Yes, I’ve checked several Mandelbrot sites, but haven’t found the answers to:
1) It is said that the particular equation that generates this fractal, and/or the shape itself, is “the most complicated thing in the universe.” Is this true?
2) Surely the interesting shape of the Mandelbrot Fractal must not be the only peculiar shape that an equation could be written for, for instance one with the cardioid touching the circle instead of facing away from it, or one with a lemniscate (figure-8 shape) instead of a cardioid, and so on. The Mandelbrot seems so arbitrary-looking --aren’t there many peculiar shapes like it with similar properties?
3) Does the equation plot just the outline of this fractal, or are all the points on the black part also involved? (Ie., is the Mandelbrot Fractal an outline or is it solid?
4) The M Fractal is layed out on the complex plane. Is there an equation for it that would give the same shape just using the X and Y axes of the regular plane?
5) I got some clay and made a three-dimensional Mandelbrot Fractal. It was the flat one revolved around the horizontal axis, to make spheres out of the black circles and a 3-D cardioid out of the cardioid. I was wondering:other than the sheer interest of such a thing as a Mystery and aesthetically, would there be any point to it, pun not necessarily intended? Would it have an equation as simple as the one that generates the usual flat Mandelbrot Fractal?
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Originally posted by don willard *
** Yes, I’ve checked several Mandelbrot sites, but haven’t found the answers to:
1) It is said that the particular equation that generates this fractal, and/or the shape itself, is “the most complicated thing in the universe.” Is this true?*
The set is generated by iterating a very simple equation: Take a complex number. Square it, and add the square to the original number. Square the result, add the original number. Keep going. If the result takes off towards infinity, the original number is not in the Mandelbrot set. Otherwise, it is.
The shape itself, however, is infinitely complex, in the sense that no matter how closely you look at it, the amount of detail stays the same. That’s the thing that people are interested in: how such a simple process can generate such a complicated pattern.
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2) Surely the interesting shape of the Mandelbrot Fractal must not be the only peculiar shape that an equation could be written for, for instance one with the cardioid touching the circle instead of facing away from it, or one with a lemniscate (figure-8 shape) instead of a cardioid, and so on. The Mandelbrot seems so arbitrary-looking --aren’t there many peculiar shapes like it with similar properties?
**
Yes, there are many such “fractal” shapes. They’re discovered by starting with the equation and seeing what shape it generates, not the other way around as you suggest above.
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3) Does the equation plot just the outline of this fractal, or are all the points on the black part also involved? (Ie., is the Mandelbrot Fractal an outline or is it solid?
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It’s solid.
**
4) The M Fractal is layed out on the complex plane. Is there an equation for it that would give the same shape just using the X and Y axes of the regular plane?**
No. The equation only works on complex numbers, using the rules for multiplying and adding that are peculiar to complex numbers.
**
5) I got some clay and made a three-dimensional Mandelbrot Fractal. It was the flat one revolved around the horizontal axis, to make spheres out of the black circles and a 3-D cardioid out of the cardioid. I was wondering:other than the sheer interest of such a thing as a Mystery and aesthetically, would there be any point to it, pun not necessarily intended? Would it have an equation as simple as the one that generates the usual flat Mandelbrot Fractal?
**
You can’t make the real Mandelbrot set out of clay - there is too much detail. I don’t think you can define a 3-D Mandelbrot set because the complex plane is inherently two-dimensional. However, if there is some kind of 3-D generalization out there, I’m sure some genius will say so in a few posts.
The only practical application for fractals I know of is using them to generate irregular-looking patterns for use in computer animation, but I’m sure there are other uses that someone else will mention. I think at this point in its mathematical development, it is mainly of theoretical and philosophical interest rather than something with everyday practical applications.
- The equation to generate the Mandelbrot set is really quite simple. Pick a complex number, square it and add it to itself. Do the same thing with the new number. And again. And again. And so on… If the numbers get larger and larger without bound, the original number is not in the Mandelbrot set. Otherwise, it is in the Mandelbrot set.
