Parker mentioned the existence of these measuring tapes in his talk at Gathering 4 Gardner 12 and he gave every attendee a D-tape that he made (see below).

That got me thinking. If you know the circumference, you can compute the diameter. But you can also compute the area. Also, if you wrap the tape around the equator of a sphere, you can compute the the volume and the surface area of the sphere.

Thus, inspired by this, I made measuring tapes to measure the diameter and area of a circle and the volume and surface area of a sphere. The top of each tape is marked off in centimeters, so it measures the circumference. The bottoms can be used to measure the diameter of the circle, the cross-sectional area, the volume of the sphere, and the circumference of the sphere.

I’ve also created a printable pdf of these measuring tapes, which should, I hope, print to the right scale so that 1 cm on the measuring tape is actually 1 cm. (Note that although a 25 cm ruler looks long, when you wrap it around a circle, you find that the diameter is only 8 cm. Not very big.)

It was a fun exercise to figure out how to mark and make the rulers. Except for the diameter ruler, the relationships aren’t linear. (I set everything up in Excel, then I moved the info over to a Geogebra spreadsheet. I used GeoGebra to draw everything. I exported the images as pdfs. Then I cleaned them up and added the text in Adobe Illustrator.)

]]>

- Twist the paper zero times, and tape the ends (making a cylinder). Cut down the midline.
- Give the paper one half-twist, and tape the ends (making a Möbius band). Cut down the midline.
- Give the paper two half-twists ,and tape the ends. Cut down the midline.
- Give the paper three half-twists, and tape the ends. Cut down the midline.
- Twist the paper zero times, and tape the ends. Cut into thirds.
- Give the paper one half-twist, and tape the ends. Cut into thirds.
- Give the paper two half-twists, and tape the ends. Cut into thirds.
- Give the paper three half-twists, and tape the ends. Cut into thirds.

In fact, these activities are fun for people of *any* age. My senior math majors enjoy it, and my kids’ kindergarten classes have too.

Last week I attended the 12th biennial Gathering 4 Gardner conference—a wonderful meeting of people interested in mathematics, puzzles, games, magic, and skepticism. One of the speakers (Iwahiro Hirokazu Iwasawa) suggested making zip-apart Möbius bands. Genius! And perfect timing (since I’m teaching topology this semester).

When I got home I bought zippers and Velcro and used them to make reusable strips. Half of them were just one zipper with Velcro on the ends. These can be used to do activities 1-4 above. Then I took identical pairs of zippers and joined them side-by side to make strips for activities 5-8.

Some tips:

- The zippers I used were 12″ or 14″ long. They seemed to work well.
- The Velcro has adhesive on the back, so I could just stick them to the ends of the strips. I did it so that the velcro folded over the ends and so was on both sides. That gave me flexibility if I gave an even or an odd number of half twists.
- For the doubled zippers, align them so that both zippers start at the same end of the strip and so that one is on the top side and one is on the bottom side. Then, when you join with an odd number of half-twists, one begins where the other ends and they’re both on the same “side” (locally) of the Möbius band.
- Ideally I would have sewn the two zippers side-by-side. But we don’t own a sewing machine. So I stapled them.
- I learned the hard way that there are separating zippers (think of the zipper on a parka that comes apart at the bottom) and closed-end zippers (think the zipper on your pants that stops at the bottom). You want the former type for this activity.
- The zippers were about $3 apiece, but my wife told me about a local fabric store that was going out of business, so I got them for 70% off!

]]>

My friend Dan Lawson came to the rescue—he posted the following lovely proof on Twitter. Thanks Dan!

]]>

I spent a good chunk of last week reading about David Johnson Leisk (1906–1975), who is better known by his nom-de-plum Crockett Johnson. Johnson is most well known as the author of *Harold and the Purple Crayon, *a children’s book from 1955, and its sequels. Johnson was also the author of the 1940s comic *Barnaby.*

Later in his life Johnson became interested in mathematics. He was particularly interested in geometry, and most specifically in the problems of antiquity (squaring the circle, trisecting the angle, doubling the cube, and constructing regular polygons). He turned many geometric theorems into works of art. Eighty of his paintings are now at the Smithsonian.

