Take a shape. Make smaller copies. Arrange them according to a rule. Repeat the procedure. The result may show detail at more than one scale without requiring a different instruction at each scale.
Some mathematical forms are exactly self-similar. A coastline or a plant may instead show approximate or statistical repetition over a limited range. Real material has limits. A useful drawing does not give nature permission to repeat forever.
The triangle below divides each retained triangle into three smaller ones at half the side length. After n steps there are 3ⁿ retained triangles, each with side length 2⁻ⁿ of the original. Its ideal similarity dimension is log(3)/log(2), about 1.585. The drawing shows a finite construction, not infinity.
Now consider a habit: a pause before one sentence, a pause before one meeting, a pause before one decision. That is an analogy across scales. It may be useful. It is not a proof that a person, a plant, and the cosmos share one physical mechanism.
what if the room lives at its edge? →