Tessellation revelation

This article was originally published in Scientific American on November 19, 2024.

How few corners can a shape have and still tile the plane?” mathematician Gábor Domokos asked me over pizza.

His deceptively simple question was about the geometry of tilings, also called tessellations—arrangements of shapes, called tiles or cells, that fill a surface with no gaps or overlaps. Humans have a preoccupation with tessellation that dates back at least to ancient Sumer, where tilings featured prominently in architecture and art. But in all the centuries that thinkers have tinkered with tiles, no one seems to have seriously pondered whether there’s some limit to how few vertices—sharp corners where lines meet—the tiles of a tessellation can have. Until Domokos. Chasing tiles with ever fewer corners eventually led him and his small team to discover an entirely new type of shape.

It was the summer of 2023 when Domokos and I sat at a wood picnic table at the Black Dog, a cozy spot for pizza and wine just a few blocks from the Budapest University of Technology and Economics, where Domokos is a professor. He reached across the table to grab a paper pizza menu and flip it over, revealing a blank underside, and gestured to me to grab a pen. The midsummer sky was taking on shades of orange and indigo as I filled the menu with triangles. Domokos watched expectantly. “You’re allowed to use curves,” he finally said. I started filling the page with circles, which of course can’t fill space on their own. But Domokos lit up. “Oh, that’s interesting!” he said. “Keep going, you can mix shapes. Just try to keep the average number of corners as low as possible.”

I kept going. My page of circles filled with increasingly desperate, squiggly forms. Domokos’s pizza Margherita had long since disappeared, but he wasn’t quite ready to leave. A quick glance at my crude drawing wasn’t enough to determine its average number of corners, let alone the minimum possible. But the right answer must have been something less than the triangle’s three—otherwise, the question would be boring.