Geometrical Quotes & Sayings

A geometrical theory in physical interpretation can never be validated with mathematical certainty ... ; like any other theory of empirical science, it can acquire only a more or less high degree of confirmation. — Carl Gustav Hempel

The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure. — Joseph-Louis Lagrange

String theorists have found special pairs of geometrical shapes for space that have completely different features when each is probed by unwrapped strings. They also have completely different features when each is probed by wrapped strings. But-and this is the punch line-when probed both ways, with wrapped and unwrapped strings, the shapes become indistinguishable. what the unwrapped strings see on one space, the wrapped strings see on the other, and vice versa, rendering identical the collective picture gleaned from the full physics of string theory. — Brian Greene

The original route, whether called U. S. 40 or something else, will furnish the main-traveled road until someone disproves the geometrical proposition that a straight line is the shortest distance between points. — George R. Stewart

General relativity is the cornerstone of cosmology and astrophysics. It has also provided the conceptual basis for string theory and other attempts to unify all the forces of nature in terms of geometrical structures. — Paul Davies

Population, when unchecked, increases in a geometrical ratio. — Thomas Malthus

Everything is composed of small particles of itself and they are flying around in concentric circles and arcs and segments and innumerable other geometrical figures too numerous to mention collectively, never standing still or resting but spinning away and darting hither and thither and back again, all the time on the go. These diminutive gentlemen are called atoms. Do you follow me intelligently? — Flann O'Brien

CHAPTER VII Geometrical Details. - Calculation of the Capacity of the Balloon. - The Double Receptacle. - The Covering. - The Car. - The Mysterious Apparatus. - The Provisions and Stores. - The Final Summing up. — Jules Verne

Astronomy was the cradle of the natural sciences and the starting point of geometrical theories. — Cornelius Lanczos

The game created a parallel world, Sidney thought. It was drama; it was excitement; it was a metaphor for the vicissitudes of life. It was also quintessentially English: democratic (there were teams with all levels of ability), communal (the cricket 'square' was often at the centre of the village green), and convivial (the game was full of eccentric characters.) It was the representation of a nation's cuisine, with its milky tea, cucumber sandwiches, Victoria sponge and lashings of beer. It was also beautiful to watch, with fifteen men, dressed in white and moving on green, creating geometrical patterns that looked as if they had been choreographed by a divine choreographer. As — James Runcie

In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it came to us from Euclid.
As belonging to these imperfections, I consider the obscurity in the fundamental concepts of the geometrical magnitudes and in the manner and method of representing the measuring of these magnitudes, and finally the momentous gap in the theory of parallels, to fill which all efforts of mathematicians have so far been in vain. — Nikolai Lobachevsky

Remembering ... that Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun, the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces. — Marcus Vitruvius Pollio

It is impossible to anticipate all of the misdeeds engendered by the universal conflict of human passions. They multiply at a compound rate with the growth in population and the interlacing of particular interests that cannot be directed with geometrical precision towards the public utility. — Cesare Beccaria

Bees... by virtue of a certain geometrical forethought, knew that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material. — Pappus

At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, "out of a curiosity to see what there was in it." He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid's Elements of Geometry, and began to read. Two years later he invented the differential calculus. — Carl Sagan

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas. — David Hilbert

Pure geometrical regularity gives a certain pleasure to men troubled by the obscurity of outside appearance. The geometrical line is something absolutely distinct from the messiness, the confusion, and the accidental details of existing things. — T. E. Hulme

No Roman ever died in contemplation over a geometrical diagram. — Alfred North Whitehead

A planar geometrical figure with more than three vertices can be decomposed into a set of triangles, and it can be reconstructed from a set of triangles. — Vladimir Prelog

Being perpetually charmed by his familiar siren, that is, by his geometry, he neglected to eat and drink and took no care of his person; that he was often carried by force to the baths, and when there he would trace geometrical figures in the ashes of the fire, and with his finger draws lines upon his body when it was anointed with oil, being in a state of great ecstasy and divinely possessed by his science. — Plutarch

The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth. This — Marcus Du Sautoy

Population, when unchecked, goes on doubling itself every 25 years or increases in a geometrical ratio. — Thomas Malthus

