exupero's blog

Divisibility graph identicons

In the previous post I showed a sampling of square, checkerboard-style identicons of a size that could generate almost 69 billion patterns even before color was added. Here I'll revisit another earlier post and create circular identicons based on divisibility graphs. The divisibility graph for 7 seven looked like this:


We can simplify the graph by stripping it of numbered nodes, arrows, and loops:

Here's a sampling of patterns generated using bases between 2 and 22, and numbers between 2 and 52:

There's definitely some visually interesting patterns, though even in this small sample set three (almost 20%) look very similar. Given the base and number constraints listed above, we only have 20 × 50 = 1,000 distinct combinations, though not all of those generate unique identicons (such as 3 and 9 in base 10, which both generate circles without interior lines).

We can liven up both the images and the possibility space by adding color. There's no obvious approach to how to color such patterns, but one way is to pick a chord at random and color each side of it differently:

Which chord to use to bisect the color does expand the possibility space by about an order of magnitude. We can expand it further by using multiple coloring strategies. Here's a strategy that colors one side of every chord:

We can also use semi-transparent colors:

I like the muted colors and subtle gradations of this strategy.

Even with many coloring strategies, the possibility space is still far smaller than than tens of billions of distinguishable images, but the combination of symmetric design and asymmetric color does create a pleasing artistic tension that might make these self-intersecting-designs useful something more interesting than identicons, though what I'm not sure yet. If you have an idea, email me.

In the next post we'll try a different strategy for generating symmetrical identicons.