Candlesticks

This pattern  I stumbled upon when I was playing with hexagons and octagons.  After a bit of massaging, I ended up with an outline of the candlestick (here it is the white 3-layered  pattern between the stars).  In the original play, the 3 triangles were part of a hexagon that intersected with the others.  From there, I connected 6 together to make large hexagons with octagons in the corners.  I was surprised at first that I could make 3 into the triangular shape and more surprised that a small triangle exactly fit inside it.  After some thought, it made more sense.

 Image a loop of hexagons with octagonal spacers between them.  Then connect each of the hexagons with a rectangle to other hexagons from other loops.  Voila, this tiling with less decoration.  The reason for the triangle exactly fitting is a little more involved.  You have to take the loop of polygons on the left and separate the rhombus with two perpendicular lines.  The bottom perpendicular is easy--going from a 4-sided figure to an 8-sided.  The top perpendicular is trickier--going from a 12-side figure (the square and triangle make the 150 degree angle of a 12-sided polygon) to a 24-sided polygon (the 135 and the 60 of the octagon and the triangle make 195 which is the exterior angle to 165, the angle of a regular 24-sided polygon).

I'm not sure about how close the pentagons are to each other.  It looks like they touch at the vertices...  I may have to get a pencil and paper to check it out....

Bats

Here's a crazy tiling that reminded me of bats.  I did a bit of playing with the image to try to capture the eye a bit more. 

Essentially, this pattern is completely constructed from alternating regular heptagons and equilateral triangles.  You can scan the image and you will see there are no two heptagons that are connected by an edge and the same goes for the triangles. 

The gaps between the regular polygons resemble bats.  The bats spiral around and have two different appearances.  I find the most visually interesting spot near the centre of the spirals where it's tricky to find the pattern of the bats.  The regularity increases further from that centre.

I framed the image with one of the spiral arms. It seemed a natural choice.