If you just want to purchase a set right this minute, then click the above link! If you want to learn more, keep reading.
As explained in this original blog post from 2012 and this follow-up post, the basic idea behind factorization diagrams is to visualize the prime factorization of a positive integer by taking dots and recursively grouping them according to the prime factors. For example, can be visualized by making two groups of three groups of five dots, as seen in one of the cards above. You can find a lot more information about factorization diagrams here, including links to related things people have made, posters for sale, and so on.
Very early on I heard from teachers who had printed the diagrams, cut them out into cards, and used them successfully in their classrooms. After hearing that, I decided that there really ought to exist a high-quality deck of factorization diagram cards for purchase. It’s taken four years for that idea to come to fruition, but they are finally here!
So, what’s in a deck, you ask? Each deck contains 54 large (3.5 inch) square cards. The front of each card has a factorization diagram, and the back has the corresponding number and factorization written out. Numbers with multiple distinct prime factors have multiple cards with different diagrams, one for each distinct permutation of the prime factors. (For example, has three different cards, as illustrated below.)
You can buy your very own deck through The Game Crafter for $13.99 (see below for an explanation of the price). The images used to make the cards are freely available here (in case you’d rather just print them yourself, or do something else with them), and the source code is on github. Everything is released under a Creative Commons Attribution 3.0 license, which basically means you can do whatever you want with the source code, images, design, etc., as long as you credit me as the source (preferably by linking to https://mathlesstraveled.com/factorization).
If you end up using these cards in a classroom and come up with any fun activities/games/puzzles using the cards, please let me know! I would like to collect a big list of suggested activities for different ages and eventually be able to publish the list along with the cards.
As mentioned above, I have published the deck through The Game Crafter, who have done a great job. The process was easy and professional, and I am pleased with the final product—and I hope you will be too. You might think $13.99 is steep for a deck of cards, but (a) the print-on-demand model means there are no economies of scale to be had, and (b) these are definitely high-quality cards (3.5" square, high-quality card stock, with a linen texture and UV coating that reduces glare). I think a deck should last you a while, even if young kids are handling it.
Speaking of economies of scale though, if you are—or know—an educational/game publisher who would be interested in publishing this deck at a lower price point, please contact me! Until something like that happens though, I don’t have the resources—temporal or financial—to be able to coordinate a larger print run. The print-on-demand model means that I can get these cards out without a huge commitment.
Here’s a simple way to think about how the picture is made, as noted by Fergal Daly. The th circle (starting with ) has evenly spaced radial spokes, which we think of as being numbered clockwise from through , with spoke always lying on the positive x-axis. Then spoke has a blue dot at its end if and only if is relatively prime to , that is, . So, for example, the tenth circle has dots on spokes , , , and , since every other number shares a factor with . Note in particular that , and in general (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 -axis. (Though as noted by Macbi, we can also think of the blue dots on circle as highlighting the generators of the cyclic group —this is almost the same as the definition in terms of , 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 :
As noted by Naren Sundar, prime-numbered circles always have blue dots: one on every spoke except the th.
The dots always have reflection symmetry across the -axis, since if is relatively prime to , then so is .
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 , spoke . Then if a spoke ever appears in exactly the same spot later, then it must be spoke on circle for some multiple —but then . For example, note the blue dots on spokes and on the third circle. Those same spokes show up on circle —but now they are spokes and , so they have no dots. Likewise, the same spokes show up again on circle , as spokes and .
We can use this last fact to make some cool pictures: for example, we can give a different color to each 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 to have radius :
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 is divisible by some with , 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.
The changes I made include:
I’d love to hear any and all feedback! Modulo any final tweaks I plan to make sets available for purchase soon.
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!
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:
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 , which does not evenly divide . And anything with more than six sides will have angles bigger than , so more than two of them will not be able to fit around a vertex.
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:
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.
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!
(Click for a larger version.) I didn’t make it, and I have no idea where I got it from (do you know?). But in any case, wherever it comes from, I think it’s a really great puzzle. I did find the number that can make it through the diagram, but I never did completely finish proving that the solution is unique.
Can you solve it? Let’s see if we can prove it together. Please don’t post the number in the comments. But please do post proofs that certain combinations of nodes are impossible. For example, you might post a proof that no triangular number can be one more than a prime; that would mean the leftmost path is impossible.