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Many of our products are based on ideas we have serendipitously bumped into and found attractive. But we also hear from people who help us create a new product, or who pass along a great idea. We mention them in our blog posts, but as the number of contributors has grown, it seems appropriate to also gather them up in a single post.

We would like to thank everyone on this list for their inspiration and assistance. And if you have an idea for us, please let us know!

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Tesselating with Ammann-Beenker tiles

Most tilings we see around us are periodic tesselations. They are tesselations because there are no gaps, and periodic because they repeat like wallpaper. In fact, until the 1960s, people thought that any finite set of tile shapes that could tesselate could be used to tesselate periodically. Then researchers discovered tiles that only tesselate non-periodically. Such tiles are called aperiodic.

Robert Ammann, an amateur mathematician, found sets of aperiodic tiles. One set has two simple shapes: a square, and a rhombus.

Of course, you can tile the plane like wallpaper with rhombi and squares. But Ammann added edge-matching features to these tiles. These features must form one of two shapes where tiles meet.

We’ve made those features look like hearts and arrows. To ensure that you connect all edges correctly, we provide the edge markings as red and black key tiles, that fit into the original tiles.

Each of Ammann’s tiles can be flipped over. The square–based tile has reflective symmetry, meaning it is identical to its mirror image. So flipping it over does not affect where you can place it.

The rhombus–based tile, on the other hand, is different from its mirror image. Since our tile set is magnetic and the magnet is on one side of the tile, we provide both orientations.

As you put the tiles together, you will often see that there are spaces that can’t be filled by a key tile. Here, the arrow won’t fit.

However, you can rotate the right tile 180°, or replace the right tile with its rotated mirror image:

The tiles can tesselate the plane, so a natural objective is to tile outward from the center in all directions without leaving any gaps. Or, you can deliberately leave gaps, or negative spaces, in which case you have a packing.

You can find the Ammann-Beenker tiles in our store.

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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?

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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. Note: the image below takes time to load.

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. Note: the image below takes time to load.

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!

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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?

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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!

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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.

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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
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Solve the Golden b puzzle!

At a JMM conference, mathematician Keith Cobenhaver called our attention to Robert Ammann’s Golden b tiling. It 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!