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Don’t get stuck on Socolar–Taylor Tiles!

A set of tile shapes is aperiodic if, no matter how you try, the tiles refuse to allow a wallpaper like pattern of repetition. The first known set of aperiodic tiles appeared in the 1960s, and used over 20,000 different tile shapes. To our knowledge, nobody ever made those tiles. Ideally, a set of tiles requires very few distinct shapes. Roger Penrose discovered tile sets that only required two shapes (see P2 and P3), as did amateur mathematician Robert Ammann (see the Golden B and Ammann-Beenker).

Is there a single tile that only tiles aperiodically? Joan Taylor, an amateur mathematician from Tasmania, and Joshua Socolar provide an answer: a qualified yes. Visually, their tile is fragmented. Think of the fragments as part of a disconnected whole.

Socolar-Taylor tile

You cannot create such a tile: you need the fragments to magically maintain their relative position as you move the piece! So we simulate the tile with a simple hexagon that has markings, and impose some rules.

the tile and…
…the tile flipped over.

The Rules

  1. Where edges meet, dark lines must touch.
  2. Triangles at opposite ends of a tile edge must point in the same direction.

Look at the rule illustration below. The arrows point to the triangles at opposite ends of a tile edge. And those triangles point in the same direction:

Triangles at opposite ends of a tile edge must point the same direction.

Here the orange triangles violate the second rule:

Triangles at opposite ends of a tile edge are pointing in different directions.

Dead ends

With only a single tile, you might imagine that putting the Socolar–Taylor tiles together is straightforward. Socolar and Taylor provide substitution rules that allow you to start small, and then tear down and build up successively larger patches of tiles. But as something to explore and puzzle over, we like to put it together one piece at a time, looking for structures and (non-wallpaper) patterns. Despite having just one tile, it is a challenge to follow the rules. You may not easily visually recognize what to do next, and in fact, you might not be able to do anything! In the example below, it is not possible to place a tile in the empty spot, and so we are at a dead end. This means you need to change your tiling in order to continue.

As you build up your tiling, triangular patterns miraculously emerge from the lines on the tiles, reminiscent of Sierpinski triangles. In the photo below, you can see that we clear–coated the reverse side of the tiles so that you can recognize patterns in the flipped tiles.

You can find them in our online store.

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Cipra City Tiles: a map on a Donut

We’ve been captivated by the Cipra Loops tiles, and we’ve added another Barry Cipra invention to our lineup. Cipra has again found inspiration in the work of Sol Lewitt, who often explored the possible combinations of a graphic idea. For these tiles, that graphic idea is vertical, horizontal, and diagonal lines on a square, and Cipra created a tile for each unique combination. The tiles are oriented. In the image below, the orientation is indicated by the blue triangle. All blue triangles will point in the same direction when composing the 16 tiles. Can you convince yourself that all possible combinations of lines are represented?

Barry Cipra’s oriented tiles with horizontals, verticals, and diagonals

Cipra’s goal is to arrange the tiles in a 4×4 grid, so that all lines run from edge to edge. We thought of the lines as roads, and created tiles that look like this:

Cherry Arbor’s interpretation of Cipra’s tiles

This is the kind of puzzle that starts out seeming easy, but can get difficult as you try to finish. Here is a potential start:

a possible part of a Cipra City map

This, on the other hand, is off to a bad start. Can you see why this cannot be part of a Cipra City map?

not part of a Cipra City map

Here’s one that’s a bit more subtle. Can the arrangement below be part of a map?

What about this arrangement?

Remarkably, if you find a solution, it will also work as a map on a torus (see Cipra Loops for more about the torus). In that case, the roads do not end at the edge of the map. Instead, the road appears on the side opposite where it seems to exit, and continues in the same direction. So every road is actually a loop on the torus, and every loop passes through exactly 4 tiles.

Cipra City tiles are available in our store.

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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 2000 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.