Flower of Crowns

This one has an interesting centre. As it expands out it becomes clear that there are only 15 orientations of the crown pattern.  Given the nature of the centre it was clear that there it had to be a multiple of 5.    

I am still searching for a true spiral pattern that uses these crowns.  Currently I have only found things that look spiral but really have rings instead.  Perhaps, the shape is too complex.  

I guess I'll have to give it some more thought.

So many choices...

After playing around with the sun burst in the previous entry, I remembered these 30-gons with interiors that are made from nice loops of pentagons and triangles that meet exactly at the centre of the 30-gon.  I'm sure I have already made a bunch of patterns that already have this 30-gon but I haven't been able to find a blog entry with one.

My goal for patterns is to have an interesting and symmetric way of making neighbourhoods for tiling.  These 30-gons turn out to have a fair number of choices on how they can relate to their neighbours.  Essentially, coming up with easy to follow rules for describing the pattern is a way of completely understanding the pattern.  These rules embody what I like about patterns: access to the infinite.

My rule I choose to explore here, has to do with having the internal structures back-to-back.  I'll label the structures: loops and arrows.  The loops refer to the three positive-space structures composed of six pentagons and 5 triangles.  The arrows refer to the three negative space structures (white space) that are pointy and all touch the center of each 30-gon.

To see my rule in affect, imagine the pattern without the pink-centred 30-gons.  All the purle-centred 30-gons are connected with back-to-back arrow structures.  Paths that follow these arrow structures form a hexagonal grid with the purple centres being at the vertices.  A similar story occurs for the loops structures but there is a hitch--there are two types of purple-centred 30-gons: ones to the right of a pink-centred 30-gon and ones to the left.  If we ignore all the purple-centred 30-gons on the right, we end up with similar hexagonal grid formed by paths following the loops.

Seeing the above structures makes my rule seem complete.  However, pairing the pink-centred 30-gons with the purple-centred 30-gons on the right leaves us with only back-to-back pairs of loops with arrows.   Still I am fairly content with this patterns symmetry and it's overall appeal.

I have included this blurred patch that can be used as a background.

Sun burst tiling

Here is the pattern I mentioned last entry.  In this pattern, the E-shaped white spaces radiate out from the central circle.  I guess a better label for these E's are described as 3-point crowns since here there are also 4-point crowns.  Should the pattern be extended indefinitely there would many more 4-point crowns than 3-point crowns. The spiral effect seems to be produced by all the points pointing in the clockwise direction around the centre of the the circle.  Then again, maybe it has more to do with the curved lines that drawing the eye in a clockwise direction.  





Although the repetitive aspect of the outer rings is rigidly defined, the inner part of the circle has many options.  I have explored a few possibilities in a different context.  But I have one that incorporates the 4-point crown.  I'm not sure if I should include the middle 3 pentagons.  The three humpback whale shapes in white space can be replaced by a shape that has nice rotational symmetry. Here's an alternative centre.

 

Maybe this is a better colouring....

Playing with pentagons and triangles

On a very long train ride from Buffalo to Albuquerque, I ended up chatting with a rapper on his way back to LA.  We both had a lot a hours to fill so I showed him some of my polygon programs.  Nearly every time I play with TileLand I end up creating something new (at least to me).  This time was no different as you can see from the picture on the left.

This pattern with pentagons and triangles started out as a big 30-sided polygon (constructed by alternating pentagons and triangles) and slowly turned into a triangular grid of green triangles that has been augmented with zig-zag spacers that resemble purple and red dresses. 

I'm not yet finished with playing with the fruits of this play.  I'm going to attempt to make something different with the E shaped white space.  The interesting thing about this E is that it can be completely surrounded by polygons by inserting another pentagon where you have the two kissing red triangles. Check back in a couple of days and I should have the new pattern completed.

 Here is a blurred version to use as a background.  One of these days I'll have to eventually address the idea of colours--but that day is not here yet. :)

Playing with old program

I was about to progam a PolygonR&D program to compute with Fibonacci codings but I got distracted with an old pattern. I get so sucked in with all things pentagon. I wonder if there is some occult connection... pentagram etc. Kidding.

OK, here's my procrastination pattern.  It's related to leaves.  I thought I had a similar pattern posted but I couldn't find it (I probably should have a better index of images).   Each square in leaves is replaced with a pair of triangles.  The colouring adds a little bit of complexity as well but nothing too crazy.

With a small modification I'll post the pattern that I thought I had previously posted.... If you are the one person following this blog and happen across this  pattern, like me know.

rhombic dodecahedron balloon

Here is the design of the rhombic dodecahedron that I promised for a long time. Since it's tricky to follow I have a diagram below that may help you create it. The structure probably looks best when you use one to two inch bubbles rather than the longer ones I used in the video.

1.2.8.9.10.11_a.13.0_b.10.4.1.5.13^*.7.8
12.11.14.7.6.2_a.3.0_b.6.5.12.4.3^*.9.14

Here is the psuedo-gauss code for this design and the pattern. Probably I need to include a sequence of diagrams to really make this dodecahedron. I find these useful too. The diagrams perhaps are more useful in the design phase rather than the construction phase....

The * is there to indicate a bit of a tricky spot. For these you must pull the balloon through a square. It's not that tough on the balloon but it is important to know that it must come through the square an not just wrap around--the geometry changes when the balloon is complete and it would come apart otherwise.

Also note that 0 is mentioned but it is not in the diagram. The reason for this is that it would clutter up the diagram (it would turn the graph into a non-planar graph). The 0 point represents the centre of the dodecahedron.

crazy zigzag

It's been a while since I made a zigzag pattern (a post for that matter). This octagon grid interlaced with a zigzag of triangles and heptagons leaves very small holes or gaps. You can see the alternation of the colours and polygons by following a loop around one of the holes. Notice that there are two types of holes: one that goes up into the right and another that goes up and to the left. These are mirror images of each other.

The slight skew produced by the heptagon-triangle spacers makes the pattern shift slightly up. Perhaps I should rotate the image a little so that pattern can be used as wallpaper....

Hmmmm probably I should make PolygonR&D have a save as background options. So many ideas so little time.

I originally made this pattern with TileLand but later switched to PolygonR&D. I've included the program below.

zig {
  sides 3
  paint black
  sides 7 
  paint olive
  sides 8
  2 right
  ifon none {
    paint gray
    create zag
    4 right
    create zag
  } { }
}

zag {
  sides 3
  paint olive
  left
  sides 7
  paint black
  left 
  sides 8
  2 right
  ifon none {
    paint green
    create zig
    4 right
    create zig
  } { }
}

I'll have to make my next pattern with brighter colours. Perhaps I'll return to some winding and unwinding patterns...