Convex Quotes

And so I went through the looking glass, stepped into the netherworld, where up is down and food is greed, where convex mirrors cover the walls, where death is honor and flesh is weak. It is ever so easy to go. Harder to find your way back.

The summer demands and takes away too much. /But night, the reserved, the reticent, gives more than it takes

I am the daughter of Earth and Water, And the nursling of the Sky; I pass through the pores of the ocean and shores; I change, but I cannot die. For after the rain when with never a stain The pavilion of Heaven is bare, And the winds and sunbeams with their convex gleams Build up the blue dome of air, I silently laugh at my own cenotaph, And out of the caverns of rain, Like a child from the womb, like a ghost from the tomb, I arise and unbuild it again.

Things can harden meaningfully in the moment of indecision

I tried each thing, only some were immortal and free.

Learning is rooted in repetition and convexity, meaning that the reading of a single text twice is more profitable than reading two different things once.

And so i went through the looking glass, stepped into the nether world, where up is down and food is greed, where convex mirrors cover the walls where death is honour and flesh is weak. It is ever so easy to go. Harder to find your way back.

Its a bit mad. Too bad, I mean, that getting to know each just for a fleeting second Must be replaced by unperfect knowledge of the featureless whole Like some pocket history of the world, so general As to constitute a sob or wail

The beauty of quantum machine learning is that we do not need to depend on an algorithm like gradient descent or convex objective function. The objective function can be nonconvex or something else.

And the same things look bent and straight when seen in water and out of it, and also both concave and convex, due to the sight’s being mislead by the colors, and every sort of confusion of this kind is plainly in our soul. And, then, it is because they take advantage of this affection in our nature that shadow painting, and puppeteering, and many other tricks of the kind fall nothing short of wizardry.

On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realised that this movement was an illusion created by the dizzying world it bounded. The Aleph's diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror's face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; I saw a splintered labyrinth (it was London); I saw, close up, unending eyes watching themselves in me as in a mirror; I saw all the mirrors on earth and none of them reflected me; I saw in a backyard of Soler Street the same tiles that thirty years before I'd seen in the entrance of a house in Fray Bentos; I saw bunches of grapes, snow, tobacco, lodes of metal, steam; I saw convex equatorial deserts and each one of their grains of sand; I saw a woman in Inverness whom I shall never forget; I saw her tangled hair, her tall figure, I saw the cancer in her breast; I saw a ring of baked mud in a sidewalk, where before there had been a tree; I saw a summer house in Adrogué and a copy of the first English translation of Pliny -- Philemon Holland's -- and all at the same time saw each letter on each page (as a boy, I used to marvel that the letters in a closed book did not get scrambled and lost overnight); I saw a sunset in Querétaro that seemed to reflect the colour of a rose in Bengal; I saw my empty bedroom; I saw in a closet in Alkmaar a terrestrial globe between two mirrors that multiplied it endlessly; I saw horses with flowing manes on a shore of the Caspian Sea at dawn; I saw the delicate bone structure of a hand; I saw the survivors of a battle sending out picture postcards; I saw in a showcase in Mirzapur a pack of Spanish playing cards; I saw the slanting shadows of ferns on a greenhouse floor; I saw tigers, pistons, bison, tides, and armies; I saw all the ants on the planet; I saw a Persian astrolabe; I saw in the drawer of a writing table (and the handwriting made me tremble) unbelievable, obscene, detailed letters, which Beatriz had written to Carlos Argentino; I saw a monument I worshipped in the Chacarita cemetery; I saw the rotted dust and bones that had once deliciously been Beatriz Viterbo; I saw the circulation of my own dark blood; I saw the coupling of love and the modification of death; I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon -- the unimaginable universe. I felt infinite wonder, infinite pity.

I also think pronunciation of a foreign tongue could be better taught than by demanding from the pupil those internal acrobatic feats that are generally impossible and always useless. This is the sort of instruction one receives: 'Press your tonsils against the underside of your larynx. Then with the convex part of the septum curved upwards so as almost but not quite to touch the uvula try with the tip of your tongue to reach your thyroid. Take a deep breath and compress your glottis. Now without opening your lips say "Garoo".' And when you have done it they are not satisfied.

Some day we will try To do as many things as are possible And perhaps we shall succeed at a handful Of them, but this will not have anything To do with what is promised today, our Landscape sweeping out from us to disappear On the horizon.

...as he finally sheds his body's suit, ------ finds his gut and throat again and newly whole, clean and unimpeded, free, catapulted home over fans and the Convexity's glass palisades at desperate speeds, soaring north, sounding a bell-clear and nearly maternal alarmed call-to-arms in all the world's well-known tongues.

Concave and convex, we fit together like two praying hands. A prayer that begged for absolution and time. We were both sinners and the truth of that hurt. But having her in my arms, her mouth on mine, I couldn’t understand how I’d ever be forgiven. And I didn’t want to be.

He passes, struck by the stare of truculent Wellington but in the convex mirror grin unstruck the bonham eyes and fatchuck cheekchops of Jollypoldy the rixdix doldy.

The very idea of exercise is to gain from antifragility to workout stressors—as we saw, all kinds of exercise are just exploitations of convexity effects.

On turning to the Work in Progress we find that the mirror is not so convex. Here is direct expression--pages and pages of it. And if you don’t understand it, Ladies and Gentlemen, it is because you are too decadent to receive it. You are not satisfied unless form is so strictly divorced from content that you can comprehend the one almost without bothering to read the other. This rapid skimming and absorption of the scant cream of sense is made possible by what I may call a continuous process of copious intellectual salivation. The form that is an arbitrary and independent phenomenon can fulfil no higher function than that of stimulus for a tertiary or quartary conditioned reflex of dribbling comprehension. . . Mr. Joyce has a word to say to you on the subject: “Yet to concentrate solely on the literal sense or even the psychological content of any document to the sore neglect of the enveloping facts themselves circumstantiating it is just as harmful; etc.” And another: “Who in his hearts doubts either that the facts of feminine clothiering are there all the time or that the feminine fiction, stranger than facts, is there also at the same time, only a little to the rere? Or that one may be separated from the orther? Or that both may be contemplated simultaneously? Or that each may be taken up in turn and considered apart from the other?” Here form is content, content is form. You complain that this stuff is not written in English. It is not written at all. It is not to be read--or rather it is not only to be read. It is to be looked at and listened to. His writing is not about something; it is that something itself.

What if heredity, instead of linear, is branching? What if it's not arousal that's so finitely circumscribed? What if in fact there were only like two really distinct individual people walking back there in history's mist? That all difference descends from this difference? The whole and the partial. The damaged and the intact. The deformed and the paralyzingly beautiful. The insane and the attendant. The hidden and the blindingly open. The performer and the audience. No Zen-type one, always rather two, one upside down in a convex lens.

You bad birds, But God shall not punish you, you Shall be with us in heaven, though less Conscious of your happiness, perhaps, than we. Hell is a not quite satisfactory heaven, probably, But you are the fruit and jewels Of my arrangement . . .

