Penrose sketches, 19.6.11: Five fold symmetry in nature and with tiles.

Five fold symmetry appears in many shapes in nature. The most famous is probably the starfish, but many flowers like the one pictured above have a pattern that starts at a central point and radiates out just about equally in five directions, each one a 72 degree turn away from its nearest neighbor.

Here is a simple Penrose tile pattern with five fold symmetry that resembles a five fold symmetry flower. I used fifteen yellow kites and topped of that central pattern with ten blue darts just to highlight the yellow more, since the pattern is currently fixed to my off-white refrigerator. Right now, I am experimenting with this fifteen kite pattern to see what can be made with it. Recall that we can make larger shapes similar to the kite using Penrose tiles.

This is my favorite design using fifteen kites. Depending on my mood, I see a gear in a machine or a modern logo. More whimsically, I see a hitchhiking cartoon bird whose head, feet, tail feathers and hitchhiking fist with thumb are all the same size.

Here is a sketch for a larger work using the fifteen kite shape, this time with fifteen Daddy Kites, each made of five kites and three darts. My first idea was to make the outside of the pattern all purple, but with my 216 tiles, it was not possible.

I call this a sketch because some time this week, I should be getting another 432 tiles from SeriousPuzzles.com, as well as some other toys I will be playing with. Then I will be able to make the final version of my original concept using Grandaddy Kites, which should be made of thirteen kites and eight darts, but instead are composed of twelve kites, seven darts and a gap the shape of a large dart, which is not possible to make, as proven here last week.

I'll be giving SeriousPuzzles.com some link love on all my Penrose tiling posts, and they will be using some pictures of my shapes and patterns on their website.

Mutual backscratching aside, I've been very happy with the customer service from them, whether things were immediately in stock or on back order. My thanks to Chris Dillon and all the rest of the staff.

Penrose sketches, 16.6.11: Five fold symmetry with three kites.

Here are three kites put together. If there were five kites centered around the angle at the bottom, it would create a regular decagon, a ten sided polygon. The angle for the gap at the bottom is 144°, which is an angle that can be filled many ways. The idea is to take five of these three kite shapes, twist them by the same angle each time to make the patterns below. The first is using the regular kite shape and a two color pattern, the second is using the larger kite I call the Mama Kite, again using just two colors, and the last is using the Papa Kite and three colors.

Completed: 15 June 2011
15 tiles, all kites
5 yellow, 10 purple

Completed: 16 June 2011
45 tiles
30 kites and 15 darts
20 yellow and 25 blue

Completed: 16 June 2011
120 tiles
75 kites and 45 darts
40 yellow 40 purple and 40 blue