# TEACHING SUGGESTIONS

The chapter-by-chapter suggestions below pertain mostly to teaching a 1-semester course for not-necessarily-math majors.

CHAPTER 1: Strongly emphasize to students the need to stay on top of the vocabulary (perhaps using flashcards). I recommend structuring the lectures around the Ch. 1 PowerPoint presentation. Visual aids are very helpful: cardboard polygons and stars, a plexiglass "plane" (on which you can draw with a dry erase marker), a wooden dowel (for a pointer and a reflection line), a picture of a clock and right hand copied into an overhead transparency sheet, and an actual wall clock modified so you can see through the back to observe the second-hand turning counterclockwise.

One of the PPT slides has a question about the number of rotations and flips and the minimum rotation angle of a regular n-gon. Take time to fully discuss and answer this, since the answer will be crucial later. The explanation of why there are n reflection lines depends on whether n is even or odd. Another PPT slide asks whether an object with a translation symmetry must be unbounded. Take time to discuss and prove the answer. It is a good context for the students to see their first proof (and in the follow-up question, to discuss the contrapositive).

PowerPoint can help students decide whether an object is oriented. Switch from presentation mode to editing mode, double-click an object, duplicate the object via copy-and-paste, and then choose "Flip Horizontally" to flip the duplicate. Does the flipped copy look different than the original?

This chapter is a good place to emphasize the importance of precise language. In the PowerPoint, most key vocab words are first introduced informally via examples before they are given precise definitions. For example, the students see a few wallpaper patterns and hear some vague words about "a repeating pattern stretching indefinitely left, right, up and down." At this point, they will believe that they know what a wallpaper pattern is, and you need to convince them that there's a need for a precise definition. This science analogy is good: if I tell you what a mammal is by showing you a few examples, then you will vaguely know what a mammal is, but you won't be able to decide whether a platypus is one.

CHAPTER 2: I recommend structuring the lectures around the Ch. 2 PowerPoint presentation. Take time to have students construct the Cayley table for a square. To make this go smoothly, make a copy of this handout for each student (printed 2-sided in color with the bottom of the first page cropped to make it a square). Pause during the PPT lectures to discuss the proofs of one or two of the key theorems mentioned (The Sudoku Theorem, the All-or-Half Theorem, the Rigid-Motion-Detector Theorem).

Avoid over-emphasizing the abstract. Students will never be asked to prove that an arbitrary group has a particular property. Avoid proving things that belong in an abstract algebra class, even simple things like the uniqueness of inverses. It is enough if students understand the goal: to formalize what addition/multiplication of numbers has in common with compositions of symmetries (and later with composition of permutations). Keep the focus not on general groups but on symmetry groups, and the beautiful things we learn about symmetry with this new point of view. For example, students might have conjectured the All-or-Half Theorem in Chapter 1, but one needs the group viewpoint of Chapter 2 in order to prove it.

CHAPTER 3: I recommend structuring the lectures around the Ch.3 PowerPoint presentation. Motivate this chapter by peeking ahead to the next chapter, in which we will prove that any object is "symmetric in the same way" as one of a list of model objects. For this, we'll need to decide precisely what "symmetric in the same way" should mean.

Avoid over-emphasizing the abstract; in particular, avoid function notation. For a class of humanities majors, I recommend NOT writing things like "f(x)*f(y) = f(x*y)".

CHAPTER 4: I recommend structuring the lectures around the Ch.4 PowerPoint presentation. For border patterns, I devote some class time to this handout. Give each student a second copy of the handout printed on transparency (for performing rotations and flips). The handout could lead to a discussion of parts of the proof of the border pattern classification theorem. I do not spend much time discussing wallpaper patterns.

Although I've never tried this in class, there are several online Java applets for creating your own border and wallpaper patterns. You draw in a fundamental domain, and the applet does the rest. To locate such programs, a good starting point is the Wikipedia article on "Frieze Groups".

CHAPTER 5: I recommend structuring the lectures around the Ch. 5 PowerPoint presentation. For generated subgroups of cyclic groups, have 10 students stand in a circle passing a ball by 1s and by 2s and so on. Then I add an 11th student and then a 12th. I use this activity simply to help them better understand generated subgroups; I do NOT mention more complicated patterns (the order of each member divides the order of the group, and <a> = <gcd(a,n)> in C_n) unless the students bring it up, which they sometimes do.

One way to minimize abstraction is to SKIP the section about product groups here, in which case you must also later skip the "Proper versus Full Symmetry Groups" section of Chapter 7.

CHAPTER 6: I recommend structuring the lectures around the Ch. 6 PowerPoint presentation. A set of refrigerator magnet letters is an indispensable visual aid. Each student should receive a set of letters to permute. If you don't have enough magnet letters, just give each student one row of this letter handout (let them separate the 6 letters in the row themselves). I devote class time for them them to fill in this Cayley table (using their cut-out letters A,B,C).

