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Dimension

maximum number of independent directions within a mathematical space
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A dimension is a conceptual means of designating and measuring many aspects of the complexity of Reality or any mathematical systems, observable within any world, Universe or Cosmos. In physics and mathematics, the dimension of a mathematical space, object, or event is informally defined as the minimum number of coordinates needed to specify any points within it. A line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude is required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. The four common dimensions of spacetime consists of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. 10 or 11 dimensions are used in string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical spaces and times of our Universe. In general usage, the word dimension can refer to any specifiable aspects or qualities of reality or perceptions.

Like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity. ~ Edwin Abbott Abbott

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My loyalties will not be bound by national borders, or confined in time by one nation's history, or limited in the spiritual dimension by one language and culture. ~ Edward Abbey

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  • Eternity isn't some later time. Eternity isn't a long time. Eternity has nothing to do with time. Eternity is that dimension of here and now which thinking and time cuts out. This is it. And if you don't get it here, you won't get it anywhere. And the experience of eternity right here and now is the function of life. There's a wonderful formula that the Buddhists have for the Bodhisattva, the one whose being (sattva) is illumination (bodhi), who realizes his identity with eternity and at the same time his participation in time. And the attitude is not to withdraw from the world when you realize how horrible it is, but to realize that this horror is simply the foreground of a wonder and to come back and participate in it.

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  • In a sensible theory there are no [dimensionless] numbers whose values are determinable only empirically. I can, of course, not prove that ... dimensionless constants in the laws of nature, which from a purely logical point of view can just as well have other values, should not exist. To me in my 'Gottvertrauen' [faith in God] this seems evident, but there might be few who have the same opinion ...

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Faith is sensitiveness to what transcends nature, knowledge and will, awareness of the ultimate, alertness to the holy dimension of all reality. ~ Abraham Joshua Heschel
  • Bailey reached up and shook his arm. "Snap out of it. What the hell are you talking about, four dimensions? Time is the fourth dimension; you can't drive nails into that."
Teal shrugged him off. "Sure. Sure. Time is a fourth dimension, but I'm thinking about a fourth spatial dimension, like length, breadth, and thickness. For economy of materials and convenience of arrangement you couldn't beat it. To say nothing of the saving of ground space—you could put an eight-room house on the land now occupied by a one-room house. Like a tesseract—"
  • And He Built a Crooked House by Robert Heinlein [www.math.union.edu/~dpvc/...11/.../crooked-house.pdf]
  • The supremacy of expediency is being refuted by time and truth. Time is an essential dimension of existence defiant of man's power, and truth reigns in supreme majesty, unrivaled, inimitable, and can never be defeated.
  • Faith is sensitiveness to what transcends nature, knowledge and will, awareness of the ultimate, alertness to the holy dimension of all reality. Faith is a force in man, lying deeper than the stratum of reason and its nature cannot be defined in abstract, static terms. To have faith is not to infer the beyond from the wretched here, but to perceive the wonder that is here and to be stirred by the desire to integrate the self into the holy order of living. It is not a deduction but an intuition, not a form of knowledge, of being convinced without proof, but the attitude of mind toward ideas whose scope is wider than its own capacity to grasp.
    Such alertness grows from the sense for the meaningful, for the marvel of matter, for the core of thoughts. It is begotten in passionate love for the significance of all reality, in devotion to the ultimate meaning which is only God.
    • Abraham Joshua Heschel, in "The Holy Dimension" in Moral Grandeur and Spiritual Audacity : Essays (1997), p. 330

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  • How sickness enlarges the dimension of a man's self to himself!

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Though she could not put it clearly even into thoughts, the peculiar and exact lines of the tunnel had been carefully angled to lead through poly-dimensional space as well as through the underground — perhaps through time, too. ~ C. L. Moore
  • Lois Lane: I can’t describe what Mxyzptlk then became. He had height, width, depth, and a couple of other things, too.
    • Alan Moore, Whatever Happened to The Man of Tomorrow?
  • She was no scholar in geometry or aught else, but she felt intuitively that the bend and slant of the way she went were somehow outside any other angles or bends she had ever known. They led into the unknown and the dark, but it seemed to her obscurely that they led into deeper darkness and mystery than the merely physical, as if, though she could not put it clearly even into thoughts, the peculiar and exact lines of the tunnel had been carefully angled to lead through poly-dimensional space as well as through the underground — perhaps through time, too.

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Look, Cooper, they're creatures of at least five dimensions, to them the past might be a canyon they can climb into and the future a mountain they can climb up... but to us it's not, okay? ~ Amelia Brand in Interstellar
  • Time is relative, it can stretch and squeeze, but it can't run backwards. The only thing that can move across dimensions like time is gravity. … Look, Cooper, they're creatures of at least five dimensions, to them the past might be a canyon they can climb into and the future a mountain they can climb up... but to us it's not, okay?

