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The Fourth Dimension Simply Explained
A Collection of Essays Selected from Those Submitted to the Scientific American's Prize Competition
By Henry P. Manning
Dover Publications, Inc.Copyright © 2005 Dover Publications, Inc.
All rights reserved.
AN ELUCIDATION OF THE FOURTH DIMENSION.
THE PRIZE-WINNING ESSAY.
BY "ESSAYONS" (HEUT.-COL. GRAHAM DENBY FITCH, CORPS OF ENGINEERS, U. S. A.)
It is impossible to form a mental picture of the fourth dimension. Nevertheless, it is not an absurdity, but a useful mathematical concept with a well-developed geometry involving no contradictions. To gain a partial and symbolic idea of its meaning, resort must be had to analogy with dimensions of a lower order.
An aggregate is said to be one, two, or three-dimensional according as one, two, or three numbers are necessary to determine any one of its elements. Considering space as an aggregate of points, a line is a one-dimensional space, because to determine the position of any point on it one number, giving its distance from some fixed point, suffices. Similarly, a plane is a two-dimensional space, and the point aggregate of ordinary space is three-dimensional. Thus, the exact position of any point of the earth is known when its latitude, longitude, and elevation above sea level are given. Now, if we have four variable, related quantities, each capable of assuming, independently of the others, every possible numerical value, we obtain a four-dimensional aggregate. Such an aggregate, if of points, constitutes four-dimensional space.
If we connect all points of our space (a 3-space) with an assumed point outside of it, then the aggregate of all the points of the connecting lines constitutes a 4-space (hyperspace). Again, just as a point moving generates a line, just as a line moving outside itself generates a surface, and a surface moving outside itself generates a solid; so a solid moving outside of our space generates a hypersolid, or portion of hyperspace. Or hyperspace itself may be conceived as generated by our entire space moving parallel to itself in a direction not contained in itself, just as our space may be generated by the similar motion of an unlimited plane, which may itself be generated by an unlimited right line. Any space is that which forms the boundary between two portions of a higher space, and just as every unlimited plane divides our space into two equal infinite parts, so every 3-space divides hyperspace into two equal infinite regions between which that 3-space forms a boundary of an infinitely small thickness in the fourth dimension.
Two plane figures (say triangles) if in the same plane may partially coalesce, but cannot intersect unless in different planes; similarly two volumes (say cubes) if in the same 3-space may partially coalesce but cannot intersect unless in different 3-spaces. In hyperspace we have the following possible intersections: A hypersolid and a 3-space intersect in a solid, two 3-spaces in a plane, three 3-spaces in a right line, four 3-spaces in a point, a 3