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 LESSON 4.   Finding Patterns Scientists and mathematicians are always looking for connections in math and science.  They collect and analyze data to see if a pattern emerges.  If a mathematical model of an experiment can be developed, it usually reveals what a scientist may have discovered.   We see tiling all the time from brick walls to tiles on your bathroom walls.  Artist and brick workers have used tiles in design throughout humanity.  However, mathematicians and scientists have developed properties and rules to classify patterns.  This information may help understand materials especially at the microscopic level. Tessellations are an assemblage of regular polygons that do not overlap and creates an arrangement or repeatable pattern.  The word “tessellate” is derived from a Greek derivation which refers to the four corners of the tiles in a mosaic.  Only three regular polygons tessellate in the Euclidean plane: triangles, squares and hexagons. The pattern caused by all triangles needs to “flip flop” the triangles to make them fit.  A square and a hexagon stack up without inverting one of the shapes. Equilateral triangles have three-fold symmetry, squares have four-fold symmetry, and hexagons have six-fold symmetry. A tile from the Roman times (Pompeii) Alhambra in Granada, Spain
 Square tilings Hexagonal stacking Tessellating triangles
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