He approaches a cockroach in only one respect: his coloration is brown. That is all. Apart from this he has a tremendous convex belly divided into segments and a hard rounded back suggestive of wing cases. In beetles these cases conceal flimsy little wings that can be expanded and then may carry the beetle for miles and miles in a blundering flight. Curiously enough, Gregor the beetle never found out that he had wings under the hard covering of his back. (This is a very nice observation on my part to be treasured all your lives. Some Gregors, some Joes and Janes, do not know that they have wings.)

What if in fact there were ever only two really distinct individual people walking around back there in history's mist? That all difference descends from this difference? The whole and the partial. The damaged and the intact. The deformed and the paralyzingly beautiful. The insane and the attendant. The hidden and the blindingly open. The performer and the audience. No Zen-type One, always rather Two, one upside-down in a convex lens.

It is the lumps and trials That tell us whether we shall be known And whether our fate can be exemplary, like a star.

The soul establishes itself. But how far can it swim out through the eyes And still return safely to its nest?" Self-Portrait In a Convex Mirror

The convex surface of any segment of a sphere is, to the entire surface of the sphere itself, as the versed sine of the segment to the diameter of the sphere.

There remains a mirror, on the hall wall. If I turn my head so that the white wings framing my face direct my vision towards it, I can see it as I go down the stairs, round, convex, a pier-glass, like the eye of a fish, and myself in it like a distorted shadow, a parody of something, some fairytale figure in a red cloak, descending towards a moment of carelessness that is the same as danger. A Sister, dipped in blood.

An hour passed and another and then, like a glass of water overfilled - the meniscus inverting, going convex, gravity pulling at the edges, the overflow finally giving way - he could not longer suffer his own cowardice.

Having shaved, washed, and dexterously arranged several artificial teeth, standing in front of the mirror, he moistened his silver-mounted brushes and plastered the remains of his thick pearly hair on his swarthy yellow skull. He drew on to his strong old body, with its abdomen protuberant from excessive good living, his cream-colored silk underwear, put black silk socks and patent-leather slippers on his flat-footed feet. He put sleeve-links in the shining cuffs of his snow-white shirt, and bending forward so that his shirt front bulged out, he arranged his trousers that were pulled up high by his silk braces, and began to torture himself, putting his collar-stud through the stiff collar. The floor was still rocking beneath him, the tips of his fingers hurt, the stud at moments pinched the flabby skin in the recess under his Adam's apple, but he persisted, and at last, with eyes all strained and face dove-blue from the over-tight collar that enclosed his throat, he finished the business and sat down exhausted in front of the pier glass, which reflected the whole of him, and repeated him in all the other mirrors. " It is awful ! " he muttered, dropping his strong, bald head, but without trying to understand or to know what was awful. Then, with habitual careful attention examining his gouty-jointed short fingers and large, convex, almond-shaped finger-nails, he repeated : " It is awful. . . .

My eyes traced the lines of my hips, my convex stomach, the legs beneath my jeans. What did the world see in me?

Still my soul is in its boyhood; Nor of year or changes recks. Though my scalp is almost hairless, And my figure grows convex.

To day we made the grand experiment of burning the diamond and certainly the phenomena presented were extremely beautiful and interesting... The Duke's burning glass was the instrument used to apply heat to the diamond. It consists of two double convex lenses ... The instrument was placed in an upper room of the museum and having arranged it at the window the diamond was placed in the focus and anxiously watched. The heat was thus continued for 3/4 of an hour (it being necessary to cool the globe at times) and during that time it was thought that the diamond was slowly diminishing and becoming opaque ... On a sudden Sir H Davy observed the diamond to burn visibly, and when removed from the focus it was found to be in a state of active and rapid combustion. The diamond glowed brilliantly with a scarlet light, inclining to purple and, when placed in the dark, continued to burn for about four minutes. After cooling the glass heat was again applied to the diamond and it burned again though not for nearly so long as before. This was repeated twice more and soon after the diamond became all consumed. This phenomenon of actual and vivid combustion, which has never been observed before, was attributed by Sir H Davy to be the free access of air; it became more dull as carbonic acid gas formed and did not last so long.

Fortune accepted that he could not become a helpless, blameless baby again and yet he was strangely satisfied at the prospect of being carried upwards into the receptive convexities of the clouds.

That bear hug of yours embraces worlds in its squishy compass, concavity nested in convexity, simplicity vanquishing complexity, in folds of flesh enfolding time, enmeshed eternal in love sublime.

...[T]wo of you can be no match for the three giants, I will find you, if I can, a third brother, who will take on himself the third share of the fight, and the preparation...I will show him to you in a glass, and, when he comes, you will know him at once. If he will share your endeavors, you must teach him all you know, and he will repay you well, in present song, and in future deeds.' She opened the door of a curious old cabinet that stood in the room. On the inside of this door was an oval convex mirror...we at length saw reflected the place where we stood, and the old dame seated in her chair...at the feet of the dame lay a young man...weeping. 'Surely this youth will not serve our ends,' said I, 'for he weeps.' The old woman smiled. 'Past tears are present strength,'said she.

On his convex forehead his namam is a vertical, three-pronged fork, the center stripe red and the flanking two white. It signals he’s a Vaishnavite, holding Vishnu to be the supreme god in the pantheon.

I have always held that it is only when one sees one's own mistakes with a convex lens, and does just the reverse in the case of others, that one is able to arrive at a just relative estimate of the two.

Man's mind, what is it but a convex glass Wherein are gathered all the scattered points Picked out of the immensity of sky, To re-unite there, be our heaven for earth, Our known unknown, our God revealed to man?

Thanks to the printing press, the Continent was suddenly populated by people who were experts at manipulating light through slightly convex pieces of glass. These were the hackers of the first optical revolution.

looked excitingly purposeful, with large video screens ranged over the control and guidance system panels on the concave wall, and long banks of computers set into the convex wall. In one corner a robot sat humped,

Why wouldst thou that God should in power, in act and in effect (which in him are identical) be determined as the limit of the convexity of a sphere, rather than that he should be as we may say the undetermined limit of the boundless? The limit I say, without limit, that I may differentiate the one infinity from the other. For He is the whole, comprehensive [26] and complete totality of the infinite, but the universe is the explicit though not the all-comprehensive totality (if indeed we may in any wise use the term totality where there is neither part nor boundary). Therefore the nature of the one doth comprehend boundaries; that of the other is bounded. And this is not the distinction between infinite and finite. The distinction is rather that the one is infinite, while the other doth limit according to the nature of the totality and of the whole being thereof. So that although it is entirely infinite, the infinity thereof is not completely comprehensive, for this would be repugnant to dimensional infinity.

