Charles Darwin's Tree of Life

"Tree of Life", Charles Darwin (from "On the Origin of Species", 1859)

 This is one of the most influential diagrams ever made. It's Charles Darwin's published diagram of the branching relationships between species, from his book "On the The Origin of Species" (1859).

" As buds give rise by growth to fresh buds, and these, if vigorous, branch out and atop on all sides many a feebler branch, so by generation I believe it has been with the great Tree of Life, which fills with its dead and broken branches the crust of the earth, and covers the surface with its ever branching and beautiful ramifications."

At the time, we'd already had "family tree" diagrams of our relatives, and Linneus' classification system had given rise to tree diagrams for the organisation of plant types, but Darwin is supposed to be the first known example of anyone had suggesting that your personal family tree could be extended outwards, and outwards, and backwards, to encompass all life on Earth and every living creature that exists or has ever existed on the planet. In other words, in four dimensions, we're all part of a single fractally-branching organism.

"The Fractal Universe", Pecha Kucha talk

The Fractal Universe 28-Nov-2011: All slides

I just gave a talk at the Lighthouse, Brighton, on fractals ("The Fractal Universe). 
It was part of the Brighton Digital Festival. Pecha Kucha is a 20×20 format where a series of presenters stand up and talk in front of a slideshow of exactly twenty slides, that are shown for exactly twenty seconds each. I was on first.

I hadn't been to one of these before, and with hindsight, I think I kinda missed the brief. The other presenters were using the talk to describe their recent work and their approach using digital technology ("this is me, here's some of my stuff, and here's how I make it").

That would have been much easier to prepare for and present (and probably easier to watch) than what I did, which was to attempt to compact an all-encompassing talk on the fundamental nature of fractals that could have been used as the basis of a major twelve-part BBC series, into a mere six minutes and forty seconds. It ended up as an exercise in talkingasfastasIpossiblycould for just under seven minutes as the slides clicked past. If you've ever tried explaining recursion in architecture, or the apparent large-scale fractal structure of the universe and its possible implications in twenty seconds flat, then you'll know what I mean. :)

On the upside, if anyone at the Beeb does want to turn "The Fractal Universe" into a TV series, give me a call...

The Jerusalem Cube

Jerusalem Cube

The Jerusalem Cube fractal is a little odd. Although it seems simple enough — it's just a cube repeatedly penetrated by crosses — for it to work properly, the ratios of the cube and sub-cubes don't have whole-number integer, or even fractional integer ratios. We're talking irrational numbers, here, and while you might expect irrationals to show up when you're assembling shapes at funny angles, in this case, they appear when we connect simple cubey blocks together, face-to-face.

It can't be built using a simple integer grid, and that's probably why you probably haven't come across it before. Where the Menger Sponge can be visualised as the result of applying discrete logic within a simple "base three" number system, the Jerusalem Square and Jerusalem Cube correspond to the same sorts of orderly processes being performed on number systems that aren't based on integers.

Tony Bomford's Hyperbolic Rugs

Tony Bomford (1927-2003) made a series of hooked rugs based on the Coxeter/M.C. Escher hyperbolic tiling patterns. He started the series in 1981.

You can find a useful biography and listing of the rugs (and further discussion) on Doug Dunham's pages as a series of PDF files (e.g. http://www.d.umn.edu/~ddunham/dunham04.pdf ).

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.

The Fractal Library

Library-space

A well-designed library’s architecture has a fractal structure: the books form a perimeter with maximum length surrounding a cascading series of open spaces, with a central atrium spawning secondary spaces, which in turn spawn additional offshoot spaces, and so on. Library architects try to avoid the use of corridors, and keep as much floorspace visible from the centrally-placed staffed desks as possible ... this also makes it easier to distribute natural ambient daylight around the structure from large windows that typically shine into the atrium-space.  The branching-space structure also makes it easier for library staff to make sure that patrons aren't Getting Up To No Good with the books - even if your location is out of a direct line-of-sight with a staffed desk, someone might come around the corner at any moment ...

The distance from the centre to any book should be as short as possible, and systems like the Dewey Decimal System, which categorise all books into a single sequence, ideally lead to the shelves being arranged into a single (crinkly) perimeter with maximum perimeter and minimum area, enclosing a series of access and study spaces that lead back to the central information desk. Library architects are solving something that is essentially a fractal problem - the fractal organisation extends down through the system to sections, subsections, and arguably even down to the indexes on individual books.

Thanks to Hove Library for letting me take the interior picture.

Fractal Heart

· fractal heart ·  (2011)

This is a three-dimensional fractal network based on Golden Section ratios. It's based on the shapes in Figures 8-5 and 32-4 (pp. 59, 154 and bc) of the fractals book.
The shape reminded me of those anatomical models where they inject an organ with latex and then dissolve away the tissue to leave a network of blood vessels and capillaries. The texture and bright red colour also reminded me of that guy who makes sculptures out of frozen blood.