Pentagons and Isosceles Trapezoids

This dot pattern diagram from the series, Self-similar Boundaries, is based on pentagons and isosceles trapezoids. The trapezoids are the scaling polygons. They share non-parallel sides with a pentagon and parallel sides with scaled pentagons. The trapezoids generate one pentagon that scales down and one that scales up. The initial pentagon generates one trapezoid, with two scaled pentagons. Each subsequent iteration could double the number of pentagons, except that many are duplicates. In this example I follow three scaling iterations which might have produced 1 + 2 + 4 + 8 = 15 different pentagons, but since some are equal the pentagons produced are 10.

Pentagons and Scalene Triangles

This dot pattern diagram from the series, Self-similar Boundaries, is based on pentagons and scaling scalene triangles. Neither the design of the tile set nor the arrangement of tiles makes any attempt to create symmetry or even a tiling that fills the plane. Nevertheless, the tile set and tiling are interesting in that they can be used to generate tendrils that extend indefinitely. In this case the tile set scales down seven times.

Scaling Triangles

These two new dot patterns from the series, Self-similar Boundaries, are based on different tiles sets. They have in common triangles for scaling tiles. In the first example below the tile set is three rhombi and an isosceles triangle scaling tile. The second image tile set is a pentagon and two mirror image scalene triangle scaling tiles. I've included the tilings for the dot patterns.