The start of a Penrose tiling from pentagons. Three types of gaps left between the pentagons (stars, diamonds, boats) are usually filled with colored tiles, but I just wanted to explore how the pentagons determine the tiling, and the resulting five-fold symmetry. I need to clean it up a bit, make the tiles line up better. Let me know of any thoughts.
Triangles can be tiled, without gaps, on a wall. Squares can too. So can hexagons (because made up of triangles). But pentagons cannot be tiled on a wall (2D surface) without gaps. Roger Penrose made up a tiling by filling in the gaps with three other shapes (stars, diamonds, boats). But these shapes emerge between pentagons, when you tile the pentagons as best you can. Penrose went on to show how the tilings, if matched according to his rules, formed patterns that were non-repeating, aperiodic. This is the real genius of his tilings, and he had many more aperiodic tilings. My very modest draft projects is just a start to show the five fold symmetry based on a pentagon. Interestingly, what I see as the dominant motif, 5 pentagons surrounding one, if you cut two out of paper and connect the edges, forms a 3-dimensional dodecahedron, so the pentagon can tile in 3-dimensions.
