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What regular polygons can be used to make a regular tesselations?
A regular polygon's angles are measured by the formula 180 * (n - 2) / n. Regular polygons will only tesselate if m * ( 180 * (n - 2) / n ) = 360, where m is an integer. Let's go through all the possible regular polygons. 3 sided polygon: m * ( 180 * 1 / 3 ) = 360 -> 60m=360 -> m=6, Able to tesselate 4 sided polygon: m * ( 180 * 2 / 4 ) = 360 -> 90m=360 -> m=4, Able to tesselate 5 sided polygon: m * ( 180 * 3 / 5 ) = 360 -> 108m=360. Not able to tesselate 6 sided: m * ( 180 * 4 / 6 ) = 360 -> 120m=360. Able to tesselate We do not need to check more, for the polygons that are able to tesselate have a decreasing m value, from 6 to 4 to 3. The next possible m value would be 2, and we know this cannot happen, because if m = 2, then the polygon would have to have angles of 180 degrees; impossible. Therefore, we can only tesselate using triangles, squares, and hexagons.
Can you tesselate a regular octagon?
No, you cannot.
Does a decagon tesselate?
yes of course * * * * * A decagon does NOT tessellate. All triangles and quadrilaterals do, there are 14 tessellating pentagons and a number of hexagons (including regular hexagons). There are no tessellations which use polygons of the same shape - regular or irregular - for polygons with 7 or more sides.
Does a regular 20 sided polygon tesselate?
No
Can a pentagonal prism tesselate?
Yes. Can it tesselate R^3? Only if its base pentagon is non-regular.
