What Else Exists Beyond the Universe

If you followed my argument that numbers exist, and that humans discovered them (we didn't invent them), then a logical next step is to ask what else exists, without physical form, ready for intelligent life to discover.

Let's start with a simple equation :

I've already shown you the formula for a circle :

x² + y² = 1

This only use real (i.e. not complex) numbers for x and y, but as I've already shown, complex numbers are the true nature of numbers, and real numbers are a subset. If we start with only real numbers we will be forced to embrace complex numbers as soon as we notice that square, so another way of defining a circle is using the "magnitude" operator, which is denoted within vertical bars like so :

|z| = 1

You can read this a the magnitude of the complex number z equals 1. On the 2d plane of numbers, the set of points, whose magnitude is one defines a circle of 1 centred on the number zero.

Now lets look another equation :

a = b² + c

It looks similar to the first equation for a circle, but it isn't so similar, because from now on, I'll be using complex numbers throughout. i.e. a, b and c are all complex (2 dimensional) numbers.

Lets make b = 0 and c = 1 (I could also write this as b = 0 + 0i and c = 1 + 0i, which makes it clearer that b and c are complex numbers).

In this case a = 1

Now I'm going to perform a trick called "iteration". Let b become the previous value of a. So now that b = 1, we can calculate another value for a. The new answer is 2.

Repeat this again, setting b to 2, and calculate a new value for a.

If we continue this process we get the sequence :

1, 2, 5, 26, ...

Let's now say that b is always 0 for the first iteration, but we can choose any value for c. So let's choose a different value for c and do it again.

Letting c = −1 gives the sequence −1, 0, −1, 0, ...

Letting c = 2 gives the sequence 2, 6, 38, ...

Letting c = -2 gives the sequence -2, 2, 2, 2, ...

Try putting in other values for c. You will notice that some values give sequences that stay small, and other sequences get bigger and bigger. Mathematicians use the terms bounded and unbounded.

It turns out that if |b| > 2 (the magnitude of b is greater than 2), then the sequence is unbounded, otherwise it is bounded.

So, if I draw a dot for all of the bounded numbers, then we have :

.   _____________________
   -2   -1    0    1    2

But this diagram only shows real values of c, but we are free to choose any complex number for c, so we need to extend this diagram into 2 dimensions.

Letting c = i gives the sequence i, -1 + i, -1 -i, -1 +i, ...

If you'd like to calculate other sequences for yourself but you don't want/know how to calculate using complex numbers, then consider using Wolfram Alpha to do the hard work for you ;-) (You can use it as a calculator, but it can do a whole lot more - it's great!!) Note. To square a number use ^2, which means "to the power of two", which is another way of saying "squared". But when squaring a complex number, remember to use brackets, because 1+i ^ 2 is NOT the same as (1+i) ^ 2

So, now let's plot the bounded points in the 2D complex plane :

.
 			
                                    #### 
                                   #######
                                    ####  
                            ##   ###########    
                             #################
                    ###       ##################
                 #########  ####################
                #################################
          # ####################################
  #############################################
          # ###################################
                ################################
                 #########  #####################
                    ###       ##################
                             ##################
                            ##   ###########  
                                    ####
                                   #######
                                    #### 
                                      

You may recognise this shape as the Mandelbrot Set.

So far so good, but there is another trick we can do to make the picture more interesting. For the values of c, which lead to unbounded sequences, instead of colouring them white, we can pick a colour to indicate how many iterations it takes before |a| > 2 (the magnitude of a is greater than 2).

Here's an example plot which uses dark blue for 1 iteration, and gradually getting lighter as more iterations are required before |a| > 2. Black is used for points which are bounded (i.e. whiere |a| is never > 2).

This diagram looks quite complicated, there are lots of tendrils, which is quite surprising for such a small equation. Don't forget, we started with a very simple equation :

a = b² + c

which resulted in a beautiful and complex picture. But how complex is it? If we zoom in on any of the tendrils, something surprising happens. We see more and more detail, and there is no end, just more and more complexity. Also, no two areas are the same (ignoring the obvious horizontal symmetry). Many areas are quite similar to others, but never exact. There are many other interesting features. For example, the bounded (black) area is connected. i.e. there is just one single island of black. If you see two black lake, there is always a river connecting them.

