I spent a while thinking about an example I could show them where this is the case. I found this Math Overflow link on this same question. They gave a few examples. Here’s one of them—the multi-function product rule for derivatives. The base case requires using the definitions of continuity and differentiability and the theorem that every function differentiable at a point is continuous at the point. It is not a simple “1=1” base case. The inductive step is much easier, although it is still not totally trivial because it requires using both the base case and the inductive hypothesis.

**Theorem.** *Let be functions that are differentiable at some point . Then *

**Proof. **This is a proof by induction on Let be the statement that given differentiable functions it follows that

Base case: Let and be differentiable at Recall that because is differentiable at it is continuous at Then

The penultimate equality follows from the continuity of at and the final inequality follows from the differentiability of and at Thus holds.

Inductive step. Suppose holds for some integer Let be functions that are differentiable at some point . Then By the base case, this equals By the inductive assumption, this equals

Thus, holds.

It follows that holds for all #

If you have another example—especially one that would be suitable for into-to-proof students (who don’t have an extensive math background)—post them in the comments.

]]>In 1667, James Gregory did the same, but he used areas: He discovered the following beautiful double-recurrence relation that can be used to compute the areas of inscribed and circumscribed *n*-gons:

**Gregory’s Theorem. ***Let I _{k} and C_{k} denote the areas of regular k-gons inscribed in and circumscribed around a given circle. Then for all n, I_{2}_{n} is the geometric mean of I_{n} and C_{n}, and C_{2n} is the harmonic mean of I_{2n} and C_{n}; that is,*

and

We can use these formulas to approximate π. For instance, a square inscribed in a unit circle has area *I*4=2 and a square circumscribed about the unit circle has area *C*_{4}=4. Applying the recurrence relations, we obtain the following sequence of bounds:

n |
In |
Cn |

4 | 2 | 4 |

8 | 2.828427125 | 3.313708499 |

16 | 3.061467459 | 3.182597878 |

32 | 3.121445152 | 3.151724907 |

64 | 3.136548491 | 3.144118385 |

128 | 3.140331157 | 3.14222363 |

256 | 3.141277251 | 3.141750369 |

512 | 3.141513801 | 3.141632081 |

1024 | 3.14157294 | 3.14160251 |

2048 | 3.141587725 | 3.141595118 |

This summer I tweeted this theorem:

Let I_n and C_n be the areas of the inscribed and circumscribed regular n-gons for the unit circle. In 1667James Gr… twitter.com/i/web/status/1…

—

Dave Richeson (@divbyzero) June 20, 2018

My friend Tom Edgar—a mathematician at Pacific Lutheran University and a master at finding “proofs without words”—emailed me to see if I wanted to try finding a proof without words of Gregory’s theorem. This is what we came up with.

The two parts of Gregory’s theorem follow from the two parts of the following lemma. We give the proof… without words.

**Lemma. ** and

*Proof.*

]]>

Möbius bands have the surprising properties that they are one-sided and have only one edge. This inspired me to write the words MÖBIUS and BAND as ambigrams—in particular, as ambigrams that appear the same after a 180° rotation.

Then I created a Möbius band with the words running along the band.

If you want to make your own, you can download this printable pdf.

However, after I made the Möbius band, it occurred to me that we don’t actually need a Möbius band to achieve this effect with the ambigram. In particular, if you tape the band together as a cylinder, it still looks the same when it is flipped over.

Now, if you were really good, you could create a version of MÖBIUS and BAND that have the 180° rotation and look the same in a mirror. Then you could write the words on a transparent Möbius band. . . .

]]>It got a lot of interest, so I thought I’d try making my own. Here’s the final product.

You can download a printable pdf if you want to make your own.

]]>We do not know who discovered the cardioid. In 1637 Étienne Pascal—Blaise’s father—introduced the relative of the cardioid, the limacon, but not the cardioid itself. Seven decades later, in 1708, Philippe de la Hire computed the length of the cardioid—so perhaps he discovered it. In 1741, Johann Castillon gave the cardioid its name.

Got your coffee? Turn on the flashlight feature of your phone and shine the light into the cup from the side. The light reflects off the sides of the cup and forms a caustic on the surface of the coffee. This caustic is a cardioid.

The Mandelbrot set is one of the most beautiful images in all of mathematics. It is the set of complex numbers *c* such that the number 0 does not diverge to infinity under repeated iterations of the function *f _{c}*(

Cardioids even show up in audio engineering. Sometimes engineers need a uni-directional microphone—one that is very sensitive to sounds directly in front of the microphone and less sensitive to sounds next to or behind it. When they do, they reach for a *cardioid microphone.* The microphone is so-named because the graph of the sensitivity of the microphone in polar coordinates is a cardioid.

