Bilateral symmetry (mirror-image symmetry): From a distance, nearly all animals have bilateral symmetry, with a left side looking like a mirror-image of the right side.
Vi Hart: Doodling in Math: Spirals, Fibonacci, and Being a Plant:
Radial symmetry: Many plants and a few animals (starfish, sea anemone, etc) have radial symmetry.
Fractal symmetry: Many structures -- trees, ferns, lungs, blood vessels -- have a fractal symmetry. From a distance, the appearance of a large branch on a tree looks similar to a closer look at a smaller branch on that large branch.
Shape recognition:
Many enzymes and drugs have a specific geometric shape -- molecular shape -- that exactly fits into and matches some specific biological molecule. (The "lock and key" theory).
Structure based drug design allows people to design drugs that affect only one specific molecule, rather than having unwanted side effects on other parts of the body. Optimum geometric structures:
The shapes of many structures in living things are often very close to some mathematically optimum geometric shape.
For example:
The cornea and lens of the eye have a shape that is extremely smooth, less than 400 nm of irregularity. Otherwise it would not focus light properly.
Many bones have a solid outer layer, and a more sponge-like inner region, rather than having a uniform structure throughout. This is similar to the way structural I beams have a broad top and bottom surface, but a narrow interior web, rather than a simple uniform rectangle from top to bottom. Both these shapes give a much better strength-to-weight ratio than a uniform structure.
Geometric scaling and power laws: Larger animals have many features different from smaller animals that can be explained by geometric scaling.
For example:
An animal twice as large requires feet that can handle 8 times as much weight. So an animal twice as long as another need feet that are more than twice as wide or made of stronger substances or both.
In situations where a small surface/volume ratio is useful -- such as keeping a warmblooded animal warm in cold place -- larger animals with smaller ears have an advantage.
In situations where a large surface/volume ratio is useful -- such as staying cool in hot places -- smaller animals with larger ears have an advantage.
... something about aerodynamics and wings here ...
... something about sunflowers and Fibonacci here ...
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