Euclidean Geometry: Introduction to Tilings

Math 308

Fall 2021

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 structure is undergoing an adaptation following on a fully-online version of a closely related course; click here to see this course structure in action last term. This structure enables maximum flexibility as the term evolves around the ongoing Covid 19 pandemic. The course is divided into unites 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
  Virtual office hours Tuesdays 12:30 to 1:30, or by appointment. Meetings are scheduled in Canvas.
  Note that, in addition, I tend to spend ~20 minutes after every class answering student questions outside LSK.

• Where, when...
  The class meets Mondays, Wednesdays, and Fridays from 9:00 to 9:50 in LSK 460.

• 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 (revised) detailed formula from which your final grade will be calculated:
  Assignment 0 is worth 1% of your final grade; and then of the remaining assignments
  Your lowest assignment score is worth 4% of your final grade;
  Your second lowest assignment score is worth 15% of your final grade; and
  Your remaining four (4) assignments are each worth 20% 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.
  
  The decision to pick a required text is not one that I took lightly. It is a great book and at less than $50 I think that it is of great value. However, I recognize that this cost may be a hardship for some students and, to try and help mitigate this, I have put the library's copy on course reserve (you need to go to the IKB circulation desk and request it).

• Piazza
  Piazza is provided as a resource for you to interact with your classmates. This is a moderated forum; we have a TA for the course (Pamela Shah) who will answer questions. It is your responsibility to ensure that this is a respectful space that does not stray from the topic of the course. Note that if you have questions about assignment grades, Piazza is NOT the appropriate forum.

• 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. As I have said in class and to many of you in person: 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 my posted notes, for instance). I appreciate that there are very good reasons that in this particular term attendance might be lower. But when you can come to class, I encourage you to do so.

              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 17 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 24 by 11:59pm.
• Assignment 2 (linked here) is due on Friday October 8 by 11:59pm.
• Assignment 3 (linked here) is due on Wednesday October 27 by 11:59pm.
• Assignment 4 (linked here) is due on Tuesday November 9 by 11:59pm.
• Assignment 5 (linked here) is due on Friday November 26 by 11:59pm.
• Assignment 6 (linked here) is due on Tuesday December 7 by 11:59pm.

              Course Units

• Unit 1: Foundations
  Reading: Introduction, Sections 1.1 and 1.2
• Lecture 1 (September 8)
  Thinking geometrically
  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 different ways, and led us to think about tilings.
  Notes by Pamela Shah.
  
  Lecture 2 (September 10)
  What is a tiling?
  We sorted through the definition of a tiling proposed by Grünbaum and Shephard, and constructed some basic examples including tilings based on regular hexagons. In some cases, for constructing these tilings, it was useful to think about collections of tiles all at once; we called these patches.
  Notes by Pamela Shah.
  
  Lecture 3 (September 13)
  Contraints on tiles
  The tilings we have thought about so far have been relatively simple, given the tilings we see 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.
  Notes by Pamela Shah.
  
  Lecture 4 (September 15)
  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). Next time we will delve into congruences more deeply; today 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.
  Notes by Pamela Shah, which correct the typo in the scaling to λ=2 in the example.
  
  Lecture 5 (September 17)
  Linear transformations of the plane
  We recalled some transformations of the plane in the language of linear algebra: dilations, rotations, reflections, and shears.
  Notes by Pamela Shah.
  
  Lecture 6 (September 20)
  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.
  My notes.
  
  Lecture 7 (September 22)
  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.
  My notes.
• Assignment 1 (linked here) is due Friday September 24 by 11:59pm.

• Unit 2: Symmetries
  Reading: Sections 1.2 and 1.3
• Lecture 8 (September 24)
  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. In fact, we saw that a right angle triangle with unit base and height admits many different tilings, and that by deleting certain edges from these we arrive at some interesting dihedral tilings. Some of Escher's works are generated by planar monohedral tilings.
  My notes.
• Lecture 9 (September 27)
  Symmetries of tilings
  Today 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.
  My notes.
• Lecture 10 (September 29)
  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?
  My notes.
• Lecture 11 (October 1)
  Isohedral versus monohedral
  Today we discussed an answer to the problem 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, which we will look at in detail next time.
• Lecture 12 (October 4)
  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.
  My notes for lectures 11 and 12.
• Assignment 2 (linked here) is due Friday October 8 by 11:59pm.

