Quotes about cube

A collection of quotes on the topic of cube, square, likeness, time.

Quotes about cube

“Jack Reacher ordered espresso, double, no peel, no cube, foam cup, no china, and before it arrived at his table he saw a man's life change forever. Not that the waiter was slow. Just that the move was slick. So slick, Reacher had no idea what he was watching. It was just an urban scene, repeated everywhere in the world a billion times a day: A guy unlocked a car and got in and drove away. That was all. But that was enough.”

Lee Child book The Hard Way

The Hard Way, Ch. 1 (2006).

Life, World, Cars, Guy

“He could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."”

G. H. Hardy (1877–1947) British mathematician

Source: Ramanujan (1940), Ch. I : The Indian mathematician Ramanujan.

Hope, Way, Personality

“Do you have a lot of other profound thoughts like that? Blood is blood? A toaster is a toaster? A Gelatinous Cube is a Gelatinous Cube?”

Cassandra Clare book City of Fallen Angels

Source: City of Fallen Angels

“Want a sugar cube?" he asks in his old seductive voice.”

Suzanne Collins book Mockingjay

Source: Mockingjay

“Is that too much to expect? That I would name the stars
for you? That I would take you there? The splash
of my tongue melting you like a sugar cube?”

Richard Siken (1967) American poet

Source: Crush

Star

“Is that how you're going to take me? Scare me into voluntarily coming aboard, then steal my Ice Cube?"

"It's always cubes with you," noted Foaly, somewhat randomly. "What's wrong with a nice sphere?”

Eoin Colfer (1965) Irish author of children's books

Source: The Atlantis Complex

“If I want to play mind games, I'd buy a Rubik's cube. ~ Acheron, a character.”

Sherrilyn Kenyon (1965) Novelist

Variant: If I wanted to play mind games, I'd buy a Rubik's cube
Source: Acheron

Game

“This was the tricky bit. The really tricky bit, trickiness cubed.”

Hugh Laurie book The Gun Seller

Source: The Gun Seller (1996)

“We are frankly amused to think that many a novice may perhaps pay for his too literal comprehension of the remarks of one cubist, and his faith in the existence of an Absolute Truth, by painfully juxtaposing the six faces of a cube or the two ears of a model seen in profile.”

Albert Gleizes (1881–1953) French painter

1910s, Du Cubisme (1912)

Truth, Faith, Thinking

“CIMOSA provides a Reference Architecture (known as the CIMOSA cube) from which particular enterprise architectures can be derived. This Reference Architecture and the associated enterprise modelling framework are based on a set of modelling constructs, or generic building blocks, which altogether form the CIMOSA modelling languages.”

Peter Bernus (1949) Hungarian-Australian computer scientist

Peter Bernus, Kai Mertins, Günter Schmidt (1998) Handbook on Architectures of Information Systems. p. 244

“We all have our little solipsistic delusions, ghastly intuitions of utter singularity: that we are the only one in the house who ever fills the ice-cube tray, who unloads the clean dishwasher, who occasionally pees in the shower, whose eyelid twitches on first dates; that only we take casualness terribly seriously; that only we fashion supplication into courtesy; that only we hear the whiny pathos in a dog's yawn, the timeless sigh in the opening of the hermetically-sealed jar, the splattered laugh in the frying egg, the minor-D lament in the vacuum's scream; that only we feel the panic at sunset the rookie kindergartner feels at his mother's retreat. That only we love the only-we. That only we need the only-we. Solipsism binds us together, J. D. knows. That we feel lonely in a crowd; stop not to dwell on what's brought the crowd into being. That we are, always, faces in a crowd.”

David Foster Wallace (1962–2008) American fiction writer and essayist

"Westward The Course Of Empire Takes Its Way", Girl With Curious Hair
Short stories

Love, Dogs, Feeling, Fashion

“Some… agreeing with Philolaus, believe that the proportion is called harmonic because it attends upon all geometric harmony, and they say that 'geometric harmony' is the cube because it is harmonized in all three dimensions, being the product of a number thrice multiplied together. For in every cube this proportion is mirrored; there are in every cube 12 sides, 8 angles and 6 faces; hence 8, the [harmonic] mean between 6 and 12, is according to harmonic proportion…”

Nicomachus (60–120) Ancient Greek mathematician

this harmonic proportion may be expressed as <math>\frac{12}{6}=\frac{12-8}{8-6}</math> or inversely.
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

“One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them…these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension—between the simple rules that describe the world and the complex world we see—is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain—or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create?—are the ones found in the world around us.”

