The Jerusalem Square

Jerusalem Square

The Jerusalem Square is the "shadow" of the Jerusalem Cube. The ratio between the sizes of a piece of the shape and its copies is an "irrational" number,  1 : ( 1 + root 2 ), or ~2.414213562... .
The square root of two (1.4142...) is the diagonal corner-to-corner distance across a square that has sides of length "one", so if you take a square and butt it up against a copy rotated though 45 degrees, you get the length of the side of the next size up.

As with Fibonacci packings, if you start with a proportion that's way off (say, you try to build the shape with two initial sets of squares of ratio 1:2), then the shape converges on the correct ratio by itself as you add more iterations.
 

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If you're wondering where the name comes from, here's the Jerusalem Cross as used in the heraldic shield of the Kingdom of Jerusalem (1099 -1291) : 

The shield is notable in heraldry for its "illegal" use of precious metal-on-metal colours (gold on silver), which is said to symbolise the idea that the Knights considered themselves above conventional laws. However, it may also be a reference to the quantity of precious metals that went into the area during the period, and which the Knights Templar then hauled away with them when they scarpered.

The design persists in the Vatican-based Equestrian Order of the Holy Sepulchre of Jerusalem (1099-), and alternative versions of the Jerusalem Cross, with different proportions and embellishments, appear in other organisations' logos as a reference to the Kingdom, including the masonic orders and the National Flag of the Country of Georgia.

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.

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.

A Koch Snowflake Sponge

Koch Snowflake Sponge

After the little triumph of the Koch Snowflake Solid, here's a better one: this beastie doesn't just give you the Koch Snowflake profile, it's also punctured by an infinite number of little Koch-snowflake-shaped copies, as holes.

It's a sponge!

The method's pretty much the same as before: divide your source cube into a 3×3×3 grid and delete the eight corners, but then also delete the central cubelet.
When you carry out the very first division it seems that deleting the centre shouldn't change the shape in any way, because the centre cube is totally isolated from the outside world by its six face-adjacent neighbours, but as the number of iterations increases, the cubelets get their corners progressively nibbled away and you get to peek through the holes created by the missing corners into the central void (and out of the other side).

At each iteration the sponge opens up a new set of hexagonal holes, and the existing holes open out and grow more Koch-Snowflakey detail.

This is a way cooler shape than the last one ("It's a sponge!").

And again, for the ultimate verification, I have a real one sitting on my desk made out of plastic.

The Impossible Snowflake

3D Koch Snowflake

Since the 2D Sierpinski Carpet projects nicely into 3D to give the Menger Sponge, and the 2D Sierpinski Triangle similarly up-dimensions to give the Sierpinski Pyramid, it seems obvious to try to get a 3D version of another famous 2D fractal construction, the Koch Snowflake.

A number of people have probably tried this over the years, but I haven't seen anyone manage it. The snag is that the "obvious" solution doesn't work. We're taught that the Koch Snowflake outline is created by assembling triangles, but if we try to use the most obvious triangle-based solid, the tetrahedron, we fail ... starting with a single tetrahedron and adding half-scale copies to each side initially produces a six-pointed profile, but after that it all goes horribly wrong (book, page 20). People have tried offsetting the positions of the daughter pieces to try to keep the shape looking interesting, but it's kinda cheaty.

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So the secret to creating this "impossible" solid is not to use a standard approach. "Step One" is to understand that the Koch Snowflake doesn't have to be made out of triangles, it can also be built from hexagons (book, Figure 3.13, page 26), and "Step Two" is to remember that the simplest Platonic Solid with a hexagonal profile is the cube  ... when viewed corner-on.

The rest turns out to be simple. Take a cube, apply a 3×3 grid to each face to divide it up into 27 smaller cubes, and throw away the eight corner-pieces. Then do the same thing for each of the smaller remaining  cubes, and repeat.

The resulting fractal solid (diagrammed as Figure 37 in the book) has a crosslike fractal pattern on each of its six faces, and shows a perfect Koch Snowflake silhouette when viewed from each of the original cube's eight corners.

A 3D Koch Snowflake ... paperweight

I have one of these that I'm using as a paperweight. It's perhaps not the prettiest of fractal solids, but I suppose that it's not bad for something that wasn't really supposed to exist.