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All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child  especially a baby might see more. ~ Lewis Padgett
  • "Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
    "Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child  especially a baby might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."

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There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone. ~ Rod Serling
  • You are about to enter another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination. Next stop, the Twilight Zone!
    • Rod Serling, in The Twilight Zone (1959 TV series), Season 1 alternate opening
  • You're traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination. Next stop, the Twilight zone!
    • Rod Serling, in The Twilight Zone (1959 TV series), Season 2 (first three episodes)
  • You're traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land whose boundaries are that of imagination. That's the signpost up ahead — your next stop, the Twilight Zone!
    • Rod Serling, in The Twilight Zone (1959 TV series), Season 2, main intro
  • You are traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land whose boundaries are that of imagination. Your next stop, the Twilight Zone!
    • Rod Serling, in The Twilight Zone (1959 TV series), Season 3
  • You unlock this door with the key of imagination. Beyond it is another dimension—a dimension of sound, a dimension of sight, a dimension of mind. You're moving into a land of both shadow and substance, of things and ideas. You've just crossed over into the Twilight Zone.
    • Rod Serling, in The Twilight Zone (1959 TV series), Season 4 & 5
  • In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system.
    • Charles Proteus Steinmetz, as quoted in Electricity will keep the world from freezing up; noted expert, Dr. C.P. Steinmetz, talks of the future wonders of scientific discovery and ridicules many prophecies in The New York Times (11 November 1911)

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  • He sat back down again in his armchair. “Of course, that name really isn’t accurate. I suppose a pentaract should really be a four-dimensional pentagon, and this is meant to be a picture of a five-dimensional cube.”
“A picture?” It didn’t look like a picture to me.
“Well, it couldn’t really have five-dimensionality—length, width, breadth, ifth and oofth—or I don’t think it could.” His voice faltered a little at that. “But it’s supposed to illustrate what you might call the layout of an object that did have those.”
“What kind of object would that be?” I looked back at the thing in my lap and was mildly surprised to see that I had folded a good many of the cubes together.
“Suppose,” he said, “you put a lot of points in a row, touching; you have a line—a one-dimensional figure. Put four lines together at right angles and on a plane; a square—two-dimensional. Six squares at right angles and extended into real space give you a cube—three dimensions. And eight cubes extended into four physical dimensions give you a tesseract, as it’s called—”
“And eight tesseracts make a pentaract,” I said. “Five dimensions.”
“Exactly. But naturally this is just a picture of a pentaract, in that sense. There probably isn’t any ifth and oofth at all.”
  • The Iifth and the Oofth by Walter Tevis, Science Fact / Fiction (1974), [johnesimpson.com/pdf/Ifth_of_oofth-waltertevis.pdf]
  • When you listen to a thought, you are aware not only of the thought but also of yourself as the witness of the thought. A new dimension of consciousness has come in.
  • Make it your practice to withdraw attention from past and future whenever they are not needed. Step out of the time dimension as much as possible in everyday life.

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  • Well, I do not mind telling you I have been at work upon this geometry of Four Dimensions for some time. Some of my results are curious. For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensioned being, which is a fixed and unalterable thing.
  • If I am recalling an incident very vividly I go back to the instant of its occurrence; I become absent minded, as you say. I jump back for a moment. Of course we have no means of staying back for any length of time any more than a savage or an animal has of staying six feet above the ground. But a civilized man is better off than the savage in this respect. He can go up against gravitation in a balloon, and why should we not hope that ultimately he may be able to stop or accelerate his drift along the Time Dimension; or even to turn about and travel the other way?
  • There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it.
  • Perhaps the first to approach the fourth dimension from the side of physics, was the Frenchman, Nicole Oresme, of the fourteenth century. In a manuscript treatise, he sought a graphic representation of the Aristotelian forms, such as heat, velocity, sweetness, by laying down a line as a basis designated longitudo, and taking one of the forms to be represented by lines (straight or circular) perpendicular to this either as a latitudo or an altitudo. The form was thus represented graphically by a surface. Oresme extended this process by taking a surface as the basis which, together with the latitudo, formed a solid. Proceeding still further, he took a solid as a basis and upon each point of this solid he entered the increment. He saw that this process demanded a fourth dimension which he rejected; he overcame the difficulty by dividing the solid into numberless planes and treating each plane in the same manner as the plane above, thereby obtaining an infinite number of solids which reached over each other. He uses the phrase "fourth dimension" (4am dimensionem).
    • Gerald James Whitrow, "Why Physical Space has Three Dimensions," British Journal for the Philosophy of Science, 6 #21 (May 1955)

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