John Barnes book Mother of Storms

Source: Mother of Storms (1994), pp. 470-471

Pleasure, World, Law, Way

“I'm an ice sculptor - last night I made a cube.”

Mitch Hedberg (1968–2005) American stand-up comedian

Do You Believe in Gosh?

Night

“Maltesers have the less-fattening centre. Well, yes, but they are covered in chocolate! That's like saying 'I'll have a mineral water, please. Can you put some cubes of lard in it?”

Linda Smith (1958–2006) comedian

Packet of Three, Channel 4, 1991
Stand-up

Water

“[I] do not like poems that resemble hay compressed into a geometrically perfect cube. I like it when the hay, unkempt, uncombed, with dry berries mixed in it, thrown together gaily and freely, bounces along atop some truck—and more, if there are some lovely and healthy lasses atop the hay—and better yet if the branches catch at the hay, and some of it tumbles to the road.”

Yevgeny Yevtushenko (1932–2017) Russian poet, film director, teacher

New York Times (2 February 1986).

Love, Perfection

“Then I gathered the éléments of what people call my symbolism. I do not understand anything about long words and theories. But I am willing to be a symbolist, if that defines the ideas that Michael Angelo gave me, namely that the essence of sculpture is the modelling, the general scheme which alone enables us to render the intensity, the supple variety of movement and character. If we can imagine the thought of God in creating the world, He thought first of the construction, which is the sole principle of nature, of living things and perhaps of the planets. Michael Angelo seems to me rather to derive from Donatello than from the ancients; Raphaël proceeds from them. He understood that an architecture can be built up with the human body, and that, in order to possess volume and harmony, a statue or a group ought to be contained in a cube, a pyramid or some simple figure. Let us look at a Dutch interior and at an interior painted by an artist of the present day. The latter no longer touches us, because it docs not possess the qualities of depth and volume, the science of distances. The artist who paints it does not know how to reproduce a cube. An interior by Van der Meer is a cubic painting. The atmosphere is in it and the exact volume of the objects; the place of these objects has been respected, the modem painter places them, arranges them as models. The Dutchmen did not touch them, but set themselves to render the distances that separated them, that is, the depth. And then, if I go so far as to say that cubic truth, not appearance, is the mistress of things, if I add that the sight of the plains and woods and country views gives me the principle of the plans that I employ on my statues, that I feel cubic truth everywhere, and that plan and volume appear to me as laws of all life and ail beauty, will it be said that I am a symbolist, that I generalise, that I am a metaphysician? It seems to me that I have remained a sculptor and a realist. Unity oppresses and haunts me.”

Auguste Rodin (1840–1917) French sculptor

Source: Auguste Rodin: The Man, His Ideas, His Works, 1905, p. 65-67

Life, God, People, Truth

“The work Was begun in 1913, but the bulk of it was written, as a distraction, during the first three years of the war, the hideous course of which seemed day by day to enforce the profound truth conveyed in the answer of Plato to the Delians. When they consulted him on the problem set them by the Oracle, namely that of duplicating the cube, he replied, 'It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another'.
Truly,Greece and her foundations are
Built below the tide of war,
Based on the crystàlline sea
Of thought and its eternity.</center”

Thomas Little Heath (1861–1940) British civil servant and academic

A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

God, War, Truth, Mathematics

“A spade may be made of any size, and if the same number of strokes be made in the hour, the requisite exertion will vary nearly as the cube of the length of the blade.”

William Stanley Jevons The Theory of Political Economy

Source: The Theory of Political Economy (1871), Chapter V, Theory of Labour, p. 173.

“There is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes quoted by Eutocius…
"Eratosthenes to King Ptolemy greeting.
"There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way,Too small thy plan to bound a royal tomb.
Let it be double; yet of its fair form
Fail not, but haste to double every side.But he was clearly in error; for when the aides are doubled, the area becomes four times as great, and the solid content eight times as great. Geometers also continued to investigate the question in what manner one might double a given solid while it remained in the same form.”

