As 3D animals get bigger, their surface area generally increases more slowly than their volume. The surface-to-volume ratio impacts the energy requirement and metabolic rate of an animal. For example, a small warm-blooded animal (hummingbird) loses more heat through its relatively larger surface than a gigantic one (whale) and needs to make up for the lost energy. Some animals have developed traits (fur, blubber, and feathers) to counteract some of the effects. However, according to West’s network theory of metabolic scaling, there are some universal constraints at work related to power laws and fractals. When you change the angle so that the fractal looks more like a tree, the smaller branches of the tree start to overlap. This limits how many iterations of the fractal you can perform within a certain space. Examples of this constraint in the natural world include: trees or other plants that can only produce a limited number of leaves for photosynthesis; blood vessels that can only supply a limited amount of blood to the surface area of animals’ skin; or lungs that only have a limited number of alveoli to facilitate oxygen / CO2 exchange.

Adjust the angle between adjacent branches by dragging the first slider. Then drag the second slider to watch the fractal evolve over time. In the first iteration, we begin with three branches. In each subsequent iteration, each branch’s tip sprouts three new branches. Observe how the angle between the branches affects how quickly the branches begin to overlap.