Fractals

A Julia set, a fractal related to the Mandelbr...
A Julia set, a fractal related to the Mandelbrot set (Photo credit: Wikipedia)

Now and then I fire up one of those programs that displays a fractal on the screen. These programs use mathematical programs to display patterns on the screen. Basically the program picks the coordinates of a pixel on the screen and feeds the resulting numbers to the program. Out pop two more numbers. These are fed back to the program and the process is repeated.

There are three possible outcomes from this process.

Firstly, the situation could be reached where the numbers being input to the program also pop out of the program. Once this situation is reached it is said that the program has converged.

Convergent light beam passing through a square...
Convergent light beam passing through a square hole (Photo credit: Wikipedia)

Secondly, the numbers coming out of the program can increase rapidly and without bounds. the program can be said to be diverging.

Thirdly, the results of the calculation could meander around without ever diverging or converging.

English: The Markov chain for the drunkard's w...
English: The Markov chain for the drunkard’s walk (a type of random walk) on the real line starting at 0 with a range of two in both directions. (Photo credit: Wikipedia)

A point where the program converges can then be coloured white. Where it diverges, the point or pixel can be coloured black. A point where the program seems to neither converge nor diverge can then be coloured grey. A pattern will then appear in the three colours which is defined by the equation used.

Anyone who has seen fractals and fractal programs will realise that a three colour fractal is pretty boring as compared to other published fractal images. Indeed the process that I have described is pretty basic. A better image could be drawn by colouring points differently depending on how fast the program converges to a limit. This obviously requires a definition of what constitutes convergence to a limit.

Fractal Art
Fractal Art (Photo credit: Wikipedia)

Convergence is a tricky concept which I’m not going to go into, but to compute it to say in a computer program you have to take into account the errors and rounding introduced by the way that a computer works. In particular the computer has a largest number which it can physically hold, and a smallest number. Various mathematical techniques can be used to extend this, but the extra processing required means that the program slows down.

[Fractal]
[Fractal] (Photo credit: Wikipedia)
I’m not going to explain how this difficulty is circumvented, since I don’t know! However the fact is that the computer generated fractals are fascinating. Most will allow you to continually zoom in on a small area, revealing fantastic “landscapes” which demonstrate similar features at all the descending levels. Similar, but not the same.

fractal landscape
fractal landscape (Photo credit: Wikipedia)

The above far from rigorous description describes one type of fractal of which there are various sorts. Others are described on the Wikipedia page on the subject.

Another interesting fractal is created on the number line. Take a fixed part of the number line, say from 0 to 1, and divide it into three parts. Rub out the middle one third. This leaves two smaller lines, from 0 to 1/3 and from 2/3 to 1. Divide these lines into three parts and perform the same process. Soon, all that is left is practically nothing. This residue is known as the Cantor set, after the mathematician Georg Cantor.

English: A Cantor set Deutsch: Eine Cantor-Men...
English: A Cantor set Deutsch: Eine Cantor-Menge Svenska: Cantordamm i sju iterationer, en fraktal (Photo credit: Wikipedia)

This particular fractal can be generalised to two, three, or even higher dimensions. The two dimensional version is called the Sierpinski curve and the three dimensional version is called the Menger sponge.

One of the fractal curves that I was interested in was the Feigenbaum function. This fractal shows a “period doubling cascade” as shown in the first diagram in the above link. If you see some versions of this diagram the doubling points (from which the constant is determined) often look sharply defined.

English: A very old ficus tree in São Paulo, B...
English: A very old ficus tree in São Paulo, Brasil. Deutsch: Ein sehr alter Feigenbaum in São Paulo, Brasilien. Português do Brasil: Uma figueira muito antiga nas ruas de São Paulo, Brasil. (Photo credit: Wikipedia)

I was surprised the doubling points were not in fact sharply defined. You can see what I mean if you look closely at the first doubling point in the Wolfram Mathworld link above. Nevertheless, the doubling constant is a real constant.

English: Bifurcation diagram Česky: Bifurkační...
English: Bifurcation diagram Česky: Bifurkační diagram Polski: Zbieżność bifurkacji (Photo credit: Wikipedia)

Another sort of fractal produces tree and other diagrams that look, well, natural. A few simple rules, a few iterations and the computer draws a realistic looking skeleton tree. A few tweaks to the program and a different sort of tree is drawn. The trees are so realistic looking that it seems reasonable to conclude that there is some similarity between the underlying biological process and the underlying mathematical process. That is the biological tree is the result of an iterative process, like the mathematical trees.