Next comes the realist phase ("After all, from a purely geometrical point of view a cat is only a tube with a door at the top."). — Terry Pratchett

Boldwood looked at her - not slily, critically, or understandingly, but blankly at gaze, in the way a reaper looks up at a passing train - as something foreign to his element, and but dimly understood. To Boldwood women had been remote phenomena rather than necessary complements - comets of such uncertain aspect, movement, and permanence, that whether their orbits were as geometrical, unchangeable, and as subject to laws as his own, or as absolutely erratic as they superficially appeared, he had not deemed it his duty to consider. He — Thomas Hardy

In this science the illustrations and examples are not confined in their effect merely to the practice they afford in the analytical art, but [ ... ] they also store the mind with independent geometrical and physical knowledge. Besides, it should be considered, that the only effectual method of impressing abstract formulae and rules upon the memory, and, indeed, of making them fully and clearly apprehended by the understanding, is by examples of their practical application. — Dionysius Lardner

Many transition states have a well-defined preferred geometrical requirement. — Derek Harold Richard Barton

Surprise quality can be achieved in many ways. It may be produced by a certain stimulating geometrical relationship between elements in the picture or through the human interest of the situation photographed or by calling our attention to some commonplace but fascinating thing we have never noticed before or it can be achieved by looking at an everyday thing in a new interesting way. — Alexey Brodovitch

With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series the sum of which has been rigorously determined. In other words, the things which are the most important in mathematics are also those which have the least foundation. — Niels Henrik Abel

The effect was that of a Cyclopean city of no architecture known to man or to human imagination, with vast aggregations of night-black masonry embodying monstrous perversions of geometrical laws and attaining the most grotesque extremes of sinister bizarrerie. — H.P. Lovecraft

Whether we like it or not, if we are to pursue a career in science, eventually we have to learn the "language of nature": mathematics. Without mathematics, we can only be passive observers to the dance of nature rather than active participants. As Einstein once said, "Pure mathematics is, in its way, the poetry of logical ideas." Let me offer an analogy. One may love French civilization and literature, but to truly understand the French mind, one must learn the French language and how to conjugate French verbs. The same is true of science and mathematics. Galileo once wrote, "[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to understand a single word. — Michio Kaku

Deep down, it's all baseball, no matter what kind of geometrical shape you play it with. — Vernon D. Burns

When Leonardo da Vinci wanted to create a whole new style of painting, one that was more lifelike and emotional, he engaged in an obsessive study of details. He spent endless hours experimenting with forms of light hitting various geometrical solids, to test how light could alter the appearance of objects. He devoted hundreds of pages in his notebooks to exploring the various gradations of shadows in every possible combination. He gave this same attention to the folds of a gown, the patterns in hair, the various minute changes in the expression of a human face. When we look at his work we are not consciously aware of these efforts on his part, but we feel how much more alive and realistic his paintings are, as if he had captured reality. — Robert Greene

Certainly, the terror of a deserted house swells in geometrical rather than arithmetical progression as houses multiply to form a city of stark desolation. The sight of such endless avenues of fishy-eyed vacancy and death, and the thought of such linked infinities of black, brooding compartments given over to cob-webs and memories and the conqueror worm, start up vestigial fears and aversions that not even the stoutest philosophy can disperse. — H.P. Lovecraft

I know death hath ten thousand several doors
For men to take their exits; and 'tis found
They go on such strange geometrical hinges,
You may open them both ways: any way, for heaven-sake — John Webster

Another means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions or property in geometrical progression as they rise. Whenever there are in any country uncultivated lands and unemployed poor, it is clear that the laws of property have been so far extended as to violate natural right. The earth is given as a common stock for man to labor and live on.
Letter to James Madison, October 28, 1785 — Thomas Jefferson

In his ... 'Geometrical peculiarities of the Pyramids', Ballard shows the relationship between the equal area theory and the golden number. After checking Herodotus' statement via dimensions Ballard concludes: 'I have therefore the authority of Herodotus to support the theory which I shall subsequently set forth, that this pyramid was the exponent of lines divided in mean and extreme ratio. — Roger Herz-Fischler

