On the equian­gu­lar spi­ral, and how the nar­whal got its horn

The Irish Times - Weekend Review - - ENVIRONMENT - Michael Viney Michael Viney’s Re­flec­tions on An­other Life, a se­lec­tion of col­umns from the past four decades, is avail­able from irish­times.com/irish­times­books viney@anu.ie

Once, walk­ing the tide­line on my first Greek is­land, I came upon a most beau­ti­ful oceanic arte­fact. It cov­ered the palm of my hand and there weighed al­most noth­ing: a glossy white fan of a shell, ribbed in a spi­ral curve to its cor­nu­copia throat, and so frag­ile that the sun in­fused it, gleam­ing through ev­ery tight fold.

I’d found the paper nau­tilus, Arg­onauta argo, nest and back­pack for a lit­tle seabed oc­to­pus. Aris­to­tle, in His­tory of An­i­mals, pic­tured the crea­ture ris­ing to the sur­face “with its shell in­verted, in or­der that it may go out more eas­ily and nav­i­gate”, hoist­ing a mem­brane as a sail and thrust­ing out two ten­ta­cles as rud­ders. His whimsy lin­gered to in­spire ro­man­tic po­ets, By­ron among them.

I took my trea­sure back to Lon­don, packed in cot­ton wool. It sur­vived for years, a pre­cious curve of light on sundry bach­e­lor win­dow sills, un­til it fell, its magic shat­tered.

What brought this back was find­ing draw­ings in a book that an­a­lysed the geo­met­ric growth of such a shell: “If, in­stead of trav­el­ling with a uni­form ve­loc­ity, our point moves along the ra­dius vec­tor with a ve­loc­ity in­creas­ing as its dis­tance from the pole, then the path de­scribed is called an equian­gu­lar spi­ral.”

This is not re­ally my kind of lan­guage. Nor, for that mat­ter, is the log­a­rith­mic spi­ral, the Mi­raldi an­gle or the te­tra kai dec ah ed ron.

I was the wrong kind of school­boy for math­e­mat­ics and ge­om­e­try, let alone log­a­rithms or even, sadly, physics. So why am I wrestling with a 100-year-old book so full of the stuff (along with, as is widely agreed, some of the finest English prose in sci­ence)?

On Growth and Form, by D’Arcy Went­worth Thomp­son (1860-1948), has been kept in print by Cam­bridge Univer­sity Press since 1917, in­spir­ing fol­low­ers in sci­ence and the arts as var­i­ously em­i­nent as Alan Tur­ing, Jack­son Pol­lock, Stephen Jay Gould and Lud­wig Mies van der Rohe. I had been quite ig­no­rant of it un­til a re­cent BBC ra­dio pro­gramme con­vinced me that the book be­longed on the same shelf as Dar­win’s On the Ori­gin of Species. A bit late in life, I bought the pa­per­back.

Thomp­son’s sci­en­tific chal­lenge to Dar­win is still, as the journal Na­ture puts it, “mak­ing waves in re­search” and driv­ing ne­glected fields of study.

The im­pos­si­bly eru­dite Scot­tish pro­fes­sor, math­e­ma­ti­cian, clas­si­cal scholar and zo­ol­o­gist at St An­drew’s Univer­sity was mostly con­tent with the the­ory of evo­lu­tion. But he ar­gued there was much more to the shapes of life than the process of nat­u­ral se­lec­tion.

Se­lec­tion ac­counts for the myr­iad forms of species by an on­go­ing win­now­ing that favours forms and struc­tures best fit­ted to sur­vive. Thomp­son, writ­ing be­fore the sci­ence of ge­netic in­her­i­tance, pre­ferred to set­tle for forces af­fect­ing growth in the here and now.

“Cell and tis­sue, shell and bone, leaf and flower,” he wrote, “are so many por­tions of mat­ter and it is in obe­di­ence to the laws of physics that their prin­ci­ples have been moved, moulded and con­formed.”

To­day, for ex­am­ple, re­searchers in “mechanobi­ol­ogy” find his the­ory borne out in cells that “squeeze, flex, stretch and pull on their sur­round­ings and each other, ex­ert­ing force as they do so”, ac­cord­ing to Na­ture.

Even when not shaped di­rectly by phys­i­cal forces, Thomp­son the­o­rised, species can take the ideal forms to suit struc­tural prob­lems. As Stephen Jay Gould put it, Thomp­son’s equian­gu­lar spi­rals of paper nau­tilus and other mol­luscs ap­ply also to “ram horns and paths of moths fly­ing to light, as the only way to coil and main­tain the same shape as size in­creases . . .”

An­other kind of horn, the sin­gle tusk of the nar­whal, pre­sented a dif­fer­ent puz­zle. As Thomp­son wrote, “the only tooth in the crea­ture’s head to come to ma­tu­rity, it grows to an im­mense and ap­par­ently un­wieldy size, say to 8 or even 9 ft long; it never curves or bends, but grows as straight as straight can be – a very sin­gu­lar and ex­cep­tional thing.

Thomp­son, writ­ing be­fore the sci­ence of ge­netic in­her­i­tance, pre­ferred to set­tle for forces af­fect­ing growth in the here and now

“It looks as though it were twisted, but re­ally car­ries on its straight axis a screw of sev­eral con­tigu­ous low-pitched threads [that] wind evenly and con­tin­u­ously from one end of the tusk to the other, even ex­tend­ing to its root, deep-set in the socket or alve­o­lus of the up­per jaw . . .

“I can­not see how to avoid be­liev­ing that the nar­whal’s tooth must re­volve . . . very slowly on its lon­gi­tu­di­nal axis, all the while it grows – how­ever strange, anoma­lous and hard to imag­ine such a mode of growth may be.”

Thomp­son found a pos­si­ble an­swer in the way a dol­phin swims. Each stroke of the nar­whal’s tail, driv­ing for­ward, he de­cided, trans­mits a re­peated and ul­ti­mate twist to the horn. “The horn does not twist round in per­fect syn­chro­nism with the an­i­mal, but the an­i­mal (so to speak) goes slowly, slowly, lit­tle by lit­tle, round its own horn!”

The paper nau­tilus: nest and back­pack for a lit­tle oc­to­pus. ILLUSTRATION: MICHAEL VINEY

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