So regular pentagons can't tessellate.įor a shape with more than 6 sides, the number of vertices would have to be less than 3, which is impossible. So for a pentagon, the number of vertices would have to be greater than 3 but less than 4, so it can't be an integer. This table shows how this works for the three shapes: Shapeįor a tessellating square, 4 vertices meet, and for a tessellating hexagon, 3 vertices meet. If we try this with an octagon(8 sides) or any shape with more sides, this will also fail for the same reason.įor regular tessellation, the vertices of the shapes must meet at a point, which means that the internal angles of all the shapes that meet at a vertex must add up to 360°. If we try the same thing with regular heptagons (7 sides) we see a slightly different issue:Īfter joining two heptagons, the remaining angle is too small to add an extra heptagon. The angle is too small for another pentagon to fit, so it is impossible to fill the plane without the shapes either overlapping or leaving a gap. If we place three regular pentagons (5 sides) together, it leaves a small angle. To understand why this is, we will look at a couple of regular shapes that don't tessellate. The only regular polygons that tessellate are those with 3, 4 or 6 sides. Why will no other regular polygons tessellate? Here is a regular tessellation made up of regular hexagons: Rectangles, or any other quadrilaterals, will tessellate, but if the quadrilaterals are not squares it won't be a regular tessellation. Here is a regular tessellation made up of equilateral squares: Here the base of the red triangle meets the edges of two different blue triangles. This means that in some cases an edge of one triangle meets two edges of other triangles: Here notice that the horizontal bands of triangles are shifted relative to each other, so the edges of some triangles do not line up. The example below is not edge-to-edge tiling: There are no edges that join two or more other edges. In the scheme above, every edge of any triangle is joined to a complete edge of a different triangle. Here is a regular tessellation made up of equilateral triangles:Īny triangles will tessellate, but if the triangles are not equilateral triangles it won't be a regular tessellation. We will look at these cases, and also learn why no other regular tessellations are possible. There are only three regular shapes that can form regular polygons. Regular tessellations also need to use edge-to-edge tiling, as we will see in a moment. A tiled floor is a real-life example of a tessellation.Ī regular tessellation is a special case where the plane is covered by shapes that are all regular polygons of the same size. An alternative name for a tessellation is a tiling. A tessellation is created when one or more shapes are used to completely cover a plane, with no gaps or overlaps.
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