Thomas Little Heath (1861–1940) British civil servant and academic

Apollonius of Perga (1896)

Way, Error, Quotes, History

“We are less than a decade away from the medical lab the size of a sugar cube.”

Philippe Kahn (1952) Entrepreneur, camera phone creator

Founding speech for Fullpower, 2003, focusing in particular on the power of MEMS and Nanotechnology and its applications to life sciences.

“Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type x^3 + bx = c; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. …With respect to the irreducible case… we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic… before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2</sup”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1925) Vol.2, pp.461-464

Idea, Question, Time

“Prostrating myself mentally before my Guide, I cried, "How is it, O divine ideal of consummate loveliness and wisdom that I see thy inside, and yet cannot discern thy heart, thy lungs, thy arteries, thy liver?" "What you think you see, you see not," he replied; "it is not given to you, nor to any other Being to behold my internal parts. I am of a different order of Beings from those in Flatland. Were I a Circle, you could discern my intestines, but I am a Being, composed as I told you before, of many Circles, the Many in the One, called in this country a Sphere. And, just as the outside of a Cube is a Square, so the outside of a Sphere presents the appearance of a Circle."Bewildered though I was by my Teacher's enigmatic utterance, I no longer chafed against it, but worshipped him in silent adoration. He continued, with more mildness in his voice. "Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland. By degrees they will dawn upon you. Let us begin by casting back a glance at the region whence you came.”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART II: OTHER WORLDS, Chapter 18. How I came to Spaceland, and What I Saw There

Wisdom, Understanding, Thinking, Teachers

“The seventeenth century witnessed the birth of modern science as we know it today. The science was something new, based on a direct confrontation of nature by experiment and observation. But there was another feature of the new science—a dependence on numbers, on real numbers of actual experience.
…The ancients knew a few numerical laws… But prior to the Scientific Revolution, the goal of science (or the study of nature) was not to seek laws of nature expressed in terms of numbers or number relations. …the new science …not only found laws based on numbers but they were also willing to express these laws in terms of higher powers of numbers—squares and cubes.”

I. Bernard Cohen (1914–2003) American historian of science

The Triumph of Numbers: How Counting Shaped Modern Life (2005)

Law, Birth, Nature, Study

“Gilt-tooled on yard-square panels of green leather—imitation, of course—the zodiacal signs looked down from the walls of the executive lunch-room. The air was full of the chatter of voices and the clink of ice-cubes. Waiting to be attacked when the president of the company joined them (he had promised to show at one sharp) was a table laden with expensive food: hard-boiled eggs, shells intact so that it could be seen they were brown, free-range, rich in carotene; lettuces whose outer leaves had been rasped by slugs; apples and pears wearing their maggot-marks like dueling scars, in this case presumably genuine ones though it had been known for fruit growers to fake them with red-hot wires in areas where insects were no longer found; whole hams, very lean, proud of their immunity from antibiotics and copper sulphate; scrawny chickens; bread as coarse as sandstone, dark as mud and nubbled with wheat grains...”

John Brunner book The Sheep Look Up

December “HOUSE TO HOUSE”
The Sheep Look Up (1972)

Food, Waiting

“The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
Achimedes (1920)

Problem

“Vacuum stands and remains a mathematical space. A cube placed in a vacuum would not displace anything, as it would displace air or water in a space already containing those fluids.”

Robert Grosseteste (1175–1253) English bishop and philosopher

Commentarius in VIII Libros Physicorum Aristoteles (c. 1230-1235)

Mathematics, Water

“Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent, the finding of two mean proportionals in continued proportion between two given straight lines. …if all the recorded solutions are collected together, it is much easier to see the relations, amounting in some cases to substantial identity, between them, and to get a comprehensive view of the history of the problem. I have therefore dealt with this problem in a separate section of the chapter devoted to 'Special Problems,' and I have followed the same course with the other famous problems of squaring the circle and trisecting any angle.”

Thomas Little Heath (1861–1940) British civil servant and academic

A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

History, Problem, Parting

“What does a scanner see? he asked himself. I mean, really see? Into the head? Down into the heart? Does a passive infrared scanner like they used to use or a cube-type holo-scanner like they use these days, the latest thing, see into me—into us—clearly or darkly? I hope it does, he thought, see clearly, because I can’t any longer these days see into myself. I see only murk. Murk outside; murk inside. I hope, for everyone’s sake, the scanners do better. Because, he thought, if the scanner sees only darkly, the way I myself do, then we are cursed, cursed again and like we have been continually, and we’ll wind up dead this way, knowing very little and getting that little fragment wrong too.”

