— The in­t­rig­ing ge­om­e­try of bor­ders

The length of a bound­ary de­pends on the scale at which it is mea­sured.

Cosmos - - Contents -

WHEN MATTHEW FLIN­DERS com­pleted the first cir­cum­nav­i­ga­tion of Aus­tralia in 1803, he read­ily es­tab­lished it as the world’s largest is­land. How large ex­actly is harder to an­swer. The 1978 Year Book of Aus­tralia, for ex­am­ple, gives the length of the coun­try’s coast­line as 36,735 km. The Aus­tralian En­cy­clopae­dia, pub­lished from 1925 to 1996, quotes the wildly dif­fer­ent fig­ure of 19,658 km. What is go­ing on?

Lewis Fry Richard­son was an English physi­cist and me­te­o­rol­o­gist. He was also a Quaker and paci­fist. Ap­palled by the slaugh­ter of World War I, he de­cided to an­a­lyse con­flict math­e­mat­i­cally. Fol­low­ing a hunch that the risk of a flareup might de­pend on the length of two coun­tries’ common bor­ders, he ploughed through some sta­tis­tics, and no­ticed that many coun­tries gave highly dis­crepant es­ti­mates. Richard­son car­ried out a care­ful study to get to the bot­tom of the con­fu­sion. He soon put his fin­ger on the key point that the length of a bound­ary de­pends on the scale at which it is mea­sured.

When Flin­ders sailed around Aus­tralia, you might think his jour­ney would be some­what longer than the length of ac­tual coast­line, given the ship was out at sea. In fact the op­po­site is true. If a ship vis­ited ev­ery lit­tle bay and in­let, and hugged the coast around ev­ery promon­tory, it would clock up many more kilo­me­tres than sim­ply sail­ing by off­shore. If a sur­veyor walked along ev­ery beach and coastal path, mea­sur­ing the dis­tance around each rocky out­crop, the length would be greater still. The length just seems to go up and up the smaller the scale used to mea­sure it, be­cause the wig­gli­ness doesn’t di­min­ish. Con­trast this with a smooth curve, like a sag­ging rope, where the line gets straighter and straighter on smaller scales, and the to­tal length con­verges to a def­i­nite an­swer as the seg­ment size of each mea­sure­ment shrinks to zero.

So does it make any sense to even talk about the length of a coast­line? Richard­son recog­nised the bound­aries of coun­tries with very wig­gly fea­tures, like Nor­way, would in some sense be longer than those, like South Africa, that have rel­a­tively smooth coast­lines.

How to make this pre­cise? Al­though all coast­lines are longer the smaller the scale on which they are ex­am­ined, the rate at which that length grows as one zooms in to ever finer scales varies from coun­try to coun­try. Richard­son de­ter­mined that if the ruler length is l, then the to­tal length varies like ld, where D is a num­ber de­pend­ing on the de­gree of wig­gli­ness. For a smooth curve, like a sag­ging rope, D = 1. But for a line that gets ever longer on smaller and smaller scales, D will be greater. A care­ful anal­y­sis shows that D = 1.52 for Nor­way and 1.05 for South Africa, con­firm­ing one’s in­tu­ition that South Africa is some­how ‘smoother’ than Nor­way. Us­ing the same for­mula, Bri­tain comes out at 1.25 and Aus­tralia at 1.13.

Richard­son’s pro­posal went largely ig­nored un­til 1975, when the math­e­ma­ti­cian Benoit Man­del­brot recog­nised his pre­de­ces­sor had tapped into some­thing math­e­mat­i­cally pro­found. He ar­gued that Richard­son’s scal­ing pa­ram­e­ter D could be in­ter­preted as the di­men­sion of the line. In ele­men­tary ge­om­e­try, smooth lines (e.g. sag­ging ropes) have di­men­sion 1 and ar­eas have di­men­sion 2. An in­fin­itely wig­gly line, how­ever, is some­how try­ing to fill out an area but fail­ing. D, a num­ber be­tween 1 and 2, is a mea­sure of how close the line gets to be­ing an area.

Man­del­brot coined the term ‘fractal di­men­sion’ for D. Thus Aus­tralia has a coast­line with fractal di­men­sion 1.13 – big­ger than a smooth curve but less than Nor­way’s coast­line.

Fol­low­ing Man­del­brot’s work, frac­tals be­came all the rage, in­spir­ing works of art as well as ad­vances in sci­en­tific ar­eas such as chaos the­ory. The con­cept can be ex­tended to any di­men­sion, such as ar­eas that strive to be­come vol­umes, or solids full of holes try­ing to be­come ar­eas.

Once you start look­ing, frac­tals (or at least good ap­prox­i­ma­tions) crop up ev­ery­where in na­ture, wher­ever there are ir­reg­u­lar­i­ties over many scales of size – the shapes of fern leaves, the fili­gree pat­terns of cap­il­lar­ies or the trib­u­tary sys­tem of river deltas, the jagged out­lines of moun­tain ranges, and the spiky path­ways of light­ning, snowflakes and sponges. The con­cept also has prac­ti­cal value across en­gi­neer­ing and medicine, from im­age data com­pres­sion to reti­nal dam­age in di­a­bet­ics.

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