PolygonCraze

Fibonacci Fern

2012-02-15T20:07:00.001-08:00

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

2011-10-12T00:30:00.000-07:00

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

2011-10-09T21:56:00.000-07:00

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

2011-06-08T04:30:00.000-07:00

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

2011-05-27T22:26:00.000-07:00

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

2011-05-25T20:33:00.000-07:00

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.

More estimates

2010-11-27T14:52:00.000-08:00

Last time, I talked about the height of a hexagon; the strategy I used works with any shape with a pair of parallel sides. The parallel sides allow you to stack the polygons and add up the heights without having to worry about angles. For instance, it will work with a regular octagon. The picture to the left shows 5 stacked octagons that are slightly more than 12 square (otherwise in TileLand the top purple square would be wiped out). That means that 5 octagons > 12 squares or that 1 octagon > 12/5 =2.4 (since they are unit squares).

For those who like to exact lengths, you can break the height of an octagon up into three pieces. This diagram below demonstrates the breakdown: two half squares of length 1/ root 2 (since the diagonal is 1) and a rectangle with the long side of length 1. Root 2 plus 1 is 1.41421+1 = 2.41421.

Look forward to when I tackle odd sided polygons that don't have parallel sides.

Root 3

2010-11-16T16:38:00.000-08:00

Who said approximating can't be fun? This is an activity that I designed to get geometry students to come up with estimates for the height of a hexagon. The tool I asked them to use was Tileland. Tileland is a world of polygon paths (well known to those who follow this blog regularly) that doesn't allow for polygons to overlap (the top green squares replaced vertical squares that were there previously). The following polygon path demonstrates an estimate for the height of a hexagon with unit length sides that is pretty good. Here the height of four hexagons is slightly less than seven squares; this means that the height of one hexagon is slight less than 7/4 = 1.75. Using the diagram of the hexagon composed of equilateral triangles and the Pythagorean Theorem, it is easy to show that the hexagon's height is actually root 3. Thus root 3 must be slightly under 1.75. It is pretty close since root 3 is about 1.732--so within 0.02.

The reason I prefer to use types of approximations like this has to do with having a concrete physical focus to talk about. It allows for variations and better discussions. These two diagrams are more of the conclusions to the discussions and experiments. Also I like reducing and/or delaying using the magical calculator. To me, it seems to be an ender rather than a promoter of thought. In fact, I prefer to have students multiply 1.75 by itself to show how close it is to 3.

For better estimates, you can construct paths corresponding to these fractions 12/7, 19/11, or 26/15 fairly easily in Tileland. To obtain 45/26, you have to use a more wiggly approach....

Flower of Crowns

2010-10-28T16:26:00.000-07:00

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...

2010-10-20T10:25:00.000-07:00

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

2010-10-19T09:47:00.000-07:00

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

2010-10-18T11:23:00.000-07:00

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

2010-01-29T09:58:00.000-08:00

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

2009-06-11T15:08:00.000-07:00

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 stop. 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

2009-06-11T03:27:00.000-07:00

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...

The Balloon Dodecahedron

2008-08-24T21:10:00.001-07:00

The dodecahedron is a tricky structure to make out of a small number of balloons. There are twenty degree 3 vertices and thirty edges. For symmetry, we need the number of balloons to divide the number of edges but as well figure out how to deal with the odd degree vertices. I have designed a 2-balloon dodecahedron but it was not very aesthetically pleasing. The design I currently prefer is a 3-balloon design. The trick was to instead of using all the ends of the balloons to each take care of an odd degree vertex leaving fourteen (which is not divisible by three), was to have two vertices where three ends come together leaving eighteen vertices of degree 3 (which is divisible by three).

In this design, I use nine doubled edges. The doubled edges stabilize the shape as well as locking the balloons local geometry. To make the weaving of the balloons easier, the doubling of the edges are delayed until the other balloons are in place. This delay reduces the tension on the twisting.

Below is a video of the construction of a dodecahedron.

Here is the psuedo-gauss code for this design.

