Why can't some shapes tessellate?

Answer 1

A "tessellation" (also called a "tiling") of a plane region is a covering of that 2-dimensional region using shapes that don't overlap and don't leave any gaps uncovered. Typically, we are interested in trying to use shapes that are congruent (all the same size and shape) regular polygons (the angles and sides of each polygon are the same), such as an equilateral triangle, a square, a regular pentagon, etc. This is called a "regular tessellation". It has been shown that the only regular polygons that tessellate are equilateral triangles, squares, and hexagons. So for example, a regular pentagon can't be used to tile a floor, because the angles don't match up as needed and will leave gaps on the floor that would need a different shape to fill them in. Consider, for example, a regular octagon. Each interior angle is 135o. So if you put two octagons next to each other, sharing a common side, then the two interior angles would combine to be 270o. But that leaves only another 90o of the full 360o at the point the two edges meet and need another shape to complete the tiling, which is not enough room to squeeze in another octagon that would take up 135o. The 90o does allow enough room for a square, however, and in fact octagons and squares can be combined to tile a floor in what is called a "semiregular tessellation" (using more than one shape).