I have heard the Mandelbrot set itself described as the most complex thing in mathematics, but that’s just hyperbole, in my opinion. While the set is certainly complicated, I can think of other mathematical objects that are more complicated, such as the Stone-Cech compactification of the natural numbers. “This is the most complex thing in mathematics,” is just a cool thing to say when showing someone a picture of it, but I don’t really think it’s true.
- There are bunches of fractals that are studied, the Mandelbrot set is just one of them. Another common one is the Newton basin fractal. Take an equation like x[sup]4[/sup] - 1, use Newton’s method to approximate solutions. Depending on your initial choice, you’ll get one of the four different solutions in the complex numbers. “Color” each initial choice one of four colors, depending on which solution it goes to through Newton’s method, and voila, you’ve got a fractal. Here’s a good freeware program to play around with fractals called Fractint. It’s got dozens of different fractals:
http://spanky.triumf.ca/www/fractint/fractint.html
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The equation of M I described above give the Mandelbrot set as that whole black set, not just the outline. The colors you generally see lying on its border are not actually part of the M set, they’re colored according to how fast they go to infinity, basically.
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Sure, just take an ordered pair (x,y), and do the operations I described in 1) just as you usually would if it was the complex number x + iy. In a sense, there’s no major difference between the complex plane and the xy-plane, anything you do in one can be done in the other, under the proper interpretations.
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It wouldn’t be much more involved than the original M set to develop a similar system to describe this 3D M set. Picture a real number line going perpendicularly through the origin of the complex plane. (In other words, we’re just setting up an xyz-coordinate space). Any point in this can now be described with the coordinates (x,y,z). Pick any of these points, use trig functions to rotate it back onto our original complex plane, then take that point and do the original operations I described in 1) to determine if our starting point was part of the 3D M set or not.
Anyway, hope that helps; let me know if any of it is unclear.
Well, I can take a stab at this:
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Not having seen everything in the universe, I’m probably not qualified to answer this one. The Mandelbrot set is pretty darn complicated, though. The equation that generates the Mandelbrot set isn’t very complicated at all; see the answer to the next question…
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You bet there are other shapes like it. Consider the way the Mandelbrot set is generated: for a complex number c, construct the sequence z[sub]0[/sub]=0, z[sub]1[/sub]=z[sub]0[/sub][sup]2[/sup]+c, z[sub]2[/sub]=z[sub]1[/sub][sup]2[/sup]+c, and so on. If the sequence z[sub]n[/sub] goes to infinity as n goes to infinity, then c is not in Mandelbrot set; otherwise it is.
The important bit is the equation z[sub]n+1[/sub]=z[sub]n[/sub][sup]2[/sup]+c which describes how to build the sequence. But if you replace that equation with another one, say z[sub]n+1[/sub]=z[sub]n[/sub][sup]2[/sup]+z[sub]n[/sub]+c, you’ll get a different shape, but one that still has many of the Mandelbrot set’s fractal properties.
One of my undergraduate professors studied this kind of thing; check out his web page.
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The Mandelbrot set is “solid”; the points inside the boundary are also part of the fractal.
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You could break the equation z[sub]n+1[/sub]=z[sub]n[/sub][sup]2[/sup]+c into it’s real and imaginary parts. That would give you two equations to iterate, namely x[sub]n+1[/sub]=x[sub]n[/sub][sup]2[/sup]-y[sub]n[/sub][sup]2[/sup]+a and y[sub]n+1[/sub]=2x[sub]n[/sub]y[sub]n[/sub]+b, where a+bi=c.
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I’m not 100% clear on what you’re describing, but I don’t think there would be a simpler mathematical description than the one you’ve already used: “take a Mandelbrot set and rotate it around the x-axis”.
There are some mathematically significant 3-D extensions of the Mandelbrot set that I can think of, but they’re a real pain to visualize. For example, instead of setting z[sub]0[/sub]=0 in the construction of the set, let z[sub]0[/sub]=r where r is any real number. This gives you three real parameters (two for c, one for r), and so you can construct a 3-D fractal which has a Mandelbrot-shaped cross section at r=0. Or you can use a complex parameter for r and get a four-dimensional fractal which has a two-dimensional Mandelbrot-shaped cross section, as well as two-dimensional cross sections which look like any possible Julia set, but at that point my brain starts to hurt.
