit in the course.
• Assignment 5 (linked here) is due Friday November 24 by 11:59pm.

• Unit 6: Penrose tilings
  Reading: Chapter 10 (mainly 10.3 and 10.4)
• Lecture 19 (November 21)
  Kites & Darts
  A tiling that one might find in a google search about Penrose tilings is dihedral consisting of tiles known as kites and darts. These are closely related to tilings that are admitted by the set of 6 tiles from last day, and we saw how to mark kites and darts so as to move between tilings of the two types. For now, this is our proof of existence of kite-dart tilings. One way to see that a kite-dart tiling is non-periodic is to observe that the kite-to-dart ratio that occurs is not rational. (In this case, amazingly, the golden number appears as ratio.)
  
  Lecture 20 (November 23)
  Rhombs: guest lecture by Pamela Shah
  The figure at the top of this page is a portion of a tiling by Penrose rhombs. We looked at the matching conditions on these rhombs and made some observations about similarity, where the golden number shows up once more as the inflation factor.
  
  Lecture 21 (November 28)
  Forbidden configurations
  Continuing on last day, we looked at compositions and saw that in a tiling by Penros rhombs we expect compatibility at all scales. Here is a great set of notes by David Austin.
  
  Lecture 22 (November 30)
  Up-down generation
  The first direct construction of a Rhomb tiling we'll consider is by up-down generation, which makes use of a series of compositions followed by a series of decompositions inorder to create a patch. The more we compose, the larger the patch, so this approach, due to de Bruijn, allows us to tile an arbitrarily large portion of the plane by Penrose rhombs. Here is a follow-up set of notes by David Austin.
  
  Lecture 23 (December 5)
  Pentagrids
  Still following the set of notes by David Austin closely.
• Assignment 6 (linked here) is due Thursday December 7 by 11:59pm.