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.