Hello Peter,

This is an exhibit within an Arboretum, learning people to look at plants, how they are build etc, e.g. the repetition of phytomers, the symmetry of monocots versus dicots, spirals for more primitive plants, whrolds and circles for advanced flowers....  So it is an invitation for botanists to look in a geometric way and for mathematician or similar to look from a botanic perspective. 

Geometry is everywhere, not just in allometry.

Karl Niklas' book is on my book shelf, yes, and I am very familiar with the work in recent decades of Karl with B. Enquist.  From a geometric point of view, the allometric laws are related to the multiplication of monomials (some variable to some powers) while the shapes themselves are very often Lamé-like, or superformula like, and thus related to addition of such monomials.  The search for harmony is,  in the inequalties between arithmetic and geometric means. 

The Geometry Garden inside the Arboretum is starting now, but it is a dynamic project, building up towards the future, aiming at different levels of entry, from Kindergarten to Univ. Level.   

Johan

Johan,

It seems to me that all scaling and symmetry laws are examples of the log-log linear regression known as Huxley's Law.  In fact, I believe that allometries are a universal BAUPLAN for development and have written dozens of papers over decades with just this concept in mind. But, I do think that Enquist's work is valuable, too.

Hello Peter,

In my view allometry is part of an inequality: development will depend on the relation between some 'intrinsic' characteristics (e.g. allometry) and the 'extrinsic'  (the shape the forms assume in their environment. Geometrically speaking submanifold theory.

Apart from that, bridging the gap between biology and mathematics / geometry is a challenge, to say the least. 

Yours,

Johan

Hi there, Johan,

I understand what you mean about a submanifold being more appropriate for models involving the phenotypical response to environment during ontogenesis. It is a difficult approach. I have taken a different route and have been circling around.. Hilbert's 4th Problem... for years! It asks for determination of all geometries in which straight lines( the allometries) are the shortest paths. Stated about 100 years ago, it remains largely unsolved, accept in dimension 2. Such geometries are far from Euclidean. They are called Finslerian  and over 20 years I have come to know a little about them. To us, the environment is an external source, forcing adaptation in the interaction schemes between cell populations in the developing embryo along the lines of Leo Buss. These populations produce biochemicals, including hormones, whose production is modulated/controlled by genetic switches which are situated on the epigenetic landscape(sensu, C.H. Waddington).We have applied our methods to forestry and coral reef ecology where the developmental sequence is replaced by succession along a sere. But we are currently working in development proper studying diabetes in humans.