Pythagorean Tiling

The pattern can be made with the following PolygonR&D program

go{
 scale 0.4
 right
 scale 2.5   
 right
 sides 4
 ifon none {
  paint purple
  4 { 
   create go
   right }
  } { }
}

This tiling often appears on many floors. It is made with two different sizes of square tiles. The sizes of tiles doesn't matter--they will always form a pattern without gaps. A related fact about this pattern is that it demonstrates a proof of the Pythagorean theorem. The combined area of a purple and a yellow square is the same area as one of the squares constructed by the grid to the left.

Dodecagons R Us

This week was a long one--you may have noticed the reduction in entries. Well I believe that this will be the new norm for a while. But not because of work but because I dislocated one of my fingers while playing basketball. Typing is slower but the real bottle neck is creating patterns which can take a while in front of the computer which probably is bad for my recovery.
Anyway, this is a simple pattern with two types of vertices: 3.4.6.4 and 4.6.12 where the numbers take the place polygons with the specified number of sides. Still it's quite pretty.

Fifteen Turns

Fifteen turns mostly--the five squares in the middle occur only every third arm. This one is just a big cut and paste job with a square add on the third paste. Then a bigger cut was pasted five times.

Catching Zeds

Well there's a lot of zeds here but there are also a bunch of ens (mirror images of the zeds). This uses the same strategy as the shurikens posts. Notice between each octagon there is a pentagon and triangle inserted. In fact, this pattern is also similar to Candies--the difference is that the local connections of the octagons. Here an octagon is connected to two pentagons that are on sides 90 degrees apart whereas in Candies they are 180 degrees apart. As well, this pattern has extra olive triangles for decoration.

More Shurikens

I decided that it was probably useful to add a visual explanation of the three set of Shurikens below, which can be a bit disorienting because it feels like there should be more symmetry to it than it has. Here the set with only orange and green are highlighted. The other sets have purple and pink in their colours. One has orange in the middle (the boxy shurikens) and the other has pink in the middle, which is the mirror image to the highlighted one. Another thing that is clearly shown in the image is that I need a colour consultant.

Assorted Shurikens

This pattern is another variation on the triangular grid (black triangles). The difference is that the polygons inserted between the triangles have less symmetry than the previous ones. Notice that there are three types of holes where the original grids vertices were--hence "assorted". I'm calling the holes shurikens because to the rotational symmetry making them look a little like throwing stars. Two sets of shurikens are mirror images of each other and the third is the one with boxy ends. I like this pattern because of the not-quite-rightness of the repeating pattern. Perhaps I'll change the border of one of the "shurikens" to a different colour to emphasize the differences...

Not the Zipper

Well this is the pattern that is related to yesterday's zipper. The program to create this pattern is fairly simple. It has two different local patterns: one that looks like a propeller spinning clockwise and one that looks like a propeller spinning counter-clockwise. In the program below I have labeled them zig and zag.

zig {
  sides 5
  paint blue
  sides 3
  ifon none {
     paint orange
     create zag
     left
     create zag
  } { }
}
zag {
  sides 5
  paint blue
  left
  sides 3
  ifon none {
     paint orange
     create zig
     left
     create zig
  } { }
}

Zipper

This pattern has a vertical meshing of that reminds me of a zipper. The pattern is based on a grid orange triangles, which are separated by pentagons and squares. The holes made in the middle of the triangles looks something like elongated hexagons that were sheared. The pattern looks very different when only pentagons are used. That'll be tomorrow's post.

Swirl

The negative space makes the swirls. The main pattern is a grid of triangles with pentagons and squares that separate the triangles. The green triangles are surrounded by triangles and the orange ones are surrounded by three pentagons. The extra triangles and squares make the swirl pattern of the holes in the middle.

Almost

This pattern almost works. It looks like it fits together perfectly but alas the gray hexagons loosely fit. Each hexagon is connected to only one orange triangle and not touching the two. But because of the closeness it's hard to see. The way the pattern was designed from a triangular grid that had a double zig-zag of triangles and pentagons inserted between the grid triangles (red and green). Maybe it's best just to omit the misleading hexagons....