There are lots of beautiful videos on YouTube, zooming deeper and deeper into the Mandelbrot set. You can also download programs for you PC/tablet/phone which let you pan and zoom around the set, and in a short space of time you will see part of the set that nobody has ever seen before.

Why?

You may be asking why I've spent so long explaining the Mandelbrot set, when my initial question was "What exists beyond the physical universe". Every picture of the Mandelbrot set (and there are infinitely many of them) was generated from a very simple equation. They existed before Benoit_Mandelbrot discovered them, and they will continue to exist after all humans have died. It is a discovery and not an invention. Other races other than humans may discover the same images (though they may render them differently, for example, I can imagine creatures without sight to render it using sound). If there were other universes with different laws of physics, beings in that alternative reality could discover it. It exists outside of the physical universe.

How Much Data is Contained in the Mandelbrot Set?

There are two answers, the first is "very little". After all, the equation a = b² + c is very small. The second answer is "infinite" (you can zoom in as much as you wont, and pan around where you want, and no two regions are exactly the same).

For those who understand a little about Information theory will see a flaw in the second answer. If we plot the infinite set as a picture and save it as a bitmap image, it would take a large (actually infinite) number of bytes. But when we save photos, we often use jpeg, which compresses the image, so that it takes up fewer bytes. We could use compression to save the Mandelbrot set too. Jpeg compression wouldn't work, but there is a compression routine that is capable of compressing the image from an infinite amount to a finite amount... That compress would be the program that created the beautiful images in the first place!

However, there's a flaw in the compression algorithm... It would take an infinite take to decompress. To put it another way, the decompression would never finish.

Even if you aren't concerned that in theory, you could compress the entire set (but could never practically decompress it in a finite time), that doesn't concern me. Here's why...

We look out at the night sky, and see millions, nah, billions of stars. Until we created powerful telescopes all the stars we could see were in our own galaxy, the milky way. Some bright spark at NASA decided to point the hubble telescope at a tiny, boring, black patch of the sky, and instead of seeing nothing, we see yet more complexity of distant galaxies.

The universe is unimaginatively huge, the number of galaxies is staggering. The number stars in a single galaxy is staggering. The number of atoms in each star is staggering. The amount of data in our universe must be staggeringly large. No! Maybe not.

If there is infinite complexity in the image of the Mandelbrot set, and yet it can be compressed down to a mere six symbols, then who is to say that the unimaginable complexity of our universe cannot also be boiled down to a tiny number of symbols. This is the ultimate goal for physicists.

Imagine that we lived inside a Mandelbrot set, instead of our own universe, then we would see lots of swirly patterns, all quite similar, but all subtly different. Physicists would try to understand this strange world. If they had telescope-like objects, which allowed them to look at distant parts of the set, they may discover that there are lots of black lakes and that the lakes are all joined to each other by thin rivers. If they discovered the simple equation a = b² + c, then their goal would be complete; physics would be solved. They would have a Theory of Everything.

They would know how their entire universe worked, and if theists asked them how such a complex and infinite universe could spring from nothing, the answer would be simple. The infinite complexity is contained within just six symbols :

a = b² + c

I find it funny that our universe and the Mandelbrot set both contain swirly patterns, all quite similar but ultimately unique. Galaxies are swirly pattens of stars. I could even stretch the point by saying that tornadoes look similar to galaxies, with the eye of the tornado being analogous to the black hole at the centre of galaxies.

Alternative Sets

The Mandelbrot set isn't the only game in town. It's known as a fractal, and there are many (infinite) other fractal patterns to choose from. For example, about 30 years ago, while learning A-level maths, I used Newton's Method to find the roots of z³. If you count the number of iterations it takes to get "close enough", and assign colours in the same way, you get an infinite fractal pattern with three fold rotational symmetry. The image isn't a compelling, and there's a problem defining "close enough" - the further you zoom in, the smaller that number has to be to maintain the fractal pattern. There is no clear dividing line as there is with the bounded/unbounded region of the Mandelbrot set.

