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Showing posts from 2006

Patch Quilt

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This quilt has three sizes of squares: the small white square, the medium olive square, and the rest. The pattern can take a while to understand but the colours help. Since this quilt doesn't have a four-way rotational symmetry, the program written isn't as short as I would like. When I get a little more free time I'll experiment with some more unravelling of loops. This pattern like the last is from "Tilings and Patterns".

Tri-triangles

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Three different triangles fitting nicely together. This one came directly out of the wonderful book Tilings and Patterns (p. 74). There are a few ones similar to this one as well that I may put up there. I'm more a fan of the patterns that come from polygon loops that are combined rather than ones like this. Still these ones are easily described since they have only one local neighbourhood. The PolygonR&D program has only one subprogram that describes the path from a purple triangle it's neighbour. The smallest triangles are not drawn since they are white.
tri {
right
scale 0.625
left
sides 3
paint orange
scale 1.6
sides 3
ifon none {
paint purple
3 { create tri
left
}
} {}
}

A New Checker Board

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This one has a lot of patterns in it. At some vertices you can see three differently coloured teardrop shapes that form a propeller. In fact, each teardrop shape belongs to only one propeller. The pattern can be thought of as a particular colouring of a hexagonal grid where certain hexagons are broken into six triangles. Let me know if you want to see more patterns like this one.

Eggplant

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Well this one is simply labeled eggplant because of the colours. The pattern demonstrates some local three-way rotational symmetry and some local three-way reflectional symmetry. The program that created it only has two sub-programs: one that describes the neighbourhood of nonagon with four triangles around it and another one that describes the neighbourhood of the nonagon with only three triangles around it. Those are the only local neighbourhoods in the pattern.

Non Play

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I'm still playing with nonagons. I'd like to have PolygonR&D create a similar pattern to this one but I'd like it to be a be less regular. Right now when I extend, I end up with six wedges that look too regular and have too many three point stars. I'll keep up the non playing until I get a pattern that has pattern and not too many three point stars. I hope to make some non sense soon.

More Cats

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Here is the promised dancing cats with colours that highlight the spiral nature. Close to the edge it becomes easier to see the straight edges which form an overall polygonal nature of the pattern. Unfortunately, I achieved this pattern by triplicating the original code that generated the pattern. On further reflection I may try to rewrite this another way so that this cutting and pasting isn't required...

Slippery Triangles

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These triangles have slipped a bit. They used to be all lined up then they started to slip. Now they have these gaping hexagonal holes. This pattern is related to the pattern called Pythagorean Tiling where the same thing happened to squares. The big Squares slipped and made little square holes. I wonder why I ended up choosing the same colour. I get the feeling the more of these patterns I make I'll end up repeating some days and not even know it.

Dancing Fat Cats

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Perhaps I should call this dancing cat slugs. I'd be happy to entertain some alternative names for this one. A close examination of this pattern may reveal a connection to the spiral with nonagons. Notice the same three interconnected spirals appear here--different colouring would help see the spirals but I think it would destroy any catness that the pattern has (slug or otherwise)...

Wings

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Busy times call for quick and dirty variations. This one is entitled wings for the shape of the gaps on one side of the gray squares. It comes from a variation of a simple pentagon pattern. A zig-zag of squares is inserted into the simple pattern. Half of the squares are not visible but the ninety degree angles can be seen from in each wing. Perhaps I'll have to revisit the way program deals with colour so that I can have more choices....

Dodecagon Elaboration

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Here we have octagons and triangles making up the majority of the pattern. The blue hexagons just fill in the centres of the patterns. The pattern comes from combining a number of discs like the one found in sun-spot. At a larger scale the discs can be thought of as dodecagons and the large triangular shaped that have six point triangles in them can thought as triangles. With just the dodecagons and triangles we are left with a simple pattern where there are no gaps and the vertices are all the same: two decagons and one triangle. The orange and purple of the octagons makes it more difficult to see the elaboration especially since the discs eight-way symmetry is highlighted not the twelve-way symmetry.

Decagon Web

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A web of decagons woven with pentagons. The holes are irregular hexagons made from two overlapping pentagons. Perhaps I should have made the green pentagons pink but sometimes it's hard to say no to green. The pattern derived from the Degagon Zig-Zag: using pentagons instead of triangles and filling in a bunch more pentagons. The rotation symmetry of the purple pentagons is a remnant of the zig-zag construction.

Decagon Zig-Zag

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This is a variation on the last pattern. Perhaps I needed to have a colour upgrade. Maybe tomorrow I'll try for the same repeating structures on the outside but with the same central loop as the last pattern. The zig-zag part of the title refers to the triangles that are inserted between the decagons--the first go right then left.