Русский: Ещё одно фрактальное дерево. Фракталь...
Русский: Ещё одно фрактальное дерево. Фрактальное дерево. (Photo credit: Wikipedia)

I’ve mentioned natural objects, trees, which show fractal characteristics. Many other natural objects show such characteristics, the typical example which is usually given is that of the coastline of a country. On a large scale the coastline of a country is usually pretty convoluted, but if one zooms in the art of the coastline that one zooms in on stays pretty much as convoluted as the large scale view.

Mandelbrot fractal. Rendered as an island with...
Mandelbrot fractal. Rendered as an island with Terragen, a fractal-based landscape generator. (Photo credit: Wikipedia)

This process can be repeated right down to the point where one can see the waves. If you can imagine the waves to be frozen, then one can take the process even further, but at some point the individual water molecules become visible and the process (apparently) reaches an end.

If you want a three dimensional example, clouds, at least clouds of the same type, probably fit the bill. Basically what makes the clouds fractal is the fact that one cannot easily tell the size of a cloud if one is simple given a photograph of a cloud. It could be a huge cloud seen from a distance or a smaller cloud seen close up. Of course if one gets too close to a cloud it becomes hazy, indistinct, so one can use those clues to guess the size of a cloud.

Fractals were popularised by the mathematician Benoit Mandlebrot, who wrote about and studied the so-called Mandlebrot set, wrote about it in his book, “The Fractal Geometry of Nature”.  I’ve read this fascinating book.

English: Topological model of Mandelbrot set( ...
English: Topological model of Mandelbrot set( reflects the structure of the object ) Polski: Topologiczny model zbioru Mandelbrota ( pokazuje strukturę obiektu) (Photo credit: Wikipedia)

While I was searching for links to the Mandlebrot Set I came across the diagram which shows the correspondence of the period doubling cascade mentioned above and the Mandlebrot set. This correspondence, which I did not know about before, demonstrates the interlinked nature of fractals, and how simple mathematics can often have hidden depths. Almost always has hidden depths.

English: Paths of correspondence between scien...
English: Paths of correspondence between scientists (Photo credit: Wikipedia)

 

The Negative Universe

Dandelion(negative)
Dandelion(negative) (Photo credit: tanakawho)

 

If cosmologists are to be believed the Universe came from nothing and is likely to return to nothing. This is odd as there seems to be an awful lot of it! There are between ten to the power 78 and ten to the power 82 atoms in the observable universe, according to some estimates. There’s also a huge amount of energy out there in the universe, and as Einstein said, this is equivalent to matter, according to his famous equation.

Maxwell's Equations
Maxwell’s Equations (Photo credit: DJOtaku)

 

It is likely that the enormous amount of matter and energy that we see out there is balanced by an equivalent amount of “negative” matter and energy somehow. “Negative” is in scare quotes because it may not describe what is actually going on. Anyway, the negative matter and energy may be incorporated into this universe somehow, which means that on average half the universe is this sort of energy. We can’t see it anywhere so far as I know, so it is a bit of a puzzle.

Large Format Doha Panorama Portra 400
Large Format Doha Panorama Portra 400 (Photo credit: Doha Sam)

We can see evidence everywhere for “normal” matter and energy, and we should be able to see evidence of “negative” matter if it is anywhere near us. As I understand it, “negative” matter would behave differently to “normal” in various ways and should be detectable. I’m not sure in what ways it would be different – I can guess that there could be a gravitational attraction between particles of “negative” matter, just as there is between particles of “normal” matter, but there could be a gravitational repulsion between “negative” matter and “normal” matter for example.

Galaxy NGC 720 (NASA, Chandra, 10/22/02)
Galaxy NGC 720 (NASA, Chandra, 10/22/02) (Photo credit: NASA’s Marshall Space Flight Center)

But my ignorance is almost total. I do believe it is true that “negative” matter should be detectable.) Since we can’t see or detect “negative” matter within our locality (ie “the observable universe“) it may be grouped elsewhere in the universe. If so, it may not have any observable effect in our neck of the woods, but inevitable it will have an effect at some time in the astronomical future.

