Several very zoomed in sections of the Mandelbrot set using software I wrote a while back (and an image of the whole set) - they don’t seem to have the psychedelic qualities of most online images we see and I’m not sure whether I’m doing something wrong or if that just reflects the artistic license of others…. Critical feedback/guidance welcomed.

The Mandelbrot set explained:1. The Mandelbrot set - the usually black bug-like shape - is a set of points which under some straightforward steps essentially don’t run off to the hills but stay local!

2. OK - with an ordinary real number, say 2 or -0.5, let’s see what happens - we need to multiply the number by itself then add the very first number we started with, then the answer to that gives us the next number to do the same process to, and so on:

Starting number is2: 2x2 +2 = 6, so6: 6x6 + 2 = 38, so38: 38x38 + 2= … And so on, getting bigger and bigger so clearly running to the hills (towards infinity).

Starting number is-0.5: -0.5x-0.5 - 0.5 = -0.25 so-0.25: -0.25 x-0.25 - 0.5 = -0.4375, so-0.4375: … and if you repeat a hundred or more times you will still get small numbers, so -0.5 is a ‘stay local’ number and would be a black dot in the set.

3. No, this does not give us the 2D bug-like picture…. However, in schools we have a thing called a number line, with zero in the middle, negative numbers on the left and 1, 2, 3, 4…. heading off to the right.This is so well ingrained with us that adults can sometimes be seen subconsciously gesturing to the left and right when adding and subtracting, even though this line is a thing we’ve invented - so for example a couple of thousand years ago, before even zero was invented* by the way, if you had eight sheep there was no sense of that being any further to the right than if you had three (*and if you did not have any sheep at all you did not bother counting them, so zero was not needed).

So we have a way of showing a number as a place on a line. If we tested a section of the number line to the left of zero we’d get a few bits of black line which essentially are the spine of the bug in the above picture.

4. Now we need to introduce imaginary numbers, which I’ll hope to make clear as you may not have met them before or it might be a bit of a fog if you have. Part of doing maths involves trying out new ground to see what happens. So we have square roots - what number multiplied by itself gives 9, say? (3), or 2, say? (1.41…), the square root of 100 is 10 - but a second square root of 100 is -10, because -10 x -10 =100 (10 x -10 is -100 though) - so for 9 it’s really +3 or -3, and every number it seems has two square roots positive and negative. So we’ve invented negative numbers and used them here, but can we find a square root of a negative number? Well no, because every number multiplied by itself gives a positive result (or zero).

5. There is no square root of -1, but let’s pretend there is, say mathematicians, and see what happens. What happens amazingly is that some problems in engineering get solved and we end up with things like the Mandelbrot set.

6. Let’s call them imaginary numbers, they say, with the letter i as the square root of -1, so i x i = -1, and so 2i x 2i = -4. Now let’s mix them up with real numbers, 2 + 3i say, which can’t mix any more than that as apples are not oranges, but we can do all the usual things in maths which all give answers with a real bit and an imaginary bit, including equations which solve things in engineering like I said - generally with the imaginary bits all cancelling out in these cases. To multiply these combined (‘complex’) numbers is a bit like long multiplication or if you did that (2 + x) stuff in algebra, but I’ll not go into that here.

7. Now our number line is looking a bit weak to try to show these numbers with two parts, so we use up/down for how many imaginary i it is, and the usual number line left to right. In the same way I suppose we could show 3 apples and 2 oranges (or 3 sheep and 2 goats) as a place on this grid of 3 steps across and 2 steps up, so we can show the number 3 + 2i as a place on this grid rather than just on a number line.

8. So now if we take one of these combined numbers on the grid, multiply it by itself and then add the number we first thought of (itself on the first go), and then repeat…, then we’ll be able to see whether that starting number leads to a series heading for the hills or staying local - if local then we put a black dot where that number is and if not then we leave it blank. Do that across the grid and the bug-shaped Mandelbrot set will appear.So there it is - from a simple process of ‘multiply by itself and add the number you first thought of’ we get something with amazing limitless complexity like this. A major characteristic of the set is the infinite detail, which however much you zoom in there is always more intricate - ugly or beautiful as you may decide - detail at the edges.

We can never draw a perfect circle as the pencil thickness should really be zero but then we would not see it - and we cannot get a computer to draw a perfect Mandelbrot set either because how do you know that a particular combined number won’t hang around locally for 100 or 200 rounds and then head off to the hills, as we can’t test each dot for an infinite number of rounds? This does happen at the very edge points we zoom in on, hence the yellow to red sections which would appear as black if a slightly smaller number of rounds was used, so they show how ‘firm’ a particular edge is.

In the software to produce the above pictures, the computer selects an ‘interesting’ looking bit of edge (like finding a fjord on a map) and zooms in. It starts with 100 rounds to test each dot, but when more red to yellow starts to appear it ups the number of rounds which has an ‘acid bath dip’ effect of wiping off the red etc. but also etching more into the black to find a firmer edge, and then zooms in again and again until it reaches a computing digit limit, in the above at scales smaller than an atom, or if the above zooms are considered full size then the whole bug is on the scale of our solar system.

There is just one Mandelbrot set though there are other ‘fractals’, and anyone in the universe doing some version of mathematics would very likely come up with/discover/invent circles, zero, negative numbers, imaginary numbers and the Mandelbrot set.