Monday, October 22, 2012

triskelion spiral

Maybe inspired by looking at those snake limits or water-bomb twists, I was looking at spirals again today (see a post here from a bout a year ago). This time, looking at spirals that are generated when you take a polygon, rotate it, and grow it several times. Here's one made up of about sixty triangles:

This spiral reminded me of the Manx flag - which is a motif that is sometimes called the triskelion.

Here's a similar spiral, using pentagons instead of triangles:

Thursday, October 18, 2012

Borges, Escher, & Origami Tessellations

The Argentinian writer Jorge Luis Borges is a favorite of mathematics enthusiasts - many of his short stories and essays have overtly mathematical themes, and much of his writing plays with structure and paradox in a way that appeals to readers that have a mathematical bent (see this Wikipedia article on Borges and mathematics).

I just recently read his short-story collection, a Universal History of Iniquity, which does not have any direct mathematical overtones (that I can recall), but whose cover in the Penguin Classics edition plays homage to Borges' mathematical ways with one of M.C. Escher's limit engravings Snakes.

In keeping with the mathematical theme, a  new cover for the same book features the origami tessellations of Eric Gjerde (see his post about the cover here).

The tessellation is Gjerde's water-bomb tessellation, and is one of the more easily folded models from his book. See this video on Happy Folding for a demo of how to make the water-bomb tessellation - it is fun to fold, and can be playfully integrated into other origami projects... for example, Beth Johnson has turned it into a sheep.

Friday, October 12, 2012

hex life

One of mathematics' best known recreations is Conway's Game of Life, popularized by Martin Gardner in his Mathematical Games column (see this recent post on Math Munch for another classic Gardner recreation: flexagons). John Conway first came up with his Game of Life using a square grid and (as described in this nice wikipedia overview) four basic rules:
  1. Any live cell with fewer than two live neighbours dies, as if caused by under-population.
  2. Any live cell with two or three live neighbours lives on to the next generation.
  3. Any live cell with more than three live neighbours dies, as if by overcrowding.
  4. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.
Things like Conway's Game of Life are examples of what are more generally called Cellular Automota - there are lots of great websites that offer overviews of these, and many include interactive demonstrations.

I've been wanting to play around with life-like automata for a while, so for fun I decided to try out the standard 4 rules (above) on a grid made of hexagonal close-packed disks, instead of the usual square grid. On a hexagonal grid, every cell has only 6 neighbours, instead of the usual 8. This little experiment showed me how delicate and perfect Conway's original life is - and how changing the geometry a little can have a big impact.

In the experiments that I tried, hex-life, quickly dies down to a few stable formations (patterns that don't change) and oscillators (patterns that move through a set cycle of formations). These were pretty, but I didn't observe any of the explosive or travelling patterns that makes the usual Game of Life so interesting.

One oscillator that emerged was a simple "windmill" (you can see one in the middle of the screen capture above) that has three dots that seem to rotate around a central dot. This is actually an oscillator of period 2 that is governed by rules 1, 2, and 4. The diagram below shows how many neigbours each cell has - with live cells coloured blue and dead cells coloured white.

Another oscillator was the blinking diamond shown below. The two cells across the middle of the diamond blink in and out of the live state, governed by the rules 2, 3 (blinking off) and 2, 4 (blinking on).

I found a some other oscillators, and quite a few stable patterns, but nothing too exciting - I am sure there is a joke I could make here about having a stable, nice, but not too exciting hex life, but I won't. :) 

BTW - just as I was writing this up I came across a new post about a continuous version of Life - definitely worth looking at.