In this blog post, we present a few favorite places that cardioids appear. In particular, we will look how we can use lines to construct the curved cardioid. At the end of the blog post, we provide a template that you can use to make your own cardioid. And we provided printable pages that can be used to make a cardioid flip book.

A common kids math doodle is to draw a set of coordinate axes and then draw line segments from (0,10) to (1,0), from (0,9) to (2,0), and so on. This procedure magically produces a suite of lines that, when viewed together, has what appears to be a curved boundary. This curve is called the *envelope* of the family of lines.

Let *C _{t}* denote a family of curves parametrized by

Let us look at some features of this envelope. First, each line *C _{t}* is tangent to the curve. Second, if we take two nearby lines

In the following definition we let denote the partial derivative of *F* with respect to *t*.

**Definition.** Let be a differentiable function. The *envelope* of the set of curves *F*(*x,y,t*)=0 is the set of points (*x,y*) such that both *F*(*x,y,t*)=0 and *F _{t}*(

This is a mysterious definition. Why does it produce the envelope? For a fixed *t* and any the curves *F*(*x,y,t*)=0 and *F*(*x,y,t+h*)=0 (that is, *C _{t}* and

Returning to our “kids doodle” example, *F _{t}*(

It turns out that we can construct the cardioid as the envelope of curves, and we can do so in a number of different ways. For instance, pick a point *P* on a circle (the blue circle below, say). Draw circles with centers on the original circle that pass through *P.* The envelope of these circles is a cardioid.

But we will focus on a different example. Begin with a circle (the red circle below). Mark a certain number of evenly spaced points around the circle, *N,* say, and number them consecutively starting at some point *P*: 0, 1, 2,…, *N*-1. Then for each *n,* draw a line between points *n* and 2*n* (mod *N*). In our example, *N*=54, so we would join points 5 and 10, 19 and 38, and 31 and 8 (since 8 is 62 mod 54). The envelope of these lines is a cardioid.

Let’s see why this is the case. Suppose our circle has center (1,0) and radius 3 and that *P*=(4,0). Now, starting at *P,* find points *t* and 2*t* radians around the circle from *P,* and draw the line segment joining them. We will show show that the envelope of all such lines is the cardioid with polar equation

The two points on the circle—corresponding to *t* and 2*t*—have coordinates and The line joining them is After some some algebra and some applications of double angle formulas, we can express this line as In particular, the expression on the left is our function *F*(*x,y, t*). Taking the partial derivative of *F* with respect to *t* we obtain

Now, we want to show that the *x* and *y* coordinates at which *F*(*x,y,t*)=*F _{t}*(

It turns out that this analysis explains the cardioid in the coffee cup. We can view the caustic as an envelope of lines. As we see below, if we draw lines emanating from a single point *P* on the circle and allow them to reflect off the circle (the angle of incidence equalling the angle of reflection), then the cardioid is the envelope of these lines.

If the light source is located at point *P,* then a beam of light will reflect off a point *Q* on the circle and strike the circle again at *R* (see figure below). Since arc *PQ* equals arc *QR* arc *PR* is twice arc *PQ.* But then segment *QR* is a line that we would have drawn in the previous construction.

[Update: When I wrote this post I debated to myself whether to include the following info. Thanks to the nudge by Rick Wicklin in the comments, I decided to add it.] The coffee cup example requires one final comment. In practice, the light source is not at the edge of the coffee cup, but rather, far away. So the rays of light are roughly parallel when they reach the cup. In this case, the curve won’t be a cardioid, but its cousin—a *nephroid*. This is the envelope of lines one obtains by joining *n* and 3*n*. In particular, as we see below, arc *QR* is twice arc *PQ*. (So in our numbering, *n*=0 sits at the point *P*.)

This printable pdf has a circle with 60 numbered points. Connect each number *n* to the number 2*n* mod 60 to obtain a cardioid. For a little extra fun, try connecting *n* to 3*n* or 4*n* or 5*n* to see what shapes you obtain.

This 12-page pdf is a printable flip book. Print the pages double-sided. The pages are designed so that the mathematical figure is on one side and the flip book page number is on the reverse side. Cut out each page, put in numerical order, and secure with a binder clip. Flip through the pages and see the animation in action!

]]>My son is now in Algebra 2, and for the first time, he showed me something that I’ve never seen before—the relationship between polynomials and finite differences.

