Here’s the seventh match in Round 1 of The Big Internet Math-Off. Today, we’re pitting Fran Herr against Tom Edgar.
Take a look at both pitches, vote for the bit of maths that made you do the loudest “Aha!”, and if you know any more cool facts about either of the topics presented here, please write a comment below!
Fran Herr – Modular Multiplication and Dancing Planets
This is a story about fantastical patterns emerging from a pair of integers, and it has more to do with geometry than number theory (*sighs with relief*). Start with a positive integer
You may know these pictures by another name: “string art,” “light caustics,” “spirographs,” or “curve stitching,” for example. They are popular across the math communication world. The following question has been partially answered, but today we dive deeper than you have likely been before!
Given the values for
From the small selection of modular multiplication tables below, one can see that they are an eclectic bunch.
I encourage you to draw some of these tables for yourself as you read! There are many tools to do so online (made by people who know how to make webapps); one that I like is this one. You can also download my python code or write a quick script yourself.
We see our first pattern by fixing
This curve, to which all the chords are tangent, is called the
envelope of the chords. As we repeat this experiment with
These curves are called epicycloids. The epicycloid with
When we look at higher multipliers, we see the large variety of
patterns showcased earlier. We notice that the numerical relationship
between
Of course, you should not take me at my word! I encourage you to check by drawing more tables from the three families above. Bonus: Can you explain why we get these three patterns?
Another way to write these three families is to assume
A natural generalization is to replace ‘2’ by an arbitrary ‘
The resulting designs for
There are some beautiful patterns to understand here, and I encourage you to run some experiments yourself. But here we press forward. A few years ago, I became obsessed with modular multiplication tables and I went down this same line of exploration. The next question I asked was, “what if I dropped the assumption that
Also, for each
Figure 1 is a “table of tables”. Each table is of the form
I encourage you to pause and ponder this array. What patterns do you notice? Do the designs remind you of other curves we have looked at? Why do you think these patterns might be emerging?
Let’s look at one particular example from this collection. The table
As we plot the chords one at a time, we notice that we can form the same pattern by moving the initial endpoint of the chord 3 spaces and the terminal endpoint 2 spaces. This observation unlocks a new perspective on these modular multiplication tables.
Imagine two planets, named A and B, on the same circular orbit around a central sun. Suppose an infinitely stretchy tether connects the two planets. At any one moment this forms a chord of their circular orbit. This setup is very similar to an animation by Matt Henderson, posted on Twitter, except in our case both planets are on the same orbit. We will call this system a planet dance.
Formally, a planet dance, denoted by
We can realize a modular multiplication table
Planet dances open up a whole new world; we can choose
Our observations above indicate that
Now, finally, we can enter into topology land! A directed chord on
the circle is uniquely determined by the position of the two endpoints,
planet A and planet B. Hence, the space of all such possible chords is a
circle times a circle:
We model our torus as the unit square in
When we graph our two favorite planet dances (
For a modular multiplication table
This phenomenon is a visual example of aliasing, a
well-studied aspect of signal processing (hello applied
mathematicians!). The
Fran Herr is a PhD student in mathematics at the University of Chicago. She studies low dimensional topology and geometric group theory. You can follow her on YouTube and X.
Tom Edgar – A Rational Day at the Hilbert Hotel
I want to tell you about a curious conundrum I encountered during my time working at the Hilbert Hotel. Maybe you’ve heard about the standard problem from the mystical lodge before, but in case not, here’s a quick recap of the classical story. The Hilbert Hotel contains infinitely many rooms, one for each natural number
The classic Hilbert Hotel dilemma helps us learn to make sense of one-to-one correspondences between infinite sets. The last scenario succeeds because the set of natural numbers
I expected working at the hotel to be an easy gig because I thought I knew all the necessary tricks. But the job tested me immediately. On the first day after the hotel had been closed for renovations, an infinite collection of people, one for each positive rational number, requested a room. The positive rational numbers are numbers that can be written in the form
:
The idea is that you can write down pairs of positive integers in a two dimensional grid, and then you can line them up in an ordered list (corresponding to the natural numbers) by traversing the anti-diagonals starting at the top left. The resulting ordered list is
I used this ordered list to assign rooms to the rational guests and each headed to their room. Easy-peasy.
Later that day, the hotel owner, David, came by visibly frustrated. He commended me for accommodating the infinite collection of guests, but he expressed disappointment that I left so many rooms vacant. After all, the slogan for the hotel was “The Hilbert Hotel: Where every room is full but we always have space for you.” I had broken the cardinal rule. Each rational number has many representations; for instance the number 1 can be written as 1/1, 2/2, 3/3, etc, So by my list, the rooms I assigned for 2/2, 3/3, etc. remained empty since the guest 1 occupied only room 1. In fact, each rational number has infinitely may representations so I had left infinitely many rooms vacant!
I realized my mistake: the listing of rational numbers from the two-dimensional grid does not create a one-to-one correspondence. It only guarantees the existence of such a one-to-one correspondence! Instead the grid argument provides a one-to-one correspondence between the natural numbers and the set of ordered pairs of positive natural numbers. One way to get the correspondence between natural numbers and rationals is to skip the rationals you have already written in the list earlier. But I wanted to fix my mistake in a more prescriptive way, finding a formulaic way to write down the list of positive rationals so that each appears once and only once in the list.
I turned to an unexpected friend, one of my favorite mathematical objects: Pascal’s triangle (which, perhaps should be attributed to Omar Khayyam or even Halayudha). To construct this triangular array, begin and end each row with a 1, and, for each other entry, add the two entries in the previous row directly above the entry. I reduced the arithmetic triangle modulo 2, that is, I replaced odd numbers with
:
This process results in a famous integer sequence called Stern’s diatomic sequence:
which can also be defined recursively by
Stern’s diatomic sequence has a truly amazing property that solved my resort riddle. By taking successive ratios of terms in the sequence, we get the following list of rational numbers:
The magic? Each positive rational number appears once and only once in this list.
While I don’t have time to discuss the proof of this fact, suffice it to say that I used this fascinating list to completely fill the Hilbert Hotel with the positive rationals. My solution pleased David, and I kept my job for a little while longer. But eventually I quit that job to investigate more properties of this intriguing sequence. If you find yourself wanting to know more, I recommend “On Stern’s Diatomic Sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4,…” by Sam Northshield or “Recounting the Rationals” by Neil Calkin and Herbert Wilf.
Tom Edgar is a math professor and the outgoing editor of Math Horizons. He enjoys thinking about and animating so-called “proofs without words.” You can follow him on YouTube and Instagram, or look at his homepage.
So, which bit of maths has tickled your fancy the most? Vote now!
Match 7: Fran Herr vs Tom Edgar
- Fran with dancing planets
- (57%, 79 Votes)
- Tom with too many hotel guests
- (43%, 60 Votes)
Total Voters: 139
This poll is closed.
The poll closes at 08:00 BST tomorrow. Whoever wins the most votes will get the chance to tell us about more fun maths in the quarter-final.
Come back tomorrow for our eighth match, the last in round 1, pitting Dave Richeson against Kit Yates, or check out the announcement post for your follow-along wall chart!
I enjoyed those very much.
I like the humor in Christian’s entry, but I had to vote for Fran’s.
I love mathematical art, but the kicker is that it reminded me of string art at my grandmother and I did when I was a child. We bought kits that had intricate templates for where to put nails and where to place the strings. The one I got to help with was an owl that we hung in my mother’s room.
The unexpected places that pascal’s triangle appears continues to amaze me!!