Mirrorspace

'. mirrorspace .' a fractal optical space using mirrored spheres, (c) Eric Baird 2011 

This is a colour reworking of the black-and-white image on page 218 of the book.

As "physical" fractals go, it’s one of the simplest to describe: it’s simply a stack of mirrored spheres.

However, an //optical// description, of what we actually //see//, gives a very different, fractal result: the surface of each spherical mirror is a boundary that seems to contain a complete perfect mirror image of its own outside universe, turned inside out. That apparent inverted "fishbowl universe" contains copies of the spheres outside to the first sphere, and each of those also appears to contain a copy of //their// own outside universes, including copies of the first sphere, and copies of each other. And each of the copies of copies contains copies, which contain copies of copies, which contain copies …

If we had just four spheres loosely arranged around a common position, then in observerspace, each sphere’s interior volume would //seem// to be connected topologically, surface-to-surface, to four others -- three additional spheres apparently embedded "inside" the volume and one "outside" (which is ours). We also see four different volumes facing us (the four spheres) and each of those four volumes in turn sees four connections – three inward connections to its own internal mirrored images of its neighbouring spheres, and a fourth connection, outward to us. Each of those connections adds a further three connections to the network, which each add a further three, which each add a further three …

The resulting observerspace map describes a negatively-curved space with a recursive pattern that gets progressively more distorted (as seen from "here") the deeper you go, and the corresponding optical space seems to consist of an infinite series of four-surfaced cells that are each touching four neighbours, and where every cell in the network is topologically exactly equivalent. We can argue that this isn't the //real// physical situation, and that our own space is “real” and all the others are fakes, but while we’re saying that, all of our reflected counterparts are saying the same thing about their spaces. We can also try to prove that our own space is is the "original" by taking measurements and showing that our universe is obviously larger than the others, and that there’s a timedelay in "their" reflected light that shows that all the signals in the network originated “out here” and not “in there” … but again, our reflected counterparts with their reflected instruments seem to get exactly the same readings that we do – they seem to be pointing at us (and each other) from within their mirrored prisons, and claiming that they've proved that //we're// time-lagged, not them.

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The mirrorballs also illustrate the deep connection between fractals and tilings.

To us, the mirrorball image is clearly fractal, it shows repeating self-similar patterns across different scales and across different locations. It’s also a fractal that’s only infinite in one direction (you can zoom in infinitely far, but there’s an outer limit), and the shapes become progressively more twisted at deeper levels.

To a topologist, the full shape isn’t fractal at all. It’s a set of identical tiles of the same size shape and dimensions that just happen to perfectly tile a space that’s slightly non-standard. It’s a simple non-fractal tesselation or tiling problem. What we’d say was an obviously "wibbly" fractal, the topologist could claim as an artificial perspective effect caused by our trying to take an integer-dimensionality projection of simple repeating tiling in a noninteger-dimensionality space. What we see as an obvious change in size across the network, the topologist coudl argue us a lensing effect caused by curved space -- every cell in the network shows the same locally-observed size. They have to, because they're effectively all the same cell.

So some fractals are topologically equivalent to simple non-fractal  tesselations (tilings), and some tesselations have corresponding fractal configurations.

Phew.