Take any polynomial, such as and any arithmetic sequence, such as 0, 2, 4, 6,… Plug these values into the polynomial. Take the neighboring pairwise differences. So, for instance, Then take the neighboring pairwise differences of those values, and so on. It turns out that the *n*th level will consist entirely of the same nonzero value if, and only if, the polynomial has degree *n*. Wow! That’s so cool!

Here’s my worked-out degree-3 example. Notice that after three levels, we yield the constant value 336:

My son’s textbook (*Algebra 2*, by Larson, Boswell, Kanold, and Stiff) shared this fact but provided no explanation for why it is true (which, as a mathematician, I find very disappointing). So, I had to work it out myself.

It turns out that **more** is true—if the polynomial has degree *n *with leading coefficient *c, *and *a *is the difference between terms in the arithmetic sequence, then the final value is In particular, if *a*=1 and *c*=1, then the pairwise difference process will terminate with *n*!. Notice that in my example, the final value is

Why is this true?

Here’s a proof by induction on the degree of the polynomial. As the base case, consider a degree-1 polynomial: *p*(*x*)=*cx*+*b*. Then,

so the base case holds.

Now, assume that the result is true for any polynomial of degree *n*-1, for some *n*≥2. We will prove that it is true for a polynomial of degree *n.* Let where c≠0 and “l.o.t.” means “lower order terms.” We see that

Let us call this polynomial *q*(x).

Notice that because the leading coefficient *acn** *is nonzero, *q*(*x*) has degree *n*-1. By our inductive hypothesis, after *n*-1 pairwise differences, the polynomial *q*(*x*) will yield a constant value Thus, for *p, *the process terminates after *n *steps with the constant value This proves the theorem.

After playing around with this, I googled it, and—no surprise—the mathematics of finite differences has a long history. Also, it is not difficult to see the resemblance of these calculations to the calculation of the derivative (using the definition of the derivative).

]]>I am interested in the so-called “problems of antiquity”—squaring the circle, trisecting the angle, doubling the cube, and constructing regular polygons. If you look in reference books, we *now* know that three of the four problems (all but squaring the circle) were proved impossible in 1837 by a French mathematician named Pierre Wantzel (1814–1848). I emphasized the word “now” because his contribution was largely ignored for over 100 years!

There has not been much written about Pierre Wantzel, especially in English. For instance, he does not appear in the massive 27-volume *Complete Dictionary of Scientific Biography, *and his Wikipedia entry is very minimal. The MacTutor site has the most thorough treatment I’ve seen. About 100 years ago Florian Cajori wrote an 8-page bio in the *Bulletin of the AMS*. In it, he cites three 19th century French sources. So far I’ve tracked down two of them.

I have a passable knowledge of French. It has been many years since I’ve studied French, so I can slowly go through and puzzle out a French math document. But it takes a while and I’m not always confident in the outcome. Fortunately, online translators like Google Translate and Microsoft Translator have gotten much better, so they provide a quick way to get the gist of a document in another language.

So, I took the two French articles, copied-and-pasted them into a Word file, cleaned up any errors, used the online translators to get a first approximation of the translation, and then started cleaning them up. Then it occurred to me: Maybe this would make a good crowdsourcing project. This is where you come in.

I made Latex files of these two documents, split them into two columns (French on the left, English on the right) and posted them to ShareLatex. I made them publicly editable. If you are interested in working on this translation project—go for it. I have no idea what is going to happen. It may be a huge success, it may be a disaster, or it may be somewhere in between. We may have one person do almost all the work or we may have an end-product that is the work of many people. If you work on this, leave your name as a comment at the start of the document. We’ll see how this goes! You can post comments, questions, and suggestions in the comment section at the end of this blog post.

Here’s the first document: the original 11-page article and the ~~editable latex~~. It was written by one of Wantzel’s collaborators, Jean-Claude Saint-Venant in 1848 shortly after Wantzel died. [UPDATE! I found out that there’s already a translation of this one online! So, I copied and pasted that version into my Latex file and gave the proper citation; here is a pdf version.]

Here’s the second document: the original 3-page article and the editable Latex. It was written by A. de Lapparent in 1895.

Thank you! I look forward to watching this take shape.

]]>I came across this neat pdf by Troy Jones about using salt to do geometry. So, over Thanksgiving break I got my kids and their cousins together to do a little mathematics. We cut various shapes out of paper, propped them up on glasses, and poured salt over them. The salt is a natural bisector. The ridges can be used to bisect angles and to find the locus of points equidistant from two curves. We had fun making triangle centers, Voronoi diagrams, and conic sections. I had a good time thinking about why this worked (it is a fun exercise to see why these ridges form the various conic sections).

