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Every body knows that identical regular triangles or squares can tile thewhole plane. Many people know that identical regular hexagons can tile theplane properly as well. In fact, even the bees know and use this fact! Is thereany other convex domain which can tile the Euclidean plane? Yes, there is along list of them! To find the list and to show the completeness of the list isa unique drama in mathematics, which has lasted for more than one century andthe completeness of the list has been mistakenly announced not only once! Up tonow, the list consists of triangles, quadrilaterals, three types of hexagons,and fifteen types of pentagons. In 2017, Michael Rao announced a computer prooffor the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made aseries of unexpected discoveries in multiple tilings in the Euclidean plane.For examples, besides parallelograms and centrally symmetric hexagons, there isno other convex domain which can form any two-, three- or four-fold translativetiling in the plane; there are only two types of octagons and one type ofdecagons which can form five-fold translative tilings.

Can You Pave the Plane Nicely with Identical Tiles