parallelogram

To build the lantern, the team began with a thin polymer sheet cut into a diamond-shaped parallelogram.

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!