# Golden Section

12 13 PhylloTaxis PaTTerns leaves on a stem Emerging as a science in the 19th century, phyllotaxis has been extended to the spiral patterns of seeds in a sunflower head, petals in the daisy, scales of pine cones, cacti areoles, and other patterns exhibited in plants. In the 15th century Da Vinci 14521519 observed that the spacing of leaves was often spiral in arrangement. Kepler 15711630 later noted the majority of wild flowers are pentagonal, and that Fibonacci numbers occur in leaf arrangement. Appropriately, in 1754 Charles Bonnet coined the name phyllotaxis from the Greek phullon leaf and taxis arrangement. Schimper 1830 developed the concept of the divergence angle of what he called the genetic spiral, noticing the presence of simple Fibonacci numbers. The Bravais brothers 1837 discovered the crystal lattice and the ideal divergence angle of phyllotaxis 137.5o 360o2. The diagram by Church top row opposite shows the main features of spiral phyllotaxis. As the seed head expands, new primordia are formed at angles of 137.5o. In the seventh item we can see the Archimedean spiral which connects the growth. The diagrams below after Stewart show primordia plotted at angles of 137.3o, 137.5o and 137.6o. Only the precise angle produces a perfect packing. 137.3o 137.5o 137.6o 13 138 phyllotaxis thirteen spirals one way, eight the other way, known as parastichies. 3421 phyllotaxis, with dots indicating the smaller numbered more highly curved spirals. a seed head displaying 3421 spiral phyllotaxis 138 spiral phyllotaxis in a section of monkey puzzle tree spiral phyllotaxis an Archimedean or Fermat spiral with a new element every 137.5 degrees.