Some tiles associated with the 6th unit cubic Pisot number

Order 7 Paraallosymmetric Tiles

VI. Tiles with large areas of 50% coverage

There are two sets of paraallosymmetric tiles with dissection equation c+c2+3c3+c4+c5. These are some of the complex teragons from one of those sets. Unlike the previous tiles (groups I-V), at the plotted resolution these tiles have a mixture of regions that cover 100% of the plane, and regions that cover 50% of the plane. To show this, in addition to the dissections of the tiles, plots of their measure functions are displayed below. (Purple is 100% and green 50%.)

paraallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cell paraallosymmetric tileparaallosymmetric tile unit cell

Because of the nature of these tiles plotting the unit cell in conventional fashion (a separate colour for each copy in the unit cell) doesn't effectively show that each point in the unit cell is covered by one and only copy of the tile. To do this requires plotting the measure function instead, which shows that the density of points is uniform across the unit cell.

paraallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cellparaallosymmetric tileparaallosymmetric tile unit cell paraallosymmetric tileparaallosymmetric tile unit cell

© 2017 Stewart R. Hinsley