Beauty is not beyond description——a physico-mathematical equation to describe flower morphology

Besides the well-known rule of genes, there are still many known or unknown rules for the structure of living organisms. It has been shown that the morphology of flowers and the arrangement of leaves known as phyllotaxis well fit certain rigorous mathematical rules, while the corresponding underlying physics remains to be explored. This paper begins with a brief introduction to the beautiful morphology of nature, such as snow flakes, flowers and phyllotaxis which well demonstrate bio-mathematical phenomena. This is followed by a brief historical perspective of the discovery of the above phenomena, especially the Fibonacci numbers appearing in phyllotaxis. To describe plant growth a second order differential equation is then derived, based on a continuous fluid model at the cellular level which regards osmotic pressure as the main driving force for growth, and the steady-state solution under the boundary condition of symmetry breaking is obtained. The co-evolution model is discussed for the flower pattern, and under the condition that the Pythagorean numbers dominate the selection in the evolution, it can be concluded that the basic Pythagorean numbers 3,4 and 5 are the mostly populated numbers for the branching numbers of flowers. The inherent difference between the Fibonacci numbers for phyllotaxis and the Pythagorean numbers in flower evolution is addressed.