Turtles all the way down

Turtles all the way down!

Turtles, all the way down

#063 – Turtles all the way down!

There is this new hype, wide-spreading in town,
A tad irritating, enough to make me frown,
That there’s no God as such, it’s only a noun,
That any ass could claim the jewels of the crown,
And the man who believes, you should call him a clown.

Well, listen carefully, my little Charlie Brown:
Choice is yours, between blue and white, for your gown.
In fiery black holes, you can let yourself drown.
On your big bang mythos, there’ll be no crackdown.
For you, it’s about turtles, all the way down!

Quad Damage

They say the world is born by a giant cosmic turtle. At least such are the faint echoes we get from ancient myths in China, Japan, North America and India. What if it where true?

So let’s assume there’s this big turtle holding the whole universe upon its shell. Well, the next question comes rather legitimately: What is it that the turtle stands upon? Another larger turtle? And if so, how many like those is there? Are there turtles all the way down?

This story was briefly and humorously narrated by Stephen Hawking back in 1988 in his best seller “A brief history of time”. Although he gave credit for it to Bertrand Russell, a beautiful British mind, multi-talented in various scientific fields, and incidentally a Nobel laureate for Literature in 1950.

It poses the problem of the “first cause” and “prime mover”, as coined by Aristotle, also known in other philosophical systems, as the “anavastha” in Indian philosophy, or more trivially today as the “chicken and egg problem”. The funny expression “turtles all the way down” asks the question of the infinite regress problem in cosmology posed by the “unmoved mover” paradox: How can there be large enough a turtle to hold all that stuff out of thin air?

Well … the fractal dimension might be the answer, and disarmingly so. There’s an infinite number of turtles, but the formula driving them is one. As translated in Fragmentarium, this is how it goes:

uniform float MinRadius; slider[0,0,10]
uniform float Scaling; slider[-5,0,5]
vec2 formula(vec2 z,vec2 c) {
float m =dot(z,z);

if (m<MinRadius) {
  z = abs(z)/(MinRadius*MinRadius);
} else {
  z = abs(z)/m*Scaling;
return z+c;

And this is what it gives, leaving aside technicalities regarding colouring and lighting:

Fractal Turtles

Kali formula in Fragmentarium 1.0.22

This formula has been inspired by Kali’s works.

Other variations of the Kali formulae can be found here:



Turtles, all the way down


Fragmentarium source code