That is not quite true, however. While it is true that if we were to color the sides of the trihexaflexagon, we would need three colors, each colored side can appear on the front or on the back of the flexagon. And when the face moves from front to back, the arrangement of the rhombus-shaped pieces change orientation relative to each other. So in a sense, the trihexaflexagon has six faces.

I took advantage of this property to make the following Halloween trihexaflexagon. You can download a printable pdf.

Here are the six faces.

Here is a video of the flexagon in action.

Instructions for making the FLEX-A-GHOUL:

- Print the template and cut out the flexagon pattern.
- Color the template (better now so that the ink—if you use markers—doesn’t bleed through to the other side).
- Fold in half lengthwise and glue or tape back-to-back.
- Although the pattern shows rhombuses, really you should think about it as a string of 10 equilateral triangles. Fold and unfold along the edges of each triangle so they are creased.
- Fold the “TRICK” rhombus in half along its diagonal, and do the same with the “TREAT” and “OR” rhombuses.
- Finally, the two end triangles are both blank on one side. Glue or tape these blank sides together.

Akiva Weinberger (@akivaw) tweeted back to me saying that this is the same number of “preorders” on a set with *n *elements. I admitted that I’d never heard of a preorder. Then he and Joel David Hamkins (@jdhamkins) filled me in on what a preordered set is.

It found it to be pretty cool—especially the connection to topologies of finite sets. So I thought I’d share it here on my blog.

**TOTAL ORDERS, PARTIAL ORDERS, AND PREORDERS**

We are all familiar with sets that are *totally ordered* (like the integers or the real numbers). A set *S* is totally ordered with a relation ≤ if it has the following properties for all elements in *S*.

*Reflexivity*:*Transitivity*: if and then*Antisymmetry*: if and then*Comparability*: either or (or both).

A set *S *with and relation ≤ that satisfies properties 1–3 (it is reflexive, transitive, and antisymmetric), but not necessarily property 4, is called a *partially ordered set*. For instance, we can put a partial order ≤ on the set of positive integers as follows. We’ll say that provided *a *divides *b. *Using this ordering, we have and because both 2 and 3 divide 6. However, not all positive integers are comparable. For instance, and because 5 doesn’t divide 6 and 6 does not divide 5. So this relation gives a partial order and not a total order of the positive integers.

Now we are ready to talk about preordered sets. A relation ≤ gives a *preorder* on a set *S * provided it satisfies properties 1 and 2 (it is reflexive and transitive) but not necessarily properties 3 and 4. If we extend the divisibility relation to the full set of integers, we get a preorder that is not a partial order. The ordering is reflexive and transitive, but it is not antisymmetric. For example, because –2 divides 2, and since 2 divides –2, However,

One way to think about preorders is via *directed graphs* (mathematical graphs in which edges have a direction). Define a relation ≤ on the vertices of the graph as follows. We say that provided *a *is *reachable *from *b*; that is, we can get from *b* to *a* along a path of directed edges (including the trivial path of not going anywhere).

For instance, the graph below has vertex set We see that because we can get from vertex *a *to vertex *e*. It is not too difficult to see that the relation is reflexive and transitive. However, notice that we can’t get from vertex *b *to vertex *e *nor from *e *to *b*, so they are incomparable (that is, property 4 fails). But also we have and but so the relation is not antisymmetric (property 3 fails). Thus, the relation is a preorder.

**TOPOLOGIES OF FINITE SETS**

Let’s see how preorders are related to the topologies of finite sets. First, let’s give the definition of a topology.

Let *X* be a set. A set of subsets of *X*, *T*, is a *topology* for *X* provided.

- The intersection of any finite number of sets in
*T*is in*T*. - The union of any collection (finite or infinite) of sets in
*T*is in*T*.

The sets in *T *are called *open sets.*

So, for instance, the three-point set has nine different topologies. (By “different” we mean that any other topology can be obtained from one of these just by renaming the letters *a, b, *and* c.* Or, using more technical terminology, any other topology is homeomorphic to one of these.)