1_a 2 5 2^d 6 12 6^e 7_b 14 15_e 16 17 16 20
1_a 3 7_b 3 8 15 8^e 9_c 17 18_e 19 11 19 20
1_a 4 9_c 10 18^e 10 5_d 11 12_e 13 14 13 20

* the superscript means a delayed connection. For example in the first balloon, the first line, the sequence 2 5 2^d 6 12 6^e 7_b can be created all at once as a string of 7 segments like a string of sausages. Even though the 2 is repeated, it is not joined until the third balloon reaches the d (see the 5_d). Then the two segment 2-5 and 5-2^d can be twisted together to form a double edge.

The graph has a three-fold rotational symmetry (C₃). To keep the symmetry of the graph, node 20 has three copies but really it is a single node at infinity.

The Balloon Icosahedron

2008-08-22T07:32:00.000-07:00

The icosahedron is another platonic solid that has all of its vertices with odd degree. What makes it a little trickier than the cube is that it has twelve degree 5 vertices and 30 edges (instead of 8 vertices of degree 3 and 12 edges of the cube). To ease in complexity of remembering the balloon design, symmetry in the use of balloons is preferred. Essential this means that each balloon could rotated into the position of an other balloon. Symmetry dictates the number of balloons should divide the number of edges so 3 balloons is a good choice. This would have ten segments per balloon.

With three balloons, we have six ends that can take care of six of the twelve degree 5 vertices. The other six vertices with odd degree can be turned into degree 6 vertices by joining them to an interior point. The video below demonstrates the construction but perhaps the psuedo-gauss code is more explicit.

From the graph of the balloon, you can see that symmetry of the balloons since you can rotate the graph 120 degrees and the shading of the lines (which represent the colour of the balloon) all become swapped. A further symmetry is that it can be inverted from start to end. This describes a D3 symmetry group.

Here is the psuedo-gauss code:

1^z 3 9 10 5_a 6 9_c 8_d 2 6 11 12_z

2^z 1 5_a 11 7_b 8_d 0 5 4_e 3 8 12 10_z

3^z 2 7_b 12 9_c 4_e 0 7 6 1 4 10 11_z

The Balloon Cube

2008-08-21T06:44:00.000-07:00

The cube balloon, like the tetrahedron balloon, has the problem of having odd degree vertices. By having two balloons, four of the problem vertices can be dealt with. The other four will be dealt with by going through the interior point once for each balloon turning those vertices into degree 4 vertices. To symmetrically space these degree 4 vertices, I used the vertices so that none share an edge (the points form a tetrahedron).

3 4 1 5_a 0_b 2 6 7 8
6 5 8 4_a 0_b 7 3 2 1

This design thus use two balloons with eight segments each, which allows for large segments for a large cube. Notice that the two internal segments would theoretically be 0.8660 (root 3 over 2) the length of the larger sections but I think in practice it's more like 0.7 because of the bulging properties of balloons. With only 4 internal balloons this design is very sensitive to the ratio of the external and internal balloons. This allows for some interesting patterns that are topologically equivalent but that don't look like a cube.

An alternative design that would ensure a cube look would have all 8 vertices connected to the centre point:

0^z 3 4 1 5_a 0_b 2 6 7 8 0_z
0^z 6 5 8 4_a 0_b 7 3 2 1 0_z

The Balloon Octahedron

2008-08-20T17:20:00.000-07:00

The octahedron was the first balloon I ever made. It is by far the easiest one to make since all the vertices have degree 4. There are many one balloon paths that construct the octahedron but the one I describe here is nice because it does not require a balloon to be pulled through a triangle.

To describe this I have the following youTube videos.

Here's the original with a 350 balloon.

The psuedo-gauss code for this is

1 2 3 4 5 2 6 4 1 5 6 3 1

The Balloon Tetrahedron

2008-08-20T08:10:00.001-07:00

The first balloon design is this 1-balloon tetrahedron design. It's not the easiest design but it's the easiest shape. The tetrahedron is composed of four triangles but I like to focus on the edges and vertices (I like to think in terms of graph theory). There are six edges and four vertices. The problem with the making the tetrahedron is that the vertices have degree three--this means that every vertex has a problem (see Eulerian path). A balloon comes into a vertex and leaves the vertex which takes care of two of the three edges but the remaining one has to be dealt with. So you can either have the balloon start or end at that vertex (but each balloon only has two ends) or you could cheat. There a number of ways to cheat but the one used here is to join each balloon with an interior point. With this design, each of the vertices now has degree four.

