Plants : {2-D Spirals} The seeds of sunflowers display a two-dimensional pattern involving the Fibonacci sequence. The seeds of a sunflower form two spirals, called parastichies, one set eminating from the center in a clockwise direction, the other in a counter-clockwise direction. In most cases, the number of clockwise spirals and the number of counter clockwise spirals will be adjacent numbers of the Fibonacci sequence. The parastichies image is linked from an excellent and informative web article, "Practical Procedural Modeling of Plants," by www.td-grafik.de.

 
click for full-sized image, series of sunflower images from "Fibonacci Numbers" website by Brantacan.co.uk.

What is the significance behind the double spiral pattern? One explaination describes the pattern of seed growth as occuring in offset angles of the value 222.49... degrees. This value is the angle of phi (0.618...) rotations. "phi" is the inverse of "Phi" (1.618...), also known as the golden ration. This angle of rotation causes the double spiral effect because phi is an irrational number (which the ratio of two adjacent Fibonacci numbers approaches as they increase in value). Seeds placed according to an angle of a rational number appear as only single spirals in one or the other direction. Interestingly enough, the arrangement of sunflower seeds in this double spiral is optimal for maximum exposure of the seeds to light.

Continue upwards in the tree: {3-D Spirals}
Downwards in the tree: {Plants}


Sources: sunflower clockwise and coutner-clokwise spiral image, from "Practical Procedural Modeling of Plants" article; series of sunflower images from "Fibonacci Numbers" website by Brantacan.co.uk. See biboliography.