Philip K. Dick book A Scanner Darkly

Source: A Scanner Darkly (1977), Chapter 11 (p. 185)

Hope, Way, Heart, Dead

“I will not, from henceforward, talk to any squarer of the circle, trisector of the angle, duplicator of the cube, constructor of perpetual motion, subverter of gravitation, stagnator of the earth, builder of the universe, etc.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Introductory p.9
A Budget of Paradoxes (1872)

Universe

“The new architecture considers not only space, but also time to be an architectural value. The unity of space and time will give architecture a new form of appearance, which is more complete. This is what is meant by 'active space'.... the dissimilar space-cells develop eccentrically from the center to the borders of the cube, thereby granting a new plastic quality to the dimensions of height, width, depth, and time.”

Theo van Doesburg (1883–1931) Dutch architect, painter, draughtsman and writer

Quote from his unpublished writing, 'Fundamental principles', 1930; as cited in Theo van Doesburg, Joost Baljeu, Studio Vista, London 1974, p. 203
1926 – 1931

Appearance, Time

“We laugh out loud when we think of all the novices who expiate their literal understanding of the remarks of a cubist and their faith in absolute truth by laboriously placing side by side the six faces of a cube and both ears of a model seen in profile.”

Albert Gleizes (1881–1953) French painter

Quote of Gleizes, c. 1911; as cited by Anne Ganteführer-Trier, in 'Cubism, Taschen, 2004
1910s

Truth, Faith, Understanding, Thinking

Jean Metzinger (1883–1956) French painter

Du Cubisme (1912)

Truth, Faith, Thinking

“In the year 1775, the Paris Academy found it necessary to protect its officials against the waste of time and energy involved in examining the efforts of circle squarers. It passed a resolution… that no more solutions were to be examined of the problem of the duplication of the cube, the trisection of the angle, the quadrature of the circle, and the same resolution should apply to machines for exhibiting perpetual motion. an account… drawn up by Condorcet… is appended. It is interesting to remark that the strength of the conviction of Mathematicians that the solution of the problem is impossible, more than a century before an irrefutable proof of the correctness of that conviction was discovered.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), pp. 3-4

Problem, Effort, Time

“The earth is the sphere, the measure of all; round it describe a dodecahedron; the sphere including this will be Mars. Round Mars describe a tetrahedron; the sphere including this will be Jupiter. Describe a cube round Jupiter; the sphere including this will be Saturn. Now, inscribe in the earth an icosahedron, the sphere inscribed in it will be Venus: inscribe an octahedron in Venus: the circle inscribed in it will be Mercury.”

Johannes Kepler book Mysterium Cosmographicum

Walter William Bryant, Kepler (1920), pp. 16–17
Mysterium Cosmographicum (1596)

“There is a smallest unit of space. Its minimum value is given by the cube of the Planck length… If you take a volume of space and measure it to a very fine precision… It has to fall into some discrete series of numbers, just like the energy of an electron in an atom. …we can calculate the discrete areas and volumes from the theory.”

Lee Smolin (1955) American cosmologist

"Loop Quantum Gravity," The New Humanists: Science at the Edge (2003)

“Be able to sell ice-cubes to Eskimos – you may have to!”

Nigel Cumberland (1967) British author and leadership coach

Source: Your Job-Hunt Ltd – Advice from an Award-Winning Asian Headhunter (2003), p.65

“Even children know that drafting rule is easier to bend flat than across the edges; and… they will not be much surprised if told that for the same width of the rule its resistance is proportional to the square of the thickness and the deflection is inversely proportional to the cube of the thickness. Nevertheless some modern designers seem to be unaware… since they require… beams of such slenderness that they resemble springboards…”

Eduardo Torroja (1899–1961) Spanish architect

p, 125
Philosophy of Structures (1958)

Children

“By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded pari passu with the completion of the body of the Elements.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
Achimedes (1920)

Question, Problem, Time, Parting

“If the formula for water is H2O, is the formula for an ice cube H2O squared?”

