A natural question is: can we choose the weight and the shape of the curved track so the bridge and the weight are always in equilibrium? That is, when the bridge is in any position, it is neither pulled up by the counterweight nor down by gravity. Surprisingly, the answer is yes, and it involves one of my favorite curves: the cardioid.

First, let’s introduce some notation. As shown in the figure, we will assume the length of the bridge and the height of the tower are both units. At any moment, the angle between the tower and the bridge is and the angle between the bridge and the cable attached to the counterweight is Let be the length of the cable from the top of the tower to the counterweight. The masses of the bridge and the counterweight are and respectively.

Let’s first discuss the masses. When the bridge is down and the counterweight is at the top of the tower, the system should be in equilibrium. The drawbridge is a lever with a fulcrum at the tower’s base. The bridge produces a downward force of that is applied units from the fulcrum. The force along the cable is and because it makes a angle with the bridge, the upward component is applied units from the fulcrum. Because the forces are in equilibrium,

Hence, the counterweight’s mass must be

When the drawbridge is down, the length of the cable is So, obviously, this is the cable’s length at any intermediate stage. In such an intermediate configuration, the bridge, the tower, and the cable joining them make an isosceles triangle with vertex angle So, the length of that portion of the cable is Thus,

Finally, saying that the system is always in equilibrium is the same as saying the potential energy is the same in every configuration. When the bridge is down, the counterweight holds all the potential energy, What if the bridge is partially raised? To compute the potential energy of the bridge, we assume that the entire mass is located at the bridge’s midpoint. Its height is so its potential energy is The height of the counterweight is so its potential energy is Setting these equal, we obtain

which simplifies to

A trigonometric identity gives us

When we combine the equations for cable length and potential energy by eliminating the terms, we obtain

which is the polar equation for a cardioid! As we see in the figure below, the track is only a small portion of the cardioid curve (from to to be precise).

This design has been used around the world at various times. The bridge below is the Jackknife Bridge in Buffalo, New York. It was constructed in 1897, and this photo dates to 1900–1910. (Image source: Library of Congress)

The Glimmer Glass Bridge, located in Monmouth, New Jersey, is another such example. It was built in 1898 and was placed on the National Register of Historic Places in 2008. (Image source: Wikimedia)

This topic appeared in a pair of articles in the *Mathematical Gazette* in the 1990s. In 1990, H. Martyn Cundy wrote “The bascule bridge: An unexpected cardioid” (74 (468): 124–127). In 1995, Michael Deakin followed up with the much simpler argument I have expanded in this blog post (“More on bascule bridges.” 79 (484): 107–108).

In this blog post, we show that we can also construct a cardioid by folding paper!

Begin with a circular piece of paper. Wax paper works especially well because the fold lines are easy to see. Mark one point on the circle as the base point; call it *B*. Imagine, as in the figure below, that some diameter *PQ* is drawn across the circle and the arcs *BP* and *BQ* have lengths *x* and *y*, respectively.

Reflecting the right triangle *BPQ* over the diameter produces the triangle *PQR*. It follows that arcs *BPR* and *BQR* have lengths 2*x* and 2*y*, respectively. Thus, we should fold along *PR* and *QR*. This is true for any diameter.

The figure below shows the procedure for performing these folds. First, lightly fold the circle in half, being careful not to crease the diameter. The half circle containing the base point should be on the bottom. Align the edge of a ruler from the base point to one end of the diameter, again being careful not to crease the diameter. Fold up the top layer of the circle using the ruler as a guide. You can remove the ruler once the fold is visible to make a sharp crease. We can fold the other line by running the ruler from the base point to the other end of the diameter.

Once you understand this basic fold, repeat the procedure many times with various diameters. The form of the cardioid will slowly emerge.

We found that the best technique is to start with a diameter that ends near the base point. Then repeat, choosing slightly different diameters. This is shown in the figure below. Repeat this procedure, going in the other direction.

Notice that although the figure above shows all the fold lines, the ones that end near the base point (the ones that are nearly vertical in this figure) are harder to fold. But since they are less important to the cardioid design, they can be omitted.

]]>FISH: bass, salmon, flounder, trout

FIRE ____: ant, drill, island, opal

The words in the puzzle often have several meanings and can sometimes be grouped in multiple ways, although they can be grouped in four groups of four in only one way. It occurred to me that many mathematical terms mean different things within mathematics, or they have both mathematical and nonmathematical meanings—and thus possibly good Connections words. So I decided to create a couple Connections puzzles containing mathematical words. Here are two of them. Although the puzzle has math terms, the groupings may or may not be mathematical.

I’ve added links to the answers below the puzzles. I had a hard time calibrating the difficulty level of the puzzles. I hope they are a little challenging but not unsolvable. Have fun!

