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.

Andy Lomas' "Aggregations"

Andy Lomas: DLA fractal art

Andy Lomas makes computer simulations of things that look a little like limescale deposits in water-pipes ... but really pretty limescale deposits. They're another example of the DLA (Diffusion Limited Aggregation) fractals mentioned earlier (like my iced-up fridge!), but his are computer-simulated at extremely high resolution, and can contain between around fifty to a hundred million individual particles.

• Andy Lomas' profile on CGSociety

More unusually, he's also done high-resolution plots of the tracks that particles took to reach those points, which gives a sort of inverse "ferny"-looking or lichen-like pattern. You can see some of his "Images of Aggregation" and "Images of Flow" art on his website.

Moorish-styled Tiles

'Granada' range tiles, FiredEarth.com

I think these tiles are absolutely gorgeous. They’re based on classic Moorish tiling artwork from Spain. If you know the Alhambra in Granada (widely regarded a representing the pinnacle of Islamic geometrical art), you’ll know roughly what to expect (M.C Escher made a pilgrimage to the Alhambra to study its designs before embarking on his series of famous tesselated artworks, and some of the Alhambra's designs also seem to show up in the work of Roger Penrose).

The borders between the smaller protruding glazed shapes are recessed and unglazed, so that each tile looks as if it's been assembled from a mosaic of smaller and more complicated tile shapes. It's a nice surface.

The Granada range of tiles is made in Spain and sold by Fired Earth. They also do other North African and Middle Eastern designs without the mosaic effect, like the Andalucia and Anatolia ranges.

(Thanks to Mark at FiredEarth Brighton for providing background info and letting me take the photos)

Fibonacci Textiles

Fibonacci quilt (Louise Mabbs)

Today I popped along to an exhibition in Portslade, of quilts and textiles designed around the Fibonacci Series ("Fun With Fibonacci 2"), part of a larger display called Art in Creation 3 (19-21 May 2011).

The main exhibiting designer, Louise Mabbs, also creates fabric origami, and other mathematically inspired hybrid pieces.