In 1911, Piet Mondrian painted a tree. “Gray Tree” uses only curved lines of various thickness on a gray background. No leaves, no bark texture, no ground, no sky. The image is unmistakably a tree.
In 1912, Mondrian painted the same tree again. Same composition, same curved lines, same gray background, same dimensions. Some lines occupy identical positions. But in “Blooming Apple Tree,” every line has the same thickness. Viewers see fish, scales, dancers, water. The tree disappears.
Mondrian’s sketches confirm both paintings depict the same tree. The difference is one mathematical property: scale-invariant branch diameter. In “Gray Tree,” thick lines divide into thinner lines, following the proportional relationships that real trees use. In “Blooming Apple Tree,” uniform thickness eliminates those proportions. With the scaling gone, the identity goes with it.
A research team studying fractal geometry in biological systems measured this precisely. Real trees maintain scale invariance in branch diameter because evolution fine-tuned the proportions for efficient water transport and structural integrity. The body exerts tighter control over vessel diameter than vessel length — arteries take circuitous routes through the body, but their thickness stays within 10% of the optimum. The same principle holds in branches. The ratio of parent branch to child branch follows a consistent scaling parameter.
The researchers then measured this parameter in artwork across centuries: a carved tree in a late-medieval mosque in Ahmedabad, Klimt’s “Tree of Life,” Matsumura Goshun’s 18th-century “Cherry Blossoms.” Every great depiction of trees preserves the scale-invariant proportions. The precision exceeds what casual observation would produce. These artists were encoding a mathematical relationship they likely had no formal knowledge of.
Mondrian’s pair of paintings is the cleanest demonstration. Everything matches except the scaling. One painting is a tree. The other is abstract shapes.
The same year that article appeared, I read about a separate finding. In the late 1800s, forensic examiners studying insect activity on decomposing bodies documented the predictable succession of arthropod species over time. First one set of insects arrives, then another, in a repeatable sequence determined by chemical and physical conditions. The forensic scientists described this process formally — the concept of ecological succession — before plant ecologists recognized the same pattern in forest development.
The people who first identified one of ecology’s foundational concepts were studying corpses for legal proceedings. They had no interest in ecology. They had no theoretical framework predicting what they would find. The pattern presented itself because they were looking carefully at something else.
Both stories are about the same phenomenon: essential properties become visible when you strip away enough context or approach from an angle that makes the context irrelevant. Mondrian stripped a tree down to curves on a gray field. The scaling remained because it is what the tree fundamentally is. The forensic examiners stripped away the botanical context of succession and found the same pattern in decomposition, because succession is what biological communities fundamentally do.
There is a word for the property that survives stripping: invariant. A mathematical invariant is a quantity that remains unchanged under a set of transformations. Rotate a circle — it stays a circle. Scale a fractal — the structure repeats. Remove color, depth, leaves, bark, context from a painting of a tree — the branch-diameter ratio persists, and the tree persists with it.
The question this raises is specific: for any complex thing, what are its invariants? What survives when everything else is removed? The answer is never the most visible feature. Leaves are the most visible feature of a tree. Remove them and the tree is still a tree. Remove the scaling, and the tree is shapes on canvas.
This applies broadly. A piece of music has melody, harmony, rhythm, timbre, dynamics. Remove timbre — play it on a different instrument — and the piece is still recognizable. Remove rhythm — play all notes the same length — and in most cases the identity collapses. Rhythm, not melody, carries more of a piece’s identity than we assume.
A face has eyes, nose, mouth, proportions, coloring. Remove coloring — a black-and-white photograph — and the face is still recognizable. Change the proportions slightly — move the eyes a centimeter apart — and recognition fails. Proportion, not feature, is the invariant.
The practical lesson from Mondrian is that identity lives in relationships between parts, not in the parts themselves. Branch diameter means nothing in isolation. Thick or thin is not the issue. The ratio of thick to thin at each branching point — that is the issue. The tree is made of ratios.
And the practical lesson from the forensic examiners is that the best way to identify an invariant is to encounter the same pattern in a context where everything else is different. Succession in a forest and succession on a corpse share one structure and almost nothing else. The shared structure is the invariant. The recognition required someone willing to look carefully at decomposition for reasons having nothing to do with ecology.
Both lessons suggest the same method: to understand what something essentially is, remove everything you can, or find it somewhere you did not expect. What remains in both cases is the answer.