And from a slightly less mathematical point of view (although we can’t keep math out of this kind of discussion entirely) one of the reasons the Mandelbrot set has become popular in recent years is that it is a very interesting (and pretty) demonstration of a ‘chaotic’ system.
In this usage, a chaotic system is one whose ending state is very easily affected by small changes in its initial state. So easily changed, in fact, that for these kinds of systems, you can’t predict what the outcome will be if you make a very small change. This type of system is thought to closely model many real world, not-linear systems.
In the Mandelbrot equation, you take a qualifying equation (as described by other posters here already) and test it to see which outcome you get. It either (a) gets larger and larger without end, or (b) doesn’t and stays a (relatively) small number bouncing around a range of values.
With this type of equation, there are a couple of very interesting things.
One is that we can’t look at the beginning equation and tell if it’s going to spiral forever upward, or remain small. The only way to find out is to test it by good old brute force. In other words, cycle the equation through your calculator, computer, or pencil a couple of hundred times and see what happens.
The other is that incredibly small differences in the beginning equation switch it from one outcome to the other, and those differences follow an incredibly (infinitely, it seems) complex pattern. For instance, you might find that over a wide range of values for either component of the original equation, almost every single possibility gives you an equation that doesn’t take off for infinity. But right in the middle of that wide range, there are two numbers that do. Or vice versa, right in the middle of a million different values that do spiral toward infinity, you have a handful that don’t. It seems counter-intuitive that changing the 30th decimal position of one of the initial numbers from 5 to 6 would completely change the outcome, and that then changing it from 6 to 7 would change the outcome back.
The Mandelbrot set (the bug looking picture) is generated by graphing the results of testing the Mandelbrot equation. In other words, pick a set of starting values, and put a point on a graph corresponding to those values. Test the equation to see if it goes to infinity or not. If it does, color the point on the graph with a light color. If it doesn’t, color the point with a dark color. After you test all points across a range of values, you’ll get a drawing that is the Mandelbrot set at some level of resolution. The smaller the changes you make in your numbers, the closer together the points being tested, the more points you’ll have to test to cover the range, and the more detail your graph will have. Since the changes you make in the series of numbers you test can be (in theory) infinitely small, the set has infinite possible detail, but is completely contained in a finite range.
Cool stuff.
Ugly
I am not a math person, but I am curious.
What is a complex number?
How can one number pushed through the Mandelbrot equation spin off into infinity and another not?
Well, I assume you’re familiar with the real numbers–basically, just all the possible decimal numbers you run across from day to day, like the counting numbers 1,2,3…, fractions, along with square roots and so forth, and numbers like pi, as well.
Not all the real numbers have square roots, though. -1, for example, doesn’t have a square root (no real number, when squared, can be negative). So we introduce a new number, usually called “i”, to be the square root of -1: i[sup]2[/sup] = -1.
The complex numbers are all the new numbers we get after introducing i. We now have i, 2i, 3i, i/2, and so forth (these are the imaginary numbers), and we also have combinations of real and imaginary numbers, such as 2 - 3i, pi + 3i/4, and so forth. These are the complex numbers–any number of the form a + bi, where a and b are the usual real numbers.
A useful thing about the complex numbers is that now, any polynomial you can possibly write with real (or complex) coefficients will have to have a root (in other words, will have to be equal to zero somewhere). For example, the polynomial x[sup]2[/sup] + 1 is not zero for any real number that you plug in for x, but if you plug in the complex number x=i, then it is zero. We call the complex numbers the “algebraic closure” of the reals because of this.
Anyway, as far as generating the Mandelbrot set is concerned, let’s try a couple of examples.
Let’s pick i as our starting point. We want to square it, then add it to itself:
1st step) (Picking i, which we just did).
