4. THE TUTELAGE OF…

It seems in a material with notches or corners in it, smoothing out the inside of the notch reduces the concentration of tension, or tensile stresses, somewhat. Smoothing the inside of a notch with a

quarter-circle, for instance. But, we’ve also seen that some concentration of stress still occurs in this

case, sometimes with deadly results.

If a curve helps, maybe there are other kinds of curves out there besides circles that are more

effective at channeling forces safely through materials. Any idea where we might look for ideas for

such geometries?

The tutelage of trees. As the poet William Blake (1757-1827) put it, “A fool sees not the same tree that a

wise man sees.” Photograph by Adam Baker.

Material at the base of trees is often optimized to distribute tensile stresses evenly, and avoid a

build-up of mechanical stress where a deadly crack might start. It turns out trees use these special

curves in other places, too, not just at their base. Where branches part from the main trunk, trees

often add extra wood, either above or below the branch, shaping the added wood to this same

curvature.1

And where major branches come into the trunk, or come together, trees smooth the intersecting

angles on both sides using this curvature.2

When a tree loses a branch it heals the occlusion using this same characteristic curvature. In

comparison to a circular occlusion, these tree-curved occlusions show over 50% less concentration

of stress.3

So we’ve seen that this particular curvature is quite effective at avoiding a concentration of stress

upon any one part of the curve. Trees use these curves everywhere. Throughout their lifetimes,

1 This is known as as uniaxial loading, because the stress pulls on the material in only one direction.

2 This is known as biaxial loading, because the stress pulls on the material in two directions.

3 57% less stress; Mattheck, C. and H. Kubler. 1997. Wood – the internal optimization of trees. Springer. Page 37.

trees continually alter their shape to evenly distribute mechanical force through their bodies and

avoid stress concentrations. And why? All to avoid instigating or propagating cracks. Once you are

aware of these curvatures, and why organisms employ them, you’ll start to see them everywhere.

FOLKERT GORTER

More Resources:

A free, online program for drawing triangles:

http://www.math10.com/en/geometry/geogebra/geogebra.ht

ml

The biomechanics of Venus fly traps:

http://www.seas.harvard.edu/softmat/downloads/2005-

07.pdf

“All natural forms are forces

made visible.”

- Michael Schneider, A

beginner's guide to

constructing the

universe: the mathematical

archetypes of nature, art,

and science

http://www.math10.com/en/geometry/geogebra/geogebra.htmlhttp://www.math10.com/en/geometry/geogebra/geogebra.htmlhttp://www.seas.harvard.edu/softmat/downloads/2005-07.pdfhttp://www.seas.harvard.edu/softmat/downloads/2005-07.pdf