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"Every triangle can tile the plane. Every four-sided shape can also tile the plane."

Can someone point me to a proof of this?




The triangle is pretty easy: take two of the (same size) triangles, with vertices ABC and A'B'C'. Rotate and translate the second triangle to fit the matching side of the first triangle, e.g. AB to B'A'. You now have a quadrilateral with two pairs of equal sides (sides (AB')C and (A'B)C'). The angle on the corners comprised of the two triangles (e.g., C(AB')C') will add together to be 180 degrees minus the angle of the adjacent corners, due to the three interior angles of every triangle summing to 180 degrees. Duplicate that quadrangle and fit the second to a matching side. The angles put together will form 180 degrees, i.e., a straight line. Now you have indefinitely extensible strip. Place the strips next to each other and you've tiled the plane.


Not a formal proof, but close enough I suspect. See comments as well:

http://www.quora.com/Why-do-all-quadrilaterals-tessellate




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