Lily Tomlin (1939) American actress, comedian, writer, and producer

Contributions of Jane Wagner

Water

“If the formula for water is H2O, is the formula for an ice cube H2O squared?”

Jane Wagner (1935) Playwright, actress

Other material for Lily Tomlin

Water

“I have an artificial left eye. I lost my real eye in a car accident when I was eighteen. In fact, I had to have my entire face rebuilt because I smashed it up pretty good. It took six years and thirteen surgeries. However, I did have the pleasure of freezing a plastic eyeball in an ice cube, putting it in a friend's drink, ("Eyeball in your highball?") and watching him freak completely. Okay, so maybe that's not going down on my good karma record. But it sure was fun.”

Libba Bray (1964) American teen writer

Twenty-One Things You Don't Know About Me

Pleasure, Cars, Drink

“Where do you think I was today? I stood straight in front of him (Himmler) for a whole hour and talked, and he… he played with a puzzle the whole time – you know, this glass cube with three balls on the inside… When I finished, he took off his pince-nez, wiped it with a handkerchief – he has a skull even on his handkerchief – and said, "Listen, Ernst! Have you by any chance, ever had a dream, where you're riding in the back of a ragged truck to who knows where, and some monsters are sitting around you?" I didn’t say anything. Then he smiled and said, "Ernst, you know, I know as well as you that no astral exists. But what do you think, if you, and even Canaris, have your own people in 'Annenerbe', shouldn’t I have my own people there as well?" I did not understand what he meant. "Think Ernst, think!" he said. I kept silent. Then he smiled and asked, "Whose man do you think is Kröger?" …Yes, Emma… It seems I'm too simple for all these intrigues… But I know that while the Führer needs me, my heart will keep beating… You know, Emma… Sometimes it seems to me, that it's not me who is alive, but it's the Führer who is living inside me…”

Ernst Kaltenbrunner (1903–1946) Austrian-born senior official of Nazi Germany executed for war crimes

To Emma, recorded by secret spy listening device WS-M/13 located in Kaltenbrunner's bedroom, 1/14/1935. Quoted in "Kröger's Revelation" - by Viktor Pelevin - 1991 - Page 277

People, Smile, Dreams, Understanding

“Hippocrates also attacked the problem of doubling the cube. …Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i. e. finding x, y such that a:x=x:y=y:b, where a, b are the two given straight lines. It is easy to see that, if a:x=x:y=y:b, then b/a = (x/a)3, and, as a particular case, if b=2a, x3=2a3, so that the side of the cube which is double of the cube of side a is found.”

Thomas Little Heath (1861–1940) British civil servant and academic

Achimedes (1920)

Problem

“I remember when I was doing a film with Fred Astaire, it was nothing for him to work three or four days on two bars of music. One evening in the dark grey hours of dusk, I was walking across the deserted MGM lot when a small, weary figure with a towel around his neck suddenly appeared out of the giant cube sound stages. It was Fred. He came over to me, threw a heavy arm around my shoulder and said: "Oh Alan, why doesn't someone tell me I cannot dance?" The tormented illogic of his question made any answer insipid, and all I could do was walk with him in silence.”

Fred Astaire (1899–1987) American dancer, singer, actor, choreographer and television presenter

Alan Jay Lerner in Lerner, Alan Jay. On the Street Where I Live. New York: Norton, 1978. p. 89. (M).

Music, Question, Silence, Appearance

“Try to imagine that Minkowski cube, Ray said, as a block of liquid water freezing (as contrary as this seems) from the bottom up. The progression of the freeze represents at least our human experience of the march of time. What is frozen is past, immutable, changeless. What is liquid is future, indeterminate, uncertain. We live on the crystallizing boundary.”

Robert Charles Wilson book The Chronoliths

Source: The Chronoliths (2001), Chapter 9 (pp. 114-115)

Imagination, Water, Time, Past

“Sam is a repetitive, comic process that merely marks time: he gets nowhere, but then he doesn’t want to get anywhere. Although there is no possibility of any real change in Sam, he never stops changing: Sam stays there inside Sam, getting less and less like the rest of mankind and more and more like Sam, Sam squared, Sam cubed, Sam to the nth.”

