Posted on

Corona tiles

We were recently introduced to an interesting area of tiling research by mathematician Colin Adams. This is a story about tiles that can fill the plane, tiles that clearly can’t, and tiles that occupy a role in between those two extremes. First, some background…

You don’t often see floor tiled with circular tiles. That is because if you tile with circles, you must have substantial spaces between the tiles. We can only pack them together as close as possible. Mathematicians call this a packing.

A tiling of the plane without any gaps is a tesselation. If you have a finite set of tile shapes that tesselates, it may be possible to tesselate the plane with them. The simplest case is a single tile shape. A square tesselates…

and in fact, any quadrilateral will tesselate.

A regular pentagon, on the other hand, does not tesselate, as illustrated here by Johannes Kepler.

But there are 15 types of pentagons which do tesselate. The 15th type was discovered in 2015 by Casey Mann and others. Here is one of the tiles.

You might wonder how you would tile the plane with an irregular shape like that. You might even conclude it isn’t possible, if we had not already told you it is possible. A solution in this case is to assemble 6 of the tiles into a patch…

which then tiles the plane…

In 2017, Michaël Rao showed there are only 15 pentagonal shapes that tile the plane.

Colin Adams pointed us to tiles that appear they may tesselate, but after some progress in tiling without gaps, we get stuck. Heinrich Heesch was the first to take interest in such tiles. In 1968, Heesch gave an example. He combined a square, an equilateral triangle, and a 30-60-90 triangle

to create an irregular pentagon

Starting with the black tile in the center, you can tile all the way around the black tile without gaps.

The tiles that go around the center form a corona. However, you cannot create another corona around these tiles. Heesch suggested assigning the number 1 to this tile shape, indicating you could create only one corona.

Others became interested. There are tiles with Heesch number 2, 3, 4, and 5. Casey Mann, one of the researchers who discovered the last pentagon tile, discovered the tile with Heesch number 5. He also discovered a tile that became our favorite for playing with. It has Heesch number 3.

Starting with this single tile, you can create a corona 3 times before you get stuck. In fact, getting the third corona is not easy! If you want to see how, you can watch this video…

Of course, you can also just have fun playing the with the negative spaces:

Want to learn more?

Posted on

Playing with Cipra tiles

We were recently introduced to an interesting set of tiles by Barry Cipra. Barry writes about mathematics and science, designs games, and has a keen interest in recreational mathematics.

Barry is inspired by the art of Sol Lewitt. One of Lewitt’s methods was to begin with a particular geometric idea that could be varied by a change in orientation, color, or value, and explore all the possible arrangements. If you’re not familiar with Lewitt, we recommend taking a look at the MOMA retrospective of his work.

Here are the sixteen tiles Barry showed us. We have them available in our store. Or you can download a picture of the tiles, print, and cut them out.

You can see a binary number in the corner of each, and each tile has paths that enter one edge, and exit another. There are two paths from each edge, and each path must go to an adjacent edge. The tiles are numbered, and are meant to be oriented, with the number in the lower left corner.

This already gives rise to some interesting questions:
• Why are there sixteen tiles?
• What do the numbers in the corner of each tile mean?

Cipra focuses on loops that appear in the tiles. Here is a simple example of a loop:

As you travel all the way around the loop, it enters (and exits) a tile 4 times. We say this loop has length 4.


Here are tiles with two length 8 loops. One is highlighted, the other is not.

Notice that we always have paths that seemingly end at the edges. Let’s imagine that if we have a rectangular arrangement of tiles, then the left edge of the rectangle connects to the right edge, like a piece of paper that bent until it becomes a cylinder.

This allows our paths to continue. But the path could still exit at the top or the bottom of the cylinder. So let’s stretch and bend the cylinder into a circle and connect the top to the bottom, making a donut, or torus.

Now, the paths have all becomes loops! This inspires more questions about the 16 tiles arranged in a square:

• Is there a tiling with just one loop, that uses every path segment?
• What would be the length of a loop that contained every path segment?
• Is there a tiling where all the loops have length 4?
• Is there a tiling where all the loops have length 8?
• Is it possible to create a loop of length 6?
• How many ways are there to tile the torus?

What if we consider only using a subset of the tiles to make a smaller square or rectangle? How would you modify the questions above? For example, if you use only one of the tiles, is there a tile with a loop having length less than 4? What if you use 2×3 tiles? 3×5?

There is much more to be explored here. At the 2019 MOVES conference at MOMATH, Cipra collaborated with Donna Dietz and Peter Winkler to explore some of the features of Cipra tiles. If you want to learn more, check out their article, Exploration of another Sol Lewitt Puzzle from Barry Cipra.

