## Friday, 28 March 2008

### Geometry in the News

It's spring time in Possum Trot, and my morning walks are joined by daffodils parading alone the roadside. Reminds me of the beautiful geometry of them that I wrote of some years ago in England.

I’ve been thinking about geometry a lot lately. Partly that is due to the fact that I’m going through Trig and Vectors. Partly it is probably because it has popped up in science stories I have been reading lately. On the same day I wrote about the Daffodils in the Snow, I read a note from a researcher on why they respond differently to wind than other similar flowers. The short answer is geometry.

When you watch Tulips, for instance, they will lean away from the wind. Daffodils, on the other hand, remain almost completely erect, but turn their tilted heads away from the wind. William Wordsworth must have had this in mind when he wrote,

“ Ten thousand saw I at a glance
Tossing their heads in sprightly dance.”

The reason the daffodil twists like a weather vane and the tulip bends more in the wind is the geometry of the cross-section of the stem. A tulip stem is nearly round, and so it can bend, but not twist. The same effect causes your garden hose to crimp up and cut off the flow of water when it can’t twist. They are very good at bending, very poor at twisting. The engineering types call the twisting motion torsion, and it is related to the cumulative sum of the fourth power of the distances from the center to the edge of the stem. Circular things are far away in all directions. But a Daffodil has a cross section that is more elliptical. If you pick it up you can see the difference in the long and short axis easily with the naked eye. Since the sum of the fourth powers of the distances is lower, it is more able to turn away from the wind. Scientists studied one type of daffodil and reported that up to about 22 mph, the stems stayed essentially erect, and the trumpets all turned away from the wind. After that, the flowers both turned and bent some, but they cannot bend as low as a tulip in any wind… all geometry.

The geometry of sharks popped up too. It seems that shark geometry may be to thank for the engineering advances that will result in a few more swimming records falling at the Olympics in China this year, although I assume the swimmers will want some of the credit. It seems a shark has dimples on its scales that breaks up the flow of water, reducing the drag so it uses less energy. Speedo reckons it can make better swimsuits making its swimmers go faster using something similar. The new body suits are already in the pools and having an impact.

Another bit of research shows how, with such a huge blue ocean to wander around, how exactly do marine predators like sharks find their next meal? Yeah…. You guessed it, geometry. It has been known that many land animals search for food much like a shopper in the super market searches for a particular item. The math term is called a Levy walk (actually a Levy flight), a fractal type structure from geometry where the small parts are self-similar. Ok, the actual rules are a little technical but in essence it means that the animal undergoes lots of short-distance journeys interspersed with fewer longer-distance journeys. Just as you go to an area of the store where you think the item is located, then circle around in that area looking for it. If you don’t find it, you go off to another area and begin a close search there.

Geometry, making your life better… See, because if two sides and the included angle of one triangle….