As well as a video of the construction, I'll include a diagram and a sequences of that describes the design similar to Gauss code. This is perhaps more to introduce the method of description rather than to be useful in this case. So here I'll have a planar graph with the vertices labeled but the interior point(0) omitted (which would make it non-planar).

0^z, 1, 2, 3, 1, 4, 2, 0, 4, 3, 0_z

The z superscript indicates that the 0 is a delayed connection until the z subscript.

Balloon polyhedra

2008-08-09T15:27:00.001-07:00

Since I haven't been making many polygon patterns I've decided to put my balloon designs up on my blog. I'll try to space them out a bit. These will hopefully end up in a paper soon. Here are the platonic balloons that I have designed. Aside from the tetrahedron, these are a little tricky to design. But for now enjoy the picture. Eventually I'll have a photographer take pictures of these. I'll post the designs soon.

S tiles

2008-01-23T13:01:00.000-08:00

What else was I going to do with triangles, squares, and heptagons. This pattern was based on taking a clock-wise loop of ten heptagons (it's like a loop of six pentagons--not very circular) and unraveling it with a clock-wise loop of triangles and a clock-wise loop of squares. That is what makes the octagonal shape that surrounds the "S". So then it's just a matter of a little decoration and tiling. I'm surprised no one has really blasted me on my choice of colours. Bring the flame war on!

X Bug

2008-01-13T19:49:00.001-08:00

I'm calling this one X Bug since it is a checker board of X's and Bugs. The pattern was developed by unraveling a loop of six heptagons. Six sets of a square and two triangles were interlaced with the heptagons to create the outline for the bug. The bugs were connected by overlapping the two triangles at the corners. After the checker board of bugs were in place the holes that were left were decorated with a few squares and triangles to make the holes look like X's. This pattern is similar in construction to this pattern and many others. Perhaps the asymmetry (or rotation symmetry) makes it seem more like this pattern.

It's patterns like this one that make me wonder if I need to create another interface for PolygonR&D that incorporates TileLands interface of easily making polygon paths....

Here's close up of one bug. It's the hole on the inside that I consider the bug.

Connected heptagons

2007-11-14T19:55:00.001-08:00

Perhaps I should have spent more time picking the colours. The patterns not that difficult -- seven wedges of scale shapes made from heptagons. I put an 11-gon (hendecagon) in the middle just to fill it in a bit.

I'm trying to make a few more patterns these days because my homepage automatically puts a filmstrip of my blog pictures. It was a joke to do with Picasa. Still I have to get pictures in my blog.

The program is two subprograms of about ten lines each. What I really need to do is to have PolygonR&D work with a mySQL database so I can link to it easily.
Perhaps if I wasn't teaching four classes a semester...

Subtle pentagons

2007-11-12T15:46:00.001-08:00

I was playing around with some slippery pentagons and ended up with this pattern. I was trying to get them to squeeze together a little better and was having a bit of difficulty. I didn't look carefully enough to see that the gaps were very different. I was focusing on the lines of pentagons and I missed that the gaps changed half way along the line. It's interesting how the regularity can fool you into thinking that it's more regular than it is. Perhaps it's not that subtle and I was just playing after a long day of work.

Playing with Slippery Triangles

2007-10-27T03:01:00.001-07:00

It's been a while since I have posted a pattern. I guess it's been busy. I modified Slippery Triangles to make this. The original purple triangles were replaced with limited Sierpinski's triangles I'll include the first couple of iterations. I put red triangles as the initial triangles so that it would be clearer what the progression was. I made sure that the purple triangles didn't get too small otherwise they would be difficult to see.

One of the things that I find appealing about this pattern is the slightly out of alignment of the triangles. It introduces a bit of conflict into the pattern. As well, by stopping the replacement of triangles at a coarse level I think that it suggests the a fractal pattern but stops short. Perhaps I'm over thinking this one.