Here are the solutions for first puzzle.

Here are the solutions for second puzzle.

]]>Here’s a printable pdf of the handout and the diamond-shaped graph paper shown below.

]]>[Interesting note: I asked ChatGPT to write the first draft of this policy. I gave it a paragraph description of what I wanted my technology policy to be, and it generated an enumerated list with headers like this one. The final version looks quite different than that early version, but it was an interesting experiment!]

While technology is a powerful tool that can enhance your learning experience, it’s essential to use it correctly and effectively, to maintain academic integrity, and ultimately to attain a genuine understanding of the course material. The following guidelines are intended to ensure a fair and effective learning environment.

**Appropriate use of technology.**You are encouraged to use technology, such as WolframAlpha, GeoGebra, Desmos, scientific calculators, and graphing calculators for routine calculations, visualizations, problem-solving, and troubleshooting. However, refrain from using these resources and generative artificial intelligence (AI) tools like ChatGPT and Bard to perform the nontrivial tasks we discussed in class and that are being assessed in a given assignment. All submitted materials, including written homework, online assignments, exams, and lab projects, must be your own work. Using a technology product to generate answers or whole or partial solutions and submitting them as your own work constitutes cheating and plagiarism and is a violation of Dickinson’s*Community Standards*.**Balancing technology and learning.**Without a doubt, technology can enhance, streamline, and supercharge some aspects of your mathematical work. However, it is easy for technology to become a crutch that impedes your ability to grasp fundamental concepts, prevents your mastery of important mathematical skills, and slows your acquisition of essential problem-solving skills. Over-reliance on technology may hinder your progress in the course.**Studying, learning, and caution using generative AI.**Tools like ChatGPT and Bard may be used as aids in studying and for learning outside of assignments and exams. They can provide additional insights and explanations that support, enhance, or clarify your understanding of the material. However, keep in mind that these new and largely untested products often produce incorrect, incomplete, or misleading answers, especially in response to mathematical questions. Exercise critical thinking and cross-reference the responses with course materials to ensure accuracy. When in doubt, ask the professor.**Technology during exams:**Cell phones, graphing calculators, scientific calculators, computers, and the like are not permitted during exams. They will also not be needed, as the exams are written with this restriction in mind. However, every student will be given a simple four-function calculator that can be used for arithmetic calculations.**Note-taking devices.**Tablets such as iPads or Surfaces are permitted in class for note-taking purposes only. Laptops may not be used in class without prior approval from the professor. This policy aims to foster an environment conducive to focused learning and active participation.**Cell phone policy.**To maintain focus and out of respect for your peers, limit cell phone usage to emergencies only. Texting, web browsing, and other non-academic activities can disrupt the learning experience. Please keep your device on silent mode and stowed away.**Creative applications.**If you encounter interesting uses of generative AI*or*problematic responses from these chatbots during your mathematical studies, share them with the professor or the class. Such examples can lead to engaging discussions about the benefits and limitations of this nascent technology in the fields of mathematics and education.

They called such a tile an “einstein” (roughly, “one stone”). They highlighted two of these tiles—a “hat” and a “turtle,” named for their resemblences to these objects. (Many people online thought the hat tile looked more like a t-shirt.)

This discovery was featured in many large mainstream publications, and it was even a subject in Jimmy Kimmel’s monologue.

Their discovery was truly amazing. However, there was still a clear next step for advancement. In order to tile the plane with these hats and turtles, some of the tiles had to be flipped over. There was approximately one reflected tile for every 7 ( where is the golden ratio) nonreflected tiles. So, if these were bathroom tiles, we’d still need to manufacture two tiles—like a right-facing turtle and a left-facing turtle. Was it possible to find a single einstein tile that does not require reflections?

About two months later, these same authors answered the question in the afirmative. They found a tile that did not require any to be flipped over. This one they called a “spectre.” They showed that it is possible to tile the plane with these tiles (reflections are not allowed) and that any such tiling is aperiodic. Amazing! The NY Times had to publish a follow-up article.

The tile is in the same family as the hat and turtle, but in this case, all the sides have the same length (except one side that is twice as long as the rest). One neat consequence of this geometric fact is that if we replace every unit-length side with the same curve, then the resulting shape is still a spectre tile. For instance, in the tiling below, I changed each side into a semicircle—either curving in or curving out.

I’ve enjoyed playing with these tiles. I’ve made 3D-printable versions of the hat, the turtle, and the spectre tiles.

If you want something with a lower barrier of entry, here are some printable blank pdf pages that you can color in any way you want.

• Hat/t-shirt (pdf)

• Turtle (pdf)

• Polygonal spectre (pdf)

• Curved spectre (pdf)

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.)