1.Universal Natural Shapes -- 2.Towards a geometrical theory of morphogenesis -- 3.-1,-2,-3......,Understand The Legacy -- 4.Lamé curves and surfaces -- 5.Gielis curves, surfaces and transformations -- 6.Pythagorean-compact. - 7.Generalized intrinsic and extrinsic lengths in submanifolds.

This book focuses on the origin of the Gielis curves, surfaces and transformations in the plant sciences. It is shown how these transformations, as a generalization of the Pythagorean Theorem, play an essential role in plant morphology and development. New insights show how plants can be understood as developing mathematical equations, which opens the possibility of directly solving analytically any boundary value problems (stress, diffusion, vibration...) . The book illustrates how form, development and evolution of plants unveil as a musical symphony. The reader will gain insight in how the methods are applicable in many divers scientific and technological fields.