Polygons at a point?

Question

Why is it not possible to have a point-fill set containing a triangle and a pentagon?

Answer 1

What is a point-fill set?

If you are referring to tiling a planar area with regular polygons, the 60 degree interior angles of the equilateral triangle and the 108 degree interior angles of the regular polygon do not play nicely together ... no sum 60 n + 108 m (n, m integers >= 0) adds up to 360 degrees, so you can't join a corner of several at a point without gaps and/or overlaps.