Art and science : Where numerology and mathematics touch the lung

Mathematical analysis in the physical seiences has a long 
history that reveals not only a rich harvest of fundamental 
insights but also leads to a wonderment at the in trinsically 
formal beauty in nature. Rigorous analysis or form and 
function in the biological sciences has been a more recent 
innovation. but introduction of' the principles of similarity. 
scaling, fractal geometry, harmony and proportion has 
hroadened our perspective. The lung provides an apt example 
to illustrate the usefulness of this kind or thinking.

surprising depends on Ihc depth I ir appreciat ion or how these qu al ities have been apprai sed by human s in their conL•cption or the extL'rnal ,vorld.It he lps to remember that , in the historical development of the sc iences.their beginnings almost always were based on metaphysical belicl's.usually with an esoteric va lue -alchemy before.chemistry.astrology before astronomy.Fu rther.it is import:1nt Io kee p in mind Ihe moiivation or primiti ve scientist and arlist alike to reveal God ... ,he good and wise Artisan or all things".In his lcllcr ( 1595) to Maestl in.Kepler wrote.•' ( strive.therefore, ror the glory of God. who wants to be recognized from the. boo!,.; of TENNEY Nature."The underlying question was, how is this cosmic architecture represented'/ and the answer was.that it cou ld he found in an esthctic or numher and gcomctriL• form.That viewpoint can be traced to the shadowy rigurc of Pythagoras living in the 5th century BC. quite possibly the most influential man who ever lived.
It is traditional to cite the Tinwrns or Plato l'or its mathematical view or csthetics, stress ing simplicity and symn1clry as essential features.The early attitude towards art as excelling only to the extent that it recreated natural form was based on Plato's teaching, but he has clearly drawn on the Pythagorean prescription: rorm and number must he in harmony.Indeed.Pythagoras believed "all things arc number".and by this number mysticism, he me.ant not only that they had a sacred character but also a physical reality.Pursuit of the underlying harmonics in nature constilulcd a drive towards science, hut rationality and the discovery or order did not eradicate their mystical significance.God remained the great geometer.
No hooks epilomi,.emore comprehens ively a geometric theory or the cosmos than Johannes Kepler's flumw11ices 1'v/1111di ( I 619).a work in ri vc volumes that encompasses musical theorems.structure or the solar system.mathematical proofs.a description of Lile rhombic dodecahedron and its significance for the structure of a honeycomb.and much murc.
The guiding principle of Kepler'., synthcsi.scan he anticip;1Led in a 4uotation from Proclus that he places on the Lille page nr Hur111onicc.1-Ml(lu/i,It reads: These quotes reveal a continuity of thinking about and recognition or an inherent esthctic 4uality in the mathematics or nature that , in Kepler's understanding, reached its highest form in music.Musical relations could he derived from geometry , and harmony could he defined on a simple integral ratio.It was not simply that the good.the true and the hcautiful were natural correlates of harmony , but that the physical reality of the lengths of churds on a lyre were visible manifestations or these underlying 4ualities.Kepler found musical ratios among the velocit ies of Lhc planL' .Ls , and even Newton wrote of the numerical relationship or the seven colours l\l the seven chords of the musical octave. Although the greater em phasis was given tu linding a numerological code for all things in the physic;d uni verse, the significance of harmonious ratios for human form and physiology wa.s not entirely neglected.Particularly.a. s man was seen as a microcosm.ur in the words of Leonardo da Vinci .""a model or the world", the traditions of Hermetic philosophy became hound up with the mystery of occult numbers, and their source can orten be traced to Kepler's cosmology.Nonetheless, Kepler's works arc al so genuinely scienti fi c in the modern sense and so can he regarded as representing a transitional stage in Lill' history or science.However.if the wmks or the 17th and I Xth cenlllry iatromathL'malician.s and iatromechanists arc ignored because of their limilL~d scope, it can be seen that a long Lime elapsed hcf'orc a Lntly definitive class ic in biological science appeared, a work that would establish, with a wealth of information and rigorous analysis, the fact that the laws of physical science could be applied L o living organisrns.D'A rcy Wentworth Thomps(rn's 0 11 Cm wtlz u11d Form was publi shed in I l) 17, and an expanded edition appeared in 1942 (I).The encyclopedic character of the hook is matched by a ma.stnrul literary style that makes the work as enjoyable as it is informative.It is no surprise that the author is no less remarkable than the hooJ.. he wrote.D'Arcy Thompson' s anatomy recognized "no front icr bet ween biological and chemical form " .However. when asked L o prepare a prefal'C for his Wllrk.Thompsun replied that that really ought to he unnecessary.hccause the whole hook was nothing more than a prcf'acL '' In the composition of this modern classic Thompson established its continuity with the past.Consulting the index one find s that Kepler is referred to nine times; Borelli.eight times: Py thagoras.four.Without in any way suceumhing to the lure of' occult significance in nurnhcrs.o• Arey Thompson was not loath lo recogni1.e the insights and landmark contributions of great thinkers who did: in his own unique way his major work.now more than 70 years olLL still stimulate., and still provides a rich resource or ideas wailing lo be harvested.The •principle of similitude• is one that Thompson developed extensively.