The Diamond Eye Fractal

Starting the fractal tiling process ...
(click any image to enlarge)

The "Diamond Eye" is a fractal that I snuck into the book at the last minute as a small pagespace-filler without a title or figure number (on page 31). As a result, it's probably too small to see properly.

At first sight, this fractal looks like a fairly intricate (but unsignificant) crystal growth pattern with two competing seed types (in this case, clusters of horizontally- and vertically-aligned diamonds).

Iterations 3-5

We start with a horizontal diamond-shaped space, and add our first piece, a vertical diamond smaller than the original by a ratio of the square root of three (1.732-something). It wedges exactly across the centre of the space (top diagram, middle), and then we can’t go any further, so we switch to the second configuration. Scaling down by another factor of root[3], we can fit two horizontal diamonds into the left and right corners of the original space (top, right). Switching back to vertical mode, we can then wedge in four more smaller copies of the shape, and switching back to horizontal again lets us shove in a further eight. As the number of iterations increases we end up with a single solid mass of vertical diamonds growing out from the centre, competing with a hollow shell of horizontal diamonds growing in from the perimeter.

Iterations 6-8

Here’s what you end up with when you’ve carried out so many stages that you effectively have a single, solid,  frozen block (click to enlarge).

Fractal Rhombic Mesh

The important thing here is the shape of the boundary between the two “crystal” types. You can’t really see it too well in the above diagram, so we’ll colour the horizontal and vertical diamonds differently to emphasise the boundary.

Rhombic Koch Snowflake (interior and exterior)

Aha! And this is when we realise that what this “dual diamond” construction is really doing is sneakily growing a Koch Snowflake (I would have put "Rhombic Koch Snowflake" as this post's title, but it would have given away the punchline).

The internal network of cross-crossing diamond-ey shapes are still a bit distracting, so we’ll delete one of the two components. We’ll delete all the vertical diamonds and leave just the horizontals.

Rhombic Koch Snowflake (exterior only)

Yep, definitely a Koch Snowflake!

A Tilable Space-filling Fractal Sponge

A fractal, tileable sponge ("Alt.Fractals", p.61)

Okay, so this one looks like a "stretch" version of the conventional Menger Sponge, using a 4×4×4 grid instead of 3×3×3 .. and it is ... but it's a bit more than that. It's not just an arbitrary variation on a math standard.

It's a space-filling solid.

Space-filling solids are kinda rare. A cube will tile 3D space (obviously), but the other Platonic Solids won't, at least, not individually. A small group of other regular-ish solids have the property, like the truncated octahedron (which has a mix of square and hexagonal sides), and the rhombic dodecahedron (which has twelve identical diamond-shaped faces), but you won't normally find any mentions of space-filling solids that are also fractals.

Let's look at this shape further. It's constructed by taking a cube and dividing it into a 4×4×4 grid of smaller cubes, and deleting central 2×2 columns through the centre of each face (and then repeating, ad infinitum).
The 4×4×4 grid gives us 64 smaller cubes, so when we then delete four cubes per face, and eight from the centre, we're eliminating 6×4+8=32 cubes, out of the original 64. Every time we cut a new generation of holes, the remaining volume of the sponge halves.

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Now, how it tiles.

For the "zeroth" generation, we just have a simple cube, which will (obviously) tile an infinite 3D space as an array of (infinity^3) individual pieces. Now cut the first generation of holes into each piece. The holes punch an intersecting series of aligned 2×2 tunnels through the array, and where the 1×1 edge-columns of adjacent cubes touch, they form an intersecting series of 2×2 beams. The networks of columns and tunnels have the same shape, and we've already established that they have the same volume, so after one generation, we can take our infinite array, make a copy of it, and fit the copy exactly into the holes of the original to make it totally solid.

If we then take our resulting solid "dual" array, and cut the second-generation holes into each individual piece, the volume again halves, the new holes in both arrays again line up, and again we have identical networks of solids and spaces. Again, we can fit an identical offset copy of the original exactly into the spaces.

This goes on forever. For one of these sponges with n generations of holes, you can tile space solidly with n^2 overlapping copies of the sponge coexisting in the same space, so for the third-generation sponge shown in the photograph, you can interleave eight of them together.