This humanity we would claim for ourselves is the legacy, not only of the Enlightenment, but of the thousands of European peasants and poor townspeople who came here bringing their humanity and their sufferings with them. It is the absence of a stable upper class that is responsible for much of the vulgarity of the American scene. Should we blush before the visitor for this deficiency? The ugliness of American decoration, American entertainment, American literature - is not this the visible expression of the impoverishment of the European masses, a manifestation of all the backwardness, deprivation, and want that arrived here in boatloads from Europe? The immense popularity of American movies abroad demonstrates that Europe is the unfinished negative of which America is the proof. The European traveler, viewing with distaste a movie palace or a Motorola, is only looking into the terrible concavity of his continent of hunger inverted startlingly into the convex. Our civilization, deformed as it is outwardly, is still an accomplishment; all this had to come to light.

The very sun of heaven seemed distorted when viewed through the polarising miasma welling out from this sea-soaked perversion, and twisted menace and suspense lurked leeringly in those crazily elusive angles of carven rock where a second glance shewed concavity after the first shewed convexity.

it is only when one sees one’s own mistakes with a convex lens, and does just the reverse in the case of others, that one is able to arrive at a just relative estimate of the two. I further believe that a scrupulous and conscientious observance of this rule is necessary for one who wants to be a Satyagrahi.

If we look through a piece of glass, irregularities and impurities may distort and discolor the impression of what we see. If we regard something through a convex lens, it appears to be upside down. But if we place a concave lens in front of the convex lens, we correct the distortion in the convex lens and things no longer appear topsy-turvy. Each one of us regards the world through his own lens, his own glasses. The effect of those glasses is that, even though we may be looking at the same thing, not all of us actually see the same thing. The lenses are ground by each individual’s upbringing, disposition and other factors.

Once the solar parallax is known,' they told me, 'once the necessary Degrees are measur'd, and the size and weight and shape of the Earth are calculated inescapably at last, all this will vanish. We will have to seek another Space.' No one explain'd what that meant, however ...? 'Perhaps some of us will try living upon thy own Surface. I am not sure that everyone can adjust from a concave space to a convex one. Here have we been sheltered, nearly everywhere we look is no Sky, but only more Earth. -- How many of us, I wonder, could live the other way, the way you People do, so exposed to the Outer Darkness? Those terrible Lights, great and small? And wherever you may stand, given the Convexity, each of you is slight pointed away from everybody else, all the time, out into that Void that most of you seldom notice. Here in the Earth Concave, everyone is pointed at everyone else, -- ev'rybody's axes converge, -- forc'd at least thus to acknowledge one another,-- an entirely different set of rules for how to behave.

Lucien finally dies, rather a while after he's quit shuddering like a clubbed muskie and seemed to them to die, as he finally sheds his body's suit, Lucien finds his gut and throat again and newly whole, clean and unimpeded, and is free, catapulted over fans and the Convexity's glass palisades at desperate speeds, soaring north, sounding a bell-clear and nearly maternal alarmed call-to-arms in all the world's well-known tongues.

Wars and chaoses and paradoxes ago, two mathematicians between them ended an age d began another for our hosts, our ghosts called Man. One was Einstein, who with his Theory of Relativity defined the limits of man's perception by expressing mathematically just how far the condition of the observer influences the thing he perceives. ... The other was Goedel, a contemporary of Eintstein, who was the first to bring back a mathematically precise statement about the vaster realm beyond the limits Einstein had defined: In any closed mathematical system--you may read 'the real world with its immutable laws of logic'--there are an infinite number of true theorems--you may read 'perceivable, measurable phenomena'--which, though contained in the original system, can not be deduced from it--read 'proven with ordinary or extraordinary logic.' Which is to say, there are more things in heaven and Earth than are dreamed of in your philosophy, Horatio. There are an infinite number of true things in the world with no way of ascertaining their truth. Einstein defined the extent of the rational. Goedel stuck a pin into the irrational and fixed it to the wall of the universe so that it held still long enough for people to know it was there. ... The visible effects of Einstein's theory leaped up on a convex curve, its production huge in the first century after its discovery, then leveling off. The production of Goedel's law crept up on a concave curve, microscopic at first, then leaping to equal the Einsteinian curve, cross it, outstrip it. At the point of intersection, humanity was able to reach the limits of the known universe... ... And when the line of Goedel's law eagled over Einstein's, its shadow fell on a dewerted Earth. The humans had gone somewhere else, to no world in this continuum. We came, took their bodies, their souls--both husks abandoned here for any wanderer's taking. The Cities, once bustling centers of interstellar commerce, were crumbled to the sands you see today.

Sex is the union of two things. Any two things whether concave or convex or in any combination or number in order to provide more joy for all or any concerned with the one proviso that no little stranger appear as the result of hetero high jinks. So life not death is the Big O. Write that down, Whittaker. Tattoo it on your fat ass. Drop the news into that frying pan of a brain of yours sizzling with greasy dreams of murder to be served up like McDonald's French fries with real blood for ketchup.

Let us return to the distinction between Thalesian and Aristotelian for a minute and look at evolution from the following point of view. The frequency, i.e., how often someone is right is largely irrelevant in the real world, but alas, one needs to be a practitioner, not a talker, to figure it out. On paper, the frequency of being right matters, but only on paper—typically, fragile payoffs have little (sometimes no) upside, and antifragile payoffs have little downside. This means that one makes pennies to lose dollars in the fragile case; makes dollars to lose pennies in the antifragile one. So the antifragile can lose for a long time with impunity, so long as he happens to be right once; for the fragile, a single loss can be terminal. Accordingly if you were betting on the downfall of, say, a portfolio of financial institutions because of their fragilities, it would have cost you pennies over the years preceding their eventual demise in 2008, as Nero and Tony did. (Note again that taking the other side of fragility makes you antifragile.) You were wrong for years, right for a moment, losing small, winning big, so vastly more successful than the other way (actually the other way would be bust). So you would have made the Thekels like Thales because betting against the fragile is antifragile. But someone who had merely “predicted” the event with just words would have been called by the journalists “wrong for years,” “wrong most of the time,” etc. Should we keep tally of opinion makers’ “right” and “wrong,” the proportion does not matter, as we need to include consequences. And given that this is impossible, we are now in a quandary. Look at it again, the way we looked at entrepreneurs. They are usually wrong and make “mistakes”—plenty of mistakes. They are convex. So what counts is the payoff from success.

The totalitarian systems warn of something far more serious than Western rationalism is willing to admit. They are, most of all, a convex mirror of the inevitable consequences of rationalism, a grotesquely magnified image of its own deep tendencies, an extreme offshoot of its own development, and an ominous product of its own expansion. They are a deeply informative reflection of its own crisis. Totalitarian regimes are not merely dangerous neighbors and even less some kind of an avant-garde of world progress. Alas, just the opposite: they are the avant-garde of a global crisis of this civilization, first European, then Euro-American, and ultimately global. They are one of the possible futurological studies of the Western world.

