Which regular polygons will fit together to tile a surface?

Answer 1

It depends on the shape of the surface

Flat surface can be tiled by triangles, squares, and hexagons, these are the only combinations for the regular tessellations.

semi-regular tessellations (where multiple polygons are used in the same tiling)

There are in fact an infinite number of possible tessellations. All polygons can work from triangles to approaching a circle... a circle tiling would require an infinite number of infinitesimally small polygons around it, so you may or may not consider this a possibility.

NOT all polygons can be in the same tessellations, for example triangles, heptagons, and 42-gons cannot be in a 1:1:1 ratio.

In 3 dimensions regular polygons can be perfectly assembled into only 5 regular polyhedrons (3d version of polygon) (the platonic solids - these have been used to represent the elements, fire water, earth air and space)

tetrahedron consists of 4 triangles

cube (hexahedron) consists of 6 squares

octahedron 8 triangles

dodecahedron 12 pentagons

icosahedron 20 triangles

The hexagon didn't make it... possibly an infinite number of would assemble a sphere of infinite diameter, but this has never been included in any lists I've run across.

In 4 dimensions, there are six convex 4-polytopes, called (polychorons), the smallest of which is called the pentatope, and is composes of 10 triangles, which can only be done in 4 dimensions, it can't be constructed under normal circumstances in our worlds.

In 5, 6, 7, 8, 9, and 10 dimensions that are only 3 regular n-polytopes for each respectively... this may continue indefinitely but I don't know how to prove this, it's probably been done. If it does continue toward infinite dimensions that 2 and 3 dimensions are "special" and perhaps that is why we find ourselves in such a universe.