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During those experiments, Schwartz cut open a Möbius band and realized, “Oh, my God, it’s not the parallelogram. It’s a trapezoid.”

They could arrange four hat tiles into a hexagonlike structure, two tiles into a pentagon and another combination of two tiles into a parallelogram.

Two classes of shapes work, they proved: “boring parallelograms” and “surprising cyclic quadrilaterals,” cyclic meaning that all vertexes of a quadrilateral lie on a circle.

Now, we have to get it square since at this point it’s probably a parallelogram, not a right rectangle!

In the tiling of colored dots representing where the rat was when each grid cell fired, four neighboring red dots formed the corners of a slanted rectangle known as a parallelogram.