Philotheo. I will do so. If the world is finite and if nothing lieth beyond, I ask you Where is the world? Where is the universe? Aristotle replieth, it is in itself. [1] The convex surface of the primal heaven is universal space, which being the primal container is by naught contained. For position in space is no other than the surfaces and limit of the containing body, so that he who hath no containing body hath no position in space. [2] What then dost thou mean, O Aristotle, by this phrase, that "space is within itself"? What will be thy conclusion concerning that which is beyond the world? If thou sayest, there is nothing, then the heaven [3] and the world will certainly not be anywhere. Fracastoro. The world will then be nowhere. Everything will be nowhere. Philotheo. The world is something which is past finding out. If thou sayest (and it certainly appeareth to me that thou seekest to say something in order to escape Vacuum and Nullity), if thou sayest that beyond the world is a divine intellect, so that God doth become the position in space of all things, why then thou thyself wilt be much embarrassed to explain to us how that which is incorporeal [yet] intelligible, and without dimension can be the very position in space occupied by a dimensional body; and if thou sayest that this incorporeal space containeth as it were a form, as the soul containeth the body, then thou dost not reply to the question of that which lieth beyond, nor to the enquiry concerning that which is outside the universe. And if thou wouldst excuse thyself by asserting that where naught is, and nothing existeth, there can be no question of position in space nor of beyond or outside, yet I shall in no wise be satisfied. For these are mere words and excuses, which cannot form part of our thought. For it is wholly impossible that in any sense or fantasy (even though there may be various senses and various fantasies), it is I say impossible that I can with any true meaning assert that there existeth such a surface, boundary or limit, beyond which is neither body, nor empty space, even though God be there. For divinity hath not as aim to fill space, nor therefore doth it by any means appertain to the nature of divinity that it should be the boundary of a body. For aught which can be termed a limiting body must either be the exterior shape or else a containing body. And by no description of this quality canst thou render it compatible with the dignity of divine and universal nature. [4]

Philotheo. I will do so. If the world is finite and if nothing lieth beyond, I ask you Where is the world? Where is the universe? Aristotle replieth, it is in itself. [1] The convex surface of the primal heaven is universal space, which being the primal container is by naught contained. For position in space is no other than the surfaces and limit of the containing body, so that he who hath no containing body hath no position in space. [2] What then dost thou mean, O Aristotle, by this phrase, that "space is within itself"? What will be thy conclusion concerning that which is beyond the world? If thou sayest, there is nothing, then the heaven [3] and the world will certainly not be anywhere. Fracastoro. The world will then be nowhere. Everything will be nowhere. Philotheo. The world is something which is past finding out. If thou sayest (and it certainly appeareth to me that thou seekest to say something in order to escape Vacuum and Nullity), if thou sayest that beyond the world is a divine intellect, so that God doth become the position in space of all things, why then thou thyself wilt be much embarrassed to explain to us how that which is incorporeal [yet] intelligible, and without dimension can be the very position in space occupied by a dimensional body; and if thou sayest that this incorporeal space containeth as it were a form, as the soul containeth the body, then thou dost not reply to the question of that which lieth beyond, nor to the enquiry concerning that which is outside the universe. And if thou wouldst excuse thyself by asserting that where naught is, and nothing existeth, there can be no question of position in space nor of beyond or outside, yet I shall in no wise be satisfied. For these are mere words and excuses, which cannot form part of our thought. For it is wholly impossible that in any sense or fantasy (even though there may be various senses and various fantasies), it is I say impossible that I can with any true meaning assert that there existeth such a surface, boundary or limit, beyond which is neither body, nor empty space, even though God be there. For divinity hath not as aim to fill space, nor therefore doth it by any means appertain to the nature of divinity that it should be the boundary of a body. For aught which can be termed a limiting body must either be the exterior shape or else a containing body. And by no description of this quality canst thou render it compatible with the dignity of divine and universal nature. [4]

Here I will mention one more strange thing; but whether this peculiarity was owing to my shadow at all, I am not able to assure myself. I came to a village, the inhabitants of which could not at first sight be distinguished from the dwellers in our land. They rather avoided than sought my company, though they were very pleasant when I addressed them. But at last I observed, that whenever I came within a certain distance of any one of them, which distance, however, varied with different individuals, the whole appearance of the person began to change; and this change increased in degree as I approached. When I receded to the former distance, the former appearance was restored. The nature of the change was grotesque, following no fixed rule. The nearest resemblance to it that I know, is the distortion produced in your countenance when you look at it as reflected in a concave or convex surface—say, either side of a bright spoon. Of this phenomenon I first became aware in rather a ludicrous way. My host's daughter was a very pleasant pretty girl, who made herself more agreeable to me than most of those about me. For some days my companion-shadow had been less obtrusive than usual; and such was the reaction of spirits occasioned by the simple mitigation of torment, that, although I had cause enough besides to be gloomy, I felt light and comparatively happy. My impression is, that she was quite aware of the law of appearances that existed between the people of the place and myself, and had resolved to amuse herself at my expense; for one evening, after some jesting and raillery, she, somehow or other, provoked me to attempt to kiss her. But she was well defended from any assault of the kind. Her countenance became, of a sudden, absurdly hideous; the pretty mouth was elongated and otherwise amplified sufficiently to have allowed of six simultaneous kisses. I started back in bewildered dismay; she burst into the merriest fit of laughter, and ran from the room. I soon found that the same undefinable law of change operated between me and all the other villagers; and that, to feel I was in pleasant company, it was absolutely necessary for me to discover and observe the right focal distance between myself and each one with whom I had to do. This done, all went pleasantly enough. Whether, when I happened to neglect this precaution, I presented to them an equally ridiculous appearance, I did not ascertain; but I presume that the alteration was common to the approximating parties. I was likewise unable to determine whether I was a necessary party to the production of this strange transformation, or whether it took place as well, under the given circumstances, between the inhabitants themselves.