Español: es la misma imagen que aparece en el ...
Español: es la misma imagen que aparece en el articulo en ingles: http://en.wikipedia.org/wiki/Sloan_Great_Wall (Photo credit: Wikipedia)

The reason that I say this is that the universe doesn’t seem to be expanding faster than the speed of light so any effect such as (possibly) gravity which does appear to have a “speed of light” effect will eventually affect our corner of the universe. But the situation is complex, and as the Wikipedia article says,

Due to the non-intuitive nature of the subject and what has been described by some as “careless” choices of wording, certain descriptions of the metric expansion of space and the misconceptions to which such descriptions can lead are an ongoing subject of discussion in the realm of pedagogy and communication of scientific concepts.

In other words, there are many misconceptions and misinterpretations around this topic. However any effect of the possible existence of “negative” matter on our little neck of the universe is likely to be felt a long time in the future, even on a cosmological time scale, I feel. “Negative” matter could have created a negative universe, I guess, which mirrors this universe.

Photo of a printout of the Wikipedia Copyleft ...
Photo of a printout of the Wikipedia Copyleft reflection in mirror, with pen (Photo credit: Wikipedia)

In a negative universe at least one dimension would be reversed but all other dimension would have the same polarity as our universe. If an odd number of dimensions were reversed, would all but one cancel out? I’m not sure but a cursory mathematical examination would indicate that this would not be so, but I lack the time to explore the concept in depth.

Dimensions
Dimensions (Photo credit: Wikipedia)

In our universe things tend to a state of disorder. If one partition of a closed system contains all the matter (in the simplest case, as a gas) and the partition is removed then eventually the matter is eventually dispersed through the whole system. In a negative universe, possibly the opposite would apply – gas dispersed through a system could tend to bunch up in one part of the system. Maxwell’s demon could watch benignly without lifting a finger.

Demonio Maxwell 2
Demonio Maxwell 2 (Photo credit: Wikipedia)

Our view of this is that it is extremely unlikely – would a glass spontaneously rise from the floor, gathering the scattered wine and land on the table complete with wine? Perhaps this is a parochial view, only true in our universe. In some alternate universe, this may be the normality – entropy may tend to decrease, order may tend to increase. Such an entropy twin may simple be the time reversed twin of the original universe. Or the original universe perceived from a time reversed perspective.

The Grand Canyon Time-Zero Project
The Grand Canyon Time-Zero Project (Photo credit: futurowoman)

If the universe sprung out of nothing, then the sum of the universe is zero. Any object has its anti-object. Any event has its anti-event. Maybe the universe has a partner, an anti-universe if you like, or even a mirror universe. Time in our universe runs from the zero moment into the positive future. In a mirror universe would presumably (and debatably) run in the opposite direction from the zero moment and all spacial dimensions would be reversed.

Plus-Reversed,-1960
Plus-Reversed,-1960 (Photo credit: Wikipedia)

This would correspond to a point reflection in time and space, which may or may not be the same as a rotation in time and space. I’m not sure. Some of complexities can be seen in this Wikipedia article on “parity”. In particular some interactions of elementary particles may display chirality, which means that they come in left and right handed versions, like gloves or shoes. All of the above means that if a person were to be point reflected into an anti-universe and all the elementary particles of his body were switched with their anti-particles, there may be no way for the person to tell that the switch had occurred.

different flowers from same plant
different flowers from same plant (Photo credit: ghedo)

Sure, time would be reversed, but so would literally everything else, so a left-handed glove would appear, in the point reflected world, to still be a left-handed glove even though, if we could see the glove it would appear to us, from our point of view to be right-handed. Of course I’ve assumed for much of the above that the reflection that transforms a universe to its anti-universe is a point reflection.

Axial chirality
Axial chirality (Photo credit: Wikipedia)

In mathematical terms that means that all variables are reversed. That is x is replaced by -x, y by -y, z by -z and so on. It may be that the reflection may be in a line and the x dimension stays the same while the others are negated. Or it may be a reflection in a plane (a mirror reflection) where 2 dimensions are unchanged. Or it may be a reflection in a higher number of dimensions.

English: Upper Yosemite fall with reflection
English: Upper Yosemite fall with reflection (Photo credit: Wikipedia)

As you can see, the subject is complex and I’ve not got my head around it (obviously!), but I believe that if we were switched into the anti-universe (including all out particles) it would not look any different from this universe. In fact we would probably find ourselves discussing our anti-universe, which would be our original universe. In fact it would not matter which universe we called “the original” because they both would have come into existence at the same time and there would be no meaning to the term “original”.

A face.. (the original OMG Wall)
A face.. (the original OMG Wall) (Photo credit: eworm)