Painting as a Pastime

by Winston Churchill

© Painting as a Pastime

Many remedies are suggested for the avoidance of worry and mental overstrain by persons who, over prolonged periods, have to bear exceptional responsibilities and discharge duties upon a very large scale. Some advise exercise, and others, repose. Some counsel travel, and others, retreat. Some praise solitude, and others, gaiety. No doubt all these may play their part according to the individual temperament. But the element which is constant and common in all of them is Change.

Change is the master key. A man can wear out a particular part of his mind by continually using it and tiring it, just in the same way as he can wear out the elbows of his coat. There is, however, this difference between the living cells of the brain and inanimate articles: one cannot mend the frayed elbows of a coat by rubbing the sleeves or shoulders; but the tired parts of the mind can be rested and strengthened, not merely by rest, but by using other parts. It is not enough merely to switch off the lights which play upon the main and ordinary field of interest; a new field of interest must be illuminated. It is no use saying to the tired ‘mental muscles’—if one may coin such an expression—I will give you a good rest,’ ‘I will go for a long walk,’ or ‘I will lie down and think of nothing.’ The mind keeps busy just the same. If it has been weighing and measuring, it goes on weighing and measuring. If it has been worrying, it goes on worrying. It is only when new cells are called into activity, when new stars become the lords of the ascendant, that relief, repose, refreshment are afforded.

A gifted American psychologist has said, ‘Worry is a spasm of the emotion; the mind catches hold of something and will not let it go.’ It is useless to argue with the mind in this condition. The stronger the will, the more futile the task. One can only gently insinuate something else into its convulsive grasp. And if this something else is rightly chosen, if it is really attended by the illumination of another field of interest, gradually, and often quite swiftly, the old undue grip relaxes and the process of recuperation and repair begins.

The cultivation of a hobby and new forms of interest is therefore a policy of first importance to a public man. But this is not a business that can be undertaken in a day or swiftly improvised by a mere command of the will. The growth of alternative mental interests is a long process. The seeds must be carefully chosen; they must fall on good ground; they must be sedulously tended, if the vivifying fruits are to be at hand when needed.

To be really happy and really safe, one ought to have at least two or three hobbies, and they must all be real. It is no use starting late in life to say: ‘I will take an interest in this or that.’ Such an attempt only aggravates the strain of mental effort. A man may acquire great knowledge of topics unconnected with his daily work, and yet hardly get any benefit or relief. It is no use doing what you like; you have got to like what you do. Broadly speaking, human beings may be divided into three classes: those who are toiled to death, those who are worried to death, and those who are bored to death. It is no use offering the manual labourer, tired out with a hard week’s sweat and effort, the chance of playing a game of football or baseball on Saturday afternoon. It is no use inviting the politician or the professional or business man, who has been working or worrying about serious things for six days, to work or worry about trifling things at the week-end.

As for the unfortunate people who can command everything they want, who can gratify every caprice and lay their hands on almost every object of desire—for them a new pleasure, a new excitement is only an additional satiation. In vain they rush frantically round from place to place, trying to escape from avenging boredom by mere clatter and motion. For them discipline in one form or another is the most hopeful path.

It may also be said that rational, industrious, useful human beings are divided into two classes: first, those whose work is work and whose pleasure is pleasure; and secondly, those whose work and pleasure are one. Of these the former are the majority. They have their compensations. The long hours in the office or the factory bring with them, as their reward, not only the means of sustenance, but a keen appetite for pleasure even in its simplest and most modest forms. But Fortune’s favoured children belong to the second class. Their life is a natural harmony. For them the working hours are never long enough. Each day is a holiday, and ordinary holidays when they come are grudged as enforced interruptions in an absorbing vocation. Yet to both classes the need of an alternative outlook, of a change of atmosphere, of a diversion of effort, is essential. Indeed, it may well be that those whose work is their pleasure are those who most need the means of banishing it at intervals from their minds.

The most common form of diversion is reading. In that vast and varied field millions find their mental comfort. Nothing makes a man more reverent than a library. ‘A few books,’ which was Lord Morley’s definition of anything under five thousand, may give a sense of comfort and even of complacency. But a day in a library, even of modest dimensions, quickly dispels these illusory sensations. As you browse about, taking down book after book from the shelves and contemplating the vast, infinitely varied store of knowledge and wisdom which the human race has accumulated and preserved, pride, even in its most innocent forms, is chased from the heart by feelings of awe not untinged with sadness. As one surveys the mighty array of sages, saints, historians, scientists, poets and philosophers whose treasures one will never be able to admire—still less enjoy—the brief tenure of our existence here dominates mind and spirit.

Think of all the wonderful tales that have been told, and well told, which you will never know. Think of all the searching inquiries into matters of great consequence which you will never pursue. Think of all the delighting or disturbing ideas that you will never share. Think of the mighty labours which have been accomplished for your service, but of which you will never reap the harvest. But from this melancholy there also comes a calm. The bitter sweets of a pious despair melt into an agreeable sense of compulsory resignation from which we turn with renewed zest to the lighter vanities of life.

‘What shall I do with all my books?’ was the question; and the answer, ‘Read them,’ sobered the questioner. But if you cannot read them, at any rate handle them and, as it were, fondle them. Peer into them. Let them fall open where they will. Read on from the first sentence that arrests the eye. Then turn to another. Make a voyage of discovery, taking soundings of uncharted seas. Set them back on their shelves with your own hands. Arrange them on your own plan, so that if you do not know what is in them, you at least know where they are. If they cannot be your friends, let them at any rate be your acquaintances. If they cannot enter the circle of your life, do not deny them at least a nod of recognition.

It is a mistake to read too many good books when quite young. A man once told me that he had read all the books that mattered. Cross-questioned, he appeared to have read a great many, but they seemed to have made only a slight impression. How many had he understood? How many had entered into his mental composition? How many had been hammered on the anvils of his mind, and afterwards ranged in an armoury of bright weapons ready to hand?

It is a great pity to read a book too soon in life. The first impression is the one that counts; and if it is a slight one, it may be all that can be hoped for. A later and second perusal may recoil from a surface already hardened by premature contact. Young people should be careful in their reading, as old people in eating their food. They should not eat too much. They should chew it well.

Since change is an essential element in diversion of all kinds, it is naturally more restful and refreshing to read in a different language from that in which one’s ordinary daily work is done. To have a second language at your disposal, even if you only know it enough to read it with pleasure, is a sensible advantage. Our educationists are too often anxious to teach children so many different languages that they never get far enough in any one to derive any use or enjoyment from their study. The boy learns enough Latin to detest it; enough Greek to pass an examination; enough French to get from Calais to Paris; enough German to exhibit a diploma; enough Spanish or Italian to tell which is which; but not enough of any to secure the enormous boon of access to a second literature. Choose well, choose wisely, and choose one. Concentrate upon that one. Do not be content until you find yourself reading in it with real enjoyment. The process of reading for pleasure in another language rests the mental muscles; it enlivens the mind by a different sequence and emphasis of ideas. The mere form of speech excites the activity of separate brain-cells, relieving in the most effective manner the fatigue of those in hackneyed use. One may imagine that a man who blew the trumpet for his living would be glad to play the violin for his amusement. So it is with reading in another language than your own.

