Simanaitis Says

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CATCHING A RUM RUNNER BY VIEWING A SUNFLOWER

THE FOG LIFTS MOMENTARILY. You and your Coast Guard crew spot a Rum Runner some distance off. The fog descends immediately and the Rum Runner heads off at its known full speed on some unknown straight course.

You know your Coast Guard boat is three times as speedy. And, fortunately, one of your crew is a botanist, because a sunflower provides the optimal course that catches the culprit.

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Three Sunflowers in a Vase, by Vincent Van Gogh, oil on canvas, 1888.

The common sunflower gets its name because it’s one of 70 species whose buds are heliotropic; that is, they follow the sun. The jerusalem artichoke is another member of the family. Though of Western Hemisphere origin, the plants were introduced to Europe in the 16th century. Their Italian name is particularly appropriate: girasola, loosely, sun follower.

Once in bloom, the plants cease this sun-following routine, though they generally point east toward the morning sun. However, what’s important to our Coast Guard botanist is the pattern of the buds. The capitula (flower heads) grow into a type of Archimedean Spiral.

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Helianthus annuus, the common sunflower.

To intercept the Rum Runner, you need to follow a special version of this, a Fermat Spiral. Curiously, it shares a mathematical relationship with the Golden Ratio. A full description involves polar coordinates and other mathematical specialities “beyond the scope of this course.” However, full details are given in Paul J. Nahim’s Chases and Escapes: The Mathematics of Pursuit and Evasion.

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Chases and Escapes: The Mathematics of Pursuit and Evasion, by Paul J. Nahim, Princeton University Press, 2012.

Here’s a heuristic description of the Coast Guard strategy: It begins by heading directly toward the Rum Runner’s position when the fog lifted. If the Rum Runner had haplessly chosen to head directly toward the Coast Guard boat, you’d meet and the culprit is caught.

If the Coast Guard craft reaches the point of original sighting, then it’s known that the culprit headed off in some other direction. From there, you follow a spiral path whose radius grows appropriately at the rate of the Rum Runner’s speed.

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The Coast Guard boat C heads toward the last sighting of Rum Runner R, then begins a spiral path that’s certain to intercept the culprit.

Given that your craft is speedier than the Rum Runner’s, it’s guaranteed that you’ll nab the culprit within one complete rotation of the spiral.

Neat, eh? ds

c Dennis Simanaitis, SimanaitisSays.com, 2016

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