A Little Math Can Streamline Vacation Cookie Making

A Little Math Can Streamline Vacation Cookie Making

By Clara Eugenia Garza-Hume

When our kids had been little, we spent loads of time within the kitchen collectively. We used the events to surreptitiously educate them about science: warmth and temperature, fluid circulate, density, viscosity, bodily adjustments, chemical reactions, acids and bases. We informed them how these properties mixed to magically make desserts rise with carbon dioxide; we informed them about yeasts and different attention-grabbing microorganisms.

We additionally talked about math: weights and measures, areas and volumes. Pizza is a good way to introduce fractions. We loved baking cookies collectively and used many alternative cookie cutters. Our son cherished dinosaurs so we had dinosaur cookie cutters, but in addition flowers, hearts, stars and animals. An attention-grabbing conundrum got here up once we used the cookie cutters: they left gaps between every form we lower out. To make use of up all of the dough, it needed to be rerolled. However then it warmed up and needed to be chilled earlier than we might lower it once more. This took time.

There needed to be a extra environment friendly approach to divide the dough.

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Being a mathematician, I had heard about tilings and tessellations, which confer with the protecting of a floor, utilizing a number of geometric shapes, with no overlaps and no gaps. That was the reply to the cookie cutter drawback!

So I seemed for tessellating cookie cutters however discovered solely squares and hexagons. They work however are usually not notably attention-grabbing, so far as shapes go. After which, about 15 years in the past, the primary 3D printers arrived. It was doable to make customized cookie cutters, after which it turned a matter of selecting the form. Many figures can be utilized to tessellate, comparable to common or irregular triangles, rectangles or some irregular pentagons. A number of the irregular pentagons had been found by the late beginner mathematician Marjorie Rice and might be seen in Mathemalchemy, a creative and mathematical exhibit with collaborators together with Duke College mathematician Ingrid Daubechies and textile artist Dominique Ehrmann. The exhibit additionally exhibits a tessellating cookie cutter utilizing the image for pi, suitably tweaked by Daubechies.

There are different examples; Islamic artwork consists of stunning and inventive designs utilizing pentagonal symmetry.

If we be part of two or extra figures then there are a lot of extra choices. For instance, octagons and squares, or pentagons and rhombuses, or all of the superb work by M.C. Escher. However not all make for good cookies; we made a cookie cutter within the form of considered one of Escher’s lizards however the legs broke once we separated the figures.

All these tessellations are periodic, which suggests they’ve a repeating sample. Within the Nineteen Seventies Roger Penrose discovered two figures referred to as kites and darts that tile a airplane solely aperiodically (which means that every one doable configurations are nonperiodic) if some native guidelines, specifying which sides of the dart can contact one another, are obeyed. But when the foundations are usually not obeyed then they tile periodically. And that is good for a cookie cutter as a result of the uncooked dough is tiled periodically, after which the cookies might be positioned periodically or nonperiodically. There’s extra details about Penrose tiles within the video channel Veritasium and at a Net web page titled “A Penrose-type Islamic Interlacing Pattern.”

A few years in the past, utilizing a 3D printer, with loads of assist from my buddies, I made a cookie cutter within the form of a kite and dart collectively in a approach that tiles periodically. After baking, the items might be positioned periodically or nonperiodically. This labored fantastically.

After which the Hat arrived.

For about 50 years mathematicians puzzled if there might be one single determine that tiled a airplane aperiodically. And in 2023 David Smith, Craig Kaplan, Chaim Goodman-Strauss and Samuel Myers managed to search out the elusive aperiodic monotile, which they referred to as “the Hat” or “einstein.”

My first impulse was to make a cookie cutter within the form of the Hat. It turned out to be extraordinarily troublesome to position the items with out gaps or overlaps. However the individuals who discovered the Hat really additionally realized it might be remodeled into an infinite variety of aperiodic monotiles, as reported by Craig Kaplan. Three of the monotiles can tile each periodically and nonperiodically, and they’re good as cookie cutters.

A chevron that Smith and crew referred to as “Tile (0,1)” is simple to make, straightforward to make use of on the dough, and the baked cookies are straightforward to position in stunning patterns, particularly if they’re embellished with totally different colours. One other good candidate is what the authors referred to as “Tile (1,1).” This determine tiles periodically when mixed with its mirror picture however solely aperiodically if the mirror picture will not be allowed. Subsequently, the right cookie cutter has two of those figures, one regular and one mirrored.

Smith and collaborators additionally invented a class of curved aperiodic monotiles, which they referred to as “Spectres,” which may be changed into cookie cutters, as might the not too long ago found gentle cells. In every of those situations, if you’re cautious, you possibly can lower your cookie dough with minimal rerolling and waste.

As you prepare dinner and bake over the vacations, if you’ll find a tessellating cookie cutter and if you happen to contain your youngsters, you possibly can take the time you’d have spent rerolling dough to inform them about science and math. In our case, our kids are grown, however my assortment of tessellating cookie cutters and I are excited to share the wonders of math and science when our grandchildren arrive.

That is an opinion and evaluation article, and the views expressed by the writer or authors are usually not essentially these of Scientific American.