But reading and book-love in all their forms suffer from one serious defect: they are too nearly akin to the ordinary daily round of the brain-worker to give that element of change and contrast essential to real relief. To restore psychic equilibrium we should call into use those parts of the mind which direct both eye and hand. Many men have found great advantage in practising a handicraft for pleasure. Joinery, chemistry, book-binding, even brick-laying—if one were interested in them and skilful at them—would give a real relief to the over-tired brain. But, best of all and easiest to procure are sketching and painting in all their forms. I consider myself very lucky that late in life I have been able to develop this new taste and pastime. Painting came to my rescue in a most trying time, and I shall venture in the pages that follow to express the gratitude I feel.

Painting is a companion with whom one may hope to walk a great part of life’s journey,

‘Age cannot wither her nor custom stale
Her infinite variety.’

One by one the more vigorous sports and exacting games fall away. Exceptional exertions are purchased only by a more pronounced and more prolonged fatigue. Muscles may relax, and feet and hands slow down; the nerve of youth and manhood may become less trusty. But painting is a friend who makes no undue demands, excites to no exhausting pursuits, keeps faithful pace even with feeble steps, and holds her canvas as a screen between us and the envious eyes of Time or the surly advance of Decrepitude.

Happy are the painters, for they shall not be lonely. Light and colour, peace and hope, will keep them company to the end, or almost to the end, of the day.

To have reached the age of forty without ever handling a brush or fiddling with a pencil, to have regarded with mature eye the painting of pictures of any kind as a mystery, to have stood agape before the chalk of the pavement artist, and then suddenly to find oneself plunged in the middle of a new and intense form of interest and action with paints and palettes and canvases, and not to be discouraged by results, is an astonishing and enriching experience. I hope it may be shared by others. I should be glad if these lines induced others to try the experiment which I have tried, and if some at least were to find themselves dowered with an absorbing new amusement delightful to themselves, and at any rate not violently harmful to man or beast.

I hope this is modest enough: because there is no subject on which I feel more humble or yet at the same time more natural. I do not presume to explain how to paint, but only how to get enjoyment. Do not turn the superior eye of critical passivity upon these efforts. Buy a paint-box and have a try. If you need something to occupy your leisure, to divert your mind from the daily round, to illuminate your holidays, do not be too ready to believe that you cannot find what you want here. Even at the advanced age of forty! It would be a sad pity to shuffle or scramble along through one’s playtime with golf and bridge, pottering, loitering, shifting from one heel to the other, wondering what on earth to do—as perhaps is the fate of some unhappy beings—when all the while, if you only knew, there is close at hand a wonderful new world of thought and craft, a sunlit garden gleaming with light and colour of which you have the key in your waistcoat-pocket. Inexpensive independence, a mobile and perennial pleasure apparatus, new mental food and exercise, the old harmonies and symmetries in an entirely different language, an added interest to every common scene, an occupation for every idle hour, an unceasing voyage of entrancing discovery—these are high prizes. Make quite sure they are not yours. After all, if you try, and fail, there is not much harm done. The nursery will grab what the studio has rejected. And then you can always go out and kill some animal, humiliate some rival on the links, or despoil some friend across the green table. You will not be worse off in any way. In fact you will be better off. You will know ‘beyond a peradventure,’ to quote a phrase disagreeably reminiscent, that that is really what you were meant to do in your hours of relaxation.

But if, on the contrary, you are inclined—late in life though it be—to reconnoitre a foreign sphere of limitless extent, then be persuaded that the first quality that is needed is Audacity. There really is no time for the deliberate approach. Two years of drawing-lessons, three years of copying woodcuts, five years of plaster casts—these are for the young. They have enough to bear. And this thorough grounding is for those who, hearing the call in the morning of their days, are able to make painting their paramount lifelong vocation. The truth and beauty of line and form which by the slightest touch or twist of the brush a real artist imparts to every feature of his design must be founded on long, hard, persevering apprenticeship and a practice so habitual that it has become instinctive. We must not be too ambitious. We cannot aspire to masterpieces. We may content ourselves with a joy ride in a paint-box. And for this Audacity is the only ticket.

I shall now relate my personal experience. When I left the Admiralty at the end of May, 1915, I still remained a member of the Cabinet and of the War Council. In this position I knew everything and could do nothing. The change from the intense executive activities of each day’s work at the Admiralty to the narrowly measured duties of a counsellor left me gasping. Like a sea-beast fished up from the depths, or a diver too suddenly hoisted, my veins threatened to burst from the fall in pressure. I had great anxiety and no means of relieving it; I had vehement convictions and small power to give effect to them. I had to watch the unhappy casting-away of great opportunities, and the feeble execution of plans which I had launched and in which I heartily believed. I had long hours of utterly unwonted leisure in which to contemplate the frightful unfolding of the War. At a moment when every fibre of my being was inflamed to action, I was forced to remain a spectator of the tragedy, placed cruelly in a front seat. And then it was that the Muse of Painting came to my rescue—out of charity and out of chivalry, because after all she had nothing to do with me—and said, ‘Are these toys any good to you? They amuse some people.’

Some experiments one Sunday in the country with the children’s paint-box led me to procure the next morning a complete outfit for painting in oils.

Having bought the colours, an easel, and a canvas, the next step was to begin. But what a step to take! The palette gleamed with beads of colour; fair and white rose the canvas; the empty brush hung poised, heavy with destiny, irresolute in the air. My hand seemed arrested by a silent veto. But after all the sky on this occasion was unquestionably blue, and a pale blue at that. There could be no doubt that blue paint mixed with white should be put on the top part of the canvas. One really does not need to have had an artist’s training to see that. It is a starting-point open to all. So very gingerly I mixed a little blue paint on the palette with a very small brush, and then with infinite precaution made a mark about as big as a bean upon the affronted snow-white shield. It was a challenge, a deliberate challenge; but so subdued, so halting, indeed so cataleptic, that it deserved no response. At that moment the loud approaching sound of a motor-car was heard in the drive. From this chariot there stepped swiftly and lightly none other than the gifted wife of Sir John Lavery. ‘Painting! But what are you hesitating about? Let me have a brush—the big one.’ Splash into the turpentine, wallop into the blue and the white, frantic flourish on the palette—clean no longer—and then several large, fierce strokes and slashes of blue on the absolutely cowering canvas. Anyone could see that it could not hit back. No evil fate avenged the jaunty violence. The canvas grinned in helplessness before me. The spell was broken. The sickly inhibitions rolled away. I seized the largest brush and fell upon my victim with Berserk fury. I have never felt any awe of a canvas since.

Everyone knows the feelings with which one stands shivering on a spring-board, the shock when a friendly foe steals up behind and hurls you into the flood, and the ardent glow which thrills you as you emerge breathless from the plunge.

This beginning with Audacity, or being thrown into the middle of it, is already a very great part of the art of painting. But there is more in it than that.

‘La peinture a l’huile
Est bien difficile,
Mais c’est beaucoup plus beau
Que la peinture a l’eau.’

