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.

An evolving fractal sequence, "Zero Time"

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.

2011 notes

These pages have since been rewritten, expanded, and turned
into a book, "Alt.Fractals: A visual guide to fractal geometry and design" (2011) ISBN 0955706831 .