Static vs Dynamic

An obvious problem with the analogy between our universe and a fractal pattern is that a fractal pattern is static. Every point is unchanging, there is no sense of time. I have two potential ways to overcome this. The simplest is to point out that since Einstein, space and time have been merged - they are NOT separate entities, there is only one thing called space-time, which has four dimensions. So if we create a 4 dimensional fractal pattern (with two complex numbers instead of c in the Mandelbrot set), I posit that it could feel like a three dimensional world to creatures who inhabit that world.

For those who struggle with the concept of space-time, I have a different analogy. Take a fractal (which doesn't have time or motion), and merge it with Conway's Game of Life (which does have time and motion). I've yet to see anybody attempt to write such a program, but I can't see that the concept is impossible.

Determinism and Free Will

If our universe were similar to a fractal set, it would be utterly deterministic. There is no possibility for free will. My simple answer to this is : So what? I'm looking for truth, not comfort. IMHO, there is no compelling evidence that we have free will. In fact, just the opposite. I believe our thoughts are entirely based on the goo inside our skull and the signal it receives through our senses. I don't believe in a soul, or other non-physical stuff. If you believe a soul is required for our though processes, I ask the following questions :

1. Can a brain make decisions without a soul
2. Do dogs/mice/ants/plants/amoeba/non-living lumps of matter also have souls or do they make decisions without one?
3. What difference does the soul make in decision making process. For example, if I could somehow arrange atoms using a make-believe 3d printer, so that it had atoms in the same configuration as a human (or ant's) brain, would this artificial brain behave differently to a real one.
4. At which point does the soul effect the physical world. If I choose to hit someone, the result is the movement of billions of atoms (my arm). For a soul to affect that movement implies that the laws of physical matter do NOT apply, and instead, some physical matter is governed by something else.

Souls affecting overriding physical laws is and extraordinary claim, so I think the onus is upon you to back up this claim.I've yet to find anybody claiming that amoeba have souls, and yet they make decisions based on their environment (as do higher non-human entities), so I see no reason to invoke a soul. Club together, build a kick-ass MRI machine, and try looking for the point at which physical laws are broken!

Quantum Randomness

As stated above, my fractal analogy is purely deterministic, but quantum mechanics is not. There are quantum experiments that give completely random results (unlike Newton's view of the world which was deterministic). For example, if you measure the spin of an electron, it will be measured as either up or down (but never anywhere in between). It has been proven that this randomness is NOT derived from so called hidden variables i.e. the electron doesn't have an up or down state which is merely hidden from us until it is measured.

Check out The Look Glass Universe for a glimpse of how weird spin is, and how quantum randomness isn't like the randomness of a coin toss.

At this point, I must bale out, as I don't understand this well enough to say if it is possible to create fractal images with quantum randomness. However, it is simple enough to add run-of-the-mill randomness to a fractal image (for example, just add 1 or zero at random to every calculated point). Interestingly, the resulting image would look nearly identical (as the overall structure would remain the same), but the image would become uncompress-able. There are other ways to add randomness to a fractal, which would affect the overall shape, as if distorting the image like a hall of mirrors at a fairground.

Conclusion

The amount of stuff that exists outside of our universe is an entire universe (actually an infinite number of universes). Because either :

1) Sets, such as the Mandelbrot set have infinite complexity, and exist without reference to any actual reality. or 2) Our universe, despite its seeming complexity, could actually be ridiculously lacking in data, just as the entire complexity of infinite Mandelbrot set is only 6 symbols.

Even if you see no flaws in my analogy between our universe and fractal images, there is another important question to answer. What Breathes Fire into the Equations (a question posed by Stephen Hawking).

Which I can rephrase as : If our universe is a simple equation which gives rise to the glorious universe we see around us, where is the computer on which it is rendered?

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