Dimension

“ Modern science, feeling after the mysterious, has named this world which permeates ours and yet is invisible, the Fourth Dimension because it is not subject to the three illusions of length, breadth and height which imprison most of us from the cradle to the grave. ”

“ However, if you have properly understood the fact that our dimension exists at the place of Location itself, you will see at once that our only possible point of contact with your universe is at the central, non-moving point. ”

“ To the question, where and what is the fourth dimension, the answer must be, it is here—in us, and all about us—in a direction toward which we can never point because at right angles to all the directions that we know. ”

“ That is to say, the ultimate source of all vitality is in that other dimension, or region, where the dandelion blooms, and which men have called heaven, and which now they call the fourth dimension: which is only a way of saying that it is not to be reckoned in terms of space and time. ”

“ But the very fact that a Line is visible implies that it possesses yet another Dimension. ”

“ To us, who conceive three dimensions, a surface is that which hath length and breadth, excluding thickness: and a surface may be either plain in this third dimension, or it may be incurvated: so that the notion of a third dimension enters into our conception of a surface ”

“ We have used up our dimensions of space in representing the phases of the premisses and the conclusion in respect of mood, and to represent in an analogous manner the variations in figure we require a fourth dimension. ”

“ Dimension is an arbitrary norm constructed by the mind for the determination of various positions in space. It is an accident or by-product of the process of elaborative cognition, a convenient and appropriate means of measurement for objects in space and their space-relations. ”

“ In God’s dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. ”

“ Having held before ourselves this outline of a conjecture of the world as four-dimensional, having roughly thrown together those facts of movement which we can see apply to our actual experience, let us pass to another branch of our subject. ”

“ We see a scene before us on which are various figures and colours, but the eye cannot see more. It sees length and breadth, but no third dimension. In order to know that there are solids, we must infer as well as see. ”

“ A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line ”

“ Transferring our conceptions to those of an existence in a higher dimensionality traversed by a space of consciousness, we have an illustration of a thought which has found frequent and varied expression. ”

“ The totality of event-particles will form a four-dimensional manifold, the extra dimension arising from time—in other words—arising from the points of a timeless space being each a class of event-particles. ”

“ Upon this scale of computation for the dimensions of the world, what heights and depths and breadths must there be in this universe—in comparison of which the positive universe would be itself a nihility, were it crossed—pierced—and belted about by so illimitable a wilderness of nothing! ”

“ Greatness [17] of dimension is a powerful cause of the sublime. This is too evident, and the observation too common, to need any illustration; it is not so common to consider in what ways greatness of dimension, vastness of extent or quantity, has the most striking effect. For, certainly, there are ways and modes wherein the same quantity of extension shall produce greater effects than it is found to do in others. ”

“ Such are the dimensions actually measured in the general constitution of the universe. As yet we are only at the vestibule of the edifice, on the edge of the abyss of infinitude: and we shall never penetrate very far beyond. ”

“ No one will deny that we have the representation of dimensions, without necessarily referring them to any thing in particular; but what we wish to determine is, whether these dimensions exist in reality, and what is their nature, independently of their relations to us. ”

“ If two trees, one of which was a hundred paces from us and the other ten, looked equally large and distinct, we should think they were side by side. If we perceived the real dimensions of things, we should know nothing of space; everything would seem close to our eyes. ”

“ I. And how many solids or sides will appertain to this Being whom I am to generate by the motion of my inside in an "upward" direction, and whom you call a Cube?
SPHERE. How can you ask? And you a mathematician! The side of anything is always, if I may so say, one Dimension behind the thing. Consequently, as there is no Dimension behind a Point, a Point has 0 sides; a Line, if I may say, has 2 sides (for the Points of a Line may be called by courtesy, its sides) ; a Square has 4 sides ”

“ Are we then looking out upon a third dimensional world, and realising it as such because we are mentally capable of conceiving dimensions beyond it? Our world sensibly contains one dimensional and two dimensional facts—​the first such as a time series, for which one number is sufficient to fix a point, and the second such as a plane where position can be fixed by two numbers. Does our world contain facts of other dimensions? ”

“ And the act by which we thus create space of three dimensions out of visible extent of length and breadth, is constantly and imperceptibly going on. We are perpetually interpreting in this manner the language of the visible world. From the appearances of things which we directly see, we are constantly inferring that which we cannot directly see,—their distance from us, and the position of their parts. ”

“ The "n" dimension may be employed as a lever to the imagination. Even revealed religion demands our faith, and imagination is the prime agent in the interpretation of the universe, according to the gospel of mystic mathematics. ”