Zigging with Heptagons

2007-06-23T11:41:00.000-07:00

Today's quickie is a variation of Zigging. I put heptagons in and was required to add a few more polygons to make them all fit together. Without the squares and the triangles, the heptagons would overlap with each other.

The outline of the gaps reminds me of ducks. To help make that more visible I need to use better colours. Perhaps, I'll have to update PolygonR&D to have more colours. Then again I guess post-editing in photoshop or gimp could easily fix this.... If that is the case, I guess I'll have to have an option to not use anti-aliasing so it is easier to swap colours... The work of a programmer is never done.

Zigging

2007-06-19T21:03:00.000-07:00

Here is a quick little pattern with the triangular grid only partially expanded. The diamond shapes are two triangles that did not have a spacer inserted. These last week have all been variations on a theme. I'm sure I'll switch themes soon. Perhaps I'll start introducing a few of my concrete constructions....

Hour glasses and Pythagoras

2007-06-19T04:26:00.000-07:00

There's lot's to see in this pattern. The idea was to elaborate a square grid using triangles. By using larger triangles, the pattern creates a number of interesting features to focus on.

The first features are white hour glasses made with the two white 120 degree isosceles triangles. They have two orientations that alternate. These are the holes that come from the vertices of the original square grid. The size of the orange triangles was selected to have the triangles share a vertex. A smaller size of triangle would leave a gap which could be seen as the white isosceles triangles overlapping.

The second features are large squares that are defined by a purple square and it's four neighbouring equilateral triangles. This square that also be identified as a propeller can be chunked together with similar squares of the same orientations to create a tiling that is a Pythagorean tiling. What is interesting is either orientation of a purple square can be the basis for big squares and that each produce a Pythagorean tiling. It really is just a mater of switching your point of view and chunking different polygons together.

Little squares and big squares

2007-06-18T12:26:00.001-07:00

Today I'm playing with size a bit. I have two patterns that seem a lot different but only differ by the size of the squares used. As with my recent trend of elaborating a triangular grid by inserting polygon sequences between the triangles, here we have a triangle-square-triangle being inserted. By changing the size of the square inserted the squares can be made to meet in the middle of the hole produced by the insertion.

The change in the amount of white space and the addition of the pointy features makes the pattern feel a lot different even though they are closely related.

To highlight the path of the insertions and the role of the original triangles, I have included close-ups showing the path of the new loops.

Another Triangular Grid Variation

2007-06-17T12:49:00.000-07:00

This pattern uses the same strategy as the last one. In this case, the inserted polygons are a square, a pentagon and a triangle. There seems to a little too much going on to be attractive. The holes in the pattern are reminiscent of the shurikens. If you look back at that pattern it is pretty much the same except the inserted pattern does not include a square. The three different shapes of holes are analogous.

Triangle Grid Elaboration

2007-06-15T20:11:00.001-07:00

This pattern used the same technique of expanding a loop path with alternating squares and triangles. Here, I am using an orange triangular grid of and inserting purple squares and red triangles. The difficulty arises that I cannot expand all the triangular loops in the same way: the neighboring loops affect the possible choices. To help show how the particular loops of six orange triangles are expanded, I have included a zoomed in version below. On the left, I have drawn in the augmented paths of the two loops. On the right, I have filled in the polygons involved with each new loop where the square/triangles pairs are recoloured black and/or blue so as to highlight the choices of the orientations.

This style of elaboration is more complicated when using the triangular grid as a base. This is due to the oddness in the number of sides of the triangle and the asymmetry of square/triangle inserted pieces: there is no simple (uniform) choice of fitting things together like in the octagon case below. But it does produce a more interesting pattern.

Octagons, Esses, and Zeds

2007-06-14T20:05:00.001-07:00

Here's another pattern that comes from a grid of octagons. It would be orange octagons and white square but the focus is on the loops of four octagons. Inserted between every pair of octagons is a triangle(red) and a square(blue). The pattern of insertion is alternating. For instance, if you follow a loop of polygons (clockwise) that surrounds a hole that resembles an S, then you will see a triangle-square that bends to the right where as the next triangle-square bends to the left. In fact, the original white squares (from a octagon&square tiling with vertex 488) are transformed into the distorted S's and Z's seen in the pattern.