Given a topology *T* for a finite set *X* we can obtain a preorder on *X *as follows. We say that provided *x *belongs to every open set that contains *y*.

So, the topology on the set gives a preorder in which we can only say and Similarly, the topology on the set gives a preorder in which we can only say and The following graphs give the same preorders.

Notice that with respect to topology the so-called *trivial topology*, we have all six relations and And with respect to the so-called *discrete topology*, none of the elements are comparable.

We can go the other way as well. Given a preorder of a finite a set *X*, we can produce a topology on the set *X* as follows. A set is an open set provided there is no element in *X *that is less than any element in *U.* For instance, suppose we had the preorder corresponding to the graph with vertices at the beginning of the blog post. Then a set of vertices is an open set if there are no directed edges that point out of the set. In particular, the preorder corresponding to the graph gives topology

Pretty cool!

]]>After a little exploration, I discovered that PGFPlots was the LaTeX package I was looking for. It makes drawing graphs in TikZ easy, and the graphs are highly customizable. So, for the last two days, I dove in and started playing with it. I’ve included some of my creations below.

If you would like to see the LaTeX code for these figures, you can open this Overleaf link. Feel free to copy and modify them. Since I’m a beginner, I can’t promise that I’ve created them the best and most efficient way.

There is one thing I should mention. In order to generate the contour plots (the last two figures at the bottom of this post), I had to install gnuplot on my computer. (This was a little bit involved.) If you are using Overleaf, you don’t have to do this, but if you are using a desktop LaTeX program, you will probably have to.

As a last comment: It may take a little while for the document to compile in Overleaf. Each figure has to be regenerated each time the document compiles. There is a way to keep this from happening—essentially it regenerates the figure only when there is a change to the code for that figure. You can read more about that approach on this page.

So I decided to make a cheat sheet that contains all of the essential information on trigonometry that he will need for his calculus class—reference triangles, the unit circle, some trigonometric identities, and so forth. (I also knew that I could give it to my students the next time I taught Calculus.) I wanted it to be compact. I didn’t want to go overboard with the trig identities, for instance. I was trying to think about which ones we use again and again.

Here’s what I came up with—one sheet, front and back. Feel free to download this pdf for yourself or to share with your students. If I left anything off, let me know in the comments. (I will admit that I did not include anything about inverse trig functions. Maybe that’s for version 2.0.)

Here’s my list with some other additions. I’ve linked to biographies of the individuals and linked to the reasons why I chose the number that I did.

- Archimedes: 3141
- Srinivasa Ramanujan: 1729
- Fibonacci: 1123
- Euclid: 2357
- Brahmagupta: 0000
- Giuseppe Peano: 0123
- Hippasus of Matapontum: 1414
- Leonhard Euler: 2718
- Gottfried Leibniz: 7853
- Isaac Newton: 6674
- Carl Gauss: 5050
- Luca Pacioli: 1618
- Eugène Charles Catalan: 1125
- Édouard Lucas: 2134
- Marin Mersenne: 3731
- Pierre de Fermat: 3517
- Alexander Grothendieck: 0057
- Blaise Pascal: 1331
- Mitchell Feigenbaum: 4669
- Zeno of Alea: 1248
- John Napier: 6931
- David Singmaster: 3003
- D. R. Kaprekar: 6174
- Michael Mästlin: 6180
- Lorenzo Mascheroni: 5772
- Alexander Gelfond: 2314
- D. G. Champernowne: 1234
- Wacław Sierpiński: 1585
- John H. Conway: 1303
- François Viète: 6366

