An attempt to answer a question about the layout of this pattern.

Pattern layout is most easily understood by looking for the repeating unit, the tile.The tiling can then be set up easily. Normally it is obvious after the tile is identified.Tiling depends on the symmetry of the star. The 12 point star has 12, 6, 4 and 3 fold symmetry.

Alan D Adams, Holland New York, March 2013

No perfect tiling exists for 12 fold symmetry tiles. ( Perfect tilings cover the page without gaps or overlaps. )Hexagonal tiling for the 12 star, using its 6 fold symmetry, is very common. Hexagons tile without gaps.

Hasret’s figure begins as a hexagonal tiling of a 12 point star. After the first 7 center stars, this pattern changes to another symmetry of the 12 point star, square tiling

This pattern is a mixed, or composite, tiling.

The center of the pattern.

The center 7 stars are a classic tiling; this is a standard parallel sided star.

A hexagon is drawn around the points of the 12 point star.

The hexagons are tiled edge to edge.

There is no additional layout for this block.

A Change to Square tiling.

The 12 point star has 12, 6, 4 and 3 fold symmetry ( and 2 fold, not normally useful for tiling). The block above uses the 6 fold symmetry to define the tile.

Exactly the same layout can be enclosed in a square and used for square tiling. Two corners of the hexagonal tile are clipped off.

The hexagon is shown here with the superimposed square.

This square tile has empty spaces in the corners. This is filled by standard rules to give the tile layout shown here in the upper left - and completed for the center space.

This is a common historic solution to square tiling of this star and is the same layout as the part filled with color in Hasret Brown’s composition.

Composite tiling.

The composite tiling is constructed by choosing two adjacent hexagonal tiles on the figure of 7 tiles and expanding a square tiling of those two tiles. The spacing of the

pattern is still determined by the width of the original hexagonal tile. This is continued around the original hexagonal figure with six sets of square tiled units to Give Hasret’s figure.

The actual geometric definition tiles for this figure:

In cases like this, the rather complex distorted pentagonal tile is not used for layout. The hexagonal tile is laid out at all positions, as in the earlier figures here, and the eight sided figure which fills the empty space in the square tiling sections is laid out after the hexagonal pattern unit.

The eight sided figure is laid out using very common rules with no special decisions required.

This final composite tiling re-establishes sixfold symmetry and hexagonal tiling through the center of the square tiled part of the figure. The larger figure is, therefore, again an infinite tiling layout.

When this larger tile is repeated, hexagonal tiling alternates with square tiling across an infinite figure.

Hasret’s use of color makes a very nice finished drawing.