Wednesday, April 22, 2009


Amazing Patterns

Richard Wilbur, In Trackless Woods:
In trackless woods, it puzzled me to find
Four great rock maples seemingly aligned,
As if they had been set out in a row
Before some house a century ago,
To edge the property and lend some shade.
I looked to see if ancient wheels had made
Old ruts to which the trees ran parallel,
But there were none, so far as I could tell—
There'd been no roadway. Nor could I find the square
Depression of a cellar anywhere,
And so I tramped on further, to survey
Amazing patterns in a hornbeam spray
Or spirals in a pine cone, under trees
Not subject to our stiff geometries.
Roger V. Jean, Phyllotaxis: A Systemic Study in Plant Pattern Morphogenesis (Cambridge: Cambridge University Press, 1994), p. 1:
Phyllotaxis studies the symmetrical (asymmetrical) constructions determined by organs and parts of organs of plants, their origins, and their functions in the environment. These constructions are the phyllotactic patterns, and their building blocks, in their young stage, are called the primordia. The primordia differ in number, size, position, rate of formation, and shape, thus giving considerable diversity to phyllotactic patterns. Yet the phenomenon of phyllotaxis is simple, insofar as all the phyllotactic systems showing spirality belong to Fibonacci-type sequences of integers, characterized by the rule that every term in it is the sum of the preceding two terms, as in the Fibonacci sequence <1, 1, 2, 3, 5, 8, 13>.
Id., p. 12:
The most common pattern, the spiral pattern, also involves the insertion of a single primordium; but in this case it is possible to trace two sets or families of spirals, which run in opposite directions and which appear to cross one another. In the botanical literature these spirals are called parastichies. Two families of parastichies constitute a parastichy pair. They are formed, for example, by leaves on the stems of many plants with a cylindrical surface. The spirals made by the scales on the pineapple fruit are another example.
Id., p. 16:
Pine cones are also excellent examples. Counting the spirals on cones reveals that the number of spirals running in the right-hand direction is not the same as the number of spirals running to the left. On pine cones, a common pair of spiral numbers is 5 and 8. Other cones may show the pairs of numbers 2 and 3 or 3 and 5.
See figure 13.8 on p. 131 of R.A. Dunlap, The Golden Ratio and Fibonacci Numbers (River Edge: World Scientific, 1997), which is a "Schematic illustration of the seed bearing scales on the base of a typical pinecone. Line a shows one of the 13 clockwise spirals and line b shows one of the 8 counter clockwise spirals."

It is uncertain what amazing pattern in a hornbeam spray attracted Wilbur's attention. Perhaps it was the same pattern noticed by Thoreau in his Journal (January 25, 1852):
I am struck and attracted by the parallelism of the twigs of the hornbeam, fine parallelism.
Thoreau may have been referring to the parallel lines formed by every other twig segment as shown in this photograph of a European hornbeam:

If you extend every other twig segment in your mind's eye, you can see parallel lines.

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