As we see below, if we glue only the left and right sides together, we get either a *cylinder* or a *Möbius band*. (The arrows indicate which sides are glued together and with which orientations—we want the arrows to align when gluing.) If we glue pairs of neighboring sides together, we obtain a shape topologically equivalent to a sphere. If we glue opposite sides together without a twist, we get a donut-shaped surface called a *torus*. (Here’s a nice video showing this procedure.) The last two are trickier. In one, we glue two sides together without a twist as with a cylinder but we glue the other two sides together with a Möbius band-like twist. This shape cannot be made in 3-dimensional space regardless of how stretchy our rubber is. But if we let the surface pass through itself (or briefly “hop over” itself in the 4th dimension), we get a shape called a *Klein bottle*. (Here’s a video showing the procedure and here’s a website where you can purchase a glass Klein bottle). The last shape is by far the trickiest of them. In this case, both pairs of sides are glued together with Möbius band-like twists. The shape is called the *real projective plane*.

Like the Klein bottle, the projective plane can’t be created in 3-dimensional space. But whereas it is not too difficult to visualize the Klein bottle, the projective plane is much trickier to picture.

There are a number of equivalent ways of constructing the projective plane. We are viewing it as taking a topological disk (we could make our square round) and gluing together antipodal points—points opposite from each other on the boundary of the disk. (Another way is to take a Möbius band, which has one boundary curve, and glue a disk on to the boundary.)

Here is a sequence of steps I drew for my students.

There are other ways to visualize the real projective plane. One is called *Boy’s surface *(it was discovered by David Hilbert’s student Werner Boy). Notice that it has self-intersections (in 3-dimensions) and one triple point—a point where three parts of the surface meet.

Here’s a video showing the construction of Boy’s surface. As beautiful as the video is, I still have a hard time visualizing what’s going on. In November 2007 Rob Kirby published this short item in the *Notices of the American Mathematical Society* in which he presented another way to visualize Boy’s surface. He presented Boy’s surface as a shape that can be made from three squares and four equilateral triangles. When I saw that, I knew I had to make it out of paper.

The first thing to observe in Kirby’s construction is that the three squares cross each other along their diagonals meeting at right angles. For instance, we could take the vertices of the squares to be the six points (±1,0,0), (0,±1,0), and (0,0,±1). They are the vertices of a regular octahedron. (Keep in mind that the three squares don’t actually meet along their diagonals—this is where they pass by each other in 4-dimensional space.)

I realized that I could cut slits in my paper squares so that they would fit together in this pattern.

The final step in the process is to tape the four equilateral triangles to this octahedral shape. There are eight openings in the octahedron and four paper triangles. Tape the triangles to every other opening. Said another way, no two triangles should meet edge-to-edge. They should meet only at their vertices. This is Boy’s surface.

I made the following printable template (pdf) so you can make your own. [Update: the template now has tabs so you can glue the shape together if you want to.] It is a little tricky to get all of the pieces to fit together. The first step is to cut it out by cutting all of the solid lines. Then fold along the dashed lines (as mountain folds or valley folds as described on the template). The next step is to assemble the three squares as shown above. I recommend slotting together the 126 square with the central square first. Then do the 345 square second. Finally, tape the triangles. It should be obvious how to do this if you get to this point, but the numbers on the figure indicate what is glued to what (1 to 1, 2 to 2, and so on).

One question you may ask is: How can we see that this truly is the real projective plane? That is why I numbered the sides. First, notice that we could glue the two 5s together and the two 6s together. They are neighbors, so we are left with an eight-sided polygon with sides numbered 12341234. The key observation is that this numbering implies that we are going to glue opposite sides together on this polygon. In particular, we are gluing each boundary point to its “antipodal” point. That is exactly what we would do to generate the projective plane!

[Update: This shape happened to be on the cover of the November 2007 issue of the *Notices. *In the “About the Cover” blur, Bill Casselman mentions that the smoothed version of this shape is the Roman surface, another famous way to represent the real projective plane.]

This semester I am teaching topology, one of my favorite classes to teach. The students are second-semester seniors or junior. So for many of them, this is their last mathematics class. Near the end of the semester, I often take a break from the theory and proofs and have a class where the students play with Möbius bands and related objects—making them out of paper, cutting them apart, forming conjectures, and so on. It is usually a welcome April stress-reliever. This year, they will have to do it at home on their own. I made the following video for them to follow along.

