Leaves, petals and other organs in plants are not arranged randomly, rather they seem to follow well-defined rules. This geometrical arrangement of organs is one of the main questions in plant morphogenesis, and the study of patterns thus produced is named phyllotaxis, from the greek phyllo (leaf) and taxis (order).

The regular arrangement of organs forms spirals or parastichies. Many models have been proposed to characterize this spiral arrangements. Their approach can be descriptive or explanatory. Descriptive models only attempt to capture the geometry of phylotactic patterns, while explanatory models describe the underlying biological processes controlling pattern formation.

Flat model

A very simple descriptive model generates phyllotactic patterns of elements arranged on a disk. These patterns resemble those found in daisy and sunflower flowerheads. All elements have the same size.

Flat model L system: flat phyllotaxis

Cylindrical model

In 80% of plants, leaves are arranged along the stem tracing a helix. The angle formed by two adjacent leaves remains aproximately constant and close to the Fibonacci angle, which value is 137.5281 deg. This pattern can be clearly observed in cactus's spines and pine cones.

Cylindrical model. L system: cylindrical phyllotaxis

The study of phyllotaxis is also relevant for other non-botanical structures, such as crystals, Mandelbrot set or proteins.

References

Phyllotaxis
Roger V. Jean (ed)
Cambridge University Press, 1994

Phyllotaxis: an interactive site for the mathematical study of plant pattern Formation
http://www.math.smith.edu/~phyllo
Smith College

Fibonacci numbers and nature
http://www.math.smith.edu/~phyllo
Dr. Ron Knott

A collision-based model of spiral phyllotaxis
Deborah R. Fowler, Przemyslaw Prusinkiewicz, Johannes Battjes
Computer Graphics, July 1992