Candies

Somehow, this pattern reminds me of candies twisted in colourful wrappers. I know that I could have coloured this one better but....I was more interested in the pattern. The loops, here, have a double zig-zag pattern. The octagons are separated by pentagons and triangles in a couple of ways: pentagon with a triangle to the left (pink and red) or a pentagon with a triangle to the right (orange and purple).
As well, each octagon has two pentagons and two triangles that connect to it. I will experiment some more with this double zig-zag idea.

no gaps

This pattern has no gaps--most of the patterns that I have been looking at have gaps between the regular polygons. There are a number of different types of vertices in this pattern. Can you classify all the types of vertex meetings?

Cut and Paste

This is a TileLand design that has ten copies of a path. The path goes from one red square to the next which is a big arc. The arc has one missing pentagon between the purple square and the pink pentagon which allows for the arcs to not overlap. Below, I highlighted one of the ten arcs to make it easy to see. I probably could have coloured this one a bit better....

Sun Spot

Just for a change up, I'm back to TileLand. This pattern came about by trying to fill in the middle of the loop made by the octagons and triangles. Normally, I prefer to have only edge connected polygons but to fill in the middle required a some fiddling. If eight more octagons fit on the inside, I would have liked it more. But I settled for the hexagons--I was a bit surprised they fit so nicely.

Compass Roses

Well, I guess that this is not really compass roses since I would need eight points rather than these that only have four. Compasses was the first thing that came to mind. As with most patterns I make, the original program is not as efficient as the one I create after giving it a bit of time. Here is the program that I came up with after a bit of thought.

rose {
  sides 4
  right
  sides 6
  2 right
  3 sides 3
  sides 3
  ifon none {
      paint red
      create rose
      left
      sides 3
      paint purple
      left
      create rose
   } {
   }
}

Pants on Fire

I like the local pattern but globally this pattern doesn't work out. It can make a nice grid but not a pattern that has radial symmetry. The wedges implied here have angles that doesn't divide 360 so they do not fit together. The program to generate this image is shorter than one that would create the grid and I'm lazy today...

start {
  scale 0.4
  create pent
}
pent {
  sides 3
  paint pink
sides 5
  ifon none  {
 paint  purple
      create pent
      3 right
      create delay
  }  {
  }
}
delay {
    create pent
}

Spinner Revisited

Here's conceptual way to make spinner from Thursday. Start with this pentagon pattern below. This can be generated by the following polygonR&D code.

start {
  scale 0.4
  create pent
}
pent {
  sides 5
  ifon none {
   paint green
   sides 5
   paint olive
   create pent
   3 right
   create delay
 } { }
}
delay {
  create pent
}

The delay subprogram allows the pattern to development more symmetrically; without it, there would be a radial seam that would break the symmetry (C₅). To add spice to the pattern alternating triangles can be inserted (the zig-zag construction described in Zigzag Grid). To do this, the pent subprogram must be revised like so...

pent {
  sides 3
  ifon none{
    paint olive
  } { }
  sides 5
  ifon none {
    paint green
    sides 3
    left
    paint green
    sides 5
    paint olive
    create pent
    3 right
    create delay
  } { }
}

Finally, some extra triangles are added for decorations to get the original spinner pattern. The colouring of this pattern highlights the triangle- pentagon- triangle groups. It is interesting to see how the colouring has such a dramatic effect for each pattern. A seemingly simple change of switching the green and olive completely ruins the aesthetics of this pattern. The five pointed stars completely dominate instead of being a balanced part of the pattern.

Leaves

The leaves pattern comes directly from the pattern found on Tuesday, August 15, 2006. Here the pentagons are spaced apart by squares. The programming on how and where to insert the squares (really where not to put them) is a little tricky but not overly so. The olive coloured square indicate a filling in square where as the green ones show the growth of the polygons from the centre.