In consequence of the inevitably scattered and fragmentary nature of our thinking, which has been mentioned, and of the mixing together of the most heterogeneous representations thus brought about and inherent even in the noblest human mind, we really possess only *half a consciousness*. With this we grope about in the labyrinth of our life and in the obscurity of our investigations; bright moments illuminate our path like flashes of lighting. But what is to be expected generally from heads of which even the wisest is every night the playground of the strangest and most senseless dreams, and has to take up its meditations again on emerging from these dreams? Obviously a consciousness subject to such great limitations is little fitted to explore and fathom the riddle of the world; and to beings of a higher order, whose intellect did not have time as its form, and whose thinking therefore had true completeness and unity, such an endeavor would necessarily appear strange and pitiable. In fact, it is a wonder that we are not completely confused by the extremely heterogeneous mixture of fragments of representations and of ideas of every kind which are constantly crossing one another in our heads, but that we are always able to find our way again, and to adapt and adjust everything. Obviously there must exist a simple thread on which everything is arranged side by side: but what is this? Memory alone is not enough, since it has essential limitations of which I shall shortly speak; moreover, it is extremely imperfect and treacherous. The *logical ego*, or even the *transcendental synthetic unity of apperception*, are expressions and explanations that will not readily serve to make the matter comprehensible; on the contrary, it will occur to many that “Your wards are deftly wrought, but drive no bolts asunder.” Kant’s proposition: “The *I think* must accompany all our representations ,” is insufficient; for the “I” is an unknown quantity, in other words, it is itself a mystery and a secret. What gives unity and sequence to consciousness, since by pervading all the representations of consciousness, it is its substratum, its permanent supporter, cannot itself be conditioned by consciousness, and therefore cannot be a representation. On the contrary, it must be the *prius* of consciousness, and the root of the tree of which consciousness is the fruit. This, I say, is the *will*; it alone is unalterable and absolutely identical, and has brought forth consciousness for its own ends. It is therefore the will that gives unity and holds all its representations and ideas together, accompanying them, as it were, like a continuous ground-bass. Without it the intellect would have no more unity of consciousness than has a mirror, in which now one thing now another presents itself in succession, or at most only as much as a convex mirror has, whose rays converge at an imaginary point behind its surface. But it is *the will* alone that is permanent and unchangeable in consciousness. It is the will that holds all ideas and representations together as means to its ends, tinges them with the colour of its character, its mood, and its interest, commands the attention, and holds the thread of motives in its hand. The influence of these motives ultimately puts into action memory and the association of ideas. Fundamentally it is the will that is spoken of whenever “I” occurs in a judgement. Therefore, the will is the true and ultimate point of unity of consciousness, and the bond of all its functions and acts. It does not, however, itself belong to the intellect, but is only its root, origin, and controller.

Praise the miracle body The odd and undeniable mechanics of hand Hundred boned foot, perfect stretch of tendon Praise the veins that river these wrists Praise the prolapsed valve in a heart Praise the scars marking a gallbladder absent Praise the rasp and rattle of functioning lungs Praise the pre-arthritic ache of elbows and ankles Praise the lifeline sectioning a palm Praise the photographic pads of fingertips Praise the vulnerable dip at the base of a throat Praise the muscles surfacing on an abdomen Praise these arms that carry babies, and anthologies Praise the leg hairs that sprout and are shaved Praise the ass that refuses to shrink or be hidden Praise the cunt that bleeds and accepts, bleeds and accepts Praise the prominent ridge of nose Praise the strange convexity of rib cage Praise the single hair that insists on growing from a right areola Praise the dent where the mole was clipped from the back of a neck Praise these inner thighs brushing Praise these eyelashes that sometimes turn inward Praise these hips preparing to spread into a grandmother’s skirt Praise the beauty of the freckle on the first knuckle of a left little finger We’re gone in a blizzard of seconds Love the body human while we’re here A gift of minutes on an evolving planet A country in flux, give thanks For bone, and dirt, and the million things that will kill us someday Motion and the pursuit of happiness, no garauntees, give thanks For chaos theory, ecology, common sense that says we are web A planet in balance or out That butterfly in Tokyo setting off thunder storms in Iowa Tell me you don’t matter to a universe that conspired to give you such a tongue Such rhythm or rhythmless hips Such opposable thumbs Give thanks, or go home a waste of spark Speak, or let the maker take back your throat March, or let the creator rescind your feet Dream, or let your god destroy your good and fertile mind This is your warning This your birthright Do not let this universe regret you

Though it’s best not to be born a chicken at all, it is especially bad luck to be born a cockerel. From the perspective of the poultry farmer, male chickens are useless. They can’t lay eggs, their meat is stringy, and they’re ornery to the hens that do all the hard work of putting food on our tables. Commercial hatcheries tend to treat male chicks like fabric cutoffs or scrap metal: the wasteful but necessary by-product of an industrial process. The sooner they can be disposed of—often they’re ground into animal feed—the better. But a costly problem has vexed egg farmers for millennia: It’s virtually impossible to tell the difference between male and female chickens until they’re four to six weeks old, when they begin to grow distinctive feathers and secondary sex characteristics like the rooster’s comb. Until then, they’re all just indistinguishable fluff balls that have to be housed and fed—at considerable expense. Somehow it took until the 1920s before anyone figured out a solution to this costly dilemma. The momentous discovery was made by a group of Japanese veterinary scientists, who realized that just inside the chick’s rear end there is a constellation of folds, marks, spots, and bumps that to the untrained eye appear arbitrary, but when properly read, can divulge the sex of a day-old bird. When this discovery was unveiled at the 1927 World Poultry Congress in Ottawa, it revolutionized the global hatchery industry and eventually lowered the price of eggs worldwide. The professional chicken sexer, equipped with a skill that took years to master, became one of the most valuable workers in agriculture. The best of the best were graduates of the two-year Zen-Nippon Chick Sexing School, whose standards were so rigorous that only 5 to 10 percent of students received accreditation. But those who did graduate earned as much as five hundred dollars a day and were shuttled around the world from hatchery to hatchery like top-flight business consultants. A diaspora of Japanese chicken sexers spilled across the globe. Chicken sexing is a delicate art, requiring Zen-like concentration and a brain surgeon’s dexterity. The bird is cradled in the left hand and given a gentle squeeze that causes it to evacuate its intestines (too tight and the intestines will turn inside out, killing the bird and rendering its gender irrelevant). With his thumb and forefinger, the sexer flips the bird over and parts a small flap on its hindquarters to expose the cloaca, a tiny vent where both the genitals and anus are situated, and peers deep inside. To do this properly, his fingernails have to be precisely trimmed. In the simple cases—the ones that the sexer can actually explain—he’s looking for a barely perceptible protuberance called the “bead,” about the size of a pinhead. If the bead is convex, the bird is a boy, and gets thrown to the left; concave or flat and it’s a girl, sent down a chute to the right.

In Mahmoud’s hand is a large stone. Very calmly he lifts his palm over the slither of rock and brings the stone down on its craggy surface. A gorgeous, true, deep bell-like tone rings out, amplified by the natural rocky auditorium. ‘Is it a bell?’ I ask. ‘No, it’s not a bell. It’s what we call a rock gong.’ I know these objects. I sit next to Mahmoud in the shade with a stone in my hand and we try to beat out a staccato tune together. The surface of the gong has been shaped so that different tones are produced by different areas, rather like a convex version of a modern steel drum.

If the landscape you encounter is convex, suck it.” Stephen latched his mouth briefly onto Jamie’s neck in demonstration, then moved his lips upward and suckled an earlobe. “I’m sure if you think hard, you can imagine other examples.” Jamie shivered. “If it’s concave,” Stephen whispered, “probe it—” His tongue slid into Jamie’s ear, and the young man shuddered again. “—using the instrument that fits best. Flat surfaces can be licked—actually, licking is always appropriate, as are nibbling, stroking, and rubbing—but those are the basic rules. Remember to start gently, and end vigorously, and satisfaction is guaranteed.

