Octagonal_stellationIn the first few pages of Infinity, the book I'm currently reading, mathematician and cosmologist John Barrow makes a point in part by using the endlessness of some Islamic art.

Because the faith generally bars representations of living things, infinity in Islam can be represented, rather, by the tessellation of space. In many cases, the geometry in the tiling and art is also algorithmic: the pattern avoids simple repetition as it expands.

Likewise, a limited number of polygons can be used to entirely cover an infinite plane. At scale they become fractals, which can found throughout nature in forms ranging from plants to clouds, anywhere, in fact where a large volume needs to be enclosed by a very large surface.

Here's the thing. The step from two to three dimensions is really counterintuitive.

Rather than liberating, Barrow points out that by moving from flatland to three dimensions the possibilities are radically constrained. What was endless in two dimensions becomes small and finite in three.


Photo credit: Technical paper artist Eric Gjerde, "Octagonal Stellation, step 2"