Lots of legs

I guess I have too much time on my hands. I'm playing with the idea of having 1.5 turns clockwise mixed with a half turn counter-clockwise. Perhaps I should work with 2.5 and 1.5 mixes... Must get back to Tileland...

Heptagonal flower

I was playing around with TileLand and I stumbled on some local patterns that I turned into this picture. It seems to suggest a lot of lines and make it hard to decide what you should focus on: flowers, rounded triangles (Reuleaux triangle), or triplets of arrow heads. 
     The way I generate the pattern is by connected paths heptagons an triangles.  I'll take a step back and see if I can come up with some ideas about this pattern.    With a regular polygon focus I have ways of describing the connection between the two types of loops and simpler patterns....  I'll keep you posted.
    This extra colour makes it a little more interesting perhaps...

Fibonacci Fern

I wrote a short story about this fern to try to motivate the formula to the left.  The geometry of the fern directly relates to rectangles with lengths that are Fibonacci numbers.

This square, 55x55 is composed of the following rectangles 1x1, 1x2, 2x3, 3x5, 5x8, 8x13, 13x21, 21x34, and 34x55.  Then, I take each of fold each of the rectangles in half to create the fern.  It's an interesting shape that has a different front and back.
The story is called Symposium and is still a work in progress.

Pentagonal sunburst

Here's another pattern with pentagons.  The colours don't help show the pattern but I like the look. I am getting a little faster translating patterns made in TileLand to PolygonR&D.  I often play around in TileLand, then after I find a repeating pattern, code it up in PolygonR&D so that I can easily play with colour and size.  Perhaps, I should have played around a bit more.

This style is like a leaf pattern.  Each of the 10 radial wedges has a symmetry like a leaf (technically a glide motion?).  The pseudo mirroring can be seen by looking at the white S gaps in the pattern and then looking at the white gaps that are more like a Z.

I may play around with these local patterns of Ss and Zs to form a different global pattern.  That is of course if I can get some spare time.

A pentagonal tree

Here is a fun tree.  The basic pattern starts with a single pentagon that branches into two smaller pentagons that in turn branch into smaller pentagons etc.  The right pentagon is dark the left is light.  This simple rule makes some interesting patterns in the tree: notably, the two lower branches that are horn like and only one colour.

I stop the shrinking at some cut off mark which preserves the leaf like appearance of the polygons. This makes the edge a little less regular and thus more tree like.  I may play around with a little with a slightly less symmetric choice for shrinking the polygons.

I may also perfect this image later so that it doesn't suffer so much from pixelation.So much to do so little time.

The PentHouse

This pent house was kind of fun to create.  I was stumbling around with pentagons and triangles looking for a nice pattern.  After I found this one that fills the plane, I started to playing with it in gimp (an open source photoshop knock-off) and this is what I came up with.

One of the interesting things I found out after I finished this image was that I discovered a portion of this in an ancient version of TileLand (the one on my CSD web page).  I'm still toying with a tablet version of TileLand...It's just there is so many options...

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....