Randall Jarrell (1914–1965) poet, critic, novelist, essayist

“An Unread Book”, p. 40
The Third Book of Criticism (1969)

Change, Time

“I do not consider myself a Cubist either because I have come to the conclusion that cubes are not always made for expressing the thought of the brain and of the feeling of the spirit... I capture all these impressions [the visual sensations, Picabia sensed in the modern city] without any hurry to transfer to the canvas. I let them rest in my brain and then, when I'm visited by the spirit of creation, I improvise my paintings just as a musician improvises his music.”

Francis Picabia (1879–1953) French painter and writer

1910's
Source: 'How I see New York', in 'The New York American', New York 30 March 1913, p. 11

Music, Feeling, Sense, Painting

“In the table—and in nature—we find (leaving aside the antineutrino) fifteen fundamental fermions, with diverse strong, weak, and electromagnetic charges. …They are so closely related by symmetry transformations that they are, so to speak, no more than different faces of the same cube.”

Frank Wilczek (1951) physicist

Longing for the Harmonies: Themes and Variations from Modern Physics (1987)

Nature, Strong

“While then for a long time everyone was at a loss, Hippocrates of Chios was the first to observe that, if between two straight lines of which the greater is double of the less it were discovered how to find two mean proportionals in continued proportion, the cube would be doubled; and thus he turned the difficulty in the original problem into another difficulty no less than the former. Afterwards, they say, some Delians attempting, in accordance with an oracle, to double one of the altars fell into the same difficulty. And they sent and begged the geometers who were with Plato in the Academy to find for them the required solution. And while they set themselves energetically to work and sought to find two means between two given straight lines, Archytas of Tarentum is said to have discovered them by means of half-cylinders, and Eudoxus by means of the so-called curved lines. It is, however, characteristic of them all that they indeed gave demonstrations, but were unable to make the actual construction or to reach the point of practical application, except to a small extent Menaechmus and that with difficulty.”

Thomas Little Heath (1861–1940) British civil servant and academic

Apollonius of Perga (1896)

Problem, Loss, Time

“I taught myself decimals, equations, the square, cube, and biquadrate roots. I got some knowledge of logarithms, and some of algebra. I readily got through a small schoolbook of geometry; and having an odd volume, the first, of Williamson's ' Euclid,' I attacked it vigorously and perseveringly. Williamson's is by no means the best book on the subject, yet I am still of opinion that it is the best book I could have had for the purpose of teaching myself.”

Francis Place (1771–1854) English social reformer

Source: The life of Francis Place, 1771-1854, 1898, p. 18

Books, Knowledge

“The great Cartesian invention had its roots in those famous problems of antiquity which originated in the days of Plato. In endeavoring to solve the problems of the trisection of an angle, of the duplication of the cube and of the squaring of the circle, the ruler and compass having failed them, the Greek geometers sought new curves. They stumbled on the conic sections…There we find the nucleus of the method which Descartes later erected into a principle. Thus Apollonius referred the parabola to its axis and principal tangent, and showed that the semichord was the mean propotional between the latus rectum and the height of the segment. Today we express this relation by x2 = Ly, calling the height the ordinate (y) and the semichord the abscissa (x); the latus rectum being… L. …the Greeks named these curves and many others… loci… Thus the ellipse was the locus of a point the sum of the distances of which from two fixed points was constant. Such a description was a rhetorical equation of the curve…”

Tobias Dantzig (1884–1956) American mathematician

Number: The Language of Science (1930)

Lying, Problem

“The discovery of Hippocrates amounted to the discovery of the fact that from the relation
(1)\frac{a}{x} = \frac{x}{y} = \frac{y}{b}it follows that(\frac{a}{x})^3 = [\frac{a}{x} \cdot \frac{x}{y} \cdot \frac{y}{b} =] \frac{a}{b}and if a = 2b, [then (\frac{a}{x})^3 = 2, and]a^3 = 2x^3.The equations (1) are equivalent [by reducing to common denominators or cross multiplication] to the three equations
(2)x^2 = ay, y^2 = bx, xy = ab[or equivalently…y = \frac{x^2}{a}, x = \frac{y^2}{b}, y = \frac{ab}{x} ]Doubling the Cube
the 2 solutions of Menaechmusand the solutions of Menaechmus described by Eutocius amount to the determination of a point as the intersection of the curves represented in a rectangular system of Cartesian coordinates by any two of the equations (2).
Let AO, BO be straight lines placed so as to form a right angle at O, and of length a, b respectively. Produce BO to x and AO to y.
The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i. e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i. e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have

\begin{cases}y^2 = b. ON = b. PM = bx\\ and\\ xy = PM. PN = ab\end{cases}whence\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.
In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a.”

