Joan Taylor's Tilings
Rhombus seed tilings
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 Rhombus 8sym PDF 2.1MB |
 Rhombus 12sym PDF 2.4MB |
 Rhombus 10sym PDF 2.2MB |
 Rhomb 8 & 12 sub tiles PDF 5.0MB |
 Rhomb 10 sub tiles PDF 4.3MB |
 Rhomb 14 sub tiles PDF 0.7MB |
 Rhombus 6sym PDF 11.0MB |
 Rhomb 6 sub tiles PDF 5.7MB |
Even simpler than a hexagon "seed" shape (Eve, Penrose's rhombus, Ammann-Beenker) is the rhombus seed shape. Here we have centres with 8, 10 and 12-fold rotational symmetry in Rhombus 8sym, Rhombus 12sym and Rhombus 10sym. The substitution schemes follow in Rhomb 8 & 12 sub tiles and Rhomb 10 sub tiles. Rhomb 14 sub tiles shows the substitution scheme for 14-fold rotational symmetry, which uses 10 prototiles. These tilings all adhere to unit sided tiles as a restriction. In the 14-fold tiling at the 9th inflation tile 10 will be inflated for the first time and the resulting inflated rhombus will be over 794 units along its long diameter. This is why you don't see the tiling completed to this stage - at 2cm per side it would be a little under 16 metres long! The others are on A1 paper.
Rhombus 6sym and Rhomb 6 sub tiles. This tiling was found to be describable with matching rules entailing 18 prototiles and another with 12 prototiles but a rigorous proof was never produced, 18 and 12 being very far from a single tile.