December 23, 2024
5 min learn
A Little Math Can Streamline Vacation Cookie Making
Making cookies is time- and labor-intensive. Right here’s how just a little math could make it simpler and fewer wasteful this vacation season
A cookie cutter that creates a tessellation can save time and expend extra dough.
When our kids had been little, we spent numerous time within the kitchen collectively. We used the events to surreptitiously educate them about science: warmth and temperature, fluid movement, density, viscosity, bodily adjustments, chemical reactions, acids and bases. We informed them how these properties mixed to magically make truffles 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 an effective way to introduce fractions. We loved baking cookies collectively and used many various cookie cutters. Our son liked dinosaurs so we had dinosaur cookie cutters, but additionally flowers, hearts, stars and animals. An attention-grabbing conundrum got here up after 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 option to divide the dough.
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Making cookies utilizing extra conventional cookie cutters typically leaves numerous leftover dough.
Being a mathematician, I had heard about tilings and tessellations, which confer with the masking of a floor, utilizing a number of geometric shapes, with no overlaps and no gaps. That was the reply to the cookie cutter downside!
So I appeared for tessellating cookie cutters however discovered solely squares and hexagons. They work however usually are not notably attention-grabbing, so far as shapes go. After which, about 15 years in the past, the primary 3D printers arrived. It was attainable to make customized cookie cutters, after which it grew to become a matter of selecting the form. Many figures can be utilized to tessellate, comparable to common or irregular triangles, rectangles or some irregular pentagons. Among 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 reveals 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 a part of two or extra figures then there are lots of extra choices. For instance, octagons and squares, or pentagons and rhombuses, or all of the wonderful work by M.C. Escher. However not all make for good cookies; we made a cookie cutter within the form of one in every of Escher’s lizards however the legs broke after we separated the figures.
All these tessellations are periodic, which suggests they’ve a repeating sample. Within the Seventies Roger Penrose discovered two figures known as kites and darts that tile a aircraft solely aperiodically (that means that each one attainable configurations are nonperiodic) if some native guidelines, specifying which sides of the dart can contact one another, are obeyed. But when the foundations usually are 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 tessellating cookie cutter, comparable to this one within the form of a kite and dart, creates a pleasant sample and makes use of up extra dough than different shapes.
A few years in the past, utilizing a 3D printer, with numerous assist from my associates, I made a cookie cutter within the form of a kite and dart collectively in a method 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 questioned if there might be one single determine that tiled a aircraft aperiodically. And in 2023 David Smith, Craig Kaplan, Chaim Goodman-Strauss and Samuel Myers managed to seek out the elusive aperiodic monotile, which they known 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 tough to put the items with out gaps or overlaps. However the individuals who discovered the Hat truly additionally realized it might be reworked 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 known as “Tile (0,1)” is straightforward to make, simple to make use of on the dough, and the baked cookies are simple to put in stunning patterns, particularly if they’re adorned with completely different colours. One other good candidate is what the authors known as “Tile (1,1).” This determine tiles periodically when mixed with its mirror picture however solely aperiodically if the mirror picture isn’t allowed. Due to this fact, the right cookie cutter has two of those figures, one regular and one mirrored.
A cookie cutter primarily based on the Hat, found by David Smith, makes a tessellating sample that wastes little dough.
Smith and collaborators additionally invented a class of curved aperiodic monotiles, which they known as “Spectres,” which is also became cookie cutters, as might the lately found smooth cells. In every of those situations, if you’re cautious, you may lower your cookie dough with minimal rerolling and waste.
As you prepare dinner and bake over the vacations, if yow will discover a tessellating cookie cutter and should you contain your youngsters, you may take the time you’ll 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 usually are not essentially these of Scientific American.