In a recent paper by John Sharp I learned about tying a strip of paper into a regular pentagon. It goes back to ‘Tom Tit,’ which was the pen-name of Arthur Good (1853–1928).

The American Institute of Physics gave templates to make physics-related snowflakes. I used their template to make this Isaac Newton snowflake. (As cool as it is, it doesn’t have six-fold symmetry like a true snowflake.) They also have a crystallography and a Nikola Tesla snowflake.

I have been wanting to make a mathematical flip book for a long time. Yesterday was the day that I made it happen. My 13-year old son and I figured out how to do it using the Adobe suite (my son is becoming a self-taught Adobe wiz). I started by creating the first four stages of the Koch snowflake using Illustrator. My son imported them into After Effects and had them morph from one to the other. He used Media Encoder to export them as an animated gif:

Then we exported the frames of the gif as 90 separate images. I printed them on card stock with nine per page (here’s a pdf). I cut them out and used a binder clip to hold them together. It seemed like 90 frames was perhaps too many, so I took every other frame and made two books of 45 frames. Those worked well. Lastly, I put the two books back to back and took following video of the triangle turning into a snowflake turning back into a triangle. I’m excited to make more flip books.

I found this blog post, which has instructions on how to make a fractal Christmas tree. I made one and thought it looked great. In fact, it folds up nicely into a card. So my daughter made one to give to her teacher.

After that, I tried making one of my own design. Rather than cutting the paper in half at each stage, I cut it into thirds. The result is below. It looks cool, but when it is folded as a card, part of the inside sticks out.

]]>As you can see, the tweet was widely “liked” and retweeted.

I was wondering: Is it true?

I don’t know much about the history of the division symbol ÷, although I do know that it is called an *obelus*. So I cracked open a copy of the famous 1928 text written by the math historian Florian Cajori, *A History of Mathematical Notations* (Volume 1). OK, I didn’t crack it open as much as open the web page. (All of the information in the blog post comes from pages 239–245 and 268–275 of Cajori’s book.)

From Cajori I learned that ÷ was first used for subtraction, not division—and it was not just used once in a while. It had a long life in that role. The first known such use was by Adam Riese in 1525. It was used for subtraction in a variety of countries by a number of different mathematicians in the 16th, 17th, and 18th centuries. In fact, the last known use of ÷ for subtraction was in 1915!

The first known use of ÷ to represent division was in the 1659 book *Teutsche Algebra* by the Swiss mathematician Johann Heinrich Rahn. His book was translated into English shortly afterward. However, once the notation started to catch on in England, it became known as *Pell’s symbol*, in reference to John Pell. This would not be the first instance of a mathematical object having the wrong name attached. But in this case it isn’t clear that it is misnamed; Pell had visited Rahn in Switzerland, so he may have been the originator of the symbol. (Neither mathematician claimed to be the one to introduce the ÷ symbol.)

Apparently ÷ was not the only symbol for division. Leibniz’s notation of : had its proponents too. And according to Cajori, which symbol people used depended on where they lived. He wrote

There are perhaps no symbols which are as observant of political boundaries as are ÷ and : as symbols for division. The former belongs to Great Britain, the British dominions, and the United States. The latter belongs to Continental Europe and the Latin-American countries. There are occasional authors whose practices present exceptions to this general statement of boundaries, but their number is surprisingly small.

As a final neat fact, apparently the Mathematical Association of America tried to rid mathematics of the symbols ÷ and :. In a 1923 report they wrote

Since neither ÷ nor :, as signs of division plays any part in business life, it seems proper to consider only the needs of algebra, and to make more use of the fractional form and (where meaning is clear) of the symbol /, and to drop the symbol ÷ in writing algebraic expressions.

So, what about the original question? Expressing rational numbers as fractions is old. According to Cajori, the use of the horizontal line for fractions dates back at least to al-Hassâr in the 12th century. And Leonardo of Pisa (Fibionacci) used the fractional line in his 1202 *Liber Abaci. *So in principle, fractions could be the source of the ÷ symbol. But it seems unlikely given the history that Cajori presented. And Cajori does not say that the symbol came from fractions (as far as I could tell), as @Advil asserts.

In fact, I see that the math historian Thony Christie calls @Advil’s claim “folk etymology.

]]>

I will also be the academic adviser to the students in my class until they declare a major. With this last role in mind, I decided to write up some advice for these new students. Here’s my list. [It is a slight modification of the one I gave the last time I taught a FYS.]

