A few words about PWW #10

If you still want to think more about the picture in my previous post, stop reading now!

Here’s a simple way to think about how the picture is made, as noted by Fergal Daly. The nth circle (starting with n = 1) has n evenly spaced radial spokes, which we think of as being numbered clockwise from k=0 through k=n-1, with spoke 0 always lying on the positive x-axis. Then spoke k has a blue dot at its end if and only if k is relatively prime to n, that is, \gcd(k,n) = 1. So, for example, the tenth circle has dots on spokes 1, 3, 7, and 9, since every other number shares a factor with 10. Note in particular that \gcd(10,0) = 10, and in general \gcd(n,0) = n (this is not a special case, it’s just a natural consequence of the definition of $). This is why no circle except the first has a dot on the positive x-axis. (Though as noted by Macbi, we can also think of the blue dots on circle n as highlighting the generators of the cyclic group \mathbb{Z}_n—this is almost the same as the definition in terms of \gcd, but means that possibly we should not put any dot on the first circle.)

It’s worth pointing out a few consequences of this definition in terms of \gcd:

  • As noted by Naren Sundar, prime-numbered circles always have (n-1) blue dots: one on every spoke except the 0th.

  • The dots always have reflection symmetry across the x-axis, since if k is relatively prime to n, then so is n-k.

  • As noted by Dan Kearney, once a blue dot has appeared in a specific location on a circle, no blue dot will ever appear there again. For suppose a blue dot appeared on circle n, spoke k. Then if a spoke ever appears in exactly the same spot later, then it must be spoke ik on circle in for some multiple i > 1—but then \gcd(ik,in) = i \neq 1. For example, note the blue dots on spokes 1 and 2 on the third circle. Those same spokes show up on circle 2 \times 3 = 6—but now they are spokes 2 \times 1 = 2 and 2 \times 2 = 4, so they have no dots. Likewise, the same spokes show up again on circle 3 \times 3 = 9, as spokes 3 \times 1 = 3 and 3 \times 2 = 6.

We can use this last fact to make some cool pictures: for example, we can give a different color to each n and then superimpose all the circles on top of each other. No dot will ever overlap with another dot (well, at least they wouldn’t if the dots were infinitely small). To make it more visually obvious what’s going on, I’ve also scaled circle n to have radius n:

This kind of reminds me of Paul Salomon’s Stars of the Mind’s Sky, except where we have deleted any star that can’t “see” the origin when it is exactly blocked by another star.

Now, Naren Sundar also made another conjecture: he observed that in addition to the reflection symmetry, the blue dots seem to have rotational symmetry whenever n is divisible by some k^m with m > 1, that is, whenever it is divisibe by the square of a prime. This turns out to be true, but unlike the other properties mentioned above, it is not at all obvious just from the definition! I hope to talk more about this in some future posts.

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Post without words #10

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Factorization diagram card redesign: feedback welcome!

After getting a printed set of factorization diagram cards, I decided there were a few design tweaks I wanted to make. I’ve gone through a few iterations and I think they are definitely better now. Here are some representative samples (namely, 6, 13, 21, 29, and 30):

The changes I made include:

  • Better color scheme (at least I think so!)
  • Primes now have a visual representation that does not depend on color (though the color is still meaningful). For example, 29 is represented by an outer shell with two half-circles (representing the 2) and a trio of triangles (representing 9, that is, three threes).
  • The triangle representing 3 is flipped upside down so it never intersects with anything.

I’d love to hear any and all feedback! Modulo any final tweaks I plan to make sets available for purchase soon.

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Post without words #9

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Post without words #8

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Post without words #7

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Making tessellations

I just received my copy of Tessalation!, a great new book written by Emily Grosvenor and beautifully illustrated by Maima Widya Adiputri, which I helped fund on Kickstarter. It’s about a girl named Tessa who goes exploring in her backyard and finds all sorts of patterns, represented as fun tessellations. I’ve already had a lot of fun reading it with my four-year-old.

Most of the other math blogs in the blog tour for the book release are about early childhood math education, so I thought I’d write something in a slightly more advanced vein, exploring a bit of the underlying mathematics of making tessellations. My hope is that you’ll learn some things and also come away with ideas of new kinds of tessellations to explore. There is way more than I could ever fit in a single blog post (if you want to explore more, John Golden has a great list of resources on Math Hombre), but let’s see how far we get!