Students can self-discover the concept of even/odd. Using their cut-out letters A-F, ask them to generate EADCFB with swaps, and report the number of swaps required. They will notice that all reported numbers are even. Follow up with CADEFB, which is odd.

I recommend discussing Exercises #7-9, which will help them more precisely understand parts of the next chapter.

CHAPTER 7: I recommend structuring the lecture around the Ch. 7 PowerPoint presentation. Visual aids are crucial. For the tetrahedron, cube and dodecahedron, I recommend building both a solid cardboard model AND a hollow edge model of each. Zome Tools are great for building hollow edge models, through which it is easy to stick a dowel to represent a rotation axis.

The RadioLab podcast Desperately Seeking Symmetry has an interesting discussion of chirality. Listening to it might give you ideas for supplementary material and detours you could use in class.

Origami can enrich this chapter in many ways. Search for "Sonobe origami" on YouTube. Learn by video how to fold the sonobe unit, which you can easily teach to your class. Assemble 6 or 12 or 30 sonobe units together into a beautiful polyhedron. A single sonobe unit is chiral, so the most common mistake is to accidentally fold some left and some right hand versions, which will not compatibly assemble together. This can lead to a fruitful discussion of chirality.

Bring lots of solid objects to class, and ask where each object fits in the classification theorem. For example, a Switch Pitch is a toy (made by Hoberman) that is rigidly equivalent to a chiral tetrahedron. You can re-use this toy in Chapter 8 because it "flips" into its dual, which is a nice demostration of the self-duality of the tetrahedron.

CHAPTER 8: I recommend structuring the lectures around the Ch. 8 Powerpoint presentation. Physical models of the Platonic solids are crucial for yourself and if possible for all of the students. Cardboard polygons and tape can be used to "classify" the Platonic solids, but more than 2 hands are needed to do this gracefully in front of a class. To solve this problem, I recommend Magformers instead of cardboard and tape (plastic polygons which magnetically stick edge-to-edge, and also magnetically stick to most white boards). It is quick and easy to snap together a Magformer dodecahedron or icosahedron.

Most classrooms are shaped roughly like a cube. It is entertaining to build the room's dual with streamers and tape. Or to save time, you could simply mark the dual's vertices on the classroom walls, and simply ask students to imagine the dual's edges.

The toy company Hoberman makes toys that nicely illustrate the three dualities: "Switch Pitch" for the tetrhaderon's self-duality, "Flip Out" for the cube-octahedron duality, and "Super Flip Out" for the dodecahedron-icosahedron duality.

Euler's formula for the sphere can be introduced by passing out balloons to all of the students. White board markers work well for drawing graphs on balloons. Also draw a graph on (a small portion of) an inflatable inner tube, and cut out its faces with scissors to help them see why one face is not a deformed polygon.

I do not cover in class the algebraic proof that there are 5 Platonic solids because, as mentioned in the text, this proof suffers from a disguised version of the same shortcoming that is found in the cardboard-and-tape proof.

CHAPTER 9: I recommend structuring the lectures around the Ch. 9 Powerpoint presentation. I bring to class a mop bucket full of soap solution and Zome Tools polyhedra to dip.

This chapter is beautiful but difficult to assess; unlike the other chapters, its abstract concepts and proofs are not grounded by any rote skills. I find that it works well to cover the chapter quickly, without assigning homework or holding them fully responsible for the material.

CHAPTER 10: I recommend structuring the lectures around the Ch. 10 Powerpoint presentation, but pause to work out computational problems (like converting between decimal expressions and fractions) at the board. Students are usually familiar with some of this material, but they have not seen it framed with the goal of precisely defining real numbers, which presents some subtlety. For example, in the rigorous development, one never proves that 12.750000 = 12.74999999. Instead, one builds this redundancy into the definition of "real number" and then carefully defines +,-,X,/ so as to make sure these operations are well-defined.

It is good to assign Exercise #17 if you intend to cover the proof of Cantor's Theorem in the next chapter.

CHAPTER 11: This chapter is an optional excursion away from symmetry and into the deep and beautiful topic of infinity. If time permits you to cover this chapter, the students will find it fascinating and challenging. I recommend structuring the lectures around the Ch. 11 Powerpoint presentation.

CHAPTER 12: I recommend structuring the lecture around the Ch. 12 Powerpoint presentation.

The proof of the Pythagorean Theorem can by done on a felt board (a cardboard square covered with felt, together with 4 felt triangles). Show the "a^2+b^2" tiling and then ask the the students to find a new tiling with a c-by-c square left uncovered.

The dot product "match making" game is fun to play in class, and surprisingly valuable.

CHAPTER 13: I have never had time to cover this chapter.