Thanks for purchasing the Aperiodic Monotile Spectre Cookie Cutter from Maths Gear (or, for following the link on the packet of someone else’s cookie cutter to get to this page - you’re also very welcome, and if you want to buy a cutter you can do that now).

To find out more about the maths behind the Spectre, take a look further down this page.

If you want to use the cutter to bake aperiodic tile cookies, here are some recommendations from us:

• Use a smooth dough recipe: chocolate chunks are delicious, but they can make the edges of your tiles go funny and stop things matching up

• If your dough is one which spreads out slightly during baking, and you want the cookies to fit together into a tiling, you may need to re-cut the shapes after they come out of the oven but before they cool

• Refridgerate the dough between making it and cutting it, so it holds the shape better

• Since the shapes fit together into a tiling, you should be able to just roll out one sheet of dough and cut a bunch of cookies out of it - use the diagram below as a guide. You can then separate these out for baking, or keep them on the same sheet and re-cut the lines afterwards with the same cutter once they’re out of the oven.

What is a Spectre?

This cookie cutter is made in the (approximate) shape of the Spectre tile: a new mathematical discovery, made in 2023 which solved a long-standing open question in maths called the Einstein problem. The problem concerns which shapes can tile and cover the whole plane in specific ways.

Shapes like squares and regular hexagons can tile an infinite plane and cover the whole area without gaps - but they do so in a way that’s periodic: if we take a picture of the tiling and shift it across, or down, or rotate it around through some angle, we get the same pattern back. Periodic tilings have a form of symmetry, and they repeat the same layout over and over forever.

Above: tiling by squares; below: tiling by regular hexagons

It’s possible to construct non-periodic tilings in a boring way, like taking a tiling by squares, and cutting one of the squares in half. Now the tiling can’t be slid across and matched up ever again, so it’s non-periodic.

But a question mathematicians wanted to answer was, can we find a tiling that’s aperiodic: that the shape of the tile itself forces you to tile in such a way that it never produces a repeating pattern?

Sometimes, one single shape can be used to make both periodic and non-periodic tilings (like a right-angled triangle with sides of length 1 and 2, which can either form the pinwheel tiling or a boring periodic tiling, depending on how it’s arranged).

Above: Pinwheel tiling - image by FabriceReix CC BY-SA 4.0; Below: tiling a plane periodically using the same triangle

But if we can find a shape which allows a non-periodic tiling, and give instructions about how the edges have to match up - or, make the edges of the shape curved so they are forced to fit together in specific ways - we can create a tile which has to tile non-periodically, which is called an aperiodic tile.

There had been some attempts at this, but they each had slightly different problems:

• Penrose tiles, famously discovered by physicist Roger Penrose; these can be made to produce an aperiodic tiling, but there are two tiles you need to use in an alternating pattern, not a single tile (or monotile).

• Shapes like the Socolar–Taylor tile, proposed in 2010, which is technically an aperiodic monotile, but each tile is made of disconnected pieces (and they’re a nightmare to bake).

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Above: Penrose Tiling; Below: Socolar-Taylor tile, image by Parcly Taxel CC BY-SA 4.0

Then, in 2023, a team of mathematicians published the paper  "A chiral aperiodic monotile” (Smith, Meyers, Kaplan, Goodman-Strauss). They had discovered a shape which does everything we need - called The Spectre, owing to its mildly ghost-like shape, this tile creates non-periodic tilings of the plane that never repeat, and is a single tile that’s one connected piece.

The discovery was an extension of a previously discovered shape from earlier the same year, called the Hat Tile, which Smith had found while investigating tiling shapes. This shape, made from a section of a hexagonal grid, also tiles aperiodically, but needs to include some number of copies of the tile in its mirror image form (which could be achieved by turning the biscuit over, but then you’d get icing and sprinkles all over the table).

Hat tiles. Image: from An aperiodic monotile, Smith et al 2023

The Spectre tile solves this problem, and arbitrarily large regions of tiling can be made using this one shape, all the same way up. When you’re arranging the biscuits into a tiling, bear in mind that not all combinations of the biscuits/tiles will fit together - it’s possible to put some of the pieces in so you create a gap that can’t later be filled. If you follow a diagram like the one at the top of this page, or read the full research paper, there are techniques to ensure your tiling will keep working and you can keep baking to infinity, without repeating yourself.

More information:

The Aperiodical article about the original ‘Hat tile’ discovery

Follow-up article on discovery of the Spectre

Mathematical Objects podcast - interview with Chaim Goodman-Strauss

A chiral aperiodic monotile (2023, Smith, Meyers, Kaplan, Goodman-Strauss)

An aperiodic monotile (2023, Smith, Meyers, Kaplan, Goodman-Strauss)