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This is the first of two articles describing how tessellations can be defined and drawn using Mathematica. It contains functions that can be used to define and draw tilings of various types, and gives examples of their use. Section headings include the graphics primitive of a list of complex numbers, the centroid of a polygon, numbering vertices and faces, regular polygons and stars, Kepler's monster, identifying two polygons, standard polygons other than regular polygons and stars, moving a polygon adjacent to another, polytope structure, subdividing a polygon, color arguments, painting a tiling, mappings and symmetry groups, the dual tiling, drawing a frame around a tiling, Eberhardt's tiling, tiling by isosceles triangles, regular tilings: tiling by square, triangles and hexagons, semi-regular tilings and star tilings, and mapping tessellations: similarity patterns and mapping a tiling onto a surface. In a second article, I'll relate tessellations to the symmetries of the plane.
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