Fractal Lego

Sierpinski Triangle, rendered in red Lego roof-bricks (angle view)

The Sierpinki Triangle, rendered in Lego. The basic building-unit used here (to get proportions that look like an equilateral triangle) consists of four bricks: a 1×1×4 base strip, two 1×2×1 slanted roof pieces and a 1×1×2 top strip.

The Sierpinski Triangle model, in the "Lego Digital Designer" ("LDD") program

This model has 64 of these units per side. In theory, it should take 4*3^6 = 2,916 bricks to build it, but in practice, if you were building a real one and having it standing vertical, you'd want to replace a lot of the little adjacent base strips with longer strips for reinforcement, and maybe also stagger them for extra strength (which is why Digital Designer's showing a reduced brick-count in the screenshot that's closer to  ~2700).

A "World's Biggest Lego Sierpinski Triangle" competition might be entertaining. You can imagine kids building little sections of these at educational fairs, and then getting to watch them being assembled into bigger and bigger versions. Fun.

A Tetradecahedral Fractal Sponge

Truncated Octahedral Cluster Fractal Sponge

 A plastic 3D print of a tetradecahedral fractal cluster sponge, based on Figure 9.9 of the book (it's on on page 66). The two diagrams in Figure 9.9 of the book weren't actually 3D modelled – they were "faked" 2D diagrams done in CorelDraw thanks to some clever shading (and a lot of coffee) – but I figured that a real 3D model shouldn't be too difficult.

The basic shape is a truncated octahedron: a semi-regular 14-sided polygon with six square and eight hexagonal faces. It's the shape that you get either by cutting the points off an octahedron to turn its triangular faces into squares, or by cutting the corners off a cube, but really deeply so that instead of getting new triangular facets the new faces intersect and end up as hexagons.

Anyway, you can fit a smaller copy of the solid into each of its corners, and then use a network of adjacent sibling copies to link them together using only face-to-face connections, and once you've replaced the solid with a skeleton of little copies, you can then repeat the process inside each of those little copies, and repeat.

The end-result looks a little like the sort of polyhedrally-based patterns that you find in the microscopic skeletons of radiolarians.

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.

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.

"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.

Defrosting the Fridge ...

This is a physical example of something known as a DLA (Diffusion Limited Aggregation) fractal. I found it in my fridge.


So what does DLA mean? The "aggregation" part means "collection of things stuck together"  (in this case, water molecules from the air freezing on contact with the existing ice, and "aggregating" to produce a clump). The "diffusion-limited" part means that the aggregation process is hampered in some regions by the statistical difficulty of new material being able to wander into those regions without hitting another overhanging sticky-zone before it can get there.

In two dimensions, DLA fractals tend to create long, straggly, self-avoiding strands that look like the roots of a small plant (book, figure 36.7, p182). In three dimensions it produces lumpy cauliflower-like clumps of the type found in my fridge.

Once one piece of ice starts to protrude more than the others, it gets to snag more passing material than the other regions, and grows faster. And as it grows, it "shades" the more low-lying regions, and grabs the material that would otherwise have gotten to them, so they find it progressively more difficult to catch up. It's a positive-feedback process -- small initial random imbalances in height get exaggerated, and you end up with a pronounced pattern of overhanging clumps and deep valleys. 

The reason why it counts as a fractal (rather than just an example of low-pass-filtered noise) is because once you have a nice big blob of a clump, any small peaks or protrusions on top of that start to shield the lower-lying regions, and the differentiation process continues, giving clumps-on-clumps-on-clumps, and clefts-on-clumps, and clefts-in-clefts-in-clefts.

The little pale brown speck is probably a breadcrumb.

Zaha Hadid fractal wallpaper

Zaha Hadid is an architect who designs annoyingly-good buildings.

Here's what happened when Hadid designed some wallpaper. It's fractallish. And it's really cool. Again.

Damn.

(the nice bubbly design is called "cellular", it's available from Marburg wallcoverings)
http://www.zaha-hadid.com/furniture-product-design/art-borders-wallpaper
http://www.marburg.com/en/mt/wohnwelten/opulenz/zaha_hadid.php

Fractal Coffee

This is a quick cameraphone snap of the foam imprint left on someone's coffeecup at a LikeMind meeting (800×500px).

Benoît B. Mandelbrot, 1924 – 2010

The New York Times reports that the mathematician Benoît Mandelbrot died on Thursday, aged 85.

