Back when our kids were young, we spent countless hours together in the kitchen. These moments were perfect for subtly introducing them to scientific concepts such as heat, temperature, fluid dynamics, density, viscosity, as well as physical and chemical changes. We explained the science behind baking, like how carbon dioxide helps cakes rise, and we discussed the role of yeasts and other fascinating microbes.

Math was another frequent topic, especially measurements, areas, and volumes. Pizza provided a fun way to teach fractions. We also loved making cookies using various shaped cutters. Our son was fond of dinosaurs, so we had dinosaur-shaped cutters along with ones shaped like flowers, hearts, stars, and animals. A challenge we often faced involved the leftover dough from the spaces between the cutouts. To avoid wasting dough, we had to gather and re-roll it, which meant it needed chilling before we could use it again, adding more time to our baking process.

Surely, there was a better method to handle the dough efficiently.

Supporting Science Journalism

If you find value in this article, please consider supporting our distinguished journalism by subscribing. Your subscription helps sustain the future of our impactful storytelling about the advancements and concepts that shape our contemporary world.

As someone with a background in mathematics, I was familiar with concepts like tiling and tessellation, which involve covering a surface using one or multiple shapes without any overlaps or gaps. This concept seemed like the solution to our cookie cutter dilemma.

Initially, the only tessellating cookie cutters I could find were basic shapes like squares and hexagons. They were functional but lacked excitement. Then, with the advent of 3D printing about 15 years ago, it became possible to create custom-designed cookie cutters. You could choose any shape that tessellates, including various triangles, rectangles, and some unique pentagons discovered by the late amateur mathematician Marjorie Rice, which are showcased in the mathematical and artistic exhibit Mathemalchemy, featuring contributions from Duke University’s mathematician Ingrid Daubechies and textile artist Dominique Ehrmann.

Islamic art also offers beautiful examples of creative designs using pentagonal symmetry.

Combining two or more figures opens up even more possibilities, such as octagons and squares, or pentagons and rhombuses. M.C. Escher’s work is a testament to this, although not all his designs are practical for cookie cutters—we once made an Escher-inspired lizard cutter, but the legs broke off when separating the cookies.

These tessellations are typically periodic, meaning they repeat a pattern. In the 1970s, Roger Penrose discovered two shapes, kites and darts, that can tile a plane only aperiodically if certain local rules are followed. Without these rules, they tile periodically. This discovery is ideal for cookie cutters because you can arrange the raw dough in a periodic tiling, and then place the baked cookies in either periodic or nonperiodic patterns. More information about Penrose tiles can be found on the Veritasium YouTube channel and a webpage titled “A Penrose-type Islamic Interlacing Pattern.”

A few years back, with the assistance of friends and a 3D printer, I created a cookie cutter in the shape of a kite and dart that tiles periodically. After baking, these cookies can be arranged in both periodic and nonperiodic patterns, which turned out wonderfully.

Then came the Hat.

For decades, mathematicians speculated whether a single figure could tile a plane aperiodically. In 2023, David Smith, Craig Kaplan, Chaim Goodman-Strauss, and Samuel Myers discovered such a figure, which they named “the Hat” or “einstein.”

Inspired, I attempted to create a cookie cutter shaped like the Hat. It proved challenging to arrange without creating gaps or overlaps. However, the team that discovered the Hat also found that it could be transformed into an infinite number of aperiodic monotiles, as reported by Craig Kaplan. Three of these monotiles can tile both periodically and nonperiodically, making them ideal for cookie cutters.

One such shape, referred to as “Tile (0,1)” by Smith and his team, is straightforward to make and use, and the cookies look stunning when arranged in colorful patterns. Another excellent shape is what they called “Tile (1,1),” which tiles periodically when paired with its mirror image but only aperiodically otherwise. Thus, the ideal cookie cutter features these two figures, one normal and one mirrored.

Smith and his colleagues also devised a category of curved aperiodic monotiles, dubbed “Spectres,” which could make great cookie cutters, as could the newly discovered soft cells. With careful cutting, you can significantly reduce dough rerolling and waste.

As you bake during the holiday season, finding a tessellating cookie cutter and involving your children can turn the time usually spent rerolling dough into an opportunity to teach them about science and math. Though our children are grown, my collection of tessellating cookie cutters and I eagerly await sharing these wonders with our grandchildren.

This is an opinion and analysis article, and the views expressed by the author or authors are not necessarily those of Scientific American.