# Simanaitis Says

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# LEAF MATHEMATICS

HOW COME THE common Japanese shrub Orixa japonica breaks all the rules of leaf arrangement? Basil, for instance, is decussate: Each leaf is about a quarter-turn, 90 degrees, from the last. Bamboo leaves are distichous, with leaves directly opposite each other. And spiral aloe leaves form a swirl that follows the Fibonacci sequence, the same as sunflower seeds. There’s even a phyllotaxis equation (Greek: “phyllon = leaf, “taxi” = fashion) that explains leaf arrangement.

But Orixa japonica fails to comply. Instead, its leaves follow a complex five-step pattern: The second leaf up the stem grows opposite the first one, 180 degrees away. The third leaf is 90 degrees from the second, the fourth 180 degrees from the third. And (just when you think you see a pattern) the fifth one grows 270 degrees from the fourth. Only then does the sequence repeat.

Whatever do you suppose Orixa japonica is thinking??

Orixa japonica, a common Japanese shrub. Image by Qwert1234, via Wikipedia Commons, from The New York Times, June 6, 2019.

I got all this leafy information from “Solving a Leafy Mathematical Mystery,” by Cara Giaimo, in The New York Times, June 6, 2019.

Cara Giaimo discusses the paper by Takaaki Yonekura, a plant physiologist at the University of Tokyo, and his colleagues. Its title says it all: “Mathematical Model Studies of the Comprehensive Generation of Major Phyllotactic Patterns in Plants with a Predominant Focus on Orixate Phyllotaxis.”

Whew. However, the paper’s abstract begins, “Plant leaves are arranged around the stem in a beautiful geometry that is called phyllotaxis,” a charming start to a discussion of EDC2, an expanded form of DC2, the earlier math model that Orixa japonica seemingly ignored.

DC2 was developed in 1996 by S. Douady and V. Coulder of the Laboratoire de Physique Statistique in Paris. Both this mathematical model and its expanded EDC2 are grounded on Mother Nature’s intent to give each new leaf a good chance of survival.

Giaimo says the models are based on “the assumption that each leaf exerts a chemical ‘inhibitory power’ on the area surrounding it—a sort of force field that prevents other leaves from growing. The force field peters off with distance until it disappears, allowing new leaves to form.”

Common leaf arrangements. Image by Takaaki Yonekura from cff2.earth.com. See also Yonekura’s fascinating animated images in The New York Times article.

Yonekura and his colleagues added the nuance of leaf age. That is, their expanded model gives older leaves a stronger force field than younger ones.

“This time,” Giaimo notes, “when they put in Orixa japonica stats, the right shape came out.”

Is phyllotaxis science now a done deal? Nope. Mother Nature is elusive: The perennial Costus plants exhibit “spiromonostichy,” making them look like tight spiral staircases. Why?

Latin “spira” = coil, Greek “mono” = single, “stichos” = line.

Mathematics, botany, etymology. All three are entertaining. ds