Fun with Fractals

Not the Mandelbrot Set

Okay, there's only so many times you can look at a Mandelbrot Set without getting bored. After a while, you think, is that all there is? Seen that, done that.

So I figured that I'd try to come up with a few variations and alternatives. Here's a good one:

It looks a bit like one of those three-sided boomerangs, and it's listed as a "Mandelbar", or "tricorn" fractal. 

To demonstrate that this is a genuine fractal and not just something that's been "photoshopped" up from pieces of Mandelbrot set, here's a “zoom” on one of the little three-pointed islands (just visible on the left), to demonstrate the thing's recursive nature.

Okay, so it's a bit like a Mandelbrot set (the two are related), but it's different to the old “bum and a ball” shape. For starters, there's no “bum”.

The "proper" Mandelbrot set is also closely related to the Julia set, but there's a range of similar sets and combinations of sets that we can use.

Here's a "MandelDrop". It's an "inverted" Mandelbrot set:

I think this is tidier than the standard Mandelbrot, and makes the set's deeper "theme" more obvious (try counting the number of branches on the central spike of each larger bulb). 

It also reminds me a bit of the leaves of the "Mexican hat plant"

Here's a rather nice "BubbleBrot" fractal:

    

And here's a rather sinister one ...

Faceting and space-filling fractals

When I was doing the graphics for “Relativity in Curved Spacetime”, I wondered what sorts of shapes you'd get if you cut a number of maximally-sized circular faces into the surface of a sphere, then took the remaining curved surfaces and shaved off the maximum-sized piece from each of those, and then repeated the process ad infinitum. What you get is a sort of quasi-regular solid, with an infinite number of faces that have the same shape, but different sizes. In this case, all the faces are circles!

Here's a quick 3D version of the shape that you get when you start with four large facets (the equivalent of intersecting the sphere with a tetrahedron). I used this design on page 378 of "Relativity" as an "end-of-section" symbol.

One of the interesting aspects of this family of solids is that you don't need to start with a sphere. You can start by just fitting your initial circles together, keep fitting new circles where three edges form a plane ... and find,  "accidentally", that all the new points that you've created just happen to lie on the surface of the same sphere. There's been lots of work over the last couple of thousand years on conventional solids, not so much on stuff like this. Cool.

If you then create a "logical map" of this solid's surface, you get this:

This is sometimes referred to as an Apollonian net.

I was originally slightly narked when I found that this was already listed (it turns out that it's been known for raaaaather a long time), so, having written a bit of code that could generate the things, I ran off a few variations. Any of these versions can be smoothly transformed into any other by resizing the component circles.

Since the book already used the Yin-Yang symbol "[" as an icon to represent the concept of relativity (page 34 and cover), and since the quantised nature of these fractals was reminiscent of certain aspects of quantum mechanics, I used a bilateral version as the basis of a "yin-yang" symbol for quantum gravity on the title page for Part IV (p 145).

I also snuck a "ghostly" version of this symbol into the top right hand corner of the cover.

There's also a couple of variations on the same theme on p.224, one of which looks suspiciously like a fractal version of Disney's “Mickey Mouse” symbol.

If you don't need the results to be strictly mathematical, you can start with a bit of gasket code and run amok ...

Golden ratio & Fibonacci fractals

When I was producing the Hutchinson book “The Abyss of Time”, the book layout created a few blank pages that needed filling with “Golden Section”-related images.

Here's a “Fibonacci Rose”, based on a sequence of interlocking equilateral Fibonacci-series triangles that generate a double-spiral.

If we take one of these two arms and extrapolate, we can create a Fibonacci-series fractal: There's an example on p.168 of "Abyss"

As we ascend the Fibonacci series, the ratios converge on a ratio often referred to as “phi”, or the "Golden Section". The difference between the "Golden Section" version and "Fibonacci" versions is that if you use the Golden Section, you end up zooming in forever without seeing any variation. With the Fibonacci Series, you get self-similarity, but as you approach the lower limit, the proportions diverge from the Golden Ratio, and then stop dead n their tracks. It's a bit like a “quantum mechanical” version of a fractal ... at low magnifications it looks like a perfect implementation of phi, but as you zoom in you find that all its ratios are built up from a single fundamental quantised unit of scale, and once you reach that scale, there's no more detail to be had. Things stop.

A fractal configuration that did earn its place in the “Abyss” book (without an accompanying explanation) was this exercise in subdividing a rectangle.

A version of it appears on page 10. The idea is to try to fill a right-angled triangle with an infinite number of maximally-sized squares.
Although you can try this with any right-angled triangle, there are certain critical proportions at which the dimensions of the filling-squares snap into simple quantised relationships.

The first quantised solution happens with a triangle with angles of 90°, 45°, 45°. For that ratio, the squares form a cascading series where each square is exactly half the size of the last, and the quantity of squares in each size goes up in factors of two (1, 2, 4, 8, 16, 32, ... etc.).

The next solution doesn't turn up until we use the proportions above, which turn out to be those of the Golden Section.
For this solution, the sizes of the squares form a Golden-Section series, and their quantities form a familiar pattern. There's one large square in the corner, another single square one size down alongside it, then two identical copies of the next square (alongside #2 and above #1), three of the next, then five, then eight .... 

This series runs 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ... It's the Fibonacci Series!

So while it's already well-appreciated that the ratios between consecutive Fibonacci Series numbers converges on the Golden Ratio (as do an infinite number of other similar series), it's less well appreciated that if you start with the Golden Section, you can generate the Fibonacci series from it by quantisation.

As we make the shape of the triangle "sharper", we hit an infinite number of further solutions (the next one turns up at about 25 degrees, and generates 1,1,1, 2, 3, ...). This family of series that represent “special” quantised solutions for tiling the triangle with squares gives us the Generalized Fibonacci Series (mathworld). I don't know if this counted as a “new” result or not, but I posted it on sci.math just in case. :)

One of the nice things about the integer sequences that belong to the generalised Fibonacci Series is that their members interlock in a very special way, which makes them especially useful for constructing systems of weights and measures ("Abyss"), or for tiling areas. 

Here's one of the exercises that didn't make it into the book: it's a quick study of how to use Fibonacci or Golden Section sequences of cubes to fill a larger cube. 

This design was a too off-topic to be used as an incidental page-filler for "Abyss", so it'll probably end up being used for another project.

all original material copyright © Eric Baird 2007/2008