The idea of antifragility, for which evolution is the most important example, gives us tremendous guidance: o   Seek out low-cost options in your business and look for convex payouts. o   Embrace trial and error processes, and foster entrepreneurship. o   Embrace randomness, and be aware that attempts to mask or limit its effects may seem to work in the short run but could have big consequences in the future. o   Value diversity; it is important in evolutionary processes. Finally, we reviewed unpredictable changes in systems. Don't be lulled into complacency. Grove's "Only the paranoid survive" are words to live by.

convex side of the train,” he wrote, “when both the

On turning to the Work in Progress we find that the mirror is not so convex. Here is direct expression--pages and pages of it. And if you don’t understand it, Ladies and Gentlemen, it is because you are too decadent to receive it.

Sorting Laundry" Folding clothes, I think of folding you into my life. Our king-sized sheets like tablecloths for the banquets of giants, pillowcases, despite so many washings, seems still holding our dreams. Towels patterned orange and green, flowered pink and lavender, gaudy, bought on sale, reserved, we said, for the beach, refusing, even after years, to bleach into respectability. So many shirts and skirts and pants recycling week after week, head over heels recapitulating themselves. All those wrinkles To be smoothed, or else ignored; they're in style. Myriad uncoupled socks which went paired into the foam like those creatures in the ark. And what's shrunk is tough to discard even for Goodwill. In pockets, surprises: forgotten matches, lost screws clinking the drain; well-washed dollars, legal tender for all debts public and private, intact despite agitation; and, gleaming in the maelstrom, one bright dime, broken necklace of good gold you brought from Kuwait, the strangely tailored shirt left by a former lover… If you were to leave me, if I were to fold only my own clothes, the convexes and concaves of my blouses, panties, stockings, bras turned upon themselves, a mountain of unsorted wash could not fill the empty side of the bed.

Someone with a linear payoff needs to be right more than 50 percent of the time. Someone with a convex payoff, much less. The hidden benefit of antifragility is that you can guess worse than random and still end up outperforming. Here lies the power of optionality—your function of something is very convex, so you can be wrong and still do fine—the more uncertainty, the better.

(c) If the function is convex (antifragile), then the average of the function of something is going to be higher than the function of the average of something. And the reverse when the function is concave (fragile).

I knew what was about to happen. I knew I was going to cry. It’s not even about sadness—it’s about vulnerability. Seeing into a person’s soul, even for a moment, is just too much. It’s like the convex meniscus Sylvia Plath describes in The Bell Jar—the water that clings to the sides of the glass before one tiny pulse causes it to overflow.

Maryland’s districts, portions of which appear in figure 4.3, were less compact, especially on the convex hull measure, in 2013 than a decade earlier.10 figure 4.3 features District 3, the only one of Maryland’s eight districts to make the “Worst 10” lists. Districts 2, 4, and 7 also have narrow necks and tight twists and turns. A slim, contorted finger of District 2 traces the two northern blobs of District 3. Except for the 6th District, which will be discussed in the next chapter, partisan machinations did not dictate the shape of these districts. Race did play a role as the plan designed by Democratic governor Martin O’Malley carefully allocated Democratic voters so as to maintain the 4th and 7th as majority-black districts. Chapter 5 also addresses the Pennsylvania plan invalidated by the state Supreme Court, while part of the Virginia map was found to pack African Americans, a topic considered in chapter 3. Contributing to the low compactness scores for several states, such as Rhode Island and Hawaii, are their ragged shorelines. Indiana and Nevada had the most compact statewide plans for the 2010s. Table 4.3  States Having the Lowest Average Compactness Scores as of 2013 State Reock Polsby-Popper Convex Hull Maryland 2 1 1 North Carolina 5 4 4 Louisiana 7 3 3 West Virginia 8 5 2 Virginia 4 7 13 Hawaii 18 2 25

judged it the least compact in the nation.8 The report assessing congressional districts for the 1990s prepared by the Congressional Research Service applied the convex hull to the population and not to area. The least compact district as calculated using convex hull for the population was Colorado-4. Neither this district nor the next five in terms of low population scores had to defend its plan in court. Only three districts with low scores on the population compactness measure drew a legal challenge. Of these three, Florida-23 escaped unscathed when the judge dismissed the suit. On

Least Compact Congressional Districts as of 2013 District Reock Polsby-Popper Convex Hull NC – 12 2 2 1 FL – 5 3 4 2 MD – 3 27 1 3 OH – 9 1 14 4 TX – 35 5 12 5 NC – 4 13 10 6 LA – 2 28 11 7 FL – 22 6 23 18 MD – 6 9 31 8 NY – 10 42 42 16 Entries show the ranking of the district on the measure. Source: Redrawing the Map on Redistricting, 2012 (Philadelphia, PA: Azavea, 2012).

Much of the criticism concerning partisan gerrymandering during the 2010s focused on GOP-drawn districts in states like North Carolina, Pennsylvania, and Michigan, as discussed in the next chapter. However, when all states with more than one district are considered, comparison of the average district compactness concludes that Democrats ignored this traditional districting principle more than did Republicans.11 On each of the three measures considered here, Democratic districts scored less well than those fashioned by the GOP. Legislatively drawn plans in states not controlled by one party were more compact than those drawn by Republicans or Democrats on the convex hull measure, but on the other two measures they scored between the two parties’ plans. In keeping with the suspicions of reformers, partisan plans tended to have less compact districts than plans drawn by other entities. Districts in plans prepared by independent commissions scored slightly better on Reock but were a little less compact than districts prepared by courts on Polsby-Popper and convex hull. In two of the nation’s most populous states, restricting the influence of the legislature coincided with more compact districts in 2013 than in 2003. In California, authority to redistrict was shifted from the legislature to an independent commission, while in Florida the legislature retained the authority to draw new districts but a constitutional amendment banned efforts to advance the interests of a party or incumbent.

We all need to be individuals, to stand out in the crowd, and to be heard in the convex confines of the herd. It's the price we pay for free will that keeps us apart and unwilling to read or, for that matter, be on the same page. We'll never be able to get out of our own ways long enough to see the other's point of view because we're all so fucking busy paying more attention to the sounds of our own voices than to what the other person is saying. We're always more impressed with our retorts than the rest of the conversation. The whole time the opponent is talking we're just thinking about what we're going to say in reply; we don't even hear the other side of it.

We all need to be individuals, to stand out in the crowd, and to be heard in the convex confines of the herd. It's the price we pay for free will that keeps us apart and unwilling to read or, for that matter, be on the same page. We'll never be able to get out of our own ways long enough to see the other's point of view because we're all so fucking busy paying more attention to the sounds of our own voices than to what the other person is saying. We're always more impressed with our retorts than the rest of the conversation. The whole time the opponent is talking we're just thinking about what we're going to say in reply; we don't even hear the other side of it.

A Mercedes or a BMW can’t make full use of its lustre, as it busily swerves to avoid the very convex buses along very concave roads. The existence of good roads would depend upon another type of wealth. A wealth that might serve the city.

