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.edu In 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
Some pictures were made by Elena Cherkaev

The data:
• As you may see from the photos, the angle of  spirals is typically  about 30 to the vertical, but may vary.
• The weather conditions  at Bryce Canyon are as follows:
• The elevation is 8,000-9,000 feet.
• July is the warmest month, with an average daytime high temperature of 83o F and a nighttime low of 47o F.
• Spring and fall weather is highly variable.
• Cold winter days are offset by high altitude sun and dry climate. Winter nights are sub-freezing. During some winters, Alaskan cold fronts descend on the Colorado Plateau region bringing temperatures as low as -30o F.
• Annual snowfall averages 95 inches. Much of the area's precipitation comes as afternoon thundershowers during mid to late summer.

One may guess that the spirals increase  survivability of  trees, that is its resistibility to strong winds, sharp temperature change, etc. The mechanical properties of the trunks can be reliably modeled and the loading conditions can be calculated too. Therefore, the survivability can be viewed as a problem of mechanics of fracture.
Can the optimality of spiral trunks be mathematically formulated?
The problem is open. Everyone is welcome to contribute.

 The  problem of the sense of optimality of a bio-structure is an example of  inverse variational problems.  These problems form a new type of variational problems.  It may be postulated that morphology of a bio-structure 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?

## Optimality of bio-materials and "inverse optimization"

It would be natural to apply optimization methods developed for engineering constructions to biological structures. However, the two problems are mutually inverse. In engineering optimization problems, the goal is to find a solution (structure) that minimizes a given functional. The functional itself is known.

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:

1. The structure should perform a specific mechanical function, rather than be universal.

2.
3. The morphology should be relatively simple.

4.
5. The environment - loading and the temperature  - should be reliably measurable.

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 one-dimensional 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.

NSF support is acknowledged.

Historical 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.
Back to the homepage