Of course, the best way to explore is with a set of our wooden tiles!

Posted on

Triknots

Our first knot puzzle was based on Celtic knot designs that begin with a square lattice, with diagonals representing the rope of the knot. The Celtic artist selectively replaced intersections with turns to create variations and symmetries, and refined the intersections to show the rope weaving over and under.

Contructing rectangular Celtic knots

We wondered what it would look like if we used a grid of triangles. We settled on the idea of using triangular tiles that have two ropes passing through each edge. Unlike the Celtic knot, this means you can connect any two tiles, and the ropes will always meet each other.

a single triknot tile

There are seven patterns on the triangular tiles. Each tile can be characterized by where the ropes enter and exit. Can you figure out why we have seven patterns?

the seven triknot tiles

With your tiles, you can explore some of the ideas of knot theory, which studies closed loops. You can close loops using any of the edge pieces.

triknot edges

The simplest closed loop is called the unknot. You can make one with two edge tiles:

an unknot

Below are two unknots. One of them has a twist. It is still an unknot because you can untwist it to have zero crossings.

If a knot has fewer than three crossings, it is equivalent to an unknot. The simplest knot that is not an unknot is the trefoil.

the trefoil, the only knot with three crossings.

You can use your knots to explore symmetries. When the trefoil is rotated 120 degrees, it looks the same. After three such rotations, you are back to where you started. So we say the knot has 3-fold symmetry.
M.C. Escher creatively used color to reduce symmetry. He called this anti-symmetry. Looking only at the patterns on the tiles, ignoring color, this knot has 6-fold symmetry.

If you consider color, what is the rotational symmetry?

Things to think about…

• Can you explain why the knots you create always alternate going over and under?

• How many kinds of rotational symmetry are possible?

• Why are seven tiles needed to characterize the ways the ropes can enter and exit?

• If we had used square tiles with two ropes on each edge, how many different tiles would we need to show all possible connections between edges?

• Which knots in the Knot Atlas can be made with the triknot tiles?

Posted on

The Stomachion

From 1998 to 2008, a team of researchers worked to unravel the secrets of a document first written in Byzantine Greek over 1000 years ago (check out this TED talk by the lead researcher, William Noel). Over the centuries, monks periodically cleaned off the markings on the document, and wrote fresh text, thus creating a palimpsest. The team discovered that the palimpsest contained a copy of previously unknown work by Archimedes, the greatest of ancient mathematicians and scientists. Mathematicians pored over his work.

The Archimedes Palimpsest contained what is believed to be the first dissection puzzle. The Stomachion is a dissection of a square, resembling the tangram, but pre-dating the tangram by over 1000 years. Archimedes used 14 pieces, and some believe he was using the various arrangements that make a square to study combinatorics . Mathematicians Fan Chung and Ron Graham noted that 3 pairs of his pieces appear next to each other in all the square dissections. They suggested these be merged into a single pieces, leaving 11 pieces, which they called the Stomach. We use 11 pieces as well.

There are multiple solutions with the eleven pieces. If consider two solutions to be the same if you can get from one to the other by rotating the solution, or flipping it over, then there are 268 distinct solutions. Remarkably, starting from one solution, it is possible to step through 266 of the solutions by flipping or rotating a subset of the pieces.

Check it out in our store!

Posted on

Tea Towels and German felt home goods

We are happy to introduce a new line of mathematical home goods. We have designed tea towels in a robust blend of cotton duck and linen. We have chosen bold vibrant colors for this series.

And along with the ferns we brought out earlier this year, we have developed beautiful 5 mm thick German felt fractal shapes that can be reconfigured from coaster size to trivet size or larger. Sold in color packs that are interchangeable and playful.

Posted on

O’Beirne’s Cube

Much of what we make is two-dimensional. Hugh Montgomery, a mathematician at the University of Michigan, told us about an intriguing 3D dissection puzzle by Tom O’Beirne, a Scottish author and inventor of puzzles. O’Beirne’s cube consists of six pieces that can be packed into a box. The blocks seem irregular at first glance.


However, the pieces are built from four copies of the base component: a 3x4x6 rectangular prism, or cuboid. In our version, each unit in the base cuboid is 0.25 inches, or a bit over 6 mm wide.

O’Beirne combined two of these 3x4x6 cuboids on a matching face, to produce these doubled cuboids:

He then created every possible pair of the doubled cuboids, matching half of a face on each. Below, you can see how a 3x4x12 is paired with a 4x6x6. There are two ways to do this. They are mirror images of each other.

There are three possible pairings of the doubled cuboids. Each pairing produces two mirrored pieces. This generates the six blocks of O’Beirne’s puzzle.

You can combine these six to make a cube.

