Posts

Showing posts from 2007

Connected heptagons

Image
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

Image
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

Image
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

Image
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

Image
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

Image
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 …

Little squares and big squares

Image
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

Image
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

Image
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 oc…

Octagons, Esses, and Zeds

Image
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

Image
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

Image
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

Image
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

Image
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 check…