Decagonal Flower

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This decagonal flower is based on the pattern from August 8th (which uses nonagons--nine sided polygons instead of ten). The decagons are arranged in a back and forth loop and then the triangles were inserted in a zig-zag pattern to form the inner loop. The rest of the pattern, the outside, was added as decoration. Notice that there is a bit of conflict of rotation. It seems that the ten gaps seem to infuse a of a counter spin into normal spin of the inner loop.

Shrinking Pentagons

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This star shape is constructed primarily with blue pentagons that are only touching at one point with another blue pentagon. The blue pentagons are shrinking by a half each step to the away from the centre. The purple wedges fill in the gaps between the blue pentagons. The wedges are constructed in the same way--pentagons that shrink by a half. I'm still playing around to make a better looking pattern with pentagons of different scales...

Flowers

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This pattern of flowers is generated by introducing a zig-zag of pentagons into a grid of hexagons and triangles. zig {
sides 5
paint purple
sides 3
ifon none {
paint green
3 {
create zag
right
}
} { }
}
zag {
sides 5
paint purple
left
sides 6
ifon none {
paint yellow
6 {
create zig
right
}
} {}
}
Instead of the program, the pattern can generated like the windmill pattern. Below, we have the elaboration of a hexagon/triangle pair to a the pair with a pentagon inserted between them. As with the windmills, the orientation or the elaboration is important.

A Field of Windmills

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This pattern is a variation of "Not the Zipper". The main difference is that the pentagons have sides half the length of the triangles. They share only one vertex now. The negative space of a single loop reminds me of a windmill--hence the title.

A way to understand the generation of this image is to start with a triangular checker board of orange and purple then elaborate it using the following visual replacement rule.

Pentagon Play

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This pattern is the same as the purple and blue pattern from August the 15th. The twist is that the edges don't line up. The holes that used to rhombic holes now look like odd candle sticks. The program is almost identical to the Pythagorean tiling except the "sides 4" is replaced with a "sides 5"(as well the loop must go fives times rather than four).

Fractal Pentagons

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This pattern is a fun pattern with pentagons. Each step away from the central pentagon the adjacent pentagons get smaller. Here is the code that describes this trait.go{
scale 0.5
right
scale .6180339887
right
scale 2
sides 5
ifon none {
paint purple
5 {
create go
right }
} { }
} To highlight the layers of pentagons I alternated the colours.

Pythagorean Tiling

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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…

Leaves

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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.

Spinner

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This one is a simple pattern that I added a zigzag of triangles.
I'll have to get back to this in a bit--it needs a few pictures to exaplain...

Shells

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Maybe tomorrow I'll spiral this one but today I'll stick with concentric rings. The shells made of pentagons are decorated with triangles. The program that made this is not too sophisticated I think that when I try to make a spiral design I'll have to do a little more tinkering.... You'll just have to wait and see.

Heptagon Mix

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I was thinking about doing this pattern with heptagons. The problem is that in the old pattern three pentagons fit around a vertex with room to spare but three heptagons do not fit around a vertex. To make room, I inserted two squares and was off to the races. Also, I threw in some triangles to add some variation into the mix. Perhaps this would look a bit more interesting if I added some alternation of colours.... To make an better backdrop, after I cut out the rectangular part that will generate the pattern, I load it into gimp (a linux knockoff of photoshop) and fade the colours (using about a 0.4 alpha on a white background).

Jungle

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I'm not sure why I'm calling this a jungle. This pattern is a variant of the one on Tuesday, August 15, 2006. The pattern comes from putting a zig-zag of triangles into every loop in the original. For instance, the original has loops or six pentagons that form a rhombus hole in the middle. Here the rhombus is transformed into curved triangles (the triangle in the middle is just for decoration). On close examination, There are still only two orientations of the pentagons--half the pentagons have a vertical side.

Birds

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Perhaps I'm being a bit optimistic that people can see birds. The gaps that are not rhombic (like a diamond) remind me of birds. In the pattern, I see these birds circling around in big spirals. The program that generates this is a very short recursive program. The pentagons alternate between gray and orange. I was a bit surprised at first that there was only alternating colours in the pattern. Even though I programmed the colours to alternate, I am really only programming the polygons to fan in a tree-like fashion. There could be many incidental connections that occur. But on further inspection, all the loops have an even number of polygons so it would be impossible to have two of the same colour. In fact, since each colour represents a particular orientation two pentagon of the same colour can't share just one edge (they would have to share all five or none).

Clover?

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I'm calling this pattern clover because of the negative space. Looking at the polygons though, it reminds of some sort of the three legged wheel that I saw in a weird movie... This pattern was developed using a zigzag technique similar to yesterday. The three squares zig one way then the other as they go from hexagon to hexagon.