You can rearrange the six pieces into five other cuboids:

In fact, with any one of the these six cuboids, you can split it into two parts, and recombine to create another cuboid. You can quickly cycle thorugh all six! The original version of the diagram below is described in more detail in Brian Butler’s excellent article on John Rausch’s puzzle site.

You don’t have to limit yourself to cuboids. Here are some other puzzling shapes.

We sell the O’Beirne’s Cube here. But if you have the tools and are up for it, you can make your own. At the San Francisco American Craft Council show, we met Ron Choy, who was captivated by the O’Beirne’s cube. I sent him the link to John Rausch’s site. We were delighted to hear from Ron later, with a picture of the cube he made along with his notes on the process. Now that’s some hands on learning!

Ron Choy's OBeirne's cube
Posted on

Solve the Golden b puzzle!

Robert Ammann’s Golden b tiling is unlike the tilings we see around us every day. Most tilings are periodic (see wallpaper groups). In a periodic tiling, you can find two points where the tiling looks identical. In fact, if there are two, there will be infinitely many such points.

Mathematicians discovered aperiodic tilings in the 1960s. Aperiodic tilings must be able to tessellate (i.e. fill in) the entire plane, and use only a fixed number of tile types. Most importantly, any tessellation with the tiles must be non-repeating in this sense: pick any two points in the tessellation, and the tiling must look different at those points.

Robert Ammann was an early researcher in aperiodic tilings. Remarkably, he was not a professional mathematician—he was a postal worker with a passion for discovering new tilings. One of the tilings he discovered is based on a single tile shape, in two different sizes. The ratio of the area of the larger tile to the smaller is the golden ratio, \(\phi \approx 1.618\). Note that the tile is shaped like a chunky letter b, hence the affectionate moniker “Golden b.”

If we take the shortest side of the smaller tile to be one unit long, then we get the other sides by successively multiplying by \(\sqrt{\phi}\). And we create the larger tile by scaling all side lengths by \(\sqrt{\phi}\).

The elliptical markings on the tiles enforce a matching rule. So, for example, you might combine the two tiles above to create this:

And behold! It is another, larger, Golden b. This suggests you can combine it with the larger tile to create a still larger Golden b.

As you build a tessellation with the Golden b, the ratio of the number of large to small tiles converges to—you guessed it, the golden mean.

Now available in our store!

Posted on

Create Celtic Knots

To artists and mathematicians, knots can be beautiful, interesting structures. Until recently, I hadn’t explored them much from either perspective. Then my mother-in-law gave me a copy of George Bain’s Celtic Art: The Methods of Construction for my birthday. Bain shows how to create Celtic knots, from simple to elaborate.

Let’s make one, partly based on Bain’s techniques. Celtic knots can be used to fill complex shapes, but many of the motifs fill rectangles, and this is a good place to start. You may want graph paper, pencil, and an eraser. Create a grid with an even length on each side. I’m using 8×6 here, and you’ll notice that I have accented every other line, so that there is a 4×3 grid of 2×2 cells.

Sketch diagonal lines so that each 2×2 cell has a rotated square in it:

Now, we are going to draw the border of our knot. Rotated squares on the corners will have three sides (shown in green) replaced. Rotated squares on the corners have two sides (shown in red) replaced.

Green sides are replaced with a cusp, and red sides with an arc.

It’s starting to look a bit like a Celtic knot. In fact, we could skip this next step, and produce a generic knot. But part of the art of creating a Celtic knot is varying its structure. Pick a number of interior intersections to be erased. I’ve erased two here.

After erasing each intersection, reconnect the strands vertically or horizontally. In my drawing, I’ve chosen to reconnect the broken lower left intersection with vertical lines

Next, we identify where the line passes over and under itself. Pick any intersection to start, and select which line goes under. Erase a little bit on each side of the overpassing knot. Then follow either line to the next intersection, and make sure the line does the opposite of what it did at the previous intersection. In other words, if it went over first, now it goes under. And vice versa.

Here is one of the beautiful results of such diagrams: alternating over and under always works. You cannot find a knot that cannot be completed this way. If you follow the paths, you will see that it is actually two knots that are connected. Mathematicians call this a link.

Finally, widen the lines, and you have created a Celtic knot!

We created tiles that allow you to create knots. If you look at each of the small blue squares in the knot above, you notice there are only a few types. These became our tiles. To give them more of a Celtic flavor, we added a rope motif to each tile.

Posted on

Squaring rectangles

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 have created a puzzle which we introduced at the 2019 JMM. It 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. To give you a sense of scale, the 50×50 has 10 inch sides.

Posted on

Apollonian gaskets

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?

And here is an example of an Apollonian gasket we’ve made, available from our online store.


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.