Euclidean Tessellations and Regular Polygons in the Taxicab Plane

Tessellations are planes made from repetitions of geometric shapes without gaps or overlaps. Tessellations exist in any geometry in which polygons exist, including elliptic, hyperbolic and Euclidean geometries, but tessellations in Euclidean geometry are perhaps the most widely known and explored. Of particular interest are regular tessellations, tessellations which, in Euclidean geometry, are made of equilateral and equiangular polygons. We have an intuitive notion of what it means to be equilateral under the usual Euclidean metric, but how does a change of metric affect equilateral and regular Euclidean polygons? Here we explore Euclidean tessellations viewed under a different metric, namely the taxicab metric. First, we examine how the three regular Euclidean tessellations change when viewed under the taxicab metric. After observing that regularity is not preserved by our change of metric, we redefine regular polygons to be simply equilateral polygons under the taxicab metric. With this definition in hand, we explore new options for regular tessellations, and in particular show that we are not limited to only tessellations of triangles, squares, and hexagons as we are under the Euclidean metric.