Consider the following innocent sounding problem. Can you draw a rectangle that can be cut into two or more squares of different sizes?
Give it a try, then come back…
This turns out to be remarkably difficult. It is even more difficult if the rectangle you are cutting up is itself a square.
A solution to the problem is called a perfect squared rectangle. Perfect here means that all of the squares are different sizes. If the rectangle is not composed of smaller perfect squared rectangles, then it is a simple perfect squared rectangle.
Archimedes wrote about dissecting squares over 2000 years ago. Yet the first perfect squared rectangles were not discovered until 1925. One of those rectangles (pictured below) is 33×32 and uses nine squares, which is the fewest possible. Can you determine the size of the smallest square?
There is an excellent Numberphile video describing this problem and its history, which is known as Squaring the Square. Most of the methods used to find perfect squared rectangles were developed at Cambridge University by four students in the Trinity Mathematical Society. They found a useful way to represent the rectangle and its squares as an electrical circuit, and applied Kirchhoff’s circuit laws to help find solutions. For a more detailed description of this and related problems, take a look at squaring.net.
We are creating a puzzle (to be introduced at the 2019 JMM) that uses squares with side lengths from 1×1 to 50×50. There are 64 ways you can assemble a subset of these squares into simple perfect squared rectangles. Below are some shots of the prototype set. The first version is rather large: the 50×50 is almost 10 inches on the side. Producing a smaller scale version would mean the smallest pieces are pretty tiny. But perhaps we will just omit the 1×1, 2×2, and the 3×3, and leave it to the imagination to recognize negative spaces in a solved puzzle.
The Apollonian gasket is a fractal. It begins with three circles. Each pair of circles touch each other at a single point, and the three points of contact are distinct.
There are two circles that just touch all three:
Since we have added new tangent circles, we can now take new combinations of circles three at a time that are mutually touching, and find the two circles that touch all three. We continue this, filling in the empty spaces, and creating the Apollonian gasket:
The circles invite decoration with colors or depth.
There are also interesting relationships between the curvature of the circles. A circle’s curvature gets bigger as the circle gets smaller. If it has radius 1, the curvature is 1. If the radius is 1/3, the curvature is 3. And so on. Here is an Apollonian gasket showing the curvature of each circle. If you start at the 2 on the left, and follow the circles around the 3, the numbers are 2, 6, 14,…. Do you see a pattern?
If you’re curious about the name, Apollonius of Perga, the great Greek geometer, solved the more general problem of finding tangents for three circles that aren’t necessarily touching. In that case, there can be up to eight tangent circles.
We’ve been taking typography classes at our nearby community college. Type design necessarily consists of line and curve, but some type faces are explicitly geometric, including the modern Futura fonts you are reading right now.
When movable type appeared in Europe (400 years after it appeared in Asia), humanist type designers such as Nicolas Jensen wanted to improve upon the heavy, difficult-to-read calligraphic Blackletter. Returning from a visit to Italy where he studied the Latin typefaces carved by the Romans, Jensen developed a typeface that was readable, yet retained the look of the human hand wielding a broad nibbed pen.
Luca Pacioli, collaborating with his mathematics student (and housemate) Leonardo da Vinci, studied the stroke widths and curvature of this humanist type, and captured it in his famous series of uppercase letters. Besides being beautiful graphic art, we like how it illustrates geometry at work.
These abstract, fluid shapes first made us think of water and waves:
After we chose the colors, a family member was reminded of sea serpents. So we named this shape Nessie.
Nessies allow for a remarkable variety of patterns.
The above tiling patterns have translational symmetry, which means that if you had a tracing of the pattern, you could slide the tracing to other points where it would match the new location. You can also create dramatic radial patterns with rotational symmetry.
The pattern below has 3-fold rotational symmetry. This means you can stick a pin in the center of this pattern, and as you rotate around the pin, the pattern repeats itself every 120°. It is also possible to create 2-fold and 6-fold symmetries.
One of the kits I used in the classroom was a set of parallelograms. We had a few kits left over, and when we sold them, we decided at the time not to make any more. They didn’t seem as mathematically dramatic as our new fractal tiles, although they are very interesting to play with. Recently, one of our customers commented on how her son spent a lot of time playing with his parallelogram tiles. It was all we needed to hear to bring them back.
A single, simple shape allows for a remarkable amount of exploration.
If you join pieces at the corners, you can put six together to get a star:
Note that pieces can be reflected (i.e. flipped over). By flipping every other piece, the star becomes:
You can have 3, 4, 5, or 6 tiles meet at a point, and since each tile can have one of two orientations, there are many ways to connect these. Here are three of the several ways you can connect 4 pieces:
All of that flexibility is sufficient to create countless design variations. However, this parallelogram happens to belong to a special set of objects known as polyiamonds, which are polygons that are formed by joining the edges of equilateral triangles.