NATURE MORTE You are given two things today, one is an angry nail in your side; Changing what you are able to sense is the second thing you are gifted. Nature is always a referred existence, writes Emerson, never a presence. Who knows where the time goes, sings Sandy Denny. Who can bear to hear it? One reaches the moment when one loses words—in the pastoral, in the cosmic? Think about the scars on the planets, & how patient those stars seem to be. I am painting the natural landscape with my eyes closed today. It is like writing A poem with all the cross-outs left in; an expression like never thought I’d see the day. Nature that begins with unknowable & ends with more monotonous hills. Today, I want to be the country-fried philosophe or a Hudson River School painting. When this life is over, describe to me how its concave & convex forms are & are not. We live amid surfaces, writes Emerson, & the true art of life is to skate well on them.

I’m different than other men in that I’m indifferent to the ins and outs of concave and convex relationships. I feel that women like that I’m like that.

They were proclaiming the end of the world, redemption through penitence, the visions of the seventh day, the advent of the angel, cosmic collisions, the death of the sun, the tribal spirit, the sap of the mandrake, tiger ointment, the virtue of the sign, the discipline of the wind, the perfume of the moon, the revindication of darkness, the power of exorcism, the sign of the heel, the crucifixion of the rose, the purity of the lymph, the blood of the black cat, the sleep of the shadow, the rising of the seas, the logic of anthropophagy, painless castration, divine tattoos, voluntary blindness, convex thoughts, or concave, or horizontal or vertical, or sloping, or concentrated, or dispersed, or fleetin, the weakening of the vocal cords, the death of the word. Here, nobody is speaking of organisation.

One moment he's telling you why convexity leads to philostochasticity and the next he's explaining why he doesn't eat papayas. For the record, he avoids all fruits without a Greek or Hebrew name because his ancestors would not have eaten them.

It was the cap, or rather the logo that was stitched to it. The grin was hideous. The huge teeth hung from the roof of the head like gleaming convex stalactites that descended from the middle of the crown down to the cap's brim. The cavernous mouth crowded the nose and eyes into the hairline and strained without success to close around incisors that claimed three quarters of the clownish face. Had the face been black or brown, it would have incited urban riots, so patent was its insult. But the face was red and Justice wore the cap with jaunty insouciance.

Doug has experimented with materials as diverse as crayons, convex mirrors, and tinted plaster.His rooms received write-ups in the New York Times as well as a bit in House and Garden, which quoted Doug as an expert on color and reproduced his preferred palette - one ranging from bubble gum pink and fiery reds to royal blues and brilliant purples....The write up caught the attention of Trading Spaces producers.

He approaches a cockroach in only one respect: his coloration is brown. That is all. Apart from this he has a tremendous convex belly divided into segments and a hard rounded back suggestive of wing cases. In beetles these cases conceal flimsy little wings that can be expanded and then may carry the beetle for miles and miles in a blundering flight. Curiously enough, Gregor the beetle never found out that he had wings under the hard covering of his back. (This is a very nice observation on my part to be treasured all your lives. Some Gregors, some Joes and Janes, do not know that they have wings.)

the Doric order the column has no separate base, but rises direct from the top step of the platform on which the building it belongs to stands. It is of massive form and has what is known as an entasis or slightly convex surface, it is generally fluted, that is to say, cut into parallel perpendicular channels, several rings called annulets connecting it with the capital, which consists of an echinus or rounded moulding and an abacus or unrounded slab resting on the echinus. The Doric entablature is equally simple, the architrave being perfectly plain, whilst the frieze is adorned with triglyphs or three upright projections with grooves between them, set at equal distances from each other, the spaces separating them, known as metopes, being as a rule enriched with fine sculptures of figure subjects. The frieze is connected with the cornice by narrow bands called mutules resting on the triglyphs and metopes, and the cornice itself has a plain lower band known as the corona, surmounted by more or less decorated courses of stone or marble. The Ionic and Corinthian orders are alike characterised by lightness and grace rather than massiveness and simplicity. In both, the columns, instead of rising direct from the platform, have a complex base consisting of a number of circular mouldings above another, the fluted shafts are comparatively slim and tapering, and the channels in them are divided by spaces called fillets. In the Ionic order the flat abacus of the Doric capital is replaced by two coiled volutes projecting beyond the echinus on either side, and the horizontal portion between the volutes is surmounted by finely carved leaf mouldings. The Corinthian order is specially distinguished by the ornate decoration of the capitals, that represent calices of flowers and leaves, chiefly those of the acanthus, arranged so as to point upwards and curve outwards in much the same style as they do in nature. The architrave in both the Ionic and the Corinthian orders consists of plain slabs, but the frieze—which is not divided as in Doric buildings into triglyphs and metopes—is in nearly every case enriched with a series of beautiful figure

At Göbekli Tepe there is a creature, sculted in high-relief, identified by Klaus Schmidt as a beast of prey with splayed claws and powerful shoulders, its tail bent to its left over its body. A very similar animal is seen at Cutimbo [in Peru] with the same splayed claws and the same powerful shoulders, while the tail instead of being bent to its left is bent to its right. At both Göbekli Tepe and Cutimbo, reliefs of salamanders and of serpents are found. The style of execution in all cases is very similar. At about the level of the genitals of the so-called "Totem Pole" of Göbekli Tepe, a small head and two arms protrude. The head has a determined look, with prominent brows. The long fingers of the hands almost meet. The posture is that of a man leaning down through the stone and playing a drum. This is also the posture of two figures at Cutimbo, who emerge from a large convex block on one of the circular towers. They have the same determined features and prominent brow ridges as the figure on the "Totem Pole." The two serpents on the side of the "Totem Pole" have peculiarly large heads, making them look almost like sperm. So, too, does the serpent that emerges from the dark narrow entrance of the Temple of the Moon above Cuzco. Lions feature in the reliefs at Göbekli Tepe, pumas feature in the reliefs at Cutimbo and again the manner of representation is similar.

of Ronald Reagan by holding its own Inaugural Ball. This convex-mirror version of the formal ball being held that same evening at the floodlit Kennedy Center on the Potomac was dreamed up, of course, by Ann Magnuson, who, the month before, was also the art

In every way the early Salem symptoms conformed to those of Elizabeth Knapp, the Goodwin children, and the two young women to whose bedsides Mather rushed post-Salem. We will never know what felled the girls, whether it had more to do with their souls or their chores, with parental attention or inattention. The prickling sensations, the twitching, stammering, and grimacing, the ulcerated skin and twisted limbs, the curled tongues and convex backs, the deliriums, the “furious invectives against imaginary individuals” do however conform precisely to what nineteenth-century neurologist Jean-Martin Charcot, with Freud following him, termed hysteria. Where the seventeenth-century authority saw the devil, we tend to recognize an overtaxed nervous system; what an earlier age called hysteria we term conversion disorder, the body literally translating emotions into symptoms. When sublimated, distress will manifest physically, holding the body hostage. Charcot’s drawings of convulsing hysterics agree in every detail with the scenes that left Deodat Lawson reeling.

