| Math research has come a long way from its roots. While millennia ago many of the biggest problems dealt with counting objects and sorting shapes, modern math is more concerned with abstract structures and relationships, high-dimensional spaces and intangible definitions.
But every once in a while, mathematicians come back down to earth. They do live here, after all, and the shapes and phenomena they encounter in their daily lives can give them new and important insights.
I think that's why math and crafts often go together. Same with math and games. Earlier this year, at the Joint Mathematics Meetings in Seattle, there was an art exhibit where mathematicians could display intricate paintings, computer graphics and sculptures they'd made. And on the third night of the conference, while many participants returned to their hotel rooms or went out for a drink after a long day of talks, others headed to the Sheraton's Grand Ballroom to join an open-invitation knitting circle.
Sure, even mathematicians need to take breaks. But the proliferation of proofs about the seemingly banal — from dice to knots to card games — is evidence that mathematicians' fascination with the mundane is more than recreation. Often, deep mathematical breakthroughs can be traced back to an idle question about simple, everyday things.
Real Problems
Every mathematician has jotted down a stray formula or observation on a piece of scrap paper, only to find a mistake, crumple it and toss it. So perhaps it's no surprise that the folding and wrinkling of paper has led to many a mathematical theorem. In 2022, Steven Ornes wrote for Quanta about the mathematician Ian Tobasco's efforts to understand how crumpled paper adopts one seemingly random pattern of folds out of all possibilities. He found that different curvatures in a material's initial form can determine what kinds of folds it will acquire when scrunched.
The most orderly type of scrunching — origami — is also a mathematical favorite. Tobasco used the famous parallelogram-shaped Miura-ori pattern as a frame of reference in his work on wrinkling. In 2017, the physicist Michael Assis developed a new understanding of Miura-ori by connecting it to statistical physics. He thought of the origami pattern as a crystal lattice composed of atoms, and he encoded missteps among the folds as crystalline defects. This allowed him to uncover a sort of phase transition in the origami's structure. "In a sense, it shows origami is complex," he said. "It has all the complexities of real-world materials. And at the end of the day, that's what you want: real-world metamaterials."
Steve Nadis' 2020 Q&A with L. Mahadevan also touched on the Japanese art of paper folding, as well as the natural appearance of tissue folds in the brain and intestines. Mahadevan uses the world as a mathematics laboratory, "finding the sublime in the mundane." He has also studied the shape of an apple, the cracking of mud and the clumping of cereal in milk.
Persi Diaconis began his career as a professional magician before turning to math. He has since proved a number of results about playing cards, including the famous fact that you need to shuffle a deck seven times to guarantee that its cards are fully randomized. It turns out that, by examining the nature of magic tricks, you can learn a great deal about randomness, probability and more. Erica Klarreich covered some of this work for Quanta in 2015.
More abstract areas of study, like topology, might sound exotic, but we encounter them every morning. Mathematicians explain topology using two breakfast items: It's the field where a coffee mug and a doughnut are the same. We tie our shoes with topological knots before going out for the day. Mathematicians are on an ongoing quest to construct the most extreme versions of many topological shapes, including the fattest Möbius strip and the optimal trefoil knot. A story by Kevin Hartnett last year broke down four recent such breakthroughs and explored how mathematicians are using that understanding to formulate deeper, more abstract questions.
Another race to construct a weird thing finally ended a few months back. The tetrahedron — a pyramidlike shape consisting of four triangular faces — is one of the most basic solids, but "millennia later, mysteries still surround even the simplest shape," as Quanta contributor Elise Cutts wrote this summer. Her piece describes how one of those mysteries was just solved — not only in a proof, but in the physical world. Mathematicians constructed a tetrahedron that can only rest on one of its four sides, giving them a better intuition about the nature of the fundamental shape.
In all these cases, questions about the trivial and everyday have been crucial to making the world of mathematics a richer, more interesting place. | 