I write no word in disparagement of water-colours. But there really is nothing like oils. You have a medium at your disposal which offers real power, if you only can find out how to use it. Moreover, it is easier to get a certain distance along the road by its means than by water-colour. First of all, you can correct mistakes much more easily. One sweep of the palette-knife ‘lifts’ the blood and tears of a morning from the canvas and enables a fresh start to be made; indeed the canvas is all the better for past impressions. Secondly, you can approach your problem from any direction. You need not build downwards awkwardly from white paper to your darkest dark. You may strike where you please, beginning if you will with a moderate central arrangement of middle tones, and then hurling in the extremes when the psychological moment comes. Lastly, the pigment itself is such nice stuff to handle (if it does not retaliate). You can build it on layer after layer if you like. You can keep on experimenting. You can change your plan to meet the exigencies of time or weather. And always remember you can scrape it all away.

Just to paint is great fun. The colours are lovely to look at and delicious to squeeze out. Matching them, however crudely, with what you see is fascinating and absolutely absorbing. Try it if you have not done so—before you die. As one slowly begins to escape from the difficulties of choosing the right colours and laying them on in the right places and in the right way, wider considerations come into view. One begins to see, for instance, that painting a picture is like fighting a battle; and trying to paint a picture is, I suppose, like trying to fight a battle. It is, if anything, more exciting than fighting it successfully. But the principle is the same. It is the same kind of problem as unfolding a long, sustained, interlocked argument. It is a proposition which, whether of few or numberless parts, is commanded by a single unity of conception. And we think—though I cannot tell—that painting a great picture must require an intellect on the grand scale. There must be that all-embracing view which presents the beginning and the end, the whole and each part, as one instantaneous impression retentively and untiringly held in the mind. When we look at the larger Turners—canvases yards wide and tall—and observe that they are all done in one piece and represent one single second of time, and that every innumerable detail, however small, however distant, however subordinate, is set forth naturally and in its true proportion and relation, without effort, without failure, we must feel in the presence of an intellectual manifestation the equal in quality and intensity of the finest achievements of warlike action, of forensic argument, or of scientific or philosophical adjudication.

In all battles two things are usually required of the Commander-in-Chief: to make a good plan for his army and, secondly, to keep a strong reserve. Both these are also obligatory upon the painter. To make a plan, thorough reconnaissance of the country where the battle is to be fought is needed. Its fields, its mountains, its rivers, its bridges, its trees, its flowers, its atmosphere—all require and repay attentive observation from a special point of view. One is quite astonished to find how many things there are in the landscape, and in every object in it, one never noticed before. And, this is a tremendous new pleasure and interest which invests every walk or drive with an added object. So many colours on the hillside, each different in shadow and in sunlight; such brilliant reflections in the pool, each a key lower than what they repeat; such lovely lights gilding or silvering surface or outline, all tinted exquisitely with pale colour, rose, orange, green or violet. I found myself instinctively as I walked noting the tint and character of a leaf, the dreamy, purple shades of mountains, the exquisite lacery of winter branches the dim, pale silhouettes of far horizons. And I had lived for over forty years without ever noticing any of them except in a general way, as one might look at a crowd and say, ‘What a lot of people!’

I think this heightened sense of observation of Nature is one of the chief delights that have come to me through trying to paint. No doubt many people who are lovers of art have acquired it in a high degree without actually practising. But I expect that nothing will make one observe more quickly or more thoroughly than having to face the difficulty of representing the thing observed. And mind you, if you do observe accurately and with refinement, and if you do record what you have seen with tolerable correspondence, the result follows on the canvas with startling obedience. Even if only four or five main features are seized and truly recorded, these by themselves will carry a lot of ill-success or half-success. Answer five big questions out of all the hundreds in the examination paper correctly and well, and though you may not win a prize, at any rate you will not be absolutely ploughed.

But in order to make his plan, the General must not only reconnoitre the battle-ground, he must also study the achievements of the great Captains of the past. He must bring the observations he has collected in the field into comparison with the treatment of similar incidents by famous chiefs. Then the galleries of Europe take on a new—and to me at least a severely practical—interest. ‘This, then, is how —— painted a cataract. Exactly, and there is that same light I noticed last week in the waterfall at ——.’ And so on. You see the difficulty that baffled you yesterday; and you see how easily it has been overcome by a great or even by a skilful painter. Not only is your observation of Nature sensibly improved and developed, but you look at the masterpieces of art with an analysing and a comprehending eye.

The whole world is open with all its treasures. The simplest objects have their beauty. Every garden presents innumerable fascinating problems. Every land, every parish, has its own tale to tell. And there are many lands differing from each other in countless ways, and each presenting delicious variants of colour, light, form, and definition. Obviously, then, armed with a paint-box, one cannot be bored, one cannot be left at a loose end, one cannot ‘have several days on one’s hands.’ Good gracious! what there is to admire and how little time there is to see it in! For the first time one begins to envy Methuselah. No doubt he made a very indifferent use of his opportunities.

But it is in the use and withholding of their reserves that the great Commanders have generally excelled. After all, when once the last reserve has been thrown in, the Commander’s part is played. If that does not win the battle, he has nothing else to give. The event must be left to luck and to the fighting troops. But these last, in the absence of high direction, are apt to get into sad confusion, all mixed together in a nasty mess, without order or plan—and consequently without effect. Mere masses count no more. The largest brush, the brightest colours, cannot even make an impression. The pictorial battlefield becomes a sea of mud mercifully veiled by the fog of war. It is evident there has been a serious defeat. Even though the General plunges in himself and emerges bespattered, as he sometimes does, he will not retrieve the day.

In painting, the reserves consist in Proportion or Relation. And it is here that the art of the painter marches along the road which is traversed by all the greatest harmonies in thought. At one side of the palette there is white, at the other black; and neither is ever used ‘neat.’ Between these two rigid limits all the action must lie, all the power required must be generated. Black and white themselves, placed in juxtaposition, make no great impression; and yet they are the most that you can do in pure contrast. It is wonderful—after one has tried and failed often—to see how easily and surely the true artist is able to produce every effect of light and shade, of sunshine and shadow, of distance or nearness, simply by expressing justly the relations between the different planes and surfaces with which he is dealing. We think that this is founded upon a sense of proportion, trained no doubt by practice, but which in its essence is a frigid manifestation of mental power and size. We think that the same mind’s eye that can justly survey and appraise and prescribe beforehand the values of a truly great picture in one all-embracing regard, in one flash of simultaneous and homogeneous comprehension, would also with a certain acquaintance with the special technique be able to pronounce with sureness upon any other high activity of the human intellect. This was certainly true of the great Italians.

I have written in this way to show how varied are the delights which may be gained by those who enter hopefully and thoughtfully upon the pathway of painting; how enriched they will be in their daily vision, how fortified in their independence, how happy in their leisure. Whether you feel that your soul is pleased by the conception or contemplation of harmonies, or that your mind is stimulated by the aspect of magnificent problems, or whether you are content to find fun in trying to observe and depict the jolly things you see, the vistas of possibility are limited only by the shortness of life. Every day you may make progress. Every step may be fruitful. Yet there will stretch out before you an ever-lengthening, ever-ascending, ever-improving path. You know you will never get to the end of the journey. But this, so far from discouraging, only adds to the joy and glory of the climb.