Mandelbrot is credited for practically inventing the subject of Fractals, coining the name, working with other researchers and with IBM to crank out some of the earliest examples of computer-generated fractal imagery, and writing a series of books, which, although their price tags meant that they weren't exactly “popular” as far as retail sales were concerned, were actually fairly readable in parts by people without a background in advanced math background. And if you didn't like the text at all, you could always look at the pictures.
Most people know him from the Mandelbrot Set fractal that's named after him.

Some mathematicians probably get frustrated that Mandelbrot never seemed to give a conventional ironclad definition of what actually counted as a “fractal”. This wasn't sloppiness on Mandelbrot's part, it was quite deliberate. Fractals were “broken” geometry in several ways: they'd broken the rules and definitions of existing classical geometry, and to immediately replace those definitions rules with another similarly restrictive set would have been to fail to learn an important lesson. Unlike some in the mathematical community, Mandelbrot understood the occasional importance of appropriate imprecision. He could have produced a strict definition and classification system for fractals, but he chose not to. Instead, he looked at applying fractals to cosmology, to art, to the stock market and to social behaviour in general, and in the process he expanded our knowledge and appreciation of everything that fractal arguments were applied to. Knowing the fractal characteristics of a seismic tremor or a graph of stock market prices didn't let us predict when the next earthquake or financial crash (or tornado) was going to happen, but it let us understand more about the limits of our ability to predict.

To some other mathematicians, Mandelbrot's existence (and success) was probably a bit frustrating. Here was a high-profile guy popularising his work through books rather than just peer-reviewed papers, and making great strides without necessarily doing all the careful derivational work and constructed proofs that “normal” mathematicians considered essential. Where other mathematicians took great care to make sure that their works fitted flawlessly into the existing body of previous research, and with the existing rules and conventions, Mandelbrot simply declared the existing rules to be inadequate and unfit for purpose for the sort of problems that he wanted to study. Classical geometry was based on clean and simple geometrical entities such as straight lines and regular curves … and Mandelbrot essentially tore up the rulebook.

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

Like Stephen Hawking with Hawking radiation, and Albert Einstein with gravitational time dilation, Mandelbrot took an existing breakdown in mathematical procedure and instead of accepting the conventional wisdom that these breakdowns were “bad” and not worthy of study, he dove in and treated them as legitimate. He listened to the mathematics rather than to his fellow mathematicians, and if his results sometimes seemed like gibberish according to the conventional classification systems, well, that just meant that those classification systems were inadequate.

The math was what it was, and if the community didn't like it, then their opinions were based on sociology rather than mathematics, and could be ignored.

Einstein and Hawking and Mandelbrot all saw the distinction between the imperfect social operation of science as a community endeavour, and the idea of a deeper perfect underlying truth, and if the two failed to mesh, then they sided with the math.
An expert opinion was a transient fleeting thing, but an equation was forever. If you wrote an equation in an attempt to solve a problem, and instead of settling down to give you a single stable solution, it spiralled madly out of control, then many mathematicians considered your example to be a case of "bad" or "pathological" math, something to be noted only so that experts could add the example to their list of dysfunctional equations that we knew ought to be ignored and avoided. Mandelbrot's approach was different. An equation wasn't "bad" simply because it didn't do what we hoped of it. If a function didn't break any of the fundamental laws that spawned it, then its behaviour might be inconvenient, or even perplexing, but it was still mathematics.

Einstein wasn't a great physicist … in the opinion of his early college lecturers, he probably wasn't even shaping up to be a particularly //good// physicist. But Einstein broke through the mental blocks that had imprisoned some sections of physics theory and fractured parts of the existing definitional systems that we should probably have outgrown a century earlier. According to the prevailing social norms, Einstein arguably wasn't a "great physicist", but he was a physicist, and he //was// great at what he did. This wasn't necessarily what //other// physiicists did, but that's what made him different, and valuable. Similarly, Stephen Hawking still provokes grumbles from some math folk that he's not as good as the public think he is, and as a mathematician he rushes in and builds on unproven hypotheses in a way that a mathematician really shouldn't. Hawking's work arguably didn't demonstrate that he was a "great mathematician", but he's nevertheless one of the greats -- he was able to throw away the socially-ingrained preconceptions of how a subject "ought" to behave, and see how it really functioned. Both men "saw behind the curtain", and were able to communicate what they saw, and in the process they both expanded our conceptual vocabulary.

I'd place Mandelbrot in the same category.