Koch Snowflake Arrays

Koch Snowflake Fractal Arrays (an alternate Koch Snowflake construction method)

Another couple of non-standard ways of creating Koch Snowflake outlines. These two approaches use a ring of six corner-touching hexagons, with or without an additional seventh central copy. There's an example of the first version appearing as a fractal solid's silhouette in figure 9.5 of the book (as the tiny lower-right image). The book also has face-aligned hexagon-based Koch Snowflakes in figures 3-13 and 7-5.


Note that every smaller detail is also a Koch Snowflake, and every little remaining space between the snowflakes also progressively gets nibbled away to form yet more Koch snowflakes as you apply more iterations.

The more conventional Koch Snowflake approach gives a self-similar fractal outline, but its components are triangles. In these two cases, we go one step better – we generate the same fractal outline as before, but now the whole thing is fractal, including the interior. In theory, it's a Koch Snowflake built from nothing but other Koch Snowflakes. The shape becomes its own building-block, and its own template ... no fundamental shape exists in its self-contained geometrical universe but itself.

For more examples of Koch Curve tilings, see figures 30.4 and 31.2 in the book.

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 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.