• Unit 3: Tilings with few tiles
  Reading: Sections 2.1,2.2,2.3,2.4,2.5
• Lecture 13 (October 6)
  Regular polygons meeting at a vertex
  Today we considered constraints imposed at vertices in oder 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.
  My notes.
• Lecture 14 (October 8)
  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 {4,5,20}.
  My notes.
• Lecture 15 (October 13)
  Uniform tilings
  Grünbaum and Shephard list 11 possible edge-to-edge uniform tilings by regular polyhedra. This is the result of listing 17 possible vertex species, and the 21 resulting vertex types, before showing that some of these do not lead to planar tilings. The uniformity here corresponds to the fact that every vertex is of the same type, and that any two vertices are related by a symmetry of the tiling. This leads to an interesting generalization to k-uniform tilings, where k different vertex types arise.
  My notes.
• Lecture 16 (October 15)
  2-uniform tilings, I
  After carefully checking the uniformity of a particular Archimedian tiling, we considered an example from Grünbaum and Shephard of a particular pair of tilings both of which have the same two distinct vertex types. Using the symmetry group, we saw how to distinguish these tilings.
• Lecture 17 (October 18)
  2-uniform tilings, II
  Continuing with the example from last time, we carefully collected the vertices of a (purported) 2-uniform tiling into two equivalence classes, verifying that this this was indeed a 2-uniform tiling.
  My notes.
• Lecture 18 (October 20)
  Recap
  Before moving onto the next unit, we recapped the recurring players so far in the course. This focussed, particularly, on the symmetries of a tiling and equivalence classes of objects related by a symmetry.
  My notes.
• Assignment 3 (linked here) is due Wednesday October 27 by 11:59pm.

• Unit 4: Tilings by polygons
  Reading: Chapter 9
• Lecture 19 (October 22)
  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; campare their Figure 6.2.4.
  My notes.
• Lecture 20 (October 25)
  Normal tilings
  We looked at some restrictions on tiles guaranteeing that the tiling will be relatively well-behaved.
• Lecture 21 (October 27)
  Proper tilings
  A tiling by convex polygons is proper if any two tiles meet in a subset of a single edge of each tile. As a result, many tilings by tilings polygons constructed by deformation last week are not proper tilings. Here is a neat demonstration.
  My notes.
• Lecture 22 (October 29)
  Tiling types, introduction
  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.
  My notes.
• Lecture 23 (November 1)
  Vertices versus corners
  The majority of this lecture was an answer to a question from the start of class: What is the difference between a corner and a vertex? We worked though Figure 1.1.4 from Grünbaum and Shephard towards making the distinction precise. An important takeaway is that corners are parts of a tile, while vertices only make sense once tiles are placed into a tiling. Since I don't have notes from this discussion, this is a really good time to make sure you have a classmate you can get the notes from whenever you can't attend lecture.
• Lecture 24 (November 3)
  Homothety equivalence
  A homothety is a homeomorphism of the plane that preserves the symmetry groups of tiling. 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.
• Lecture 25 (November 5)
  Explicit homotheties
  Homotheties can be complicated in general, but constructing explicit examples is possible by appealing to tools from calculus (continuous functions!) or linear algebra (linear transformations!). We constructed and compared one of each.
  My notes.
• Lecture 26 (November 8)
  Tiling types, conclusion
  To sum up the unite, we collected all of the ideas from the last few days together and introduced the definition of polygonal isohedral type. This allows us to restrict attention to homothety classes of tilings by polygons in such a way that the homotheties preserve the number of edges in any given tile.
  My notes.
• Assignment 4 (linked here) is due Tuesday November 9 by 11:59pm.

• Unit 5: Aperiodic sets
  Reading: Chapter 10 (mainly sections 10.1, 10.2, and 10.3)
• Lecture 27 (November 15)
  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 os similarity tilings, which are tilings that compose to themselves. In settings where such a similarity is unique, in a sense that you will make precise in the homework, 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.
  My notes.
• Lecture 28 (November 17)
  Robinson 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. Next time, we'll see if this process can be extended to the entire plane.
  My notes.
• Lecture 29 (November 19)
  Robinson's set is aperiodic
  Today we extended our construction from last lecture by passing to larger and larger 2ⁿ-1 by 2ⁿ-1 patches, and made some observations about the coarse behaviour of these patches. Crucially, we observed that, even as these patches grow, they always contain a central cornerness cross. With this observation in hand, we are able to prove that no tiling of the plane by Robinson tiles is periodic.
  My notes.
• Lecture 30 (November 22)
  Pentaplexity
  Penrose's article Pentaplexity, which first appeared in Eureka in 1978, gives a first-had 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.
  My notes.
• Assignment 5 (linked here) is due Friday November 26 by 11:59pm.

• Unit 6: Penrose tilings
  Reading: Chapter 10 (mainly 10.3 and 10.4)
• Lecture 31 (November 24)
  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 say 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.
  My notes.
• Lecture 32 (November 26)
  Density
  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.) In the homework, you'll show that having a rational ratio (in a dihedral tiling) is a necessary condition for periodicity.
  My notes.
• Lecture 33 (November 29)
  Rhombs
  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.
  My notes.
• Lecture 34 (December 1)
  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 35 (December 3)
  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 36 (December 6)
  Pentagrids
  Still following the set of notes by David Austin closely.
• Assignment 6 (linked here) is due Tuesday December 7 by 11:59pm.