phyllotaxis

Phyllotaxis (gr: phyllous means leaf, taxis means order) refers to the arrangement of leaves on a stem or florets in a composite flower such as a sunflower or pinecone along logarithmic spirals, or summation series, in which each term is the sum of the two preceding ones: 1,2,3,5,8,13,21,34,55,89,144 etc.. The scales form in double spirals which radiate from the center, one clockwise, the other counterclockwise. The surprising feature is that the number of spirals in one direction is related to the number in the other direction as two adjacent numbers in the Fibonacci series.

see A.H. Church, (1904) "Phyllotaxis in Relation to Mechanical Law."
see also D'Arcy Thompson, On Growth and Form, Chapt. 14


Logarithmic growth forms occur in shells, horns of horned animals, and plants. They are self-similar forms. Magnifying a logarithmic spiral is the same as rotating it. See discussion in Brian Goodwin, How the Leopard Changed its Spots, pp 115-136

the growth pattern of ammonite follows a logarithmic spiral. Marcel Duchamp, Rotorelief, 1935.
(from Peitgen, Jürgens, Saupe)


Le Corbusier's Modulor is based on the logarathmic spiral The ratio of one term to the preceding on is approx 1.6180.

see Jay Hambidge, Dynamic Symmetry, Dover Books.