Fractal pictures are a pop-science favourite. Draw bronchi — looks like a tree. A snowflake — looks like coral. The brain's neural network — looks like the cosmic web. Beautiful. Goes on Instagram. Captioned "everything is connected".
There is a problem. Looks similar is pareidolia, not fractality: the brain is trained to find similarity, and finds it even where there is none. "Jesus on burnt toast" is the classic example.
A fractal is a mathematical property you can measure. Which is where it gets interesting.
What is measurable
Fractal dimensionality is a number between zero and infinity that describes how much complexity an object has as you zoom in. A flat surface has dimension 2. A solid three-dimensional ball — 3. A fractal — a non-integer between them. The coastline of Britain — about 1.25. Bronchi — about 2.9. The surface of the brain — about 2.5.
These numbers are testable. They are measured by the box-counting algorithm: cover the object with a grid at varying scales and count how the number of non-empty cells changes with scale. No mysticism. Log-log plots. The slope of the line is the dimension.
In 2020 Vazza and Feletti compared the fractal dimensionality of two objects:
— a slice of cerebral cortex under a microscope, — a computer simulation of the large-scale structure of the universe.
Same dimensionality. Same slope. Across twenty-seven orders of magnitude in scale.
A measurement, not a metaphor. Peer-reviewed, published in Frontiers in Physics. Data open.
What it means — and what it does not
It means: two systems differing in scale by a one followed by twenty-seven zeros are structurally equivalent. Connections per node, spectral density, fractal dimensionality — match.
It does not mean: that one system is the "reflection" of the other. A match in mathematical structure does not imply an ontological relationship.
Bronchi and trees also have close fractal dimensions. Bronchi are not tiny trees, and trees are not large bronchi. Both systems are optimised for the same problem — maximise contact surface while keeping volume minimal. The same optimisation problem produces the same fractal geometry.
The brain and the cosmic web, possibly, are optimised for the same problem. Which one? Maximise the density of connections while keeping material cost low. Transmit information between nodes with the shortest possible edges.
If that holds, the dimensionality match is a consequence of a shared optimisation principle, rather than a mystical "everything is one". And that is scientifically interesting: what exactly is the universe optimising, if it gives the same fractal geometry as an evolutionarily optimised brain.
Where I stop
The brain and the universe are structurally equivalent under certain metrics. A fact worth discussing. Calling them "the same thing" steps past what the data say.
The pretty phrase "each of us is a miniature copy of the universe" goes past it too. The data say something more modest: the architecture of connections in both systems obeys the same optimisation principles.
That is enough. "And therefore we are neurons of God" contributes nothing to the information, but it stops the information from being taken seriously — the listener either dismisses it or accepts the whole package together with "everything is one".
Better to say it precisely. The precise version, on its own, is already surprising enough.