The Mighty Mandelbrot

Mandelbrot Antennae
[click for big]

I’ve heard it called “the most complex” mathematical object, but that’s a hard title to earn, let alone hold. Its complexity does have attractive and fascinating aspects, though. For most, its visceral visual beauty puts it miles ahead of the cool intellectual poetry of Euler’s Identity (both beauties live on the same block, though).

For me, the cool thing about the Mandelbrot is that it’s a computation that can never be fully computed.

It’s not alone in that, There are many computations that can never be fully computed.

For a simple example, hand a naive division algorithm the values 1 and 3, and watch it loop forever generating decimal 3s. The value can never be fully computed.

Or consider any of the many algorithms for generating digits of pi, a number that goes on forever (without repeating, like 0.333… does).

In both cases, the computation needs to be stopped at an arbitrary point by an outside agency. The computation itself never ends.^[1]

The “outside agency” can be additional (or better) code to the algorithm such that it, in the first case, recognizes a repeating sequence and terminates. Or, in either case, it can involve checking a “guard” parameter — a maximum amount of time to calculate, or a maximum amount of digits to generate.

Our inability to know, in general, whether a given computation with given input halts or not, is a formal limit on what we can know about computation. It’s called the Turing Halting Problem.

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Before I continue, let me pause to clarify how I use some terms.

A computer is a mechanism that (mechanically or electrically) performs a computation on — that is to say: changes the state of — some data according to an algorithm.

Further, a computer is a mechanism with formal system states for which there is some FSM (finite state machine) describing those states.

Data is structured information that, per Information Theory, reduces to bits (ones and zeros). Data is always representational — it always symbolizes something.^[2]

An algorithm is a set of instructions. It tells the computer how to move from state to state. An algorithm always has a TM (Turing Machine) that represents it.^[3]

In particular, algorithms are mathematical abstractions. [See Calculated Math and Coded Math for a detailed look at algorithms.]

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With that in mind, let’s turn back to the Mandelbrot.

[f you are relatively new to this fabulous and fascinating fractal, or need a refresher, please read Sideband #43: Chaos!, or read the linked Wiki article.]

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“Planck-level”! (×10³³)

In mathematics, there are two very distinct ways to generate a mathematical value.

Firstly, there is the idea of evaluation.

The numeric quantity we think of a one-eighth has an immediate value. For instance, it’s one slice of an eight-slice pizza. Or one of your eight chess pawns.

When we write 1/8 we refer to that immediate one-of-eight value.

Secondly, there is the idea of calculation.

In this case, a series of steps (an algorithm), with transitory intermediate values, results in a final value. As noted above, that value may be truncated if the calculation was stopped before it delivered a final result (which implies stopping because the algorithm is “done”).

A calculation using the algorithm of long division lets us convert the numerical quantity one-eighth to its decimal expansion, 0.125 (which is just another way to express its value).

Note that calculation here is synonymous with computation.

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For me, evaluation and calculation are Yin-Yang properties.

I see them as discrete (calculation) versus analog (evaluation), as different as communicating by passing typed notes versus using two tin cans with a taut string between them.

And, getting to the point: the Mandelbrot is calculated.

It has to be. Its definition is algorithmic:

Note that very important while condition! It speaks to the algorithmic nature of the definition. The definition has a conditional aspect — a hallmark of computation.

The n and n+1 subscripts on the Z parameter also signify the algorithmic definition. They speak to the “before” and “after” state of the data.

The right side has a value, an evaluation of Z²+C, that creates a new Z.

So long as the new Z is below the max allowed value, it’s plugged into the right side, and the evaluation is repeated (looped) to generate a new Z.

This repeats until Z “escapes” (goes above max) or until n (the number of times doing this) goes above its allowed max.

This is a typical algorithmic process: Repeat a sequence of steps until the data state meets a terminating condition.

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Mandelbrot Spines!

Whether a given point, C, is in the Mandelbrot set or not depends on whether or not it escapes.

If it does, it’s not.

Only points that never escape are in the M-set.

To be clear, the algorithm is performed on every point, C (coordinate), that appears in the desired image — which can be seen as a “window” into the Mandelbrot. Every pixel in window is tested to generate the image.

