Scientists Uncover the Universal Geometry of Geology (2020)

Summary

Unaware of this work, Domokos wrote a proof which pointed to cubes as the answer. He wanted to double-check, though, and he suspected that if an answer to the same problem already existed, it would be locked in an inscrutable volume by the German mathematicians Wolfgang Weil and Rolf Schneider, an 80-year-old titan in the field of geometry. Domokos is a professional mathematician, but even he found the text daunting. “I found someone who was willing to read that part of the book for me and translate it back into human language,” Domokos said. He found the theorem for any number of dimensions. That confirmed that cubes were indeed the 3D answer. Now Domokos had the average shapes produced by splitting a flat surface or a three-dimensional block. But then a larger quest emerged. Domokos realized that he could also develop a mathematical description not just of averages, but of potentiality: Which collections of shapes are even mathematically possible when something falls apart? Remember, the shapes produced after something falls apart are a mosaic. They fit together with no overlap or gaps. Those cut-up rectangles, for example, can easily tile together to fill in a mosaic in two dimensions. So can hexagons, in an idealized case of what mathematicians would call a Voronoi pattern. But pentagons? Octagons? They don’t tile. Samuel Velasco/Quanta Magazine. Based on graphics from doi.org/10.1073/pnas.2001037117; martian surface: NASA/JPL-Caltech/University of Arizona In order to properly classify mosaics, Domokos started describing them with two numbers. The first is the average number of vertices per cell. The second is the average number of different cells sharing each vertex. So in a mosaic of hexagonal bath tiles, for example, each cell is a hexagon, which has six vertices. And each vertex is shared by three hexagons. In a mosaic, only certain combinations of these two parameters work, forming a narrow swath of shapes that could possibly result from something falling a...