SIMILARITY, ALLOMETRY, SCALING
J\ hasic problem in hiology is that of hody si ze. and rn living organism., there is an aslonishin" range: the ratio of largest to smallest is I 0 21 .For land ma711111als -elephant to shrew -it is Io<>.but function remains sim ilar.The architect and the engineer who design bridges or buildings or inLTeasing size are familiar with the prohlem or sL•aling.vi z. till' consequences ur change in si1c.Thne are predictable limits.
and there arc constraints for any given size.The hiologist confronts the prohlcm from the opposite direction: the Ji vi ng form already exists.and the challenge for the uhserver is to analyze the inherent design on the premise that it was nccc s-sa1y for survival.The scale most convenient I'm a study of animals is their mass (i'vl).The assumption that mass and rnlume arc equivalent is usually adopted.and the relationships or linear dimensions (length.L) are examined.Ir the dimensions or a small structure are foulld all to he multiplied by the sa me factor in a large structure thL'Y arc said to maintain geometric similarity.That is.they arL' i .,•011!ctric.In geometric simi larily volumes arc proportional to the cube or linear dimension and areas arc proportional to the square or linear dimension (L 2 ).Living systems arc rarely isometric: more olkn what is found is a distortion of' shape as a function llf size.and this is tcrnll'd ul/0111elr\'.Galileo noted that the bones of l;1rgc animals were proportionally thickc: r than those nf small animals in acn>rd with a principle well known lo engineers when calculating the necessary dianK'tcr or .,upportingcolumns.The essential rcquircmc11l for a structure in a living or nonliving system is to serve well the function fnr which it was designed.but functional similitude rarely follows geometric similitude.Tho111pson credits Herbert Spencer as the first to show the usefulness or similarity analysis in biology.but it was D' Arey Thompson who explored the rich variety of' applications in hiology ranging over explanations or why the ostrich cannot fly.why the hcc • s wing vihratcs faster tha11 a bird's and why a Ilea canjump as high as a human.Lord Rayleigh.~--------------------------~ Of all the structural and functional variables in respiratory physiology subjected lo allometrie analysis.the one most extensively studied is oxygen uptake rate.Oxygen enters the organism by Jiilusion across a surface.and.therefo re. the value ought reasonably to be predictable as varying with body si ze as M 21 \ this would be consiste nt with the conventional expression of basal metabolic rate as a function of ) surl"acc area (BMR/111 -).The experimental findin gs.how-CVL' L from the smallest lo the largest organisms.both warm-,1!1d cold-blooded.arc that resting oxygen uptake rate varies as M-' 14 ( Figure 2) (5 ).Fur maximal oxygen uptake ( Y02rncix ).
ninvcntionally scaled as M 1 in Y02mcixlkg. the exponent is again kss than I (6).The exponent.i• suggests a •compromi se' bcJween metabolic processes remaining cons tant.per unit weight (the simplest bioche mical assumption) and.
therefore.increasing in proportion to body weight.ic.Ai.
and being limill'd by diffusing surface art'a or. in the case of exerci se muscl e cross-scctillnal area, prt'dictcd to increase as M 213 .Conjectures like •comprnmisc• carry no useful meaning.and the real explanation remains elusive.Bu t the question is important and illustrative llf the usefulness of allon1ctric analysis to reveal some universa l principle.even tllllugh it fails to provide any mcdianistiL' explanation.