Nobody since Newton has been able to use geometrical methods to the same extent for the like purposes; and as we read the Principia we feel as when we are in an ancient armoury where the weapons are of gigantic size; and as we look at them we marvel what manner of man he was who could use as a weapon what we can scarcely lift as a burden. — William Whewell

I sit in my chair, the wreath on the ceiling floating above my head, like a frozen halo, a zero. A hole in space where a star exploded. A ring, on water, where a stone's been thrown. All things white and circular. I wait for the day to unroll, for the earth to turn, according to the round face of the implacable clock. The geometrical days, which go around and around, smoothly and oiled. Sweat already on my upper lip, I wait, for the arrival of the inevitable egg, which will be lukewarm like the room and will have a green film on the yolk and will taste faintly of sulphur. Today, — Margaret Atwood

Sit down, keep your back straight, relax and have an object on which to concentrate; you might use a candle flame, a brightly colored rock, a yantra, which is a geometrical designed specifically for the practice of concentration. — Frederick Lenz

The universe will finally become a ball of radiation, becoming more and more rarified and passing into longer and longer wave-lengths. The longest waves of radiation are Hertzian waves of the kind used in broadcasting. About every 1500 million years this ball of radio waves will double in diameter; and it will go on expanding in geometrical progression for ever. Perhaps then I may describe the end of the physical world as-one stupendous broadcast. — Arthur Eddington

The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician's "blank form of a universe" without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition. Just as Newton first conclusively showed that this is a world of masses, so Willard Gibbs first revealed it as a world of systems. — Lawrence Joseph Henderson

Jean-Pierre Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory, Springer Science & Business Media, 2008. — Roger Scruton

To characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental. — Carl Gustav Hempel

Within the holy circle of the stage, geometrical principles are used to organize physical energy so that the beauty of the human form reveals a web of drama and truth---that is the essence of dance as I have come to know it. — Gelsey Kirkland

The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation. — James Joseph Sylvester

Philosophy [nature] is written in that great book which ever is before our eyes
I mean the universe
but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth. — Galileo Galilei

Homesickness is a great teacher. It taught me, during an endless rainy fall, that I came from the arid lands, and like where I came from. I was used to dry clarity and sharpness in the air. I was used to horizons that either lifted into jagged ranges or rimmed the geometrical circle of the flat world. I was used to seeing a long way. I was used to earth colors
tan, rusty red, toned white
and the endless green of Iowa offended me. I was used to a sun that came up over mountains and went down behind other mountains. I missed the color and smell of sagebrush, and the sight of bare ground. — Wallace Stegner

Faith--in the sense of an unreserved commitment which is never completely justified-enters the picture as soon as we leave the realm of pure geometrical ideas and have to deal
with the existing world. Each of our perceptions is an act of faith in
that it affirms more than we strictly know, since objects are inexhaustibJe and our information limited. — Maurice Merleau Ponty

Each organic being is striving to increase in a geometrical ratio ... each at some period of its life, during some season of the year, during each generation or at intervals, has to struggle for life and to suffer great destruction ... The vigorous, the healthy, and the happy survive and multiply. — Charles Darwin

There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry. — Martin Gardner

Large department stores, with their luxuriant abundance of canned goods, foods, and clothing, are like the primary landscape and the geometrical locus of affluence. Streets with overcrowded and glittering store windowsthe displays of delicacies, and all the scenes of alimentary and vestimentary festivity, stimulate a magical salivation. Accumulation is more than the sum of its products: the conspicuousness of surplus, the final and magical negation of scarcitymimic a new-found nature of prodigious fecundity. — Jean Baudrillard

I don't want my 'part' taken! I haven't 'got' a part! I hate the stupid geometrical figures by which people try to understand the emotions of others, imposing hard straight lines - or 'sides' as they call them - onto tender curvaceous human beings who have none. — Frances Partridge

Legislators cannot invent too many devices for subdividing property ... Another means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions or property in geometrical progression as they rise. Whenever there are in any country uncultivated lands and unemployed poor, it is clear that the laws of property have been so far extended as to violate natural right. — Thomas Jefferson