2nd) i[sup]2[/sup] + i = -1 + i
3rd) We want to square our last result, then add i (our original choice) again:
(-1 + i)(-1 + i) + i = 1 - i - i + i[sup]2[/sup] + i
= 1 - 2i - 1 + i
= -i
4th) Again, square our last result, then add i:
(-i)*(-i) + i = -1 + i
5th) Square last result, then add i again. But this will be exactly the same as step 3)–in both cases we’ll square -1 + i, then add i, so of course we’ll get the same result, and we’ll be back at step 4). Which, of course, will give us -1 + i again, and so we’ll just cycle between step 3 and step 4–we’ll just cycle between the results -1 + i, and -i.
So, no matter how many times we square the last result, then add i, our results will never go off to infinity–they’ll just flip flop between those two values. This shows that our original choice, i, is a member of the Mandelbrot set.
(Actually, very few members of the Mandelbrot set are this well behaved; a lot of them will just bounce around “randomly” all over the place, but still have a limit to how big they get, which is the important thing).
On the other hand, say we pick 2 as our starting point:
1st) Pick 2.
2nd) Square it, then add 2: 6
3rd) Square that, then add 2: 38
4th) Square that, then add 2: 1446
It should be clear that if we keep doing this, our results are just going to keep getting bigger and bigger without bound. Therefore, 2 is not a member of the Mandelbrot set.
A computer does this for a whole bunch of complex values, performs basic tests to attempt to determine whether the results are going to infinity or not, then plots the results to give an approximate picture of the Mandelbrot set.
Funny you should mention i, Cabbage. While it’s true that the Mandelbrot equation is remarkably well-behaved at exactly i, it’s remarkably chaotic in the immediate vicinity of i. You can go an arbitrarily small distance in any direction away from i, and hit a point which will diverge, and is therefore not a part of the Mandelbrot set. If you have a computer program to draw the set, this make i (also known as (0,1)) a fun point to zoom in on.
Is the black part of the Mandelbrot set really solid? Or could there be infinitely thin intrusions into it from the border?
Lumpy:
I’m not entirely sure what you mean, but I think this should answer your question.
It is known that the Mandelbrot is simply connected, meaning, basically, that it consists of one single piece, and has no holes in it. So in that sense, yes, it’s known to be solid. There are “intrusions” into the set from the border, but none of them extend from one side all the way to the other side, which would break it into two pieces.
It is believed to be pathwise connected, but to my knowledge that has never been proven. Pathwise connected basically means if you pick any two points in the set, you can “draw” a path between them without ever leaving the set.
OK, Cabbage, this is the sort of thing that we physicists never worry about, but wouldn’t “pathwise connected” imply “simply connected”? The usual (non-rigorous) physicist’s definition for simply connected is that you can take an arbitrary “loop” of “string”, and shrink it down to a point while keeping it constrained to the set. Wouldn’t the existence of a loop in the first place be a stricter requirement than just the existence of a single strand?
WARNING – high-level math follows.
Chronos, being able to shrink a loop to a point has to do with the homology of the space, not with the connectedness. For instance, on a sphere, you can shrink any loop continuously down to a point, remaining on the surface of the sphere. On the surface of a donut/torus, there are two types of loop that cannot be shrunk to a point (the loop around the outside, and the loop through the hole, so to speak).
Basically, path-connectedness means that any two points in the space can be connected by a continuous path (in your example, you can lay a string entirely in the space that goes from one point to the other point.)
Simple connectedness has to do with overlapping open sets being able to cover the space.
It’s been almost 25 years since I’ve thought about this stuff, but there is a difference between simple connectedness and path-connectedness. IIRC, the canonical example is the set of points in the xy-plane generated by the formula y = sin(1/x). This is the curve that oscillates infinitely many times in any neighbourhood of zero (where x = 0). The line is “simply connected” (because you can cover it with open sets) but not “path-connected”, because you can’t connect a point on the positive side with a point on the negative side with a continuous line (laying a “string” between a point on the positive side and a point on the negative side has to go through zero, where the space is undefined.
OK, I await someone more current to correct my rusty memory. I do have a Ph.D. in algebraic topology, but I got out of the field in 1976.
