Why can a regular hexagon tesselate, but a regular pentagon does not ?
At each vertex (corner), the direction of the side of a regular pentagon must change by 360°/5 = 72°. That makes each internal angle (180-72) = 108°. Draw a sketch and you’ll see what I mean. If n pentagons were to fit together around a point, to make a tesselation, then we would require the angles of the n pentagons to fill the whole 360° angle at that point. But to have 108n = 360 we need n= 3.3333, i.e. not a whole number. So 3 pentagons don’t fill the angle, and there is no room for a 4th. Hence it cannot be done.