FRACTAL GEOMETRY, FRACTAL DIMENSIONS D'Arcy Thompson's life enued before the new mathematics of fractal geometry were devclopeu.but it is a sa fe prediction that, had he lived to •ee the profound insights fractal geometry introduces for the study of natural for ms, he would have been deli ghted.Benoit Mandelbrot' s book.Th e /-"m etal Geo111t'/IT of Nature (7).has had a major impact on both science and art.If. as Plato and Aristotle insisted.the line arts imitate nature.then the geometric abstraction had to fall baL•k on a resemblance more than an imitation.because the real forms in natur niuld only ht' approximated by classical maJllcmatics.which is limited to ideals.An oran ge is not a sphere.:t volcanic mountain is not a cone.Real for m:.have bumps and all sorts of irregulariti es that make them complex and i mpossililt: lo analyze by Euclidean geomet ry.What would be the mathematical description of a cloud m. for I hat matter, of the human brain'' Fractal gcomctry come, to the rcsCUL'.Its origins (hut not called by the name: Mande lbrot coineJ till' term) reach hack to troublesome construe-!ion.,.well known to many mathcmatici::ms in the I 9th century.called monster curves -curves that were continuous.but whose gene rating function was not differentiable.They had no unique tangent at any one point.and their length could not be measured.Their dimension was only an •cffeL•ti vc Jimension ', and it had a subjective basis.Consider Mandelbrot's example of a ball of twine: At a great distance it look, like a point (dimension= 0): closer up it looks like a sphere (dimension = l ): still closer it is seen to be a mass of thread, (dimension = I): and closer .still.each thread is see n to be a column (dimension= 3): at the level where fibres arc seen.they arc interpreted as lillL'S (dimension -I l: and at the atomic level, the appearance again i. ol' points (dinll'nsion = ()).However.there are ill-defined ;ones of transition fro m rnw cl ass ical Euclidean dimension to another.and these ar~ fractal zones.There.the dimension will not be an integer.
Most fractals are found to be invari<1nt under transformations or scale: they are self-simi lar under ordinary geometric similarity.and an enlargement of a small portion resemble~ the whole structure.A landmark paper that dearly prese nted the problems of a fractal curve was published in 1961 in an obscure.and for the subject, inappropriate.periodical ( Gent'ml Srste111.1•Yeurhon/..:) (8).The author.Lewis Fry Richardson.was an ingenious eccentric with a doctorate in psychology.known for his highly original rnntributions Io weather forecasting .a Quaker who wrote extensivel y on war.and a m:1n with deep nwthcmatical insights.I lis semi na l. but long neglected.work on fractals Jcall with their peculi ar properties in regard to the problems or mensuration and dimension.Richardson• s question was deceptively simple: How long is the coastline of Britain'!The answer he gave wa, that it depends on the length of the measuring stick used.Clearl y. a metre stick would miss the perimeters of a great many indentations that would be included if a centimetre stick were used, and a millimetre stick would add still morl'.In fact.re olution to :ttomic structure is imaginable: theoretically, there is no reason to stop there.The conclusion mu:.\ be: as the measu ring stick gets shorter the Jcngth ol' the coastline increases indefinitely, but the limit is meaningless.In mathematical terms a continuous series is differentiable because it can be split into an infinite numbn of' stra ight lines; but a nondifferentiabk series.e ven though continuou\.cannot be resolved in this way , because each successive spl it merel y reveal s still more roughness.To solve the practical problem of measurement an arbitrary deei.sion has to be maJ~ regarding the measuring de vice.
How can one arrive at a dimension for a structure like a coastline?The answer is that it will depend on the degree of irregularity .Imagine pe ncilled wiggles on a plane piece of paper.So long a.-; they ,ire seen as lines their dimensi on will ti.: I, but as the paper beg ins to be fill ed. the scribb led lines hcgin to look like a plane area.and tlw dimension would he 2.The rea l dimension must lie somcwhcrL• between I and 2. For a crum pled surface the dimens ion must lie between 2 and l dependin)!on the extent to wh ich ii departs from smooth 10 m1mplcd.The rradal di111c11sio11 (alsu called the Hausdorff-Besicovich fun ction) retlccts the degree of rough ness of ~patia l data : it is a ml'asurc or complex ity.The sol ution of the problems of meas un:mcnt and di men si on is L 'X prcsscd in a ,impl e equati on.
It ca n be shown that a length.L. measured with a stick of length £ (relative to a unit stick) will be where D is the fractal dimen sion.{) rcsol ws the conceptual problem of le ngth.Although all mast lines must be o f infinit e length , the val UL' of D depends on the coa!>tlinc chosen.And.hy choosi ng a val ue of £. the observer has determined lhe panicul ar parli,il reality that wil l be revealed.Human s thus 1ct themselves above nature -,Ill interesting instance of a11thro1 ocentrism .
Fracta l curves arc generated by rt•curs i vc proced ures th,tt Jcrivc each new ge neration by repeating some specific operatio n on the preceding genera tion.The idea of dimension can he illustrated by the fo ll owing.If a li ne is stretched lu twice ii-length , it can be rnl in half to produce two copies or the original.The dimens ion or the line is the exponent of 2 (thl' magnifying fact or) that will give the number of copies= 2. The exponen t (dimens ion) must be log22= I. Likewi se. if the ,ides of a square are doubled.the exponent (di mensi on ) will he log24=2.And for a cu be. the dimension is log28=3.If these integer multi pliers arc replaced by some othe r -say.one that increases the length of a line re pea tedly by a factor of 1the lim iting curve will have a dimen sion (fractal) that is log3.J.= 1.26 .. .. The fractal dimension ind icates that !he curve i, more complex than a li ne: ii parti ally fil ls a plane space, but it ha not yet fill ed it; otherw ise, it would be an area and have dimension = 2.The limiting curve fu lfils the cri teria for a fractal.and of the hundreds of compu ter-generated frac tal -:u1-res almost all have an astoni shing, abstract beauty .The wnnection of these curves with phenomena li ke turbu lence ;111d Brow nian motion is 1101 intuiti vel y obvi ous.but the sense or wondermen t was expressed by ivlandcl brot: •• ... fractal fCn metry reveals that S(1111c or the most austere ly form al d1ap1crs 01• mathematics had a hidden face.a world uf pure plastic beauty unsuspected until now.•• The significance or fractal geometry for the for m and funct ion or the lung has not been neg lected (9).The design of the bronchi al tree is that or a bifurcating se lf-si mil ar structure ,, ith progressi ve dimin ution of tube diameter in each generation.The system functions with mini mum entropy production t 10 ).The success ive di ameters in the bronch ial system appear to be fractal.but it is necessary that thi s space-fi lling ,i ,tem .which is designed to conduct gas 10 and fro m alveoli.must reach some limit in the recur ion process --a limit undoubted ly set by some minimal volume of uni nvaded .,pace(al vcoli) necessary to lllaintai II gas exchange .Thi, ill ustrates the fact that no real system is infinitely fractal.
The respiratory surface of the lung is fracta l and 111ay be regarded as tending towards a max imal surface area in a minimal vol ume ; in fi sh the nxyge n-absorbing organ can be approximated by fractal lines .The charac teri stics of a fract al sur ace pre 1ct tort e ung t 1a1 ( o ume) -cx( Arca ) , anc that D will be greater than 2 and could approach 3. Further, the 111ca.urecl alveo lar area wil l. li ke measuri ng the length or a L'Oastlinc.be dependent on the measuring ins1ru111cnt.Indeed.alveo lar area measured with a light 111icroscopc is 80 111 2 .but with an elcc1ron 111icroscope, it is 140 m 2 • and D=2.I 7 ( 11 ).Insofar as gas exchange is concerned.irregularity of the alveo lar surface mem brane is of Jillie con sequence since alveoli ha ve a liquid lining layer.but for water and so lute exchange there may be con siderable i111 portanec.

SIZE RATIOS AND LOG-NORMAL DISTRIBUTIONS
In 1959 Eve lyn Hutch inson , in a paper ent itled ''Homage to San ta Rosa lia.ur wily are there so man y kinds orani111a1s• 1 .. ( 12), observed that silllilar speci es ex ist ing in a si mi lar ecological environme nt tended to be di sti ng ui shed by a size ratio of about 1.3.The significance of the value 1. 3 became incorporated in ecological theory as represL:nting the size difference necessary for coex istenCL'.ie.th is was a limi t below which competit ion became 100 great.The community wou ld be forced to sellle out into a dis tribution or species si ze ratios of 1.3 fo r stable survival; and the aspec t or size might be whole body mass .or beak len gth of bi rds, or any structura l fea tu re that would prevent two speci es existi ng in identical ways.Th is ca me to be known as the compet itive excl usion princip le. and values around I .J were call ed the Hutch inson •constant' .or the Hutchi nson ian ratio.
Many data filled the literature eonrirm in)! the widespread prevalence of size ratios around 1.3 in nature, and the num ber began to take on a certain myst ical significance.When ii was discove red that a similar ratio rnuld be found in the world of inan imate objects ( 13) its unique signi fi cance for community survival of living things vani shed.The ubiq ui tous prese nce of the 1.3 ratio ru le was almost comically demon strated.It could be shown for lengths of an ensemb le of rec order-.for diameters of bic yc le wheels and sk ill ets.ror cooki e cutters and ani ma l figuri nes.The las t example is rar1hcs1 removed rrom any utilitarian explanati on and might be thought of as some guiding impul se in artistic ex press ion .Mai orana ( 14) has postul ated that "an expl anation for these observations may have 10 do wi th our pcrceplllal abiliti es, a cause that may hL' .applicabk to (and perhaps derived rrum) till' natural world" .
T he underlying message appears, however, to he 111ud1 simpler.The Hutchinson i:111 ratio is nllthing more than an anii'acl of log-normal di stritiutiuns ( 15 ).The usual ass umption ado pted by experimenters is that their data are di st ributed in a normal error curve, and that stati stica l in ference can then be made with the satisfact ion of knowi ng how a normal error curve arises.The central lim it theorem or statistics says that "the addition of a large number of sma ll i11depc11dc111 random variables yields a quantity whose distributi on tends towards the normal distribution••.In many instances.the requirements for the applicability ufthe central limit theorem arc not met.and most ortcn it is because the sources of data arc not the result or simple additive combinations hut instead arc based on multiplicative combinat ions.In that in stance.the data can he made to tit a normal distribut ion curve if the logarithms ol the data arc plntted.The distribution is then log-normal.and it is apparent that there arc many murc ways in which a log-nnrmal distribution can arise in biology than is the case for normal distributi on.It is unfortunate that demonstrating a good log-normal !'it in no way is helpful in deducing the interactive processes that mw,t ha ve been re-sprn1.~ible.but size ratios in the range 1.2 to 2.0 will result.given only that si1.cs arc log-normally distributed.and that the standard deviation is less than I.
Respiratory phriologists have not been prone to make much use of log-normal distributions.hut it is noteworthy that l•lcrmann Rahn. in hi s Jl)4l) paper on ventilation-blood now relationships ( I Cl).adopted a log-nunnal plot as •simrlcst • for the Vt'Ilt i lat ion :perfusion distribution rat ins throughout the lung.This contention has stood the test ur time and experimentation ( l 7).

THE 'DIVINE PROPORTION' AND
FIBONACCI SCALING From the time or Pythagoras there has been a fascination with a ratio that seems to be associated with esthetie appre-ciatilll1 and has unusual properties th at have hcen explored in the theory or numbers.Classical references sreak of the •Golden Section• of a line.wh ich is made as follows (Figure J).

If a line ur le11gth
1\LJ is cut at C in such a way that 1W li\< • = 1\C/Cf1.the conse4uenL-cs are of considerable interest.The easiest way to see th i., is to let the segment A(' be .r. and the segment CB be unit y: then the ratios result in the quadratic equation:
The number qi appears as the lim it or an infinite series that i. , associated with the name Fibonacci .Leonardo or Pi sa was nicknamed with a conjugation of Filius Bunacci (sun ur the buffalo).Fi Hon:1cL:i.The Fibonacci snit's is O. I. I. 2. 3, 5. 8.I J. 2 1, ....Each numher is me re ly the sum of its two predecessors.or .Un+1 11m -u -->c)l /J-')oo n lt is the simplest or all continued fractions.and the succes-.

