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# A mathematical note

For those with some mathematical background:

This site explores the extension of vertex figures of regular polygons and rhombs to tilings of the plane.

Imperfect Congruence is accessible to those with only a high school mathematics background, and displays beautiful tiling patterns, many of which (those involving rhombs) do not appear to have been presented before.

The great mathematician and astronomer Johannes Kepler published the first major study of vertex figures of regular polygons in 1619 as Book II of Harmonices Mundi. In this study he showed that 11 of these vertex figures, which he called perfect congruences, could be extended to uniform tilings of the plane.

In Part 1 of this site, I review Kepler's results and present them in perhaps a clearer format than Kepler's pioneering study allowed.

In Part 2 of this site, I examine Kepler's imperfect congruences and in particular the 6 "forbidden" vertex figures of regular polygons that cannot be extended to any tiling of the plane. Inspired by illustrations by the renaissance painter Albrecht Dürer, I show that these figures can be extended to periodic tilings of the plane if rhombs are allowed in the prototile sets.

In Part 3 of this site, I look at vertex figures composed of both regular polygons and rhombs. I am able to draw some very interesting conclusions about vertex figures restricted to three polygons and I describe in detail an efficient algorithm that can construct a periodic plane tiling that includes any given regular polygon. Part of this description includes an algorithm to decompose any regular polygon of composite order into smaller regular polygons and rhombs. In the concluding sections of Part 3, I outline an algorithm that appears to be able to extend any vertex figure of rational rhombs and/or regular polygons to a periodic tiling. However, a formal proof that this algorithm actually works will take much more effort to complete.

In Part 4 of this site I look in more detail at tilings extended from the 3.10.15, 4.5.20 and 5.5.10 vertex figures that exhibit some local five-fold rotational symmetry. I show that the 5.5.10 vertex figure in particular can be extended to a set of tile patches that create quasiperiodic tilings that behave like Penrose tilings.