Which flat tilings can you build from the standard set of pattern-block tiles — the equilateral triangle, square, regular hexagon, two rhombi and the trapezoid? Below are all the substantially different repeating tilings whose pieces are regular polygons (the regular and Archimedean tilings), grouped by how many different shapes they use.
Every piece has edge length 1. Interior angles: triangle 60°, square 90°, hexagon 120°, blue rhombus 60°/120°, tan rhombus 30°/150°, trapezoid 60°/120° (its long side is length 2).
There are exactly three ways to tile the plane edge-to-edge with copies of a single regular polygon.
Mixing two of {triangle, square, hexagon} gives exactly four uniform (Archimedean) tilings.
Using all three regular pieces at once gives a single uniform tiling.
Many tilings differ only by cutting up or gluing tiles along shared edges, so they are not really new patterns. The rhombi and the trapezoid never produce a fresh arrangement of regular polygons: each of them is just a few triangles fused together.
If we require the tiling to look identical at every corner (a
uniform / vertex-transitive tiling), then yes — the number is finite.
There are only 11 such tilings of the plane in total: 3 regular and 8 semiregular (Archimedean).
Of the 8 Archimedean tilings, the other three — 4.8.8, 3.12.12 and
4.6.12 — need regular octagons or dodecagons, which are not in this set.
So with the triangle, square and hexagon you can realise exactly:
3.3.3.3.3.3, 4.4.4.4, 6.6.6 (3 tilings)3.6.3.6, 3.3.3.3.6, 3.3.3.4.4, 3.3.4.3.4 (4 tilings)3.4.6.4 (1 tiling)— eight tilings, the complete list shown above.
If we drop the “same at every corner” rule, the answer is no — there are infinitely many. Your own example shows why: between two rows of hexagons you can slip a strip of triangles. Nothing stops you from choosing, row by row, how many strips of triangles to insert between strips of squares, or which way each pinwheel of the snub patterns turns. Each choice gives a genuinely different tiling, and there are infinitely (in fact uncountably) many such choices. Mathematicians count the k-uniform tilings (those with exactly k kinds of corner): there are 20 with k=2, 61 with k=3, 151 with k=4, and the counts keep growing.
So the honest answer is: finite if you insist on one repeating corner-pattern (the eight above), but infinite if you allow tilings that mix corner-patterns or stack strips freely.
Pieces use the conventional pattern-block colours. Coordinates for the snub square, snub hexagonal and rhombitrihexagonal tilings follow their standard lattice/basis descriptions; every drawn patch is a true edge-to-edge tiling computed from those coordinates.