Johnson even created new mathematics. I would like to discuss one of his contributions here. (See his article “A Construction for a Regular Heptagon,” *The Mathematical Gazette *Vol. 59, No. 407 (Mar., 1975), pp. 17-21.)

The heptagon is notewothy because it is the regular polygon with the fewest number of sides that cannot be constructed with compass and straightedge alone. In his article, Johnson gives a way to construct the heptagon using a marked straightedge (this is called a *neusis *construction). Johnson did not give the first *neusis *construction of a heptagon—François Viète gave the first such construction in 1593. (Also, Archimedes gave an unorthodox *neusis-*like construction).

However, Johnson’s proof used trigonometry (including the law of cosines and several trigonometric identities). My question to you is: **Is there a purely geometric proof of his result?** I played around with it for a little while and couldn’t find one, and I couldn’t find a geometric proof in the literature.

The key to Johnson’s construction is producing 3:3:1 triangle; that is, a triangle in which the angles are in a 3:3:1 ratio (they would be and The three vertices of the triangle are three vertices of a regular heptagon. If we construct the circumcircle, then it is easy to construct the four remaining vertices with a compass and straightedge.

Here’s his construction of a 3:3:1 triangle using a marked straightedge—that is, an otherwise ordinary straightedge, but possessing a mark one unit from the end (or equivalently, two marks one unit apart). We begin by drawing a line segment *AB* of length one (see below, left). Construct a unit line segment *AC* perpendicular to *AB*. Also, construct the perpendicular bisector to *AB*; call it *l*. Then construct a circle with center *B* and radius *BC*. Now we perform the *neusis* construction with the marked straightedge: Construct a line *AD* so that *D* is on *l* and *D* is one unit from the circle. (That is, the end of the straightedge is at *D*, the mark is on the circle, and the edge passes through *A.*) Then is the 3:3:1 triangle.

Johnson’s proof that is a 3:3:1 triangle used trigonometry (see his article for details). Is there a geometric proof?

Boiled down to its essence, here’s the question: Suppose is isosceles and *E *is on *AD*. Moreover, suppose and , prove that

Here is a fact that may help. Johnson discovered this fact—ironically—when he was sitting in a café in Syracuse on Sicily (the birth place of Archimedes) playing with the wine and food menus and some tooth picks. If we have an isosceles triangle *ABD *with the point *E *on *BD *and *F *on *AD * such that then triangle *ABD *is a 3:3:1 triangle. (This is easy to prove. Suppose Then, because is isosceles, . So the exterior angle of is Because is isosceles, Lastly, observe that and are similar, so It follows that )

Thus, a possible route to proving the theorem is to find a point *F *in the original diagram so that

If you find a proof, I’d love to hear about it!

Here are two of the paintings Johnson made from his work with the heptagon.

]]>

Abstract: *In 1928 Henry Scudder described how to use a carpenter’s square to trisect an angle. We use the ideas behind Scudder’s technique to define a trisectrix—a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.*

I also made a GeoGebra applet to accompany the article. Give it a try.

]]>

The post made me think of a neat project for an “Introduction to proofs” class. I’ll have to save it for the next time I teach that class (Discrete Mathematics at my college).

Start with the first *n* prime numbers, Divide them into two sets. Let *A* be the product of the primes in one set and let *B* be the product of the primes in the other set (the product is 1 if the set is empty). For example, if n=4 we could have {2,3,7} and {5}, so A=42 and B=5. What can we say about A+B and A-B? Form a conjecture and prove it.

I like the problem because I could imagine the students will go through a process of (1) I have no idea. (2) Aha! I’ve figured it out. How do I prove it? (3) Wait, I found a counterexample. (4) Ahhh! I see what’s going on, here’s the theorem, and here’s the proof. (I think they would come up with a different statement of the theorem than the one in the Futility Closet post.)

]]>

Yesterday, I gave them an example of a very, very poorly written proof that I created. (It was fun trying to insert as many problems as I could into the short proof!) I had them work on the assignment in class for about 7 minutes. Then we came together and talked about all the errors that we could find, and finally we rewrote the proof in the correct way. Here’s a pdf of the assignment, and I’ve posted a screenshot below.

