Isoperimetric Pentagonal Tilings

Abstract

I n 2001 Thomas Hales ([H]; see [M1, Chap. 15]) proved the Honeycomb Conjecture, which says that regular hexagons provide a least-perimeter tiling of the plane by unit-area regions. In this paper we seek perimeter-minimizing tilings of the plane by unit-area pentagons. The regular pentagon has the least perimeter, but it does not tile the plane. There are many planar tilings by a single irregular pen-tagon or by many different unit-area pentagons; for some simple examples see Figure 3. Which of them has the least average perimeter per tile? Our main theorem, Theorem 3.5, proves that the Cairo and Prismatic tilings of Figure 1 minimize perimeter, assuming that the pentagons are con-vex. We conjecture that this convexity assumption