Can You Pave the Plane Nicely with Identical Tiles

TL;DR

This paper reviews the classification of convex domains that can tile the Euclidean plane, highlighting recent discoveries and the completion of the list of such shapes, including new findings on multiple tilings.

Contribution

It summarizes recent advances in classifying convex tiles, confirming the completeness of the list and presenting new results on multiple tilings in the plane.

Findings

Complete list of convex tiles includes triangles, quadrilaterals, hexagons, and pentagons.

Only specific convex polygons can form multiple translative tilings.

New types of octagons and decagons are identified for five-fold tilings.

Abstract

Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain which can tile the Euclidean plane? Yes, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics, which has lasted for more than one century and the completeness of the list has been mistakenly announced not only once! Up to now, the list consists of triangles, quadrilaterals, three types of hexagons, and fifteen types of pentagons. In 2017, Michael Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane; there are only two types of octagons and one type of decagons which can form five-fold translative tilings.