On the Flat Phi page, we saw that the two triangles that appear in the pentagram and pentagon were used by Roger Penrose to design tiling patterns with five-fold symmetry called Penronse tilings. Is there a three-dimensional analogue of those two-dimensional tilings? The answer, thought to be impossible until Penrose's work of the early 1970's showed that there could be structures that filled space (in the same way that tilings fill planes) that have five-fold symmetry.
Three-dimensional Solid Geometry and the Golden section
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