Embellishing Octagons

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...

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.

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.

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....

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.

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).

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.