**Advice for new college students**

**Get to class on time.****Read your email, but not during class.****Spend a summer on campus.**Work for a professor, be a tour guide, do research, get an internship, etc.**Use proper grammar and capitalization in the email messages to your professors.**The email shorthand that may be appropriate between friends is not appropriate when corresponding with your professor (e.g., “hey, prof. when r u going 2 b in yr office?”).**Call your teachers “Professor —” not “Mr. —” or “Mrs. —.”**Almost all of your professors have the highest degree in their field (usually a PhD). (Addressing them as “Dr. —” is appropriate too, although it isn’t common at our school.)**Get to know your professors and let them get to know you.**They’re nice people. Ask your professors about their research, their family, their schooling, etc. Tell them about your summer research projects, your internships, etc. Down the road you may want to ask them for a letter of recommendation and they will be able to write you a much better letter if they know you. Besides, they are human beings, if you are rude to them, they will be less enthusiastic about helping you.**Don’t skip class.**Either you won’t be able to learn the material that you missed or the “free hour” that you gained will be lost several times over trying to catch up. If you do skip class, DON’T ask the professor what you missed—get notes from a classmate.**Take classes outside of your comfort zone.****Be protective of your online identity.**Don’t post photos online that you wouldn’t want your parents, your professors, your future inlaws, or your future employers to see.**Don’t sell your books back, especially for classes in your major.****Don’t be a member of a clique.**For many of you college will be the most diverse living experience of your life. Get to know as many people as possible and not just those with the same background as you.**Be organized, use a calendar, and pay attention to due dates.****Find a good distraction-free place to study.****Learn to write well.**I’ve seen far too many mathematics and science students avoid writing courses. They are under the impression that it won’t be relevant to them. Writing is an extremely important skill that is a prerequisite for almost all careers. You will be amazed at how much you will need to write.**Learn from your mistakes.**Look over your assignments when you get them back. The professor put those comments on there for your benefit. If you don’t understand the comments, ask.**Do the assigned work.**And the related…**Don’t ask for extra credit.**I don’t give extra credit and neither do most other college professors; if they do, they would give it to the entire class not just to you individually. Extra credit is great for the strong students—it can boost their grades from an A to an A+. Weaker students who*need*a grade booster should spend their time doing the assigned work (which they often haven’t done—that’s why their grade is in trouble in the first place). Doing the assigned work is the best preparation for the exams in the class—it gives the best “bang for the buck.”**Start assignments early and start studying early.**Related: don’t email the professor late the night before (or worse, the day of) an exam or the due date for an assignment asking for help.**Admit when you are wrong.**It may be difficult, painful, or embarrassing, but it is liberating. Living with a lie or a guilty conscience is worse than coming clean.**If you choose to drink alcohol, do so in moderation.**Not all college students drink alcohol.**Stay healthy: eat well, exercise, and get enough sleep.****Take the courses you want to take, not the ones your parents want you to take.****Beware of technology such as video games, movies, social media, etc.**They can be unhealthy, addictive time sinks.**Don’t read or send text messages in class.****Try new things (clubs, sports, volunteering, etc.), but don’t spread yourself too thin.****Call home, but not too often.****Get off campus and explore the area.**Eat in restaurants, go for a hike, see a movie, visit a museum, etc.**Study abroad.****Do not beg your professor for more points on your graded work.**If you have doubts about the grading, ask the professor to explain his or her reasoning. Most likely, if there was a grading error, your professor will fix it.**Show up for appointments and be punctual.****Don’t let your parents fight your battles.**Professors cannot speak with your parents anyway (without a FERPA release).**If things start going wrong, see a counselor.**Each year the counseling center is used by 15–20% of the student body. The service is completely confidential; they won’t notify your parents, your professors, your friends, or your insurance company.**Let go of your high school anxieties.**Your classmates didn’t know you in high school. Make new friends, wear new clothes, listen to different music, and try new things.**Don’t lie to your professors; they’ve heard them all (otherwise known as the “dead grandmother rule”).**(A retired professor I know used to send a condolence card to the student’s parents every time a student informed him of a death in the family.)**Be considerate of the neighbors.**Not everyone in town is a college student. Keep this in mind when you are returning from a party at 2:00 AM.**Be a good roommate.****Don’t cheat.**The penalties are steep if you are caught. If you are not caught you will have to contend with a guilty conscience. Cheating will produce a short-term gain and a long-term loss. Besides, it is a slippery slope—this is not the way you want to conduct the rest of your life.**Become a novice.**You’ll learn more and get more out of college if you don’t hold onto the attitude that you know everything already.**Go on a road trip.****Look at your final exam schedule before scheduling your flight home.****Have a growth mindset**(not a fixed mindset)—you can improve at anything if you put in the time and effort.