yesiamjames:

Mandelbrot zoom 2^83x at 720p

Zooming in 2^83x (9,671,406,556,917,033,397,649,408) on the mandelbrot fractal. If you were to render the entire mandelbrot set at this level of detail there would be more pixels than their are atoms which make up Jupiter.

The mandelbrot set is a set of points on the complex plane for which using the formula Z = Z^2+C we don’t get a number which escapes towards infinity. Due to us not having computers with infinite power we decide that if Z exceeds the overflow value (usually 2) then it’s not part of the set as Z^2 will keep producing a higher and higher number. The colored bands represent the iteration count  at which Z became higher than the overflow. 

To make sense of this we have to understand a little bit of advanced maths. 

Real numbers are any numbers which are positive or negative.

An imaginary number is neither positive nor negative they are the square roots of negative numbers. The square of any positive number is positive and the square of a negative is also positive so the answer to sqrt -1 doesn’t exist as a real number. That being said there are times in mathematics when we need to represent such numbers so we invented the number i to represent sqrt-1. It may sound confusing to include imaginary numbers in equations but since i = sqrt-1 then i^2 = -1 so the result of squaring an imaginary number is a real number which we can do something with.

If we have to add a real number to an imaginary number for example i + 1 we are left with a number which is neither real nor imaginary and so we call this a complex number. When we are dealing with fractals a complex number can be used to represent a grid coordinate  on what we call the complex plane. Again squaring a complex number results in a real number.

In the formula Z = Z^2 + C, Z always starts as 0 and C is the complex number representing the point we’re testing on the complex plane. 

If Z is <1 then Z^2 is < than Z  so we get a force pulling Z towards 0, If Z is > 1 we get a force pulling Z towards infinity. C can also push Z lower or higher.

If Z > 2 then we can safely assume that Z will head toward infinity because Z^2 will always pull Z further towards infinity than C can counter.

It’s the chaos of Z^2 and + C either holding Z under 2 or firing it off towards infinity which give rise to the shape of the incomprehensibly complex Mandelbrot set.

The resulting visualization of this set is an image of both chaos and order, incredible beauty and absolutely infinite complexity. 

This may sound complicated but to demonstrate how simple this really is I wrote a Qbasic Mandelbrot renderer  so small it will fit in a Youtube comment and half of this code was trying to tell Qbasic how to hand a complex number.

CLS : SCREEN 12 

5 FOR y = 1 TO 640 

10 FOR x = 1 TO 480 

ty = -2 + (y / 160): tx = -2 + (x / 120) 

Rz = 0: Iz = 0 

FOR a = 1 TO 50 

Tz = Rz * Rz - Iz * Iz + ty 

Iz = 2 * Rz * Iz + tx 

Rz = Tz Z = Rz * Rz + Iz * Iz 

IF Z > 2 THEN GOTO 30 

NEXT a 

GOTO 40 

30 IF a > 15 THEN a = a - 16: IF a > 15 THEN GOTO 30 

COLOR a: PSET (y, x) 

40 NEXT x 

NEXT y

Video made using fractal eXtreme:

http://www.cygnus-software.com/

1280 * 720 resolution at 2AA

coordinates: 

Zoom = 2^83

Real -0.174,727,987,629,733,024,394,180,821

imaginary +1.072,127,607,275,012,133,225,971,930

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