1. How many regular tessellations are there?
A regular tessellation is one where regular polygons fit together with no gaps or overlapping. Only three regular polygons do this: triangles, squares and hexagons.
2. Why do regular octagons not form a tessellating pattern?
The inside angle of a octagon is 135o and so when you try to put three of them together with no gaps or overlapping the angles add up to 405o and so they do overlap.See diagram below.
Regular hexagons tessellate perfectly.
(The three angles at the meeting point add up to exactly 360o)
Regular octagons do not tessellate.
(The three angles at the meeting point add up to 405o)