[Edited by C K Dexter Haven on 01-29-2001 at 08:27 AM]
I hate to quibble after a such a good explanation of connectedness, but being able to shrink a loop to a point is more related to the homotopy group of a space. The complement space of a knotted circle is a standard example of a space with trivial homology but non-trivial homotopy.
(For what it’s worth, the sphere and the torus are the only two surfaces where these two groups are the same.)
Very rewarding to learn about all this. The books never explain it as well because they think the math is too hard for people, but I understand, at least about the equation for plotting the M fractal. I forgot to add that in rotating the M about the x (the horizontal) axis, we would get a big spheric shape with a funnel shape on the right side like, say, the stem side of an apple or other similar fruit, only moreso and without the stem. That would have on the left a sphere somewhat smaller than the 3-d cardioid, and the actual surfaces of both of these would have toruses (tori) or donut shaped rings around them, which would be the traces that the circles mounted on the 2-d version would leave in the course of generating the 3-d object. My clay model just put spheres on before I realized that they would be otherwise.
Chronos:
OK, I got that information off this website:
http://www.landfield.com/faqs/fractal-faq/section-6.html
I’ve checked out some references to try and make sure I get everything straight, but different texts were giving somewhat different definitions of the various types of connectedness. Anyway, I think I’ve got it all straight.
Dex:
Actually, your definition of “simply connected” looks to me more like the regular definition of “connected”. The page I was referring to defines simply connected the same way as Chronos defined it.
OK, so some definitions, not necessarily entirely rigorous, but good enough for our purposes:
Connected: A set is connected if it cannot be divided into two non-empty disjoint open sets. Essentially saying it can’t be broken into two separate pieces.
Simply connected: A set is simply connected if it is connected, and any closed curve can be continuously deformed down to a point. Basically what Chronos was saying about the string.
Pathwise connected: A set is pathwise connected if, given any two points x and y, you can map the unit interval continuously into the set with x at one endpoint and y at the other. Basically saying you can draw a curve from x to y without ever leaving the set.
Chronos, I think you meant to ask if simply connected implies pathwise connected. (A circle or torus would be a simple example of a pathwise connected set that is not simply connected).
Anyway, the counterexample is still the one Dex gave above, except I’m going to throw in the points on the y axis between y=-1 and 1, inclusive (otherwise, we’ve got a gap right there disconnecting it). This is one of the standard variations of what’s called the “topologist’s sine curve”. It’s connected and simply connected, but not pathwise connected.
If you map the interval continuously into it, it can’t cross between the left half and the right half. So if you have a closed curve in the set, the entire curve must be either in the left half, or in the right half (or right in the middle), exclusively (by “closed curve” I’m also allowing a straight line segment, in the sense that you can start at one endpoint, go down to the other endpoint, then retrace back to the original endpoint–that’s a closed curve). It should be fairly clear, then, that it’s possible to deform any of these closed curves down to a point. Therefore, it’s simply connected.
However, it’s not pathwise connected, simply because you can’t connect any point in the left half with any point in the right half.
Anyway, I believe all the above is correct.
I couldn’t even start to enter most of the discussion on this thread, but I thought I could help the OP with the second question.
The Mandelbrot set is a special case of a type of set called a Julia set. If you search the web for Julia sets, you’ll see lots of fractals that don’t look like the Mandelbrot set.
(BTW, Math Geek did mention this term earlier in the thread. I just wanted to mention it specifically, in case it was lost in the discussion.)
Thanks for the explanation, Dex and Cabbage. I think that I understand it now… And you’re right, I did mean simply implying pathwise, not versa-vise.
ubergoober: << I hate to quibble after a such a good explanation of connectedness, but being able to shrink a loop to a point is more related to the homotopy group of a space. >>
Yes, sorry about that, shows what 20+ years can do to the mind. I certainly meant homotopy.
Cabbage, thanks for the distinction between “simply connected” and “connected” which I had forgot (or which hadn’t been made in my day.) But do I understand that a torus (donut) would NOT be simply connected (since there are two curves that cannot be shrunk to a point?)
Yeah, that’s right, a torus isn’t simply connected precisely because of those two “holes” in it.