Try it, then, before it is too late and before you mock at me. Try it while there is time to overcome the preliminary difficulties. Learn enough of the language in your prime to open this new literature to your age. Plant a garden in which you can sit when digging days are done. It may be only a small garden, but you will see it grow. Year by year it will bloom and ripen. Year by year it will be better cultivated. The weeds will be cast out. The fruit-trees will be pruned and trained. The flowers will bloom in more beautiful combinations. There will be sunshine there even in the winter-time, and cool shade, and the play of shadow on the pathway in the shining days of June.

I must say I like bright colours. I agree with Ruskin in his denunciation of that school of painting who ‘eat slate-pencil and chalk, and assure everybody that they are nicer and purer than strawberries and plums.’ I cannot pretend to feel impartial about the colours. I rejoice with the brilliant ones, and am genuinely sorry for the poor browns. When I get to heaven I mean to spend a considerable portion of my first million years in painting, and so get to the bottom of the subject. But then I shall require a still gayer palette than I get here below. I expect orange and vermilion will be the darkest, dullest colours upon it, and beyond them there will be a whole range of wonderful new colours which will delight the celestial eye.

Chance led me one autumn to a secluded nook on the Côte d’Azur, between Marseilles and Toulon, and there I fell in with one or two painters who revelled in the methods of the modern French school. These were disciples of Cézanne. They view Nature as a mass of shimmering light in which forms and surfaces are comparatively unimportant, indeed hardly visible, but which gleams and glows with beautiful harmonies and contrasts of colour. Certainly it was of great interest to me to come suddenly in contact with this entirely different way of looking at things. I had hitherto painted the sea flat, with long, smooth strokes of mixed pigment in which the tints varied only by gradations. Now I must try to represent it by innumerable small separate lozenge-shaped points and patches of colour—often pure colour—so that it looked more like a tessellated pavement than a marine picture. It sounds curious. All the same, do not be in a hurry to reject the method. Go back a few yards and survey the result. Each of these little points of colour is now playing his part in the general effect. Individually invisible, he sets up a strong radiation, of which the eye is conscious without detecting the cause. Look also at the blue of the Mediterranean. How can you depict and record it? Certainly not by any single colour that was ever manufactured. The only way in which that luminous intensity of blue can be simulated is by this multitude of tiny points of varied colour all in true relation to the rest of the scheme. Difficult? Fascinating!

Nature presents itself to the eye through the agency of these individual points of light, each of which sets up the vibrations peculiar to its colour. The brilliancy of a picture must therefore depend partly upon the frequency with which these points are found on any given area of the canvas, and partly on their just relation to one another. Ruskin says in his Elements of Drawing, from which I have already quoted, ‘You will not, in Turner’s largest oil pictures, perhaps six or seven feet long by four or five high, find one spot of colour as large as a grain of wheat ungradated.’ But the gradations of Turner differ from those of the modern French school by being gently and almost imperceptibly evolved one from another instead of being bodily and even roughly separated; and the brush of Turner followed the form of the objects he depicted, while our French friends often seem to take a pride in directly opposing it. For instance, they would prefer to paint a sea with up and down strokes rather than with horizontal; or a tree-trunk from right to left rather than up and down. This, I expect, is due to falling in love with one’s theories, and making sacrifices of truth to them in order to demonstrate fidelity and admiration.

But surely we owe a debt to those who have so wonderfully vivified, brightened, and illuminated modern landscape painting. Have not Manet and Monet, Cézanne and Matisse, rendered to painting something of the same service which Keats and Shelley gave to poetry after the solemn and ceremonious literary perfections of the eighteenth century? They have brought back to the pictorial art a new draught of joie de vivre; and the beauty of their work is instinct with gaiety, and floats in sparkling air.

I do not expect these masters would particularly appreciate my defence, but I must avow an increasing attraction to their work. Lucid and exact expression is one of the characteristics of the French mind. The French language has been made the instrument of the admirable gift. Frenchmen talk and write just as well about painting as they have done about love, about war, about diplomacy, or cooking. Their terminology is precise and complete. They are therefore admirably equipped to be teachers in the theory of any of these arts. Their critical faculty is so powerfully developed that it is perhaps some restraint upon achievement. But it is a wonderful corrective to others as well as to themselves.

My French friend, for instance, after looking at some of my daubs, took me round the galleries of Paris, pausing here and there. Wherever he paused, I found myself before a picture which I particularly admired. He then explained that it was quite easy to tell, from the kind of things I had been trying to do, what were the doings I liked. Never having taken any interest in pictures till I tried to paint, I had no preconceived opinions. I just felt, for reasons I could not fathom, that I liked some much more than others. I was astonished that anyone else should, on the most cursory observation of my work, be able so surely to divine a taste which I had never consciously formed. My friend said that it is not a bad thing to know nothing at all about pictures, but to have a matured mind trained in other things and a new strong interest for painting. The elements are there from which a true taste in art can be formed with time and guidance, and there are no obstacles or imperfect conceptions in the way. I hope this is true. Certainly the last part is true.

Once you begin to study it, all Nature is equally interesting and equally charged with beauty. I was shown a picture by Cézanne of a blank wall of a house, which he had made instinct with the most delicate lights and colours. Now I often amuse myself when I am looking at a wall or a flat surface of any kind by trying to distinguish all the different colours and tints which can be discerned upon it, and considering whether these arise from reflections or from natural hue. You would be astonished the first time you tried this to see how many and what beautiful colours there are even in the most commonplace objects, and the more carefully and frequently you look the more variations do you perceive.

But these are no reasons for limiting oneself to the plainest and most ordinary objects and scenes. Mere prettiness of scene, to be sure, is not needed for a beautiful picture. In fact, artificially-made pretty places are very often a hindrance to a good picture. Nature will hardly stand a double process of beautification: one layer of idealism on top of another is too much of a good thing. But a vivid scene, a brilliant atmosphere, novel and charming lights, impressive contrasts, if they strike the eye all at once, arouse an interest and an ardour which will certainly be reflected in the work which you try to do, and will make it seem easier.

It would be interesting if some real authority investigated carefully the part which memory plays in painting. We look at the object with an intent regard, then at the palette, and thirdly at the canvas. The canvas receives a message dispatched usually a few seconds before from the natural object. But it has come through a post-office en route. It has been transmitted in code. It has been turned from light into paint. It reaches the canvas a cryptogram. Not until it has been placed in its correct relation to everything else that is on the canvas can it be deciphered, is its meaning apparent, is it translated once again from mere pigment into light. And the light this time is not of Nature but of Art. The whole of this considerable process is carried through on the wings or the wheels of memory. In most cases we think it is the wings—airy and quick like a butterfly from flower to flower. But all heavy traffic and all that has to go a long journey must travel on wheels.

In painting in the open air the sequence of actions is so rapid that the process of translation into and out of pigment may seem to be unconscious. But all the greatest landscapes have been painted indoors, and often long after the first impressions were gathered. In a dim cellar the Dutch or Italian master recreated the gleaming ice of a Netherlands carnival or the lustrous sunshine of Venice or the Campagna. Here, then, is required a formidable memory of the visual kind. Not only do we develop our powers of observation, but also those of carrying the record—of carrying it through an extraneous medium and of reproducing it, hours, days, or even months after the scene has vanished or the sunlight died.

