Taxicab Geometry—A New Slant

Abstract

As anyone who has played with a bit of screen wire knows, a square is not a rigid geometrical figure. A slight push to the right or left will deform a square into a nonsquare rhombus. If such a transformation is performed on the points of a plane which has been coordinatized by a square grid so that the positive y-axis makes an angle of 600 with the positive x-axis and the shorter diagonals of each rhombus are drawn, an isometric grid results. Points will still be named by ordered pairs of real numbers with respect to the x-axis and the transformed y-axis (FIGURE la). There are only three regular polygons which will tessellate the plane: the equilateral triangle, the square, and the regular hexagon. The first two of these can be subdivided into smaller similar polygons (yielding the isometric grid and the square grid). Work has already been done on taxicab geometry using the square grid; this note considers taxicab geometry using the isometric grid. Square-taxi geometry arises because, for the two points A = (xi, Y1) and B = (x2, Y2), a new distance function is chosen: