by Tim Robinson
N = b3
3. The first three stages in constructing a simple fractal; each stage is derived from the preceding one by replacing every straight bit by a reduced replica of the first stage.
Now think of a square with unit sides, divided up into smaller squares, each of which is the same as the big one but with its sides reduced by the factor 1/b. The number N of such small squares in the big one is given by
N = b2
4. The self-similar dissection of a cube, a square, and a line.
And finally think of a line of unit length divided up into smaller sections of length 1/b; the number of them in the whole line is
N = b
In each case, N = bD, where the values of D agree with our idea that the cube has three dimensions, the square two, and the line one. Suppose though we take this equation as defining what is meant by dimensionality, and ask what result it gives when applied to a fractal form. The simplest case is the jigged form described above, of which any part, however tiny, will look roughly like the last curve in Fig. 3 when suitably magnified. The section of it from A to B, say, is made up of a number of parts such as that from A to C. Each of these parts is similar to the whole section but is smaller by a factor of 1/4; thus in analogy with the cube, the square and the line, in this case we have b = 4. And there are eight of them making up the whole, so N = 8. Putting these values into the equation N = bD, we have:
8 = 4D
whence:1
D = 3/2
This way of looking at the idea of dimensions thus concurs with our normal ideas about simple shapes like squares arid cubes, and gives a result for fractals too; in this case it seems very well to express the feeling that there is more space in the ‘jigged line’ than there is in an ordinary one-dimensional line, but that it is still something less than an area with two dimensions. One can easily construct examples which have a dimensionality of less than one, or between two and three. And the idea can be extended to cover irregular natural forms like coastlines and clouds, though here the results will be approximations, arrived at by statistical methods. Coastlines typically have dimensionalities of about 1.2; my very crude estimate for an average stretch of the south Connemara shore is 1.25, and the more complicated bits of it would give a higher answer still.
The fact that coastlines are virtually infinitely long had been stated before Mandelbrot, just as the concept of non-integral dimensionality had been suggested by earlier mathematicians; but it is Mandelbrot’s collation of the two ideas that opens onto new vistas of thought. A previously obscure topic in mathematics is suddenly important in physics, the earth sciences, biology and astrophysics, is crucial to the new disciplines of chaos and complexity theory, and promises to be a rich source of metaphor and imagery in literature and art. Like all discoveries it surprises us yet again with the unfathomable depth and texture of the natural world; specifically it shows that there is more space, there are more places, on a seashore, within a forest or among the galaxies, than the geometry of common sense allows.
The hints about the subject I gleaned from the newspaper cutting prompted me to get Mandelbrot’s book, The Fractal Geometry of Nature.2 It is an unlovely work in some ways (though without the abominable faults of the other principal sourcebook on the subject, The Beauty of Fractals‚3 which misidentifies the beauty of science and that of art, and crosses them to produce a hybrid with the virtues of neither). However, Mandelbrot’s concern with assessing his own priority in the field does lead him to discuss his intellectual forebears, and I was intrigued to learn that the particular concept of dimensionality he used (and of which I have given a very crude version above) was ‘formulated in Hausdorff 1919 and put in final form by Besicovitch’; in fact his mention of the latter name led me into the fractals of personal memory, which eventually spelt out a reason for my obsessive tracings of the Connemara shoreline. The anecdote involves another mathematical construction, the Peano Curve; so I had better clear the ground by introducing this old friend first.
5. The first three stages in constructing a Peano Curve; each stage is derived from the preceding one by replacing every straight section by a reduced replica of the first stage.
We start with a line that zigzags through a square area (Fig. 5). First we replace each leg of it by a zigzag similar to, but smaller than, the first, and then do the same with each leg of the resultant more complex form, and so on, to the infinitely small. My diagram is crude, but drawing even the third stage is a maddening task and the fourth would be quite beyond my powers of attention. Of course the ultimate outcome is impossible to draw, because of the thickness of penpoints and the finiteness of human existence; it is a mathematical concept, like the perfect circle. Peano showed that such a curve actually visits every point within the square. In fact, while a line, however complicated, has a dimensionality of one, this object has the same dimensionality as the square, an area, i.e. two. Each little bit of it is a labyrinth in which each detour is a smaller labyrinth, in which each detour is a yet smaller labyrinth, and so on, beyond the dreams of Borges. At the time it was discovered – or created, according to one’s view of the nature of mathemetical objects – by the Italian mathematician Peano around the turn of the century, it was regarded as a geometrical monstrosity, an exception best disregarded in the name of logical hygiene.
I will nose briefly into just two of the labyrinthine digressions this diagram suggests, in the Irish context, before taking up my story. First: ‘How Celtic!’ Perhaps one could claim that fractal geometry is to Celtic art as Euclid is to classical art. While the mainstream of European culture has pursued its magnificent course, some other perception has been kept in mind by the Celtic periphery, all the way from La Tène Iron-Age curlicues to Jim Fitzpatrick’s kitsch-Celt goddesses. In a word, that a fascinating sort of beauty arises out of the repetitive interweaving of simple elements. The beauty of Nature is often of this sort. In Connemara, which is pre-eminently the land of ‘dappled things’ – drizzling skies, bubbly streams, tussocky hillsides – one recognizes this texture. Of course, after a thorough soaking in all this dappled Celtic bewilderment, one runs for shelter in classical temples, with relief. And secondly: ‘How Ruskin would have liked this!’ In his book Modern Painters, which is largely about how Turner achieved his version of truth to Nature, Ruskin devotes pages to analyzing such visual phenomena as a mackerel sky or the successive patterns of striations revealed on the peak of Mont Blanc as the sun moves round it. But cloud formations and geological fault systems are often fractal phenomena, and, lacking this concept, Ruskin’s numerical estimates of the number of dabs of cloud that go to make up the unified spectacle of a sunset sky are heroic but doomed attempts to render the beauty of the dappled things that classical art ignored, that it defined itself by excluding.
To return to my personal entanglement in the Peano Curve. Like various other apparently freakish inventions of turn-of-the-century mathematics, it was one of the hobby-horses of Besicovitch, who was an elderly and semi-retired professor of mathematics at Cambridge at the time I passed unremarkably through the place in the fifties. Mere crotchets on the spires of intellection; so Professor Besicovitch’s chosen topics appeared to me, a gauche would-be aesthete and week-old undergraduate, in the autumn of 1952. His best-known work was a treatise on Almost Periodic Functions, which sounded like a parody of professorial abstruosity. Besicovitch was to give an exceptional and to some degree extra-curricular series of seminars to the new intake. In the first of these, the professor, who had very little English other than mathematical terminology, outlined a number of his favourite problems, some of which still lacked solutions. An example: a line one inch long can obviously be turned around inside a circle one inch across, but what is the smallest area in which it can be turned around? It had long been accepted that the answer to this curious query was the area swept out by the line in making a three-point turn, a sort of concave-sided triangle. But the professor had demonstrated that in fact it can be turned around inside a s
hape of no area whatsoever, provided it is allowed to back off to infinity in an infinite number of different directions. A typical Besicovitch result; both ludicrous and mysterious. The image of that line’s minnow-like dartings in all possible directions still remains with me as an emblem of a certain style of thought, and I suspect it also helped me fail my driving test. Having introduced us to half-a-dozen such bizarreries, the professor suggested that some of us might like to divide these problems between us, study them further, and lecture on them to the rest of the group. The first problem on offer concerned a way of representing a Peano Curve by a formula, an infinitely long and curiously constructed series of powers of a complex variable. It was, said Besicovitch, a very beautiful solution. Then he looked off into space rather sadly, and admitted that the curve this formula generated did pass through some points outside the square; nevertheless, it was a very fine solution. Also, he confessed, apparently quite disconsolate, it did pass through some points of the square more than once. But, taken all in all, it was a very beautiful solution. The proof was long, though, so he would suggest two of us divide it between us, to study it and report back to the group. Who would volunteer? He looked at us expectantly en masse, and then, as no one responded, he appealed to us mutely, face by face. I had already glanced round the crowded lecture hall at my hundred or more unknown colleagues and decided they were a dull lot; I would have preferred to consort with the lovely and witty beings reading one of the arts subjects; but then for me the only point of art was to practise it, and I was convinced that all pedagogy stultified creativity, whereas I knew I could not climb the cliff-face of mathematics without help, and so had condemned myself to sojourn among these dullards. I recognized, all the same, that most of them had more mathematics than I, for I came from an unassuming provincial grammar school which had not previously aspired to prepare a pupil for Cambridge. But when Besicovitch said ‘Please!’, in tones of pain and embarrassment, I put up my hand out of impatience with the lot of them. To my relief someone I knew to be better educated than myself in such matters followed me into the breach.
‘The Peano Curve and Its Representation by a Lacunary Power Series’ was the title of the paper Besicovitch handed to us afterwards. It looked formidable, but such was my ignorance I did not realize how far beyond me it was. My colleague and I met that evening and struggled through a page or two of it, and later called on the professor to have it elucidated. Besicovitch silently admitted us to his small rooms, and enquired our names: Davis and Robinson; he pondered them for a time and then remarked ‘D and R!’, with a kindly smile to indicate that he was proffering a joke. (Brezhnev and Khrushchev, visiting the United Nations at that time, had been abbreviated to ‘B and K’ in newspaper headlines.) Our private sessions with him became a weekly ritual. As our incomprehension drove him deeper and deeper into the problem, I believe he invented new lines of argument that skirted around difficulties too deep for us to ford. We looked over his soft, bulky, Slavic shoulders as his pen hovered and agonized in the air for minutes before inscribing each symbol. We learned a little about him; that he was born a member of the Karaites, a tribe of Jewish faith and Central Asian ethnicity that wandered between the Caspian and Lapland; that he had studied at the University of Perm, where it was so cold that he used to get into a sack to read. What convulsions of history had brought him to Cambridge we did not think to ask.
Eventually we deluded ourselves that we understood the simplified proof that emerged from these tutorials, and the day came on which ‘R’ had undertaken to expound the first half of it and ‘D’ the second, to our assembled peers. All were silent and attentive as I blundered on into the valley of death. Besicovitch soon showed signs of concern, though, and at one point where I announced my intention of deducing a certain proposition from another one he interrupted to ask how I intended to do that. ‘By the Mean Value Theorem!’ I replied, which probably revealed to him for the first time how far out of my depth I was. He put me on the track again and let me limp to my conclusion. I returned to my bench, and ‘D’ took up the chalk. But he fared even worse than I, and the professor dismissed him from the blackboard and continued the lecture himself. I was overcome by what I self-protectively identified as the absurdity of it all, and sat there snorting hysterically; once or twice Besicovitch’s eye rested on me, forebearing, puzzled by this human predicament, before gazing off into the world of forms again.
I do not remember that anyone ever mentioned this dreadful episode to me afterwards, or that I thought much of it myself; it must have vanished into the froth of my rather belated adolescence. But the Peano Curve remained branded on my mentality; it was the original of ‘the conception filling my mind of a strange map consisting of one line, and that so convoluted it visited every point of the territory’, which I wrote of in ‘Setting Foot on the Shores of Connemara’. For a personally researched map that aspires to comprehensiveness must be, as I have said, the record of a walk that covers the ground; therefore the Peano Curve, this topological monster, is its perfect emblem. But it is also a reminder of how far any such map must fall short of success. This fractal nature of the ideal one-person map is both my excuse for the map of Connemara having taken me about five times as long as I thought it would when I set out, and my condemnation for having undertaken an absurdity, an adequate map; for any such thing would have taken not five times but an infinite number of times longer than my naive estimates. And now, when I see the name of Besicovitch in the roll of those of whom Mandelbrot writes, ‘These names are not ordinarily encountered in the empirical study of Nature, but I claim that the impact of the work of these giants far transcends its intended scope,’ also that ‘while Hausdorff is the father of nonstandard dimension, Besicovitch made himself its mother,’ I realize that I have wasted some hours of the lifetime of a genius, which is an intellectual crime, and that my penitential would-be fractal journey enacts a double apology – to Besicovitch for my inability to appreciate his self-infolded gigantism, and to the surface of the Earth for the paucity of all our attentions to it.
* * *
Let me now turn to more concrete matters, a tiny selection of the places, experiences, discoveries and surprises that my convoluted walk about Connemara led me through. These are some of the moments that made the endless trudging along muddy shores and stony tracks worthwhile; perhaps my fractal theme will secure them from being mere mementos brought home by polymorphic dilettantism.
I remember one particularly filthy day on the coast south of Ros a’ Mhíl. That area is bleak at the best of times, a bare flatland of granite, walled up into the little fields that spell poverty and labour; it used to be part of the estates of the Blakes of Tully, the worst of the rackrenting landlords. It is said that eventually the Blakes were cursed by the priest for their tyrannical ways; he wrote out Salm na Mallacht, the Anathema, against them, and put it in an envelope, which he put in another envelope, and that one in another, seven in all; and gave it to a lad to deliver to the Big House, Teach a’ Bhláca, instructing him to leave immediately he had handed it over and put as much distance between himself and the house as possible while Mr Blake was opening those seven envelopes. It seems this moral letter-bomb was expected to devastate all in its vicinity. Indeed the Blakes were soon after embroiled in misfortune, and left the house, which is now roofless and ruined, though for a long time after that the tenants continued to pay their rent to a Blake, who was in the mental asylum at Ballinasloe. It still looks as if the curse had its evil effect on the neighbourhood, which is heartbreakingly dreary, especially in winter. But on this wet, blustery day I met two elderly men there, walking up a boreen past a deserted village, puffing their pipes, having driven their cattle down to some distant field. They were delighted to stop and chat with me even though the rain was teeming down; they hitched their coats up over one ear and started telling me stories. At the time I had just about enough Irish to follow them, but because of the rain I couldn’t note anything. They told me about the wicked Blakes,
and the strange prophecies of St Colm Cille whose holy well is on the seashore nearby, and then about the giant Conán, the buffoon and glutton of the old legends about Fionn Mac Cumhaill and his warrior band. They said that Conán had emptied out his pocket of winkle shells on the shore at the foot of the boreen we were on, and that it would be worth my while going to see the heap; they called it Toit Chonáin, toit being a dialect word for a helping of shellfish. So I walked down to the shore, not very hopeful of being able to find this heap of winkle-shells. And there at the end of the boreen was a hillock of smooth green sward, very conspicuous among the ragged, blackish, seaweedy rocks. Where rabbits had tunnelled into it, thousands of winkle-shells were pouring out. It was evidently a kitchen-midden, marking a place where some prehistoric people had gathered, presumably seasonally over a long period, to live off shore-food. There are many such shell-middens in the west of Connemara, exposed wherever dunes are being eroded; two of them have recently been dated to AD 400 and 700, and it is likely that some contain very much earlier deposits. But none of them is as striking as Toit Chonáin, which is twelve or fifteen feet high, and forty feet long. ‘Vaut le détour,’ as the Michelin Guide says; the drab shores of Ros a’ Mhíl have their gifts, even if they do seem to be hidden within envelope within envelope within envelope, like the priest’s curse.
The next day I returned to the same stretch of coast to search for the holy well the old men told me had been made by Colm Cille, who had sailed across from the Aran Islands on a stone boat. The boat is still there, a big boulder like the prow of a hooker, with the saint’s huge handprint and marks left by his anchor chain on its deck, and grooves worn into its rim by his oars. The holy well is a rockpool down near the low-water mark, only accessible when the tide is out. There was no one on the shore to direct me, and I spent a long time floundering about among rocks covered in seaweed before I found it. Once seen, it was unmistakable, a big smooth basin, almost circular and with a bowl-shaped bottom, like a baptismal font; in fact (in the sort of fact I was exploring that day, at any rate) St Colm Cille had made this and other similar wells for baptism of the local heathens. However, I was unsure that I was looking at the right rockpool until I put my arm down into it as far as I could reach, and felt something wedged in a crack in the rock – an old penny coin.