I was told by a friend that when Whistler guided a school in Paris he made his pupils observe their model on the ground floor, and then run upstairs and paint their picture piece by piece on the floor above. As they became more proficient, he put their easels up a storey higher, till at last the elite were scampering with their decision up six flights into the attic—praying it would not evaporate on the way. This is, perhaps, only a tale. But it shows effectively of what enormous importance a trained, accurate, retentive memory must be to an artist; and conversely what a useful exercise painting may be for the development of an accurate and retentive memory.

There is no better exercise for the would-be artist than to study and devour a picture, and then, without looking at it again, to attempt the next day to reproduce it. Nothing can more exactly measure the progress both of observation and memory. It is still harder to compose out of many separate, well-retained impressions, aided though they be by sketches and colour notes, a new, complete conception. But this is the only way in which great landscapes have been painted—or can be painted. The size of the canvas alone precludes its being handled out of doors. The fleeting light imposes a rigid time-limit. The same light never returns. One cannot go back day after day without the picture getting stale. The painter must choose between a rapid impression, fresh and warm and living, but probably deserving only of a short life, and the cold, profound, intense effort of memory, knowledge, and will-power, prolonged perhaps for weeks, from which a masterpiece can alone result. It is much better not to fret too much about the latter. Leave to the masters of art trained by a lifetime of devotion the wonderful process of picture-building and picture-creation. Go out into the sunlight and be happy with what you see.

Painting is complete as a distraction. I know of nothing which, without exhausting the body, more entirely absorbs the mind. Whatever the worries of the hour or the threats of the future, once the picture has begun to flow along, there is no room for them in the mental screen. They pass out into shadow and darkness. All one’s mental light, such as it is, becomes concentrated on the task. Time stands respectfully aside, and it is only after many hesitations that luncheon knocks gruffly at the door. When I have had to stand up on parade, or even, I regret to say, in church, for half an hour at a time, I have always felt that the erect position is not natural to man, has only been painfully acquired, and is only with fatigue and difficulty maintained. But no one who is fond of painting finds the slightest inconvenience, as long as the interest holds, in standing to paint for three or four hours at a stretch.

Lastly, let me say a word on painting as a spur to travel. There is really nothing like it. Every day and all day is provided with its expedition and its occupation—cheap, attainable, innocent, absorbing, recuperative. The vain racket of the tourist gives place to the calm enjoyment of the philosopher, intensified by an enthralling sense of action and endeavour. Every country where the sun shines and every district in it, has a theme of its own. The lights, the atmosphere, the aspect, the spirit, are all different; but each has its native charm. Even if you are only a poor painter you can feel the influence of the scene, guiding your brush, selecting the tubes you squeeze on to the palette. Even if you cannot portray it as you see it, you feel it, you know it, and you admire it for ever. When people rush about Europe in the train from one glittering centre of work or pleasure to another, passing—at enormous expense—through a series of mammoth hotels and blatant carnivals, they little know what they are missing, and how cheaply priceless things can be obtained. The painter wanders and loiters contentedly from place to place, always on the look out for some brilliant butterfly of a picture which can be caught and set up and carried safely home.

Now I am learning to like painting even on dull days. But in my hot youth I demanded sunshine. Sir William Orpen advised me to visit Avignon on account of its wonderful light, and certainly there is no more delightful centre for a would-be painter’s activities: then Egypt, fierce and brilliant, presenting in infinite variety the single triplex theme of the Nile, the desert, and the sun; or Palestine, a land of rare beauty—the beauty of the turquoise and the opal—which well deserves the attention of some real artist, and has never been portrayed to the extent that is its due. And what of India? Who has ever interpreted its lurid splendours? But after all, if only the sun will shine, one does not need to go beyond one’s own country. There is nothing more intense than the burnished steel and gold of a Highland stream; and at the beginning and close of almost every day the Thames displays to the citizens of London glories and delights which one must travel far to rival.


Paragraphs 3 to 7 were selected as Lesson 46 Hobbies in New Concept English Book 4: Fluency in English by Louis George Alexander

Who, according to the author, are ‘Fortune’s favoured children’?


  • Winston Churchill as writer
  • Winston Churchill as painter
  • The Nobel Prize in Literature 1953 was awarded to Sir Winston Leonard Spencer Churchill “for his mastery of historical and biographical description as well as for brilliant oratory in defending exalted human values.” For the first time, a great statesman has received the Prize in Literature. Sir Winston Churchill is a recognized master of the English language, that wonderful and flexible instrument of human thought. Churchill’s political and literary achievements are of such magnitude that one is tempted to resort to portray him as a Caesar who also has the gift of Cicero’s pen. Never before has one of history’s leading figures been so close to us by virtue of such an outstanding combination.
  • Churchill also stood as the University of Bristol’s longest-standing Chancellor from 1929 to 1965, during which time he also led Britain during the Second World War.
  • Winston Churchill was a racist. He said that he hated people with “slit eyes and pig tails.” To him, people from India were “the beastliest people in the world next to the Germans.” He admitted that he “did not really think that black people were as capable or as efficient as white people.” He said why be apologetic about Anglo-Saxon superiority, that we were superior, that we had the common heritage which had been worked out over the centuries in England and had been perfected by our constitution.

LeetCode - Algorithms - 906. Super Palindromes

fair and square (palindromes whose square root is a palindrome) numbers

Problem

906. Super Palindromes

Java

1 to 18 digits

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class Solution {
public int superpalindromesInRange(String L, String R) {
long[] superPalindromes = {1, 4, 9, 121, 484, 10201, 12321, 14641, 40804, 44944, 1002001, 1234321, 4008004, 100020001, 102030201, 104060401, 121242121, 123454321, 125686521, 400080004, 404090404, 10000200001L, 10221412201L, 12102420121L, 12345654321L, 40000800004L, 1000002000001L, 1002003002001L, 1004006004001L, 1020304030201L, 1022325232201L, 1024348434201L, 1210024200121L, 1212225222121L, 1214428244121L, 1232346432321L, 1234567654321L, 4000008000004L,4004009004004L, 100000020000001L, 100220141022001L, 102012040210201L, 102234363432201L, 121000242000121L, 121242363242121L, 123212464212321L, 123456787654321L, 400000080000004L, 10000000200000001L, 10002000300020001L, 10004000600040001L, 10020210401202001L, 10022212521222001L, 10024214841242001L, 10201020402010201L, 10203040504030201L, 10205060806050201L, 10221432623412201L, 10223454745432201L, 12100002420000121L, 12102202520220121L, 12104402820440121L, 12122232623222121L, 12124434743442121L, 12321024642012321L, 12323244744232321L, 12343456865434321L, 12345678987654321L, 40000000800000004L, 40004000900040004L};
int n = 0;
long lo = Long.parseLong(L);
long hi = Long.parseLong(R);
for (int i = 0; i < superPalindromes.length; i++) {
if (superPalindromes[i]>=lo && superPalindromes[i]<=hi)
n++;
}
return n;
}
}

Submission Detail

  • 48 / 48 test cases passed.
  • Runtime: 0 ms, faster than 100.00% of Java online submissions for Super Palindromes.
  • Memory Usage: 36.8 MB, less than 97.83% of Java online submissions for Super Palindromes.

References

COVID-19 Outlooks

A reverse poem by Britt MacKinnon

I have no hope or control.

Nobody can convince me that

I still have a future.

