High-dimensional Sudoku puzzles prove mathematicians wrong on long-standing geometry problem

High-dimensional Sudoku puzzle proves mathematicians wrong on long-standing geometry problem

By Max Springer

Tiling a two-dimensional bathroom floor is a simple home improvement project, but researchers have found that in higher dimensions it can evolve into nonrepetitive, incomprehensible chaos. The new results overturn a long-held tiling hypothesis and point to another way in which disorder can emerge from structured domains in mathematics.

Generally speaking, a tiling is a way of covering a space with many small pieces (tiles) that fit together without gaps or overlaps. A never-ending bathroom floor or an infinitely large car trunk being loaded for a road trip are natural examples in two or three dimensions. A tiling is “periodic” if copies of a single shape are combined in a pattern that repeats in all directions to fill the space. This is similar to the monumental task of loading an infinite number of car trunks with equally sized luggage arranged in a pattern. The periodic tiling hypothesis addressed in this study states that any shape that can tile a space without rotating or flipping should be able to do so regularly by repeating it.

The study authors wrote: Mathematics Annual Reportdisproved this conjecture by constructing strictly aperiodic tiles — tiles that cover space completely without any regular pattern. To do so, they transformed the geometric tiling problem into an algebraic problem defined by a system of equations. Each equation captures constraints that the tiling must obey, such as no rotations or gaps between tiles, forming a kind of “tiling language,” says Rachel Greenfeld, a mathematician at Northwestern University and co-author of the study.

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Adding constraints to the language reduces the number of possible solutions, in the same way that the number of numbers you can put into Sudoku squares decreases as you fill in the puzzle. The final solution, a sequence of non-repeating numbers, can be retransformed into strictly aperiodic tiles, disproving the conjecture. “The tiling is not simple enough to work right forever, but it’s not complex enough to work crazy forever,” Greenfeld says.

By disproving this result, the researchers “found a way to turn the tile geometry into a programming language,” says Craig Kaplan, a computer scientist at the University of Waterloo. They achieved this by adding more and more constraints that lead to extra dimensions, so the counterexample turned out to work in extremely high-dimensional spaces, around 10 dimensions.^100,000 Dimension (100,000 digit number)

“Higher-dimensional tilings are extremely complicated,” says Terence Tao, a Fields Medal-winning mathematician at the University of California, Los Angeles and co-author of the study. “Things seem to work out much better in lower dimensions, but three-dimensional is the state of the art right now.” Comparing this intuitive space with higher-dimensional results, he says, we’re “on the border between order and total chaos.”