TileLand Homework

2007-03-07T18:49:00.000-08:00

Here's some tricky homework. Figure out the types of vertices of each of these patterns. Then make a large patch of one of these patterns in TileLand. The patterns are assigned as follows : Sonya, Audriea, Kelly, Bobbi-Sue, Amanda, then Loren.

Mixing Hexagons and Pentagons

2007-01-27T09:56:00.000-08:00

Here's a pattern that alternates a loop of six pentagons with the green hexagons. The loops of pentagons have two orientations. The three holes surrounded by the pentagons and hexagons are the most interesting structures to look at. I'll try to make a few variations of this pattern where I replace the hexagons with other polygons.

Unraveling a Pentagon Loop

2007-01-05T12:34:00.000-08:00

See if you can construct the process of unraveling a loop of six pentagons. This is a similar process to Jan. 3rd's pattern. For me, I enjoy looking at the holes in this pattern: the s-shapes and a shape similar to the "Cross of Lorraine" (the double cross). Let me know if you can construct this.

Unwound Heptagon Loop

2007-01-03T20:11:00.000-08:00

In this busy time of year, I've had a tough time attending to the blog. I get the feeling that the first part of this semester will be a bit crazy as well. So that is warning in advance. This pattern comes from a simple loop of six heptagons that are unwound by six hexagon-square pairs. As a visual explanation of this process, the picture below shows a rule that is applied to a pattern. The rule is placed the box; it shows a pair of red heptagons being replaced by a pair of heptagons with an orange square and an olive hexagon being inserted between the pair. When this rule is applied to the six instances of heptagons pairs in the adjacent loop of heptagons, the pattern on the right of the picture results. I call this process unwinding because of the angles introduced by the hexagons makes the loop go in the opposite direction. For another example of this, see July 10th's pattern.

To create the original pattern, a number of resulting unwound loop is connected in a checker board fashion and the opposite square are slightly elaborated with squares and heptagons.

Patch Quilt

2006-12-01T05:18:00.000-08:00

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

2006-11-26T14:05:00.000-08:00

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

2006-11-22T20:00:00.000-08:00

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

2006-11-19T13:58:00.000-08:00

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

2006-11-14T07:49:00.000-08:00

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

2006-11-12T03:54:00.000-08:00

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

2006-11-09T06:37:00.000-08:00

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

2006-11-07T09:27:00.000-08:00

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

2006-11-06T19:29:00.000-08:00

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

2006-11-01T04:01:00.000-08:00

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

2006-10-15T13:27:00.000-07:00

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

2006-10-12T07:29:00.000-07:00

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

2006-10-10T20:28:00.000-07:00

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

2006-10-07T19:44:00.000-07:00

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

2006-10-06T04:20:00.000-07:00

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

2006-10-04T19:43:00.000-07:00

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

2006-10-02T08:36:00.000-07:00

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

2006-10-01T06:20:00.000-07:00

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

2006-09-30T11:50:00.000-07:00

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

2006-09-29T19:02:00.000-07:00

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

2006-09-26T20:16:00.000-07:00

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

2006-09-25T04:10:00.000-07:00

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

2006-09-24T15:50:00.000-07:00

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

2006-09-21T05:02:00.000-07:00

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

2006-09-20T17:08:00.000-07:00

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

2006-09-19T10:02:00.000-07:00

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

2006-09-17T06:20:00.000-07:00

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

2006-09-16T12:54:00.000-07:00

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

2006-09-14T06:04:00.000-07:00

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

2006-09-12T19:08:00.000-07:00

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

2006-09-10T20:17:00.000-07:00

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

2006-09-08T05:08:00.000-07:00

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

2006-09-07T07:12:00.000-07:00

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

2006-09-05T08:01:00.000-07:00

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

2006-09-04T00:58:00.000-07:00

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

2006-09-02T23:47:00.000-07:00

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

2006-08-31T01:54:00.000-07:00

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

2006-08-29T23:19:00.000-07:00

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

2006-08-29T06:12:00.000-07:00

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

2006-08-28T04:22:00.000-07:00

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

2006-08-26T13:19:00.000-07:00

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?