I recognize that

I am safe and loved

But

I am overwhelmed by fear

This situation dictates my daily well-being. I refuse to believe that

There is a bright future ahead.

Our world is disrupted.

No longer do I feel that

We have support and help from our leaders. During self-isolation.

I am reminiscing and dreaming.

Now I cherish the good old days.

My way of life

Changing

Because of COVID-19

(Now read it from bottom to top)


Britt MacKinnon graduated from chemical engineering at Queen’s University in 2019.


LeetCode - Algorithms - 633. Sum of Square Numbers

Problem

633. Sum of Square Numbers

Java

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class Solution {
public boolean judgeSquareSum(int c) {
if (c == 0 || c == 1)
return true;
int root = mySqrt(c);
for (int i = root; i >= 0; i--) {
int re = c - i * i;
if (isPerfectSquare(re)) {
return true;
}
}
return false;
}

private int mySqrt(int x) {
if (x == 0 || x == 1)
return x;

long start = 1, end = x, ans = 0;
while (start <= end) {
long mid = (start + end) / 2;

if (mid * mid == x)
return new Long(mid).intValue();

if (mid * mid < x) {
start = mid + 1;
ans = mid;
} else
end = mid - 1;
}
return new Long(ans).intValue();
}

private boolean isPerfectSquare(int num) {
long lo = 1, hi = num;
while (lo <= hi) {
long mid = (lo + hi) / 2;
if (mid * mid == num)
return true;
if (mid * mid > num) {
hi = mid - 1;
} else
lo = mid + 1;
}
return false;
}
}

Submission Detail

  • 124 / 124 test cases passed.
  • Runtime: 242 ms, faster than 5.54% of Java online submissions for Sum of Square Numbers.
  • Memory Usage: 37.6 MB, less than 14.48% of Java online submissions for Sum of Square Numbers.

2

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class Solution {
public boolean judgeSquareSum(int c) {
if (c == 0 || c == 1)
return true;
for (int i = 0; i <= c/2; i++) {
int re = c - i * i;
if (re < 0)
break;
if (isPerfectSquare(re)) {
return true;
}
}
return false;
}

private boolean isPerfectSquare(int num) {
long lo = 1, hi = num;
while (lo <= hi) {
long mid = (lo + hi) / 2;
if (mid * mid == num)
return true;
if (mid * mid > num) {
hi = mid - 1;
} else
lo = mid + 1;
}
return false;
}
}

Submission Detail

  • 124 / 124 test cases passed.
  • Runtime: 206 ms, faster than 7.18% of Java online submissions for Sum of Square Numbers.
  • Memory Usage: 35.5 MB, less than 93.83% of Java online submissions for Sum of Square Numbers.

number theory

An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no term \( p^k \), where prime \( p \equiv 3{\pmod {4}} \) and k is odd.

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class Solution {
public boolean judgeSquareSum(int c) {
if (c == 0 || c == 1)
return true;

while (c % 2 == 0) {
c = c >> 1;
}

for (int p = 3; p * p <= c; p += 2) {
int k = 0;
if (c % p == 0) {
while (c % p == 0) {
k++;
c /= p;
}
if (p % 4 == 3 && (k & 1) == 1)
return false;
}
}
return c % 4 != 3;
}
}

Submission Detail

  • 124 / 124 test cases passed.
  • Runtime: 0 ms, faster than 100.00% of Java online submissions for Sum of Square Numbers.
  • Memory Usage: 35.7 MB, less than 74.91% of Java online submissions for Sum of Square Numbers.

References

LeetCode - Algorithms - 367. Valid Perfect Square

Problem

367. Valid Perfect Square

Java

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class Solution {
public boolean isPerfectSquare(int num) {
long lo = 1, hi = num;
while (lo <= hi) {
long mid = (lo + hi) / 2;
if (mid * mid == num)
return true;
if (mid * mid > num) {
hi = mid - 1;
} else
lo = mid + 1;
}
return false;
}
}

Submission Detail

  • 70 / 70 test cases passed.
  • Runtime: 0 ms, faster than 100.00% of Java online submissions for Valid Perfect Square.
  • Memory Usage: 35.8 MB, less than 46.67% of Java online submissions for Valid Perfect Square.

The Final Speech from The Great Dictator

Copyright © Roy Export S.A.S. All rights reserved

I’m sorry, but I don’t want to be an emperor. That’s not my business. I don’t want to rule or conquer anyone. I should like to help everyone - if possible - Jew, Gentile - black man - white. We all want to help one another. Human beings are like that. We want to live by each other’s happiness - not by each other’s misery. We don’t want to hate and despise one another. In this world there is room for everyone. And the good earth is rich and can provide for everyone. The way of life can be free and beautiful, but we have lost the way.

Greed has poisoned men’s souls, has barricaded the world with hate, has goose-stepped us into misery and bloodshed. We have developed speed, but we have shut ourselves in. Machinery that gives abundance has left us in want. Our knowledge has made us cynical. Our cleverness, hard and unkind. We think too much and feel too little. More than machinery we need humanity. More than cleverness we need kindness and gentleness. Without these qualities, life will be violent and all will be lost….

The aeroplane and the radio have brought us closer together. The very nature of these inventions cries out for the goodness in men - cries out for universal brotherhood - for the unity of us all. Even now my voice is reaching millions throughout the world - millions of despairing men, women, and little children - victims of a system that makes men torture and imprison innocent people.

To those who can hear me, I say - do not despair. The misery that is now upon us is but the passing of greed - the bitterness of men who fear the way of human progress. The hate of men will pass, and dictators die, and the power they took from the people will return to the people. And so long as men die, liberty will never perish. …..

Soldiers! don’t give yourselves to brutes - men who despise you - enslave you - who regiment your lives - tell you what to do - what to think and what to feel! Who drill you - diet you - treat you like cattle, use you as cannon fodder. Don’t give yourselves to these unnatural men - machine men with machine minds and machine hearts! You are not machines! You are not cattle! You are men! You have the love of humanity in your hearts! You don’t hate! Only the unloved hate - the unloved and the unnatural! Soldiers! Don’t fight for slavery! Fight for liberty!

In the 17th Chapter of St Luke it is written: “the Kingdom of God is within man” - not one man nor a group of men, but in all men! In you! You, the people have the power - the power to create machines. The power to create happiness! You, the people, have the power to make this life free and beautiful, to make this life a wonderful adventure.

Then - in the name of democracy - let us use that power - let us all unite. Let us fight for a new world - a decent world that will give men a chance to work - that will give youth a future and old age a security. By the promise of these things, brutes have risen to power. But they lie! They do not fulfil that promise. They never will!

Dictators free themselves but they enslave the people! Now let us fight to fulfil that promise! Let us fight to free the world - to do away with national barriers - to do away with greed, with hate and intolerance. Let us fight for a world of reason, a world where science and progress will lead to all men’s happiness. Soldiers! in the name of democracy, let us all unite!


LeetCode - Algorithms - 1491. Average Salary Excluding the Minimum and Maximum Salary

Maybe this is the easiest problem on leetcode.

Problem

1491. Average Salary Excluding the Minimum and Maximum Salary

Java

\( O(N) \)

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class Solution {
public double average(int[] salary) {
final int N = salary.length;
int min=salary[0],max=salary[0];
double sum = salary[0];
for (int i = 1; i < N; i++) {
if (salary[i]<min) min=salary[i];
if (salary[i]>max) max=salary[i];
sum += salary[i];
}
sum = sum - max - min;
return sum / (N - 2);
}
}

Submission Detail

  • 43 / 43 test cases passed.
  • Runtime: 0 ms, faster than 100.00% of Java online submissions for Average Salary Excluding the Minimum and Maximum Salary.
  • Memory Usage: 37.1 MB, less than 11.62% of Java online submissions for Average Salary Excluding the Minimum and Maximum Salary.

\( O(NlogN) \)

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class Solution {
public double average(int[] salary) {
final int N = salary.length;
Arrays.sort(salary);
double t = salary[1];
for (int i = 2; i < N - 1; i++) {
t += salary[i];
}
return t / (N - 2);
}
}

Submission Detail

  • 43 / 43 test cases passed.
  • Runtime: 0 ms, faster than 100.00% of Java online submissions for Average Salary Excluding the Minimum and Maximum Salary.
  • Memory Usage: 36.7 MB, less than 75.02% of Java online submissions for Average Salary Excluding the Minimum and Maximum Salary.

Love is Not All

by Edna St. Vincent Millay

Love is not all: it is not meat nor drink
Nor slumber nor a roof against the rain;
Nor yet a floating spar to men that sink
And rise and sink and rise and sink again;
Love can not fill the thickened lung with breath,
Nor clean the blood, nor set the fractured bone;
Yet many a man is making friends with death
Even as I speak, for lack of love alone.
It well may be that in a difficult hour,
Pinned down by pain and moaning for release,
Or nagged by want past resolution’s power,
I might be driven to sell your love for peace,
Or trade the memory of this night for food.
It well may be. I do not think I would.


Sonnet 97

by William Shakespeare

How like a winter hath my absence been
From thee, the pleasure of the fleeting year!
What freezings have I felt, what dark days seen!
What old December’s bareness every where!
And yet this time remov’d was summer’s time;
The teeming autumn, big with rich increase,
Bearing the wanton burthen of the prime,
Like widowed wombs after their lords’ decease:
Yet this abundant issue seem’d to me
But hope of orphans and unfather’d fruit;
For summer and his pleasures wait on thee,
And, thou away, the very birds are mute;
Or, if they sing, ’tis with so dull a cheer
That leaves look pale, dreading the winter’s near.


Fractals and the art of roughness - Benoit Mandelbrot - TED2010 - Transcript

Thank you very much. Please excuse me for sitting; I’m very old. (Laughter) Well, the topic I’m going to discuss is one which is, in a certain sense, very peculiar because it’s very old. Roughness is part of human life forever and forever, and ancient authors have written about it. It was very much uncontrollable, and in a certain sense, it seemed to be the extreme of complexity, just a mess, a mess and a mess. There are many different kinds of mess. Now, in fact, by a complete fluke, I got involved many years ago in a study of this form of complexity, and to my utter amazement, I found traces – very strong traces, I must say – of order in that roughness. And so today, I would like to present to you a few examples of what this represents. I prefer the word roughness to the word irregularity because irregularity – to someone who had Latin in my long-past youth – means the contrary of regularity. But it is not so. Regularity is the contrary of roughness because the basic aspect of the world is very rough.

So let me show you a few objects. Some of them are artificial. Others of them are very real, in a certain sense. Now this is the real. It’s a cauliflower. Now why do I show a cauliflower, a very ordinary and ancient vegetable? Because old and ancient as it may be, it’s very complicated and it’s very simple, both at the same time. If you try to weigh it – of course it’s very easy to weigh it, and when you eat it, the weight matters – but suppose you try to measure its surface. Well, it’s very interesting. If you cut, with a sharp knife, one of the florets of a cauliflower and look at it separately, you think of a whole cauliflower, but smaller. And then you cut again, again, again, again, again, again, again, again, again, and you still get small cauliflowers. So the experience of humanity has always been that there are some shapes which have this peculiar property, that each part is like the whole, but smaller. Now, what did humanity do with that? Very, very little. (Laughter)

So what I did actually is to study this problem, and I found something quite surprising. That one can measure roughness by a number, a number, 2.3, 1.2 and sometimes much more. One day, a friend of mine, to bug me, brought a picture and said, “What is the roughness of this curve?” I said, “Well, just short of 1.5.” It was 1.48. Now, it didn’t take me any time. I’ve been looking at these things for so long. So these numbers are the numbers which denote the roughness of these surfaces. I hasten to say that these surfaces are completely artificial. They were done on a computer, and the only input is a number, and that number is roughness. So on the left, I took the roughness copied from many landscapes. To the right, I took a higher roughness. So the eye, after a while, can distinguish these two very well.

Humanity had to learn about measuring roughness. This is very rough, and this is sort of smooth, and this perfectly smooth. Very few things are very smooth. So then if you try to ask questions: “What’s the surface of a cauliflower?” Well, you measure and measure and measure. Each time you’re closer, it gets bigger, down to very, very small distances. What’s the length of the coastline of these lakes? The closer you measure, the longer it is. The concept of length of coastline, which seems to be so natural because it’s given in many cases, is, in fact, complete fallacy; there’s no such thing. You must do it differently.

What good is that, to know these things? Well, surprisingly enough, it’s good in many ways. To begin with, artificial landscapes, which I invented sort of, are used in cinema all the time. We see mountains in the distance. They may be mountains, but they may be just formulae, just cranked on. Now it’s very easy to do. It used to be very time-consuming, but now it’s nothing. Now look at that. That’s a real lung. Now a lung is something very strange. If you take this thing, you know very well it weighs very little. The volume of a lung is very small, but what about the area of the lung? Anatomists were arguing very much about that. Some say that a normal male’s lung has an area of the inside of a basketball &#91;court&#93;. And the others say, no, five basketball &#91;courts&#93;. Enormous disagreements. Why so? Because, in fact, the area of the lung is something very ill-defined. The bronchi branch, branch, branch and they stop branching, not because of any matter of principle, but because of physical considerations: the mucus, which is in the lung. So what happens is that in a way you have a much bigger lung, but it branches and branches down to distances about the same for a whale, for a man and for a little rodent.

Now, what good is it to have that? Well, surprisingly enough, amazingly enough, the anatomists had a very poor idea of the structure of the lung until very recently. And I think that my mathematics, surprisingly enough, has been of great help to the surgeons studying lung illnesses and also kidney illnesses, all these branching systems, for which there was no geometry. So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry. And a surprising aspect of it is that very often, the rules of this geometry are extremely short. You have formulas that long. And you crank it several times. Sometimes repeatedly: again, again, again, the same repetition. And at the end, you get things like that.

This cloud is completely, 100 percent artificial. Well, 99.9. And the only part which is natural is a number, the roughness of the cloud, which is taken from nature. Something so complicated like a cloud, so unstable, so varying, should have a simple rule behind it. Now this simple rule is not an explanation of clouds. The seer of clouds had to take account of it. I don’t know how much advanced these pictures are. They’re old. I was very much involved in it, but then turned my attention to other phenomena.