True love, immortal and useless, belongs to those figures whose feelings never change, since by nature they are static. Ever since I've known the Japanese man who sits on the convex [surface] of my teapot, he has yet to make a move. He never savours the hand of the woman who is forever out of reach. Enervated colours, like those of an emptied, poured-out sun, eternally unrealize the slopes of that hill. And the whole scene observes a moment of sorrow more faithful than the one that uselessly torments the feigned fragility of my weary hours.

I really believe that beauty is completely comical. It’s goddamn ridiculous. Fabrics and textiles can define us? The convex and concave shapes that make up our bodies can devalue us? The amount of skin you have on one of your eyelids can determine a career? So dumb!

Here the hard evidence thins out. One unconfirmed report says the disk was eventually transferred to Wright-Patterson A.F. Base. Another says that the disk was either transferred after that to another unnamed base, or was taken directly to this unknown base directly from Atlanta. “The best description of the disk was that it was sixteen feet, five inches in diameter, convex on both upper and lower surfaces to the same degree, possessing no visible doors or windows. The thickness was slightly less than five feet. The color was silver, much like polished

This final but unknown “truth valid in infinity” is somehow perceived or felt by us as an ideal, for in countless years of observation we have formed a series of less and less false, more and more nearly true “ideas” about the phenomenon. The “ideas” are reflexes of the phenomenon, reflected in our midst as in a mirror; the reflexes may be distorted, as in a convex or concave mirror, but they suggest [pg 052] an ideal reflex valid in infinity. It is of the utmost importance to realize that the words which are used to express the ideas and the ideals are the materialization of the ideas and ideal; it is only by words that we are enabled to give to other human beings an exact or nearly exact impression which we have had of the phenomenon.

For ADELARD TOOK, for his VERY OWN, from Bilbo; on an umbrella. Adelard had carried off many unlabelled ones. For DORA BAGGINS in memory of a LONG correspondence, with love from Bilbo; on a large waste-paper basket. Dora was Drogo’s sister and the eldest surviving female relative of Bilbo and Frodo; she was ninety-nine, and had written reams of good advice for more than half a century. For MILO BURROWS, hoping it will be useful, from B.B.; on a gold pen and ink-bottle. Milo never answered letters. For ANGELICA’S use, from Uncle Bilbo; on a round convex mirror. She was a young Baggins, and too obviously considered her face shapely. For the collection of HUGO BRACEGIRDLE, from a contributor; on an (empty) book-case. Hugo was a great borrower of books, and worse than usual at returning them. For LOBELIA SACKVILLE-BAGGINS, as a PRESENT; on a case of silver spoons. Bilbo believed that she had acquired a good many of his spoons, while he was away on his former journey. Lobelia knew that quite well. When she arrived later in the day, she took the point at once, but she also took the spoons.

Twenty-eight years later, the Economist of November 10, 1990, put it this way: . . . non-convex problems . . . are puzzles in which there may be several good but not ideal answers which classical search techniques may wrongly identify as the best one.

A third measure, the convex hull, suggests putting a hypothetical rubber band around a district.5 The score is the percentage of the area surrounded by the hypothetical rubber band that is in the district. The measure would yield a high score for regular geometric shapes such as a square, rectangle, pentagon, and so on. However, a district in which portions have been cut away in order to avoid including certain populations would have lower scores. Each

It was a new moon, but the stars of the northern hemisphere transformed her slim sinuous home, converting the oak strips on the convex walls into quicksilver that momentarily held the frenzied shadows of the forest, slickening their inextricable shapes, and then engulfed them.

Everyone knows this fairy tale: "Mirror, mirror on the wall, who is the fairest of them all?" "Why, you are, of course." In the corner of their hearts, everyone is looking for a magic mirror. If there was a mirror that would reflect the image of them as they fervently wished to be, surely everyone would treasure such a mirror for as long as they lived. On the other hand, there are mirrors that don't do that, such as concave mirrors and convex mirrors. For a long time there have been two full-length fun-house mirrors on the observation deck of the Tsūtenkaku tower in Osaka. For people from other parts of Japan who are not familiar with Osaka's fun-loving and idiosyncratic culture, why such things are in that particular place is a complete mystery; but in any case, it is amusing to play with them. When you stand in front of the concave mirror, you appear stretched out as though you are being pulled up and down by your head and your toes—as though you have been transformed into a toothpick. In front of the convex mirror it is the reverse: you look short and fat as though you have been squashed in a mechanical press. Unsightly and with short legs, you look like a comic book character. Tourists look at their distorted appearances and laugh. But how can they laugh at such warped reflections? Is it because they can relax knowing that they could not possibly look like the twisted images in the mirror? People do not believe they really look like the grossly distorted images in fun-house mirrors, so they laugh them off. However, when a magic mirror reflects an image distorted in a beautiful way, people want to think: yes indeed, this is how I really look. All of you astute readers should understand by now. The image reflected in the mirror that I am talking about in this book is the image of Japan drawn by foreigners. However, this brings up a question. What kind of a distorted image would a Japanese accept as being him or herself? What sort of a distorted image would he or she laugh off? Where exactly is the boundary between the two?

And there, seated at the heart of all that sumptuousness and leaning forward in his chair behind the great desk, was a figure that looked like nothing so much as a dapper but exceedingly despondent frog. The very shape of his head seemed as if it had been altered by a powerful vertical vice, resulting in a symmetrical ovoid with a horizontal polar axis. His complexion was not so much sallow as lightly green. His mouth was unnaturally wide, with thick, tautly stretched lips the color and texture of earthworms. What ears he had were small and circular and somewhat recessed. His nose was broad and rather flat, as if it had been spread on his face unevenly by a butter knife, and had what looked more like nares than full nostrils. The sparse, slick tendrils of his hair were of some murkily nondescript hue and clung unguinously to his scalp. The dense convex lenses of his wire-rimmed spectacles made it seem as if his greenish-gray eyes were peering out at the world from the bottom of a shallow pond, through a thin layer of algae. If he had a jawline, it was not immediately evident where he kept it. His hunched, narrow, rounded shoulders, moreover, amplified the amphibian quality of his appearance. He was, however, dressed in the height of fashion: a high collar and pearl-colored cravat, a waistcoat of forest-green velvet, and a formal coat of lighter, lettuce-green damask with lapels of cream-white satin with pink borders.

shows why the convex likes volatility. If you earn more than you lose from fluctuations, you want a lot of fluctuations.

In other words, it was a much more linear economy—less complex—than today. And we have more nonlinearities—asymmetries, convexities—in today’s world.

If you are antifragile (i.e., convex) to a given substance, then you are better off having it randomly distributed, rather than provided steadily.

The convex (left) is antifragile, the concave (right) is fragile (has

or mistaking a profitable bet for good forecasting—not convexity of payoff and optionality).

Boyd thought about this. “No, because my eyes would be smashed down on the 2D surface. So I’d only see a convex line, or maybe a straight one.

It is an instance of what are known as convex minimization problems, which can be solved by so-called gradient descent methods; this is just algorithm-speak for “walk downhill in the steepest direction to quickly get to the lowest point in the valley.