Deterministic systems can generate complex behavior that appears random. For example, waves and turbulence are governed by precise equations, yet their complexity often requires statistical methods for analysis. Consider the function
noise(t)=Isin(
2
t
+t)I
.
This function is fully deterministic—evaluating it at any
t
yields the same result—but its output oscillates between 0 and 1 in an accelerating pattern. The quadratic term
2
t
+t
increases the oscillation frequency over time, while the absolute value ensures non-negativity. This results in structured variability: a predictable rule that produces a complex signal.
Randomness is often just complexity beyond our perception. In A New Kind of Science, Stephen Wolfram demonstrated that even simple programs, such as cellular automata, can generate behavior mimicking randomness from deterministic rules. Notably, Rule 30 produces patterns so complex that they appear random despite being generated by a simple, deterministic process. If deterministic rules alone can create chaos, is anything truly random?
In our animation,
noise(t)
modulates the sizes and positions of circles. Combined with steady frame progression and rotation, it creates an evolving pattern that is neither strictly periodic nor completely random, demonstrating how deterministic rules can yield visually engaging complexity.
In[]:=
Animate[​​Module[{circles,r,i,noise},​​noise[t_]:=Abs[Sin[t^2+t]];​​circles=Table[​​r=Mod[3(frame+36)noise[i],2];​​{Thickness[r/40],Opacity[1-r/2],​​Circle[{rCos[i],rSin[i]},r/4]},​​{i,0,6,0.03}];​​Graphics[Rotate[circles,2.78frame,{0,0}],​​Background->LightBlue,ImageSize->500,​​PlotRange->{{-1,1},{-1,1}}]],​​{frame,0,4,0.004},AnimationRate->0.06]
Out[]=

CITE THIS NOTEBOOK

Bubbles emerge from blue depth: no true randomness, determinism only​
by Vitaliy Kaurov​
Wolfram Community, STAFF PICKS, March 6, 2025
​https://community.wolfram.com/groups/-/m/t/3410669