Now, here is another thing which is rather interesting. One of the shattering events in the history of mathematics, which is not appreciated by many people, occurred about 130 years ago, 145 years ago. Mathematicians began to create shapes that didn’t exist. Mathematicians got into self-praise to an extent which was absolutely amazing, that man can invent things that nature did not know. In particular, it could invent things like a curve which fills the plane. A curve’s a curve, a plane’s a plane, and the two won’t mix. Well, they do mix. A man named Peano did define such curves, and it became an object of extraordinary interest. It was very important, but mostly interesting because a kind of break, a separation between the mathematics coming from reality, on the one hand, and new mathematics coming from pure man’s mind. Well, I was very sorry to point out that the pure man’s mind has, in fact, seen at long last what had been seen for a long time. And so here I introduce something, the set of rivers of a plane-filling curve. And well, it’s a story unto itself. So it was in 1875 to 1925, an extraordinary period in which mathematics prepared itself to break out from the world. And the objects which were used as examples, when I was a child and a student, as examples of the break between mathematics and visible reality – those objects, I turned them completely around. I used them for describing some of the aspects of the complexity of nature.

Well, a man named Hausdorff in 1919 introduced a number which was just a mathematical joke, and I found that this number1 was a good measurement of roughness. When I first told it to my friends in mathematics they said, “Don’t be silly. It’s just something &#91;silly&#93;.” Well actually, I was not silly. The great painter Hokusai knew it very well. The things on the ground are algae. He did not know the mathematics; it didn’t yet exist. And he was Japanese who had no contact with the West. But painting for a long time had a fractal side. I could speak of that for a long time. The Eiffel Tower has a fractal aspect. I read the book that Mr. Eiffel wrote about his tower, and indeed it was astonishing how much he understood.

This is a mess, mess, mess, Brownian loop. One day I decided – halfway through my career, I was held by so many things in my work – I decided to test myself. Could I just look at something which everybody had been looking at for a long time and find something dramatically new? Well, so I looked at these things called Brownian motion – just goes around. I played with it for a while, and I made it return to the origin. Then I was telling my assistant, “I don’t see anything. Can you paint it?” So he painted it, which means he put inside everything. He said: “Well, this thing came out …” And I said, “Stop! Stop! Stop! I see; it’s an island.” And amazing. So Brownian motion, which happens to have a roughness number of two, goes around. I measured it, 1.33. Again, again, again. Long measurements, big Brownian motions, 1.33. Mathematical problem: how to prove it? It took my friends 20 years. Three of them were having incomplete proofs. They got together, and together they had the proof. So they got the big &#91;Fields&#93; medal in mathematics, one of the three medals that people have received for proving things which I’ve seen without being able to prove them.2

Now everybody asks me at one point or another, “How did it all start? What got you in that strange business?” What got you to be, at the same time, a mechanical engineer, a geographer and a mathematician and so on, a physicist? Well actually I started, oddly enough, studying stock market prices. And so here I had this theory, and I wrote books about it – financial prices increments. To the left you see data over a long period. To the right, on top, you see a theory which is very, very fashionable. It was very easy, and you can write many books very fast about it. (Laughter) There are thousands of books on that. Now compare that with real price increments. Where are real price increments? Well, these other lines include some real price increments and some forgery which I did. So the idea there was that one must be able to – how do you say? – model price variation. And it went really well 50 years ago. For 50 years, people were sort of pooh-poohing me because they could do it much, much easier. But I tell you, at this point, people listened to me. (Laughter) These two curves are averages: Standard & Poor, the blue one; and the red one is Standard & Poor’s from which the five biggest discontinuities are taken out. Now discontinuities are a nuisance, so in many studies of prices, one puts them aside. “Well, acts of God. And you have the little nonsense which is left. Acts of God.” In this picture, five acts of God are as important as everything else. In other words, it is not acts of God that we should put aside. That is the meat, the problem. If you master these, you master price, and if you don’t master these, you can master the little noise as well as you can, but it’s not important. Well, here are the curves for it.

Now, I get to the final thing, which is the set of which my name is attached. In a way, it’s the story of my life. My adolescence was spent during the German occupation of France. Since I thought that I might vanish within a day or a week, I had very big dreams. And after the war, I saw an uncle again. My uncle was a very prominent mathematician, and he told me, “Look, there’s a problem which I could not solve 25 years ago, and which nobody can solve. This is a construction of a man named &#91;Gaston&#93; Julia and &#91;Pierre&#93; Fatou. If you could find something new, anything, you will get your career made.” Very simple. So I looked, and like the thousands of people that had tried before, I found nothing.

But then the computer came, and I decided to apply the computer, not to new problems in mathematics – like this wiggle wiggle, that’s a new problem – but to old problems. And I went from what’s called real numbers, which are points on a line, to imaginary, complex numbers, which are points on a plane, which is what one should do there, and this shape came out. This shape is of an extraordinary complication. The equation is hidden there, z goes into z squared, plus c. It’s so simple, so dry. It’s so uninteresting. Now you turn the crank once, twice: twice, marvels come out. I mean this comes out. I don’t want to explain these things. This comes out. This comes out. Shapes which are of such complication, such harmony and such beauty. This comes out repeatedly, again, again, again. And that was one of my major discoveries, to find that these islands were the same as the whole big thing, more or less. And then you get these extraordinary baroque decorations all over the place. All that from this little formula, which has whatever, five symbols in it. And then this one. The color was added for two reasons. First of all, because these shapes are so complicated that one couldn’t make any sense of the numbers. And if you plot them, you must choose some system. And so my principle has been to always present the shapes with different colorings because some colorings emphasize that, and others it is that or that. It’s so complicated.

(Laughter)

In 1990, I was in Cambridge, U.K. to receive a prize from the university, and three days later, a pilot was flying over the landscape and found this thing. So where did this come from? Obviously, from extraterrestrials. (Laughter) Well, so the newspaper in Cambridge published an article about that “discovery” and received the next day 5,000 letters from people saying, “But that’s simply a Mandelbrot set very big.”

Well, let me finish. This shape here just came out of an exercise in pure mathematics. Bottomless wonders spring from simple rules, which are repeated without end.

Thank you very much.

(Applause)


  1. Hausdorff dimension: In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff.
Name exact value approx.
Lung surface 2.97
Cauliflower Measured and calculated ~2.8
Koch curve \( \log_3 4 \) 1.2619
Penrose tiling 2
Julia set 2
Feigenbaum attractor 0.538
Lorenz attractor Measured 2.06 ±0.01
  1. In 1982, Benoit Mandelbrot conjectured that the fractal dimension of the outer boundary of the trajectory of a Brownian path is 4/3. Resolving this conjecture seemed out of reach of classical probabilistic techniques. Lawler, Schramm, and Werner proved this conjecture first by showing that the outer frontier of Brownian paths and the outer boundaries of the continuous percolation clusters are similar, and then by computing their common dimension using a dynamical construction of the continuous percolation clusters. Using the same strategy, they also derived the values of the closely related “intersection exponents” for Brownian motion and simple random walks that had been conjectured by physicists B. Duplantier and K. H. Kwon (one of these intersection exponents describes the probability that the paths of two long walkers remain disjoint up to some very large time). Further work of Werner exhibited additional symmetries of these outer boundaries of Brownian loops.