I made the video for my advanced mathematics students, but I hope that the video could also be enjoyed by people of all ages and mathematical backgrounds. I have done this activity with kids as young as kindergarten. Everyone seems to enjoy it. I can’t stress this enough that it is best when you do the activity. It makes much more of an impact than watching it. By the way, when I do the activity for younger kids I tell them a story (about a circus coming to town) to go along with the activity. (I wrote about it here.)

Have fun, and if you do this with your kids or students, leave a comment (or a photo!) below.

If you would like another take on this idea, you can read my post about making zip-apart Möbius bands.

Or you can read about making Möbius band ambigrams.

]]>First things first: Everyone gets nervous giving a public presentation. Throw in a tricky math problem or a proof on a topic they have just learned, and it can be a source of great anxiety. It is natural and okay to be nervous. Your professors get nervous too! The best way to calm your nerves is to be well prepared before you step in front of your classmates. Here are some specific things you should do before you set foot in class.

**All ducks in a row.**Control what you can control, and leave as little to chance as possible. Walk into class as prepared as you can be.**Do the math.**Of course, your first task is to solve the problem or prove the theorem. If you are confident with your work, you will feel less anxiety when you present it. If you are unsure of your solution, go through the mathematics with your professor or a classmate beforehand.**Dress rehearsal.**Give a sample presentation. You can do this by yourself, or you can present it to a friend, your professor, or your cat or dog. Reading over your proof ahead of time is not the same as giving a presentation. You will often find flaws, gaps, or tricky spots that need elaboration when you present your work. It is better that this happens during a trial run than during the actual presentation.**Oops!**Don’t worry about making mistakes. Everyone makes mistakes. After your presentation, the professor and the students will give you feedback on your work. Take these comments as suggestions for improvement rather than attacks on your work.**Know it, don’t memorize it.**Do not try to memorize your proof or solution. You should be well-practiced and familiar enough with the mathematics that you can present it naturally.**Don’t “wing it.”**Walking into the class you may think you can prove the theorem or solve the problem without using notes or without being fully prepared. However, mental computations and logical thinking may be trickier to carry out when you are nervous. You may have a mental lapse and forget key details when you are standing in front of the class. It happens to us all! In such a situation, it is nice to have a safety net—make sure your notes are clear enough that they can help you string the argument together. Lastly, don’t be spontaneous; you may regret trying a new approach to the problem while you are standing at the board.

Now that you are prepared, you are ready to present the mathematics to the class. Here are some comments and suggestions for how best to deliver the material.

