Spiral trees
Dear visitor, It is my pleasure to invite you to a discussion of optimal morphology of living organisms. The picture (above) shows spiral wood fibers. We can study this strange phenomenon in a challenging problem in Applied Math/Structural Optimization. The problem is open and you are welcome to contribute.
Andrej Cherkaev Email: cherk@math.utah.eduIn collaboration with Seubpong Leelavanichkul, our graduate student, we published the paper
Why grains in the tree's trunk spiral: mechanical perspective in Int J. of Structural Optimization. There, we discuss our preliminary fundings.
The Problem:
The trees of Ponderosa pine and Utah Juniper in windy areas of South Utah possess spiral wood fibers that wiggle around the trunk. The question is: Why?Pictures below are of spiral Ponderosa pine trees and logs, they were shot at the rim of Bryce Canyon, probably the weirdest place at Earth.
Trunk, a close look 
Another trunk 
A dead tree 
A tree alive 
A log of the spiral tree 
A fallen tree 
The data:As you may see from the photos, the angle of spirals is typically about 30^{o } to the vertical, but may vary. The weather conditions at Bryce Canyon are as follows:
The problem of the sense of optimality of a biostructure is an example of inverse variational problems. These problems form a new type of variational problems. It may be postulated that morphology of a biostructure is optimal with respect to some evolution goal, which simply means that it is best adapted to the environment. The question is: In what sense is the structure optimal?

In contrast, a biological structure (morphology of the trunk) is known, but it is not clear in what sense (if any) the structure is optimal. This problem is formulated as an "inverse optimization problem":
Find a goal functional of an optimization problem,
if a solution to that problem is known.
The success of the inverse optimization critically depends on
the choice of an object.
The choice of the object is not trivial since it should satisfy strict
requirements:
From this perspective, the problem of optimality of spiral wood
fibers is clearly stated. The control is the helicoidal angle of the spirals.
The wind's strength and other weather conditions are well documented; thus
the load is determined. The mathematical model of an anisotropic onedimensional
bar with helicoidal symmetry can be derived in a standard fashion. Several
criteria of the failure and cracking for the wood and their combinations
can be examined.
This project is very ambitious indeed: to find mathematical criteria
of development and adaptation of living organisms.
Back to the homepageHistorical note:
Calculus of Variations started with a discussion of the best curve  brahistohrone that allows a heavy particle reach its other end at the minimal time. Working on the challenge, the participants: Johann and Jacob Bernoulli, Newton, Leibniz and L'Hospital, developed approaches to Optimization Problems that flourished through centuries.