The mathematical thermology created by Fourier may tempt us to hope that, as he has estimated the temperature of the space in which we move, me may in time ascertain the mean temperature of the heavenly bodies: but I regard this order of facts as for ever excluded from our recognition. We can never learn their internal constitution, nor, in regard to some of them, how heat is absorbed by their atmosphere. We may therefore define Astronomy as the science by which we discover the laws of the geometrical and mechanical phenomena presented by the heavenly bodies. — Auguste Comte

The personality and style of a photographer usually limits the type of subject with which he deals best. For example Cartier-Bresson is very interested in people and in travel; these things plus his precise feeling for geometrical relationships determine the type of pictures he takes best. What is of value is that a particular photographer sees the subject differently. A good picture must be a completely individual expression which intrigues the viewer and forces him to think. — Alexey Brodovitch

Frederick consistently used his central position to concentrate against one fraction of the enemy, and he always employed tactics of indirect approach. Thereby he gained many victories. But his tactical indirect approach was geometrical rather than psychological-unprepared by the subtler forms of surprise favoured by Scipio-and for all their executive skill, these manoeuvres were narrow. The opponent might be unable to meet the following blow, owing to the inflexibility of his mind or his formations, but the blow itself did not fall unexpectedly. — B.H. Liddell Hart

In this dynamic rivalry between house and universe, we are far removed from any reference to simple geometrical forms. A house that has been experienced is not an inert box. Inhabited space transcends geometrical space. — Gaston Bachelard

Ethics is not a
geometrical calculation; "others" are never very far from "us"; "they" and "we" are co-constituted and entangled through the very cuts "we" help to enact. Cuts cut "things" together and apart. Cuts are not enacted from the outside, nor are they ever enacted once and for all. — Karen Barad

The more we progress the more we tend to progress. We advance not in arithmetical but in geometrical progression. We draw compound interest on the whole capital of knowledge and virtue which has been accumulated since the dawning of time. — Arthur Conan Doyle

'Eclipse' is a concept piece, and its concept centers on 36 large light bulbs strung from above in a geometrical pattern and at different heights, some of them at times down below the dancers' chest level. — Robert Gottlieb

I like to think that Einstein would look at string theory's journey and smile, enjoying the theory's remarkable geometrical features while feeling kinship with fellow travelers on the long and winding road toward unification. — Brian Greene

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. — G.H. Hardy

Both a priori reasoning and experience teach us that as as these funds grow larger the geometrical rate of growth by compound interest ultimately defeats itself. — Benjamin Graham

Beauty cannot be defined by abscissas and ordinates; neither are circles and ellipses created by their geometrical formulas. — Carl Von Clausewitz

If they would, for Example, praise the Beauty of a Woman, or any other Animal, they describe it by Rhombs, Circles, Parallelograms, Ellipses, and other geometrical terms ... — Jonathan Swift

I review all I know, but can synthesize no meaning. When I doze, the Fact, the certain accomplished calamity, wakes me roughly like a brutal nurse. I see it crouching inflexibly in a corner of the ceiling. It comes down in geometrical diagonal like lightning.
It says, I remain, I AM, I shall never cease to be: your memory will grow a deathly glaze: you will forget, you will fade out, but I cannot be undone.
Thus every quarter hour it puts the taste of death in my mouth, and shows me, but not gently, how I go whoring after oblivion. — Elizabeth Smart

In my sleep I had a dream and this dream was a dream of guilt. It was not human guilt but the kind of helpless, hopeless despair that would be felt by a small wooden box or geometrical cube if such objects had consciousness; it was the guilt of sheer existence. — Joanna Russ

All conceptions in the game of chess have a geometrical basis. — Eugene Znosko-Borovsky

The mathematical difficulties of the theory of rotation arise chiefly from the want of geometrical illustrations and sensible images, by which we might fix the results of analysis in our minds. — James Clerk Maxwell

This simple idea served to provide information on the geometrical shape of reacting molecules, and I was able to make the role of the frontier orbitals in chemical reactions more distinct through visualization, by drawing their diagrams. — Kenichi Fukui

The beauty of life is, therefore, geometrical beauty of a type that Plato would have much appreciated. — John Desmond Bernal

Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence. — Ibn Khaldun

We can ... treat only the geometrical aspects of mathematics and shall be satisfied in having shown that there is no problem of the truth of geometrical axioms and that no special geometrical visualization exists in mathematics. — Hans Reichenbach