**What’s the point?**The main purpose of your presentation is to communicate mathematical ideas to your audience. You are not simply “putting your proof on the board” or demonstrating to your professor that you completed the assignment. You are not trying to convince your classmates that you are smart or clever. You are there to teach them some mathematics.**Writing is important.**Keep in mind all of the mathematical proof-writing advice you have learned, and write a high-quality proof on the board. If you are solving a problem instead of writing a proof, you may be able to be a little more sketchy and informal. However, keep in mind that your classmates are taking notes, and what you write on the board is likely going to be exactly what ends up in their notes. The more clear you are, the better their notes will be.**Give the big picture.**We all know from learning new mathematics that it can be difficult to see the forest for the trees. Your job in presenting your work to the class is to help them understand the big picture. For instance, rather than simply working through a set theory proof one line at a time, you can start by saying something like, “The key to this proof is showing that the set*A*has one and only one element.”**Remember your audience.**Pitch the proof and the discussion to your audience. Do not over-explain elementary ideas that your audience can grasp easily. Also, do not breeze over complicated or technical ideas. You can often read your audience by observing their body language. But if you are unsure whether they are following a particular argument, ask them.**Let me give you a minute.**Keep in mind that your audience will be taking notes, and they will be trying to follow the logical progression of the argument. It is difficult for them to listen and write at the same time. Don’t rush through a proof so fast that they cannot keep up. Pause after each sentence and visually check in with the class. If it looks like they are confused or if they are trying to get your attention to alert you to an error, address the issue.**Notes or no notes?**An ideal situation is to present the proof without a glance at your notes. However, this is often unrealistic and unnecessary. It is acceptable and appropriate to have a page or two of notes at hand so you can check them from time to time (do not bring your entire binder with you). Notes can help you remember what the next step is, they may contain some key wording that you need to ensure is precise, and they can give you a way to check that you haven’t omitted a key detail. However, you should never simply copy work from your notes.**All the world’s a stage.**Although you are not performing as an actor, you are performing. Be conscious of the way you deliver the content. Speak clearly, loudly, and slowly. Be enthusiastic, smile, find the right pacing, and connect with the audience. Make eye contact. Move from person to person. (It would be awkward to make eye contact with the same person the entire time—including your professor!) Do not stare up at the ceiling, down at the floor, at your notes, or at the board. Be aware of your nervous actions—”ummm”s, “ahhh”s, “y’know”s, “like”s, fidgeting hand gestures, and so on.**Writing and speaking.**Find a good balance between writing, speaking, facing the room, and facing the board. If you can write and speak at the same time, great. But any time you are not writing, you should face the class. Don’t write multiple sentences in silence. Do not speak into the board unless you are speaking and writing at the same time.**Font size.**Your writing should be large enough and neat enough that it is legible at the back of the room. Print; do not write in cursive. Use a marker color dark enough that it is easy to see. Write in horizontal lines—your sentences should not slant up or down.**Move it.**Walk around if you can. Most importantly, do not stand in front of your work when you are not writing. Give the entire class a full view of your work.**Wipe out.**Erase enough of the board to have a nice, clear area to write on. Do not squeeze your work in and around other writing that is already on the board. However, do not erase the existing work until everyone has copied it. When you do erase, use an eraser, not your hands. If you do use your hands, don’t then touch your face or your clothes as it may leave a colored smudge behind. Pro tip: erase the board up-and-down, not side-to-side. If your erasing arm moves side-to-side, then, by conservation of momentum, your torso will dance back-and-forth to compensate.**Use your arm.**Writing on a chalkboard or a whiteboard is different than writing on paper. Do not rest your palm on the board, and write using your hand muscles like you would with a pencil. The only thing touching the board should be the tip of the chalk or marker; use your arm to write.**A thousand words.**Many proofs or problems have a picture that must accompany the mathematics. Obviously, in those cases, you should draw and reference the figure. Be sure it is large enough and all the key features are clearly labeled. You can use colored markers to highlight certain aspects, but keep in mind that your note-taking classmates may not have multiple colors at their disposal. There are many cases in which the theorem or problem is abstract and there is no exact figure that accompanies it. But it still may be helpful to draw a representative figure that illustrates the ideas you are presenting.**Know when to say when.**If your chalk squeaks, break it in half. If your whiteboard marker is out of ink, throw it away.**A topologist, an algebraist, and an analyst walk into a bar.**Be careful about telling jokes. A class is typically not the appropriate venue for jokes, jokes often fall flat, and an inappropriate joke or one taken the wrong way can create new problems. Also, even if you are nervous, avoid self-deprecating jokes.**No negative talk.**You are the expert when presenting a problem to the class. Proceed with confidence and do not talk badly about yourself or your abilities. (“I’ll probably get this wrong.”) However, if there is a part of the proof or problem that you are unsure about, be honest about that. It would be best to take care of the problem before class, but if it arises while you are presenting, ask the other students or your professor for help. (“I’m not sure if I can say this. Do I have to justify it further?”)

[Image credit: fauxels]

]]>Eventually, we bought some of this soap and then used it all up. Today I finally had my first opportunity to make one. In addition to the soap container, the craft required some clear tubing and hot glue. It looks pretty good, I think!

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