but the formulas of physics, especially E=mc˛ aren't recursive.

E=mc˛ is not really a law of physics from a modern point of view, but a conversion of units from Joules to kg; mass and energy are not "transformed" into each other, they are one and the same thing.

Current laws of physics are recursive and chaotic and almost every physical system is chaotic and hence has fractal attractors.

Abstractly we can describe the universe by a (large) set of numbers, call them \( z(t) \) (we don't know exactly what they are yet, but let's say all the stuff that goes into the standard model + general relativity). Laws of physics have the form

\( \frac{dz}{dt} = f(z) \), starting from \( z(0) = "the\ big\ bang" \). We don't know exactly what f(z) is yet, not do we know what \( z(0) \) is.

If you consider discrete times \( t_0=0, t_1 = h, t_2=2h, ... \ etc \) with h a small time step (say the Planck time), and call \( z(t_k) = z_k \) we get

\( \frac{dz}{dt} \approx \frac{z_{k+1}-z_k}{h} \) and the "laws of the universe" take the form

\( z_{k+1} = z_k +hf(z_k) \).

If you make a toy universe where z is just a field of complex numbers in a plane with complex coordinate c and \( f(z) = (z^2-z +c)/h \) we get the Mandelbrot equation.

The "equation of the universe" has a similar form, but a bit more complicated.

For example cosmologists have tried to figure out what the real \( z(0) \) is by trying various things and see if they could produce the observed distribution of matter. This is like making a Julia set.