In the candle's flickering light, the library's thousands of books emerged from the shadows, and for a moment Nicholas could not help admiring them again. During free time he had almost never looked up from the pages he was reading, but now he saw the books anew, from without rather than from within, and was reminded of how beautiful they were simply as objects. The geometrical wonder of them all, each book on its own and all the books together, row upon row, the infinite patterns and possibilities they presented. They were truly lovely. — Trenton Lee Stewart

I think the Greeks first took astrology to India and took from the Hindus the science of astronomy and carried it back with them from Europe. Because in India you will find old altars made according to a certain geometrical plan, and certain things had to be done when the stars were in certain positions, therefore I think the Greeks gave the Hindus astrology, and the Hindus gave them astronomy. — Swami Vivekananda

To pray is to pay attention to something or someone other than oneself. Whenever a man so concentrates his attention - on a landscape, a poem, a geometrical problem, an idol, or the True God - that he completely forgets his own ego and desires, he is praying. The primary task of the schoolteacher is to teach children, in a secular context, the technique of prayer. — W. H. Auden

Geometry ... is of much assistance in architecture, and in particular it teaches us the use of the rule and compasses, by which especially we acquire readiness in making plans for buildings in their grounds, and rightly apply the square, the level, and the plummet. By means of optics ... the light in buildings can be drawn from fixed quarters of the sky ... Difficult questions involving symmetry are solved by means of geometrical theories and methods. — Marcus Vitruvius Pollio

I quickly learned that reading is cumulative and proceeds by geometrical progression: each new reading builds upon whatever the reader has read before. — Alberto Manguel

A few rules include all that is necessary for the perfection of the definitions, the axioms, and the demonstrations, and consequently of the entire method of the geometrical proofs of the
art of persuading. — Blaise Pascal

I am conscious that an equal division of property is impracticable. But the consequences of this enormous inequality [in Europe] producing so much misery to the bulk of mankind, legislators cannot invent too many devices for subdividing property, ... [One] means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions of property in geometrical progression as they rise. — Thomas Jefferson

The concentration and reciprocal effect of industry and agriculture conjoin in a growth of productive powers, which increases more in geometrical than in arithmetical proportion. — Friedrich List

The ancient Egyptian calendar is Precessionally Sexagesimal (Besides being theologically/decanally decimal). That means that the toggling between its "enhanced" Civil Calendar (i.e., 365 days yearly) and the geometrical Original Calendar (i.e., 360 days yearly) is based on the precession of the equinoxes (rather than being solely anchored in the solar system); where 148 squared over 365 equals to 60; and 148 multiples of 360 over 365 equals to the height of the Great Pyramid. — Ibrahim Ibrahim

We may, I believe, anticipate that the chemist of the future who is interested in the structure of proteins, nucleic acids, polysaccharides, and other complex substances with high molecular weight will come to rely upon a new structural chemistry, involving precise geometrical relationships among the atoms in the molecules and the rigorous application of the new structural principles, and that great progress will be made, through this technique, in the attack, by chemical methods, on the problems of biology and medicine. — Linus Pauling

According to Thoth, because of the placement of the Great Pyramid on the Earth connecting into the Earth's huge geometrical field - specifically the octahedral field of the Earth, which is equivalent to our own fields - and because of the pyramid's mass and the geometries used in it, the white-light energy field spirals upward and becomes extremely strong, stretching all the way out to the center of the galaxy. The dark-light energy comes in from above, spirals through zero point and connects with the center of the Earth. In this way the Great Pyramid connects the center of the Earth to the center of our galaxy. — Drunvalo Melchizedek

We cannot ... prove geometrical truths by arithmetic. — Aristotle.

The couturier should be a geometrician, for the human body makes
geometrical figures to which the materials should correspond. — Madeleine Vionnet

If I should throw down a thousand beans at random upon a table, I could doubtless, by eliminating a sufficient number of them, leave the rest in almost any geometrical pattern you might propose to me, and you might then say that that pattern was the thing prefigured beforehand, and that the other beans were mere irrelevance and packing material. Our dealings with Nature are just like this. — William James