Points that don’t escape — that are in the Mandelbrot — usually get colored black. Points that do escape get colored according to some user-specified coloring algorithm.^[4]

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You might now see why the Mandelbrot can ever be fully computed.

Points that don’t escape — no matter how many loops we do — might escape if we bothered to keep on looping.

Inside the boundaries of the set (in the the black areas), it does becomes fairly obvious that points won’t escape. (The same way it’s obvious that 1÷3 loops forever, whereas 639538121÷7720610328 is a lot less obvious.)

Near the boundaries, however, it becomes impossible to predict.

Chaos on the border! (×47)

All along its boundary, the Mandelbrot experiences the Turing Halting Problem, so the border can never be fully computed. It can only be determined to some precision.

One irony of the Mandelbrot set is that we can tell for sure which points aren’t in the set (because they escape), but we can never be sure about borderline ones that seem inside.

The image to the right illustrates the border chaos. [Click any image for big.]

The black circles lower right consist of points in the M-set. They’re part of one of the little satellite “clones” of the main Mandelbrot cardioid. What looks like colored visual static around the black areas is literally mathematical chaos.

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Zooming in on the chaos resolves deeper and deeper levels of structure but also more chaos.

And as with most fractals, it’s possible to zoom in forever. It just takes longer and longer to calculate each point. [Deep zoom Mandelbrot images can take days, even weeks, to render one image.]

One thing I love about Mandelbrot deep zooms is that, if we think of the pixels of the final deep zoom image as Planck lengths — the smallest possible meaningful distance — then the main Mandelbrot, is many, many, many times larger than the visible universe.^[5]

In fact, there is no limit, which is another example of the real numbers being quite a conundrum. It also speaks to the abstract nature of the Mandelbrot.

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Another satellite. (×10⁶)

One aspect of the Mandelbrot that fascinates me has to do with how the M-set is fully, but simply, connected.

Fully connected means we can walk from any point in the M-set to any other point. All points are connected by some path within the set.

Simply connected means there are no islands inside the M-set. No path can loop around points outside the set.

Together, these properties mean the Mandelbrot is topologically a disc. It can be stretched without breaking anything such that you end up with a disc.^[6]

One thing this means is that no matter how deep we go, no matter how convoluted and twisted things get, there is still always a path to the main cardioid and a path to the “greater outside” (the area surrounding).

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It also seems to mean that the “spines” or “threads” that connect the Mandelbrot satellites have zero thickness — they are one-dimensional lines (“singularities”) on the complex plane.^[7]

The Mandelbrot, thus, has three parts:

1. The main cardioid (sometimes called the Mandelbrot “lake”).
2. An infinite (?) number of satellites “clones” of the lake.
3. Connecting threads that link all the satellites to the lake.

The one-dimensional nature is especially apparent on the real number line, from -2.0 to -1.25 or so. The point (-0.75, 0.00) is also interesting.

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But I’ve gone on long enough for one post. I may have more to say about my favorite mathematical object later, but I have some other stuff I need to focus on right now.

At least there was some Mandelbrot in May!

Stay chaotic, my friends!

∇

^[1] It can only run out of physical resources.

^[2] It can symbolize other data: the bit pattern 00110000 is one symbol for the digit value zero.

^[3] It’s not, of course, named after Al Gore. It’s named after Muhammad ibn Musa al-Khwarizmi, a Persian polymath who contributed much to early mathematical foundations.

^[4] There are also algorithms for coloring the inside. Much of the visual beauty of the Mandelbrot comes from various ways of coloring the points, which involves both the color palette and how colors are selected from the palatte.

^[5] Assuming a universe 30×10⁹ Light Years (LY) wide, that makes it mere 1.755×10⁶¹ Planck Lengths (PL) across. [1 PL = 1.616229×10^-35 meters. 1 LY = 9.4542×10¹⁵ meters.]

Mandelbrot deep zooms go into the high hundreds or low thousands of powers of ten!

^[6] Or square or hexagon or five-pointed star or any solid 2D shape you like.

^[7] In the especially chaotic parts near the border, these lines can amount to 2D space-filling curves.