For the "perfect" version of the sponge, with an infinite number of holes, you can tile space solidly with a block assembled from (2^infinity) overlapping "same-size" copies existing in the same space. Strange, but true.


So, it's Challenge Time! Can anyone find any other fractal solids that will completely tile space at a fixed size?

"Delta" wireframe

Here's a wireframe view of the basic "Delta" building block and its first iteration, which hopefully gives a better idea of how the thing is constructed ...

The Baird Delta

The "Delta" is is one of my new favoritest fractals. It was difficult to discover because it didin't have any obvious 3D siblings. It kinda seems to be a one-off. I haven't found any record of it being documented before, I used it on page 161 of the book (anf referred to it as just the "Delta" fractal), but it really needs a more distinctive and search-engine-friendly name than just "Delta" ... so now, unless anyone can find evidence of prior work, I think I'm going to start referring to it as the Baird Delta. :)

The shape's building-block is a solid with six identical triangular faces. You can then cut away material to produce three smaller rotated copies of the original (and repeat), or stack three of the rotated blocks together to produce a larger copy. Similar things happen with the Sierpinski Pyramid, but this is a leetle bit more subtle in that the component blocks (and the main shape) aren't quite regular polyhedra, they have to be twisted perpendicularly at each iteration, and the shape doesn't seem to have an obvious two-dimensional fractal counterpart. This makes the shape more difficult to visualise and more difficult to stumble across. The first time that you see the shape, it probably takes a bit of squinting before you realise that the three corner-pieces are identical smaller angled copies of the whole thing, rotated from the original baseline by 90 degrees, and rotated with respect to their siblings by 120 degrees.

As  you iterate, the area of each face gets progressively nibbled away until it's effectively a Koch Curve (although its a differently-angled version of the curve to the one that gets used for the Koch Snowflake).

All in all, a cool shape. 

Sticky Fingers

An example of "viscous fingering" between two glass plates

This is a “finger fractal” that I noticed on the pavement a few days ago, embedded in somebody's basement skylight window. The effect is sometimes known as “viscous fingering”, and it happens when you glue two plates of glass or perspex together, and then slowly prise the plates apart at one edge before the glue is properly set.

Air penetrates between the plates, but the thick surface of the glue clings to the glass and doesn't want to retract. Eventually a weak point in the wall “fails”, and the glue behind the tip of the inclusion finds it easier to retract than the glue at the sides, and a finger of air extends into the glue.

These sorts of inclusions tend not to meet up and join – in fact they seem to avoid each other and maintain a critical distance – so presumably a region of glue that has a lot of “edge” (anchored to the glass by curved meniscus surfaces on multiple sides) is more strongly connected to the glass, and more difficult to get rid of. Once an air finger penetrates within a certain radius of another inclusion or edge, it seems to be easier for further penetration to happen somewhere else, so when a finger starts getting too near to another edge, or the plate separattion within the wedge reaches a critical point, the penetrating finger's progress "stalls", and a new finger breaks through the perimeter somewhere else. What we end up with is a branching system of inclusions, and a branching network of remaining glue, interleaved. 

We don't usually think of glue as being a "clever" material, and yet here it is, unwittingly helping to create complex, self-regulating branching designs that look more like the results of some sort of encroaching lifeform's growth pattern.

A Hypercomplex 3D Mandelbrot

I've already blogged elsewhere about hypercomplex numbers and 3D Mandelbrot solids.


That post included a pic of the nicest 3D Mandelbrot that I've come across so far, and I included it in the book on page 118 (as Figure 23.1). Anyway, I thought it'd be nice to do a quick animation and put it on YouTube (although, when I say "quick", it took a spare laptop two weeks to generate the original 45 frames of video).

When it came time to upload it, I realised that YouTube user xlace had already found the same solid and uploaded an animation of it back in 2009!



I actually prefer xlace's version, but I suppose that it's nice to have two.

Book Launch

It's the 18th!

I didn't do much to commemorate the booklaunch, unfortunately. Was still recovering from piggy flu.

But I did put up a quick "flickthrough" video on YouTube (which is basically just me picking up the book and flicking through the pages). I'll have to try to retrofit a voiceover, once I find something that'll let me record audio.