2006-08-25T06:03:00.000-07:00

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.

Zigzag Grid

2006-08-24T10:17:00.000-07:00

This one is not too inspired but...deal with it. It follows a similar construction to this one except the zigzagging here is with heptagons and the grid uses hexagons in the same layout as the the triangular grid (each hexagon only has edge connections with three hexagons rather than six). This is also similar to this construction. Perhaps the decorative red triangles should be a different colour....

bunny

2006-08-23T04:13:00.000-07:00

I'm bringing it down a notch. Here's a simple bunny. The only tricky thing with this pattern is that there are unseen polygons (a hexagon and two squares) used to place the eyes, nose, and mouth. See if you can make a better bunny.

Embellishing Octagons

2006-08-22T08:01:00.000-07:00

This pattern started out as a loop of octagons that was turned inside out with two loops of two hexagons, four squares (green), and two triangles. This inside-out loop served as my main loop which is the one that doesn't have the pentagons in the middle of it. After, I fit these loops together let a checker board. I filled other loops that were constructed with some yellow squares, triangles, and some pentagons.

The images below can be used as wallpaper. There are links to images that are 323x323 pixels. Perhaps they would look better smaller...

2006-08-21T04:31:00.000-07:00

Just a simple pattern for a Monday morning. This pattern began as a triangular grid and then two pentagons were inserted between each pair of triangles. Finally, the newly created loops were decorated with some more triangles making the boxy looking stars in the middles of the loops.

2006-08-20T16:26:00.000-07:00

Here's a polygon pattern that relies on a simple recursive call. Each heptagon attaches to two scaled down heptagons and each of those heptagons attaches to two more etc. This recursive definition is easy to do in PolygonR&D. The tightness of the spiral is determined by the main shrinking factor--the difference in size of the heptagon to the next biggest heptagon.

2006-08-19T09:37:00.000-07:00

Once again I am playing with negative space--the pattern seems more about the four-point stars than the pentagons. This pattern (with different the colouring) is on the cover of the Tilings and Patterns: An Introduction (A Series of books in the mathematical sciences) by Branko Grunbaum and G. C. Shephard which is a great book. The last couple of blog entries have convinced me to slightly augment the language of the programming language of PolygonR&D to make the programs slightly easier to write. I'll let you know if it was worth it....

2006-08-18T03:46:00.000-07:00

Here's another spiral. This is made with nonagons (nine-sided polygons) and has three arms. Some difference between this one and the one yesterday is that the banana-shaped gaps are bigger and they have eighteen orientations instead of fourteen. Close to the centre of the pattern like the image to the left it appears curved like a true spiral but a bigger image would reveal the eighteen-sidedness. To show that there is no trickery about the nonagons nicely fitting together there is a close up included below.

2006-08-17T00:40:00.000-07:00

Here's a spiral pattern made of heptagons. The fours groups of banana gaps between the heptagons form the four arms of the spiral. With close examination of the gaps, one can find fourteen different orientations, which are grouped together in wedges similar to the last pattern with alternating pentagons. This pattern was created using a short PolygonR&D program (50 lines for this two colour pattern or 15 lines if it were all blue).

2006-08-15T05:46:00.000-07:00

This pattern alternates two different orientations of pentagons. The pointing up ones are blue and the pointing down ones are purple. Notice that all but one of the pentagons (the central pentagon) are connected to exactly three other pentagons. The pattern can be thought of as concentric rings of pentagons (that will approximate decagons). Another way to understand the pattern is by grouping rhombi (the white diamond slivers) in terms of orientation--there are five different orientations. The rhombi form ten wedges that meet at the central pentagon.

2006-08-14T05:55:00.000-07:00

This pattern is somewhat like Friday's pattern in that it uses some sections of a big loop inverted to make a loop with a star shape. The pattern alternates heptagons and triangles with the exception of the orange hexagons in the middle (heptagons didn't fit). It also has a flavour of August first's pattern since it has the same type of rotational symmetry. I prefer the August first because it offers a lot of the same interesting lines yet with a simpler design.