“ The world which we have so far constructed is a world of six dimensions, since it is a three-dimensional series of perspectives, each of which is itself three-dimensional. We have now to explain the correlation between the perspective space and the various private spaces contained within the various perspectives severally. It is by means of this correlation that the one three-dimensional space of physics is constructed ”

“ If the number of dimensions comes from the way we are made, there might be thinking beings living in our world, but who might be made differently from us and who would believe space has more or less than three dimensions. ”

“ Having thus seen how it is possible to describe the properties of the simplest shape in four dimensions, it is evident that the mental construction of more elaborate figures is simply a matter of time and patience. ”

“ The science of perspective gives the dimensions of objects seen in space as they appear to the eye of the spectator, just as a perfect tracing of those objects on a sheet of glass placed vertically between him and them would do ”

“ You know how on a flat surface, which has only two dimensions, we can represent a figure of a three-dimensional solid, and similarly they think that by models of three dimensions they could represent one of four—if they could master the perspective of the thing. ”

“ It makes it conceivable that apparently separate objects, e.g. distinct people, are really physically united; that things apparently sundered by enormous distances of space are really quite close together; that a person or other object might pass in and out of a closed room without disturbance of walls, doors, or windows, etc.; and if this fourth dimension were to become a factor of our consciousness it is obvious that we should have means of knowledge which to the ordinary sense would appear simply miraculous. ”

“ For the introduction into a picture of a suggested third dimension is what prevents our thinking away from a merely two-dimensional aspect by supplying subsidiary imaginary aspects susceptible of being co-ordinated to it. So perspective and modelling in light and shade satisfy our habit of locomotion by allowing us, as the phrase is, to go into a picture; and going into, we remain there and establish on its imaginary planes schemes of horizontals and verticals besides those already existing on the real two-dimensional surface. ”

“ Thus we are in the unwonted position of being obliged to construct the daily and habitual experience of a four-dimensional being, from a knowledge of the abstract theories of the space, the matter, the motion of it ”

“ If we deal with one dimension only, the figure 1, that of a segment of a straight line, is cut out of and separated from the rest of an imaginary infinitely long straight line by the two points 16 at its extremities. ”

“ The objects of Sight and those of Touch constitute two worlds, which, though they have a most important correspondence and connection with one another, bear no sort of resemblance to one another. The tangible world, as well as all the different parts which compose it, has three dimensions, Length, Breadth, and Depth. ”

“ Now tell me what part of this earth habitable, or rather of the whole globe and compass thereof, can be said far distant or remote one from the other, seeing that the mathematicians are able to prove and make demonstration by reason that the whole, in comparison and respect of heaven or the firmament, is no more than a very prick which hath no dimension at all? ”

“ The different positions of the several parts of the body with regard to the eye, when put together, make a real figure, which is truly extended in the length and breadth, and which represents a figure that is extended in length, breadth, and thickness. In like manner, a projection of the sphere is a real figure, and hath length and breadth, but represents the sphere, which hath three dimensions. ”

“ Several forms are in tri-dimensional space, and we have one elaborate description of a form stretching away from the subject, in which the more prominent numbers stand out like mountain-peaks and hide the intervening ones. ”

“ Let us, therefore, turn to a different field, and, assuming that the human soul is a four-dimensional being, capable in itself of four dimensional movements, but in its experiences through the senses limited to three dimensions, ask if the history of thought, of these productivities which characterise man, correspond to our assumption. ”

“ And because the first subject is matter, the consequence is that all other accidents are related to their subject through the medium of dimensive quantity; just as the first subject of color is said to be the surface, on which account some have maintained that dimensions are the substances of bodies, as is said in Metaph. ”

“ And because whatsoever is real here, or there, or in any place, has dimensions, that is to say, magnitude; that which hath magnitude, whether it be visible or invisible, finite or infinite, is called by all the learned a body. ”

“ '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. ”

“ When we say that space has three dimensions, what do we mean? We have seen the importance of these "internal changes" which are revealed to us by our muscular sensations. They may serve to characterise the different attitudes of our body. ”

“ As white light is seen to be composed of but three primary colours, as Music is based on the Triad, as Space is known to us in three dimensions only, and Geometry, "the Head of all Learning," is based upon the Circle, Square, and Triangle, so may we see life in its three primary aspects: the Animal, Vegetable, and Material. ”

“ So I devoted several months in privacy to the composition of a treatise on the mysteries of Three Dimensions. Only, with the view of evading the Law, if possible, I spoke not of a physical Dimension, but of a Thoughtland whence, in theory, a Figure could look down upon Flatland and see simultaneously the insides of all things, and where it was possible that there might be supposed to exist a Figure environed, as it were, with six Squares, and containing eight terminal Points. ”