Regular polygons

Let’s start with using regular polygons (that is, polygons whose sides and angles are all equal) to tile the plane. Most everyone is familiar with the idea that we can do this with regular (equilateral) triangles, regular quadrilaterals (i.e. squares), and regular hexagons:

  • Every vertex of an equilateral triangle has an angle of 60^\circ, so six triangles can meet around every vertex to make a total of 360^\circ.
  • Four squares meet around a vertex to make a total of 4 \times  90^\circ = 360^\circ.
  • Three hexagons meet to make a total of 3 \times 120^\circ =  360^\circ.

In addition, the triangle and hexagon tilings are closely related, since we can get one from the other by subdividing the hexagons:

It’s easy to see that these are the only regular polygons that will work: regular pentagons have angles of 108^\circ, which does not evenly divide 360^\circ. And anything with more than six sides will have angles bigger than 120^\circ, so more than two of them will not be able to fit around a vertex.

Modifying a square tessellation to make your own

Now, as explained in the back of Tessalation!, and as reproduced in this blog post on Kids Math Teacher, we can take a square tessellation and modify the squares to produce more intricate tessellations which still follow the same underlying pattern. In particular, if you add some shape to one side, you have to remove it from the opposite side, and vice versa. For example, beginning with a square, we might change the right side like this:

But if we do that we need to change the left side in a symmetric way:

Now the altered squares will still line up in a row:

Likewise, we can make symmetric modifications to the top and bottom, like so:

The resulting thingy can still tile the plane:

Ants on donuts

So far so good. But if we take a step back to think about what’s really going on here, a whole world of possibilities opens up.

What we’ve really done with the square is match up certain edges, so that matching edges always meet in the tessellation.

Here I’ve marked the top and bottom edge both with a single arrow, and the left and right edges with a double arrow. (I’ve also put a letter “P” in the middle; I’ll explain why later.) In the tessellation, corresponding markings always have to match up. Like this:

Now, instead of matching up the edges of a bunch of copies of the same square, we can think about taking one square and gluing matching edges together. First, we glue the top and bottom edges together, resulting in a cylinder; then bend the ends of the cylinder around to match up the left and right edges, resulting in a torus (a donut shape).

Now imagine a very tiny ant who lives by itself on the surface of the torus. The ant is so small that it can’t tell that the surface it lives on is curved. To the ant, it just looks flat. (You may know some tiny creatures in a similar situation who live on a sphere.) Unlike those tiny creatures on the sphere, however, the ant has nothing it can use to draw with, no objects to leave behind, etc., so it has no way to tell whether it has ever been to a particular location before. The ant starts walking around, exploring its world. Occasionally there is a straight line drawn on the ground, extending off into the distance. Sometimes it finds places where two lines cross at right angles. Sometimes it finds places where the ground is black, and after making some maps the ant realizes that these places are shaped like a giant letter “P”. After exploring for quite a while, the ant thinks its world looks something like this:

Or perhaps it lives on a torus? (Or an infinitely long cylinder?) The point is that there is no way for the ant to tell the difference. The ant cannot tell whether there are infinitely many copies of the letter “P”, or if there is only one letter “P” that it keeps coming back around to. So a square tessellation is “what a torus looks like to an ant”, that is, what we get if we cut open a torus and glue infinitely many copies together so that each copy picks up exactly where the previous copy left off.

But there are lots of ways to cut a torus open so it lays flat! And all of them will produce some shape which tiles the plane just like a square. This is another way to think about what we are doing when we modify matching edges of a square—we are really just cutting the torus along different lines.

Onward

This blog post has gotten long enough so I think I will stop there! But I plan to write another followup post or three, because we have only just scratched the surface. In the meantime, I will leave you with some things to think about. First, what if we match up the edges of a square in a different way?

This is almost like the square from before, but notice that the arrow on the top edge is flipped. This means that we can’t just stack two copies of this square on top of each other, because the edges wouldn’t match:

But we can stack them if we flip one of the squares over, like this:

Finally you can see why I included the letter “P”—it lets us keep track of how the square has been flipped and/or rotated.

Can you complete the above to a tiling of the whole plane? What do such tessellations look like? Is it still possible to modify the edges to make other shapes that tile the plane in the same pattern?

How about this square?

Or this one?

Or this one?

And what about triangles and hexagons? What are different ways you can match up their edges to make tessellations? (Related challenge question: when we glue opposite pairs of sides on a square, we got a torus. If you glue opposite pairs of sides on a hexagon, what shape do you get?)

Happy tessellating!

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