2006-08-13T08:04:00.000-07:00

After a big battle with aesthetics, I finally settled on this pattern that makes a nice wall paper background (the image on the right). I'll highlight some of the battle. I originally started with a two colour pattern with just pink and purple. I thought it would have more impact than it did but I think with the regularity of the colouring that the pattern was dull. I added some colour it to make its bigger loops of twelve octagons and twelve triangles stand out. Even after I added orange to the mix it seemed too regular. The I introduced a twist to the pink loops of six octagons. To show you these transitions I'll added the pictures below. The key to recognizing the differences is to examine the negative space especially the asterisk like shapes. Notice there are two orientations in the pattern above as opposed to below where there is only one.

2006-08-12T07:46:00.000-07:00

I guess that I've been a bit obsessed with pentagons and triangles recently. This is mostly an alternating pattern but has a couple of hexagons and triangles. I guess I like this one because of the central curved pentagon. I wish I was able to find a better way to fill the inside of the outer loop...

2006-08-11T04:14:00.000-07:00

Here's a little play with hexagons and pentagons. Basically, the outside loop comes from this big loop below. The rest is mere decoration. The pentagon in the middle was difficult to place(and may not be at the theoretical centre).

2006-08-10T19:19:00.000-07:00

This quickie pattern uses a simple section that combined with a bunch of mirrors creates a nice grid. Unlike the latest previous pentagon and triangle patterns, this does not have an alternating of the polygons. Below is a section that will produce this pattern as wallpaper.

2006-08-09T05:41:00.000-07:00

Sunflower

This pattern of a sunflower was a bit tricky to construct. The outer orange section (the petals) was easy given the high level of repetition but filling the insides (the seeds) was not straight forward. The pattern is built with alternating pentagons and triangles. The five-fold symmetry is present throughout the pattern.

2006-08-08T16:49:00.000-07:00

This pattern is like the August first pattern: orange nonagons are used instead of heptagons. There are four more nonagons eight more triangles. The pattern arises from wiggly loop of nonagons that have been interlaced with a straight line of triangles. The inner and outer triangles that decorate the pattern help bring out its nine-fold symmetry.

2006-08-07T06:39:00.000-07:00

This pattern came about by elaborating a loop of interleaved triangles and squares. Each of the six loops on the outside of the central hub comes of a loop of squares and triangles where three of the triangles have been replaced with hexagons. The hexagons introduces spaces in the loops so that they can easily be connected (with squares) to one another while maintaining the original angles of the replaced triangles. As well, there are additional hexagons put in the central hub to make the negative space more eye catching.

2006-08-06T13:05:00.000-07:00

This pattern applies the concept of a five point star to heptagons. The five point star
can be constructed with a pentagon path that alternates a small turns to the right followed by a larger turns to the left. For heptagons, there are fourteen polygons instead ten. As well, the purple triangles and pentagons are added to the loop fourteen heptagons for decoration. I think that these extra polygons help bring out the symmetry.

2006-08-05T03:00:00.000-07:00

Here's another unravelled loop. A graphical representation of the process of creation is below. A pentagon loop (two partial stars) is combined with two loops of six triangles. The two gaps between the adjacent red pentagons are not filled with triangles (notice that there was originally fourteen gaps in the original loop and only twelve triangles)--these gaps are filled with squares and that are decorated with purple pentagons to maintain the alternating colours.

* = ...

2006-08-04T14:30:00.000-07:00

This one began as a simple loop of six pentagons that was turned inside-out using two loops of three pentagons. Afterwards to link those inside-out loops (which look perhaps like a toothy mouth) another loop was used--the original simple loop of six pentagons was elaborated with interlacing hexagon spacers (these loops inner hole look like the outlines of eights with a vertical javelin through it). Below, I included a wallpaper version of the pattern.

2006-08-02T08:16:00.000-07:00

Sometimes it's the missing tiles that make things interesting. Here there are a number of missing heptagons. A missing heptagon makes each of the gentler curve between the three triangles. Including the heptagons would lead to many overlapping polygons. Perhaps there is an artful inclusion of a selection of the heptagons...