Here's a little play with hexagons and pentagons. Basically, the outside loop comes from this big loop below. The rest is mere decoration. The pentagon in the middle was difficult to place(and may not be at the theoretical centre).

This quickie pattern uses a simple section that combined with a bunch of mirrors creates a nice grid. Unlike the latest previous pentagon and triangle patterns, this does not have an alternating of the polygons. Below is a section that will produce this pattern as wallpaper.

Sunflower

This pattern of a sunflower was a bit tricky to construct. The outer orange section (the petals) was easy given the high level of repetition but filling the insides (the seeds) was not straight forward. The pattern is built with alternating pentagons and triangles. The five-fold symmetry is present throughout the pattern.

This pattern is like the August first pattern: orange nonagons are used instead of heptagons. There are four more nonagons eight more triangles. The pattern arises from wiggly loop of nonagons that have been interlaced with a straight line of triangles. The inner and outer triangles that decorate the pattern help bring out its nine-fold symmetry.

This pattern came about by elaborating a loop of interleaved triangles and squares. Each of the six loops on the outside of the central hub comes of a loop of squares and triangles where three of the triangles have been replaced with hexagons. The hexagons introduces spaces in the loops so that they can easily be connected (with squares) to one another while maintaining the original angles of the replaced triangles. As well, there are additional hexagons put in the central hub to make the negative space more eye catching.

This pattern applies the concept of a five point star to heptagons. The five point star
can be constructed with a pentagon path that alternates a small turns to the right followed by a larger turns to the left. For heptagons, there are fourteen polygons instead ten. As well, the purple triangles and pentagons are added to the loop fourteen heptagons for decoration. I think that these extra polygons help bring out the symmetry.

Here's another unravelled loop. A graphical representation of the process of creation is below. A pentagon loop (two partial stars) is combined with two loops of six triangles. The two gaps between the adjacent red pentagons are not filled with triangles (notice that there was originally fourteen gaps in the original loop and only twelve triangles)--these gaps are filled with squares and that are decorated with purple pentagons to maintain the alternating colours.

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