. . I l
~ 1 1 'di s1ve ratios ol t 1e nurn 1ers (-1 1 .-~• 1 • etc) rapt y approach the Golden Section (0.(l 18 J. 1he SL'rics and the ( ,olden Section appear repeatedly in nature.and no example is mmc spectacular than that or the chambered nautilus.Each Ill'\'.chamber in the growing shel l is a self-similar replica ur thl' preceding chamber.The princi ple of this construction i, based on the Golden Section of a rec tangle made in SUL'CC\-.,ive steps to reach a limiting rec tangle.which is the role (II an equiangular spiral (the .1pim 111iml1ilis of Kepler: a curw that so fasc inati:d Jacques Bernoulli that he had a logarithmic spiral engraved on his to mbsto ne with the inscription. •-cudrn1 mllfuW n '.\'1/rgo" ).The le ngth of the sides or the squares in the construction is a Fibonacci series.and he nce.any two segments arc different in si1.e but not in .,hapc.
Even in the times or Eucl id .to divide a line into an ••extreme and mean ratio•• a., a first step in the L'lmstructiun ui' a pentagon was well recognized.and from thcrt' it was pos~ible to proceed to the Pyt hagorean symbol or the cosmos.the regular dodecahedron.
In nature.the beautiful symmetry of the bct'•s honeycomb has been an inspiration dating from ancient Greece to the present day.The hexagonal array nr cells is a well known feature of close packing.hut it was alsu recognize.clas an architecture that was most econnmical in the use or \.Va\ .Then.: were those who ascribed this ortirnal prnperty to a divine guidance of bees since they could not be supposed to have highly developed mathematical skills.Such opiniom invited D' Arey Thompson's scorn .His comment was that th~ "beautiful regularity" was due only to "some automatic play of physical fo rces" .More recently the Hungarian mathematician Fejes-Toth. in a paper entitled, "W hat the hccs know and what the bees don't know" ( 18) has demonstrated that the bottom of the hce •s cell is not constructed in the most economical way.so we arc left with an example or good.hu t no t perfect.design .
The discovery that the irregular hranching pattern of the human hronchial tree can he described hy a scale of Fibonacci numbers was made in 1985 hy Goldherger et al ( 19).This growth pattern of sel f-si mi larity is Ii kc that of the sneezewort on which a new hranch springs from each axil, and more brunches grow from the new hranch in such a way that if the old and new arc added a Fihonacci sequence is ohtaincd.
These diverse examples illustrate a common theme of fractal self-similarity and Fihonacci proportionality to give structures of great heauty on the one hand .and prohably serve to minimize constructional error on the other.as D ' Arcy Thompson pointed out.The mechanisms that determine morphogenesis according to these principles remain to he di scovered, but extending from the proportions of the Parthenon to proportions in the human ()(ldy, the numhcr <jl is a persiste nt and tantalizing presence. E thetes arc 1wt likely to look to the lung as the most beautiful structure in nature.nor are pulmonologists inclined to romanticize pulmonary structure a nd function: yet.the desi gn of the lung reveals all of the classical ingredients or beauty and of purpose optimally achieved.Recollecting the characte ristics nf similarity.scaling, complexity inherent in its fractal geometry.harmony and proportion inherent in Fibon acci scaling.the temptation to see how the lung stands up under the requirements of the •aesthetiL• forlllula • is irresistible.George D Birkhoff.a prominent A1ncrican mathematician of this century.proposed in his hook Aesthetic Mea sure ( 1933 ) that if the esthctic measure.M, or any class of' ohjcl'ls, is "any quantitative index of their comparahle aesthetic effectiveness", the n it will depend on the ratio of complexity.C. of the ohjcct and the property of harmony, symmetry.and order, 0. He nce.

M = f(O/C)
and the t•sthetic experience is seen t(i he measured "hy the density of orde r relations in the aesthetic ohject".On these g rounds, the lung must receive a favourable judgment : its architecture warrants acclamation.
Since the more profound understandi ng of the mathematical principles that determine stru cture from the molecular to the organismic level remains to be discovered, these remarks appropriately close with a quote from the great 17th century Neapolitan mathematician and hiologist.G iovanni Bordli .