In case you are wondering, here are the errors in the proof. Numbers refer to sentence numbers.

- The proof should be begin immediately after the word PROOF.
- The entire proof should be written in paragraph form.
- 1. Capitalize the first word of the sentence.
- 1. Split this sentence in two: The first sentence should be “Let
*m*be an odd integer and*n*be an even integer.” In the second sentence we apply the definition of even and odd. - 1. Do not use
*k*for both the even and the odd numbers. - 1. Don’t use a symbol (#) in place of a word.
- 2. Start the sentence with a word, not a mathematical expression.
- 2. The correct expression is
*mn*^{2}, not*m*^{2}*n*. - 2. “Equivalent to” is a term we use for logical expressions, not numbers. It should be “equal to”—and in fact, we should just use “=.”
- 2. Algebra error: the term 8
*k*should be 8*k*^{2}. - 2. In a 200-level math class we do not have to show all of these algebraic details.
- 3. This is not how we show that a number is even—we must show that
*mn*^{2}satisfies the definition of even. - 3. Do not use the word “obviously.”
- 3. In a proof like this we’d have to show that the three terms satisfy the definition of even.
- 3. “There” should be “their.”
- 3. Missing dollar signs around 2
*k*in the LaTeX code. - 4. Do not use the passive voice.
- 4. Missing period at the end of the sentence.
- 4. and 5. Write in present tense.
- 5. Do not put an example in a proof.
- 5. There are missing dollar signs in the LaTeX : after
*m*=5 and before*n*=4. - 5. It should be first person plural, not first person singular (“we” not “I”).
- 5. Replace “is = to” with “=.”

Here is our corrected proof:

]]>

Are there any rational values of for which the line is tangent to the graph of

Clearly the answer is yes: But my gut feeling was that this was the only such After some head scratching, I obtained the following proof.

Suppose they are tangent at . Then the point of tangency is Moreover, the slope of the tangent line is . Thus, must satisfy the two equations:

and

.

Using a trig identity,

.

So,

Note that is an algebraic number—it is the root of a polynomial with integer coefficients.

In 1882 Ferdinand von Lindemann famously proved that is transcendental (that is, non-algebraic) for every non-zero algebraic number and a similar proof holds for the sine and cosine functions.

Thus, because and is algebraic, it must be the case that . In particular,

This concludes the proof.

]]>

The bag of gifts that I received at the end of the conference was amazing. It contained exotic dice, clever puzzles, unique playing cards, mathematical earrings, 3d printed objects, cool pencils, puzzle books, artwork, etc. Nearly all of the gifts were handmade. We haven’t received the written gifts yet. What a fantastic idea!

My gift was a set of instructions and templates for making *Gabriel’s horn* out of paper cones. Gabriel’s horn is the surface obtained by revolving the curve about the -axis. Mathematics professors wow introductory calculus students by sharing its paradoxical properties: it has finite volume, but infinite surface area. As they say, “you can fill it with paint, but you can’t paint it.”

Photos of the paper horn (in white and in rainbow) and a photo of the cones are shown below. I’m linking to a pdf copy of the article here in case you want to make one.

Note: my inspiration was this blog post by Dan Walsh in which he makes a pseudosphere out of cones.

]]>

I received quite a few links. I’m looking forward to reading them. It seemed like there was a lot of interest, so I’m posting the unedited list here.

- Visualizing Math
- Jimi Cullen
- James Campbell
- Complex Projective 4-Space
- Fix epsilon>0
- Not Only Truth But Supreme Beauty
- 1ucasvb’s lab
- Curiosa Mathematica
- Math is Fun
- Math Professor Quotes
- Math Major Problems
- Herp, Derp, Math Nerd
- Eat, Sleep, Math
- None Sequiturs
- Maxwell’s Equations
- Euler, Erdős
- Yellow Pigs
- Prof. John Golden has had his students blog. Here are links to past semesters

If you know of any others, post them in the comments.

]]>