If a polyiamond is composed of 4 triangles, then it is called a tetriamond. Besides our parallelogram, there are two other tetriamonds, the v and the triangle:
Not long after we went online, we received a note from Tim Lexen of Cumberland, Wisconsin. Tim had created an elegant, simply defined shape he calls the Tricurve. Since the there is only one tile shape, the tilings are referred to as monohedral tilings. Here are a couple of the examples Tim sent us:
The pieces are oriented, meaning that when you flip a piece over, you can still use the piece but in different ways. You can use both orientations in the same tiling, which makes it different from most of our other sets of tiles. The flexibility of the Tricurve comes from it’s spare geometric and arithmetic properties. The edges are made from a semicircle cut into 3 pieces in the ratio 1:2:3.
Now we rearrange the 3 pieces into this shape.
Because the pieces came from the same circle, all edges have matching curvature. The 1:2:3 ratio allows combining edges that add up to other edges. Moreover, the angles in the tile are also in a 1:2:3 ratio, with 30, 60, and 90 degree angles. This enables several kinds of rotational symmetry: 12-fold, 6-fold, 4-fold, 3-fold, and 2-fold. While the Tricurve can be used to create wallpaper patterns, it seems everyone who plays with it also likes to explore these rotational symmetries. Here are examples:
photo by Tim Lexen
If you want to learn more, read about Tim Lexen and Paul Bourke’s explorations with the Tricurve:
Every distinct set of tiles has certain features, and we are always interested in those that introduce something new or combine features in a new way. Dylan Thurston, who helped us determine the boundaries of the Twin Dragon introduced us to the decomposable fractal Penrose tiles first described by Bandt and Gummelt. By decomposable, we mean every tile can be represented as a combination of other tiles. The boundaries are fractal, like the Dragon and Twin Dragon tiles. And finally, like Penrose P3 tiles, there are two shapes, and they only tile the plain non-periodically (i.e. they don’t repeat like wallpaper).
There are two essential shapes. We were at a loss what to call them. Their boundaries evoke the dragon tiles. But the dragon tiles have rotational symmetry: you can rotate them 180 degrees, and they are unchanged, and that gives them a very different look and behavior when tiling. Turned a certain way, I thought these new tiles look a bit like dogs, so we came up with terriers and poodles:
Each shape can appear in any number of sizes. To get to the next larger size, you scale the dimensions by the golden ratio \(\phi \approx 1.618\).
Our prototype tiles below show how a terrier can be decomposed into two smaller terriers and a poodle. Can you also see how a poodle can be decomposed into a terrier and a poodle?
These tiles are decidedly more challenging when it comes to creating an uninterrupted tiling. At craft shows, we often encounter people who are torn between fractal tilings and Penrose tilings. Now you can have it both ways!
After creating the Dragon tiles, we thought it would be cool to do the Twin Dragon, which has much more frilly edges. All of the descriptions of the Twin Dragon that I knew of focused on the space-filling curve that defines the interior of the dragon. But I wanted to generate just the border. Dylan Thurston found an elegant procedure that enabled us to create Twin Dragon boundary at any level of detail.
Here’s a pair of finished twin dragons.
We recently made some business cards using a low-detail twin dragon.
Here’s a framed version of a complete set of our tiles. Heidi used colors that evoke mid-century modern.
We loved it when our son Max’s girlfriend, Karen, paid us a visit. Heidi set her up with a Twin Dragon puzzle.
Dylan Thurston recently connected us with Melissa Freilich at the Ann Arbor Civic Theater, who was directing Tom Stoppard’s play Arcadia. Because of the fractal themes in the play, Melissa wanted to create images of large fractal ferns, known as Barnsley ferns, for the set. We worked with her to determine the scale, and after some thinking, came up with the idea of creating a set of chipboard stencils that her set crew could use to paint 24 foot ferns.
The Barnsley fern is not drawn in the typical way. Rather, it generated from a sequence of random numbers fed into an algorithm. Here’s an example. If you refresh your browser, you can watch it recreate itself.
Here’s the completed stencil laid out on our workshop floor:
And here’s the final product that playgoers got to see. Nice work by the artists!
Whenever we are showing at art and craft fairs, people of all ages love playing with the Koch tilings. I think the appeal is that like many fractals, they evoke nature. And there is something satisfying about getting the intricate edges to meet, and have a piece slide into place.
So we decided to try another, based on one of the dragon curves. There are several choices, some much more delicate than others. This delicate edge comes from the fact that flat fractals have infinitely long edges, and a fractional dimension between 1 and 2. The closer the dimension is to 2, the more intricate, and therefore more difficult to cut and fragile. We settled on Knuth’s terdragon because the fractal dimension is high enough to be interesting, but still makes for a sturdy tile.
It is pretty easy to learn how to attach Koch tiles to each other. The tiles cannot be the same size. There are a couple of attachment points for two Koch tiles. The Dragon tiles, on the other hand, seem to take a little longer before users readily identify how to connect them. But they are actually more flexible. Any two tiles, regardless of size, can be connected, and there are several ways to attach.
We debuted our Dragon tiles at the 2018 Joint Mathematics Meeting, and as with the Koch tiles, they were a hit with the young and not-so-young. This little guy made our day:
Recently, we’ve added frames as an option for our Dragon puzzles, which is nice if you want to be able to move your work-in-progress.