Fractal pictures are a worn-out staple of popular science. Bronchi look like a tree, a snowflake like coral, the brain's neural network like the cosmic web. It is beautiful, it goes on Instagram, and it gets captioned "everything is connected".
The trouble is that looking similar proves nothing. The brain is trained to find resemblance and finds it even where there is none; this is called pareidolia, and the classic example is a face on burnt toast.
Fractality, on the other hand, is a mathematical property, and it can be measured. That is where things get interesting.
What is measurable
Fractal dimension describes how much complexity an object shows as you zoom in. A flat surface has dimension 2, a solid ball 3, and a fractal has a fractional value somewhere in between. In these terms the coastline of Britain comes out at about 1.25, the bronchial tree at about 2.9, the surface of the cerebral cortex at about 2.7.
These numbers are testable. They are found by box counting: cover the object with grids of different sizes and see how the number of non-empty boxes changes with scale. The result is an ordinary log-log plot, and the slope of its line is the dimension. No mysticism enters anywhere.
In 2020 Franco Vazza and Alberto Feletti compared two objects:
— a slice of cerebral cortex under a microscope; — a computer simulation of the large-scale structure of the universe.
They looked at how density fluctuations are distributed across scales and at how the networks are built: how many connections each node has, how tightly nodes cluster. For several of these measures the numbers came out close, even though the objects differ in size by about twenty-seven orders of magnitude.
This is a peer-reviewed measurement, published in Frontiers in Physics.
What it means and what it does not
It means that two systems differing in size by a one followed by twenty-seven zeros are similar in several structural characteristics: connections per node, the spectrum of fluctuations, the degree of clustering.
It does not mean that one system is a "reflection" of the other. A match in mathematical structure says nothing yet about a kinship in substance.
Bronchi and trees also have close fractal dimensions, yet bronchi are not tiny trees and trees are not giant bronchi. Both systems solve the same problem: get as much contact surface as possible for as little volume as possible. The same optimisation problem yields the same geometry.
Perhaps the brain and the cosmic web also solve a shared problem: connect as many nodes as possible while spending as little material as possible, and pass information along the shortest edges.
If so, the resemblance follows from a shared optimisation principle, and the mystical "everything is one" has nothing to do with it. What appears instead is a genuine scientific question: what exactly is the universe optimising, if it arrives at the same geometry as a brain honed by evolution?
Where I stop
The brain and the universe are structurally similar under certain metrics. That fact deserves discussion. Calling them "the same thing" steps past what the data say.
The pretty phrase "each of us is a miniature copy of the universe" steps past it too. The data say something more modest: the architecture of connections in both systems obeys the same optimisation principles.
That is enough. Tacking on "so we are all neurons of God" adds no information and makes the information harder to hear: the listener either dismisses the whole thing or swallows it whole, "everything is one" included.
Better to say it precisely. The precise version is surprising enough on its own.