Home >> Encyclopedia Americana, Volume 14 >> Huskisson to Icelandic Literature >> Hyperspaces_P1

Hyperspaces

space, dimensionality, element, line, ordinary, lines and hyperspace

Page: 1 2

HYPERSPACES. Dimensionality, In order to make quite intelligible the concept variously denoted by such terms as hyperspace, space of higher dimensions or dimensionality, multi-dimensional space, n-space, n-fold or n-di mensional space, it is in the first place neces sary to explain the meaning of dimensionality and to indicate the way in which the dimension ality, or number of dimensions, of a given space in a given element is determined or ascer tained. Because, in order to determine the position of a point in a curve or straight line, it is necessary and sufficient to know one fact about the point, as, for example, its distance (with algebraic sign) from a fixed point or origin; a line is said to be a one-dimensional space of points. But instead of the point, we may choose for element of the space (line) a Pair or a triplet, . . . or an n-set of points. In such cases, in order to determine the ele ment, i.e., to pick it out or distinguish it from among all others of its kind, it is necessary and sufficient to know two or three, . . . or n independent facts about it. Hence a line is a two- or three-, . . . or n-dimensional space of pairs or triplets, . . . or n-sets, of points. In like manner a flat pencil (totality of lines of a plane that have a common point) is a one-dimensional space of lines, while its dimensionality is 2 in line pairs, 3 in triplets, and so on. For like reasons a plane is a two dimensional space of points or of lines. In circles its dimensionality is 3, in conics 5, in curves of third order 9, and so on. It is at once seen that the dimensionality of a given space depends on the entity chosen for primary element, the element, i.e., in terms of which we elect to study and express the properties of the given space. Illustrations abound. A curved surface, as, say, a sphere, regarded as the envelope of (its tangent) planes, is a two dimensional space of planes, while, conceived as the assemblage of (its, tangent) lines, it is a three-dimensional space. The reader will observe that the term space is employed ge nerically to denote any unbounded continuum of geometric entities. The generalization is, however, a natural one, for, for geometric pur poses, ordinary space is viewed primarily as an assemblage of elements of one kind or another. To determine the position of a point in ordi nary space, three independent data (as the dis tances of the point from three mutually per pendicular planes of references) are necessary and sufficient. Ordinary space is, therefore,

three-dimensional in points, and that is what is meant, consciously or unconsciously, when, without specifying the element (point), it is simply said that space is three-dimensional. But tridimensionality is in no strict sense a definitive property of ordinary space. For some little understood, probably economical, certainly extra-logical, reason, the point recom mended itself to primitive man as the element par excellence with which to geometrize, and so it has become traditional and proverbial that our space is essentially, uniquely, character istically, intrinsically, exclusively three-dimen sional. Such, however, it is not It is indeed three-fold in planes as in points, but in lines it is four-dimensional. So, too, it is.four is in spheres, but in circles its dimensionality s six. In general, it is possible by proper choice of element to endow any given space with any prescribed dimensionality however high. Ac cordingly, if by hyperspace is meant a space of dimensionality greater than 3, the notion is simple and near at hand, we need not go beyond ordinary space to realize it, we detect it in the line, in the plane, in ordinary space, here, there and yonder. Well, such is one of the recognized significations of the term. But it has °another," namely, hyperspace usually means a space whose point dimensionality is four or more. Now this latter meaning is logically and conceptionally quite consistent with the other; it is indeed a special case of it ; but a hyperspace of points is difficult or im possible to picture, to realize in visual imagina tion, and it is this non-logical circumstance that renders the term hyperspace at once so tan talizing, mysterious, baffling and fascinating to the non-mathematician. To the mathema tician, however, whose activities, so far from being confined within the limits of the visual imagination, lie for the most part quite beyond them, the conception in question offers as such no difficulty whatever, and it has long since established itself among the most approved of orthodox scientific notions.

Page: 1 2