![]() ![]() And it keeps us moving toward inventing, or discovering, new mathematics that previous generations couldn’t dream of. But it is another step forward in answering a long line of questions that started when Archimedes first invented, or discovered, $latex \pi$. Scissors alone can’t produce the 10 200 pieces needed in their decomposition. Unfortunately, you won’t be able to use their result to settle any brownie bake offs. Which is where Andras Máthé, Oleg Pikhurko and Jonathan Noel came in: In early 2022 they posted a paper in which they matched Laczkovich’s accomplishment, but with pieces that are possible to visualize. It didn’t explain how to construct the pieces, only that they could exist. In 1990 Miklós Laczkovich proved that it’s possible to slice up a circle and rearrange it into a square, as long as you can use infinitely small, infinitely disconnected, infinitely jagged pieces that couldn’t possibly be produced with a pair of scissors.Īs surprising and exciting as Laczkovich’s result was, it only proved that such a decomposition is theoretically possible. ![]() A bit of rearranging resulted in audible oohs and aahs from the crowd: Gina had turned the rectangle into an exact replica of the square.īut as they always seem to do, mathematicians turned this obstacle into new mathematics. With three quick cuts she turned the 9-by-4 piece into three smaller 3-by-4 pieces. ![]() This turned the large rectangle into two smaller ones: one measuring 9-by-12 and the other 9-by-4. Gina measured 12 inches down the long side of the rectangular brownie and made a cut parallel to the short side. What was Gina to do? How could she convince the teams that their brownies were the same size if they didn’t understand how to measure area and multiply numbers? Luckily, Gina had a genius idea. A third said, “I heard about students at Complex College measuring area using numbers once, but what does that even mean?” Imaginary University was a strange place indeed, even as dreams go. “I don’t understand what you mean by ‘times,’” said one student, who had never been taught multiplication. The brownies are the same size: It’s a tie.”īoth teams looked puzzled. “The area of the square brownie is 12 times 12, which is also 144 square inches. “The area of the rectangular brownie is 9 times 16, which is 144 square inches,” she said. Gina found it strange to be arguing about this. “But the short side of your rectangle is much shorter than the side of our square,” said a representative from Team Beta. “Ours is clearly bigger, so we are the winners!” “Our brownie is much longer than yours,” said Team Alpha’s captain. Team Beta quickly followed with their square brownie, which measured 12 inches on each side. Gina pulled out a ruler and measured the brownie: It was 16 inches long and 9 inches wide. ![]() Team Alpha was the first to finish, and they proudly presented their rectangular brownie for judging. Teams of Imaginary U students were tasked with making the biggest brownie they could, and it was up to Gina to determine the winner. Gina found herself the judge of the Great Brownie Bake Off at Imaginary University, a school where students learn lots of geometry but very little arithmetic. Gina the geometry student stayed up too late last night doing her homework while watching The Great British Bake Off, so when she finally went to bed her sleepy mind was still full of cupcakes and compasses. ![]()
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