Thomas Little Heath (1861–1940) British civil servant and academic

The point P where the two parabolas intersect is given by<center><math>\begin{cases}y^2 = bx\\x^2 = ay\end{cases}</math></center>whence, as before,<center><math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</math></center>
Apollonius of Perga (1896)

Drawing

“Science doesn't always go forwards. It's a bit like doing a Rubik's cube. You sometimes have to make more of a mess with a Rubik's cube before you can get it to go right.”

Jocelyn Bell Burnell (1943) British scientist

Beautiful Minds (2010)
Context: Science doesn't always go forwards. It's a bit like doing a Rubik's cube. You sometimes have to make more of a mess with a Rubik's cube before you can get it to go right. You build up this picture of what there is and you believe it to be true and you work with this picture and you refine it but sometimes you have to abandon the picture. Sometimes you discover the picture you thought you had, that everybody thought we had, actually turns out to be wrong.

Science

“Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes — we are all liable to the same errors, all alike the Slaves of our respective Dimensional prejudices, as one of your Spaceland poets has said —”

Edwin Abbott Abbott book Flatland

Preface to the Second and Revised Edition (1884)
Flatland: A Romance of Many Dimensions (1884)
Context: p>Suppose a person of the Fourth Dimension, condescending to visit you, were to say, 'Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you INFER a Solid (which is of Three); but in reality you also see (though you do not recognize) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.' What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes — we are all liable to the same errors, all alike the Slaves of our respective Dimensional prejudices, as one of your Spaceland poets has said — 'One touch of Nature makes all worlds akin.' </p

Family, Error, Strong

“There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own — shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein?”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART II: OTHER WORLDS, Chapter 19. How, Though the Sphere Showed Me Other Mysteries of Spaceland, I Still Desired More; and What Came of It
Context: p>Those who have thus appeared — no one knows whence — and have returned — no one knows whither — have they also contracted their sections and vanished somehow into that more Spacious Space, whither I now entreat you to conduct me?SPHERE (MOODILY). They have vanished, certainly — if they ever appeared. But most people say that these visions arose from the thought — you will not understand me — from the brain; from the perturbed angularity of the Seer.I. Say they so? Oh, believe them not. Or if it indeed be so, that this other Space is really Thoughtland, then take me to that blessed Region where I in Thought shall see the insides of all solid things. There, before my ravished eye, a Cube, moving in some altogether new direction, but strictly according to Analogy, so as to make every particle of his interior pass through a new kind of Space, with a wake of its own — shall create a still more perfect perfection than himself, with sixteen terminal Extra-solid angles, and Eight solid Cubes for his Perimeter. And once there, shall we stay our upward course? In that blessed region of Four Dimensions, shall we linger on the threshold of the Fifth, and not enter therein? Ah, no! Let us rather resolve that our ambition shall soar with our corporal ascent. Then, yielding to our intellectual onset, the gates of the Sixth Dimension shall fly open; after that a Seventh, and then an Eighth —How long I should have continued I know not. In vain did the Sphere, in his voice of thunder, reiterate his command of silence, and threaten me with the direst penalties if I persisted. Nothing could stem the flood of my ecstatic aspirations. Perhaps I was to blame; but indeed I was intoxicated with the recent draughts of Truth to which he himself had introduced me. However, the end was not long in coming. My words were cut short by a crash outside, and a simultaneous crash inside me, which impelled me through space with a velocity that precluded speech. Down! down! down! I was rapidly descending; and I knew that return to Flatland was my doom. One glimpse, one last and never-to-be-forgotten glimpse I had of that dull level wilderness — which was now to become my Universe again — spread out before my eye. Then a darkness. Then a final, all-consummating thunder-peal; and, when I came to myself, I was once more a common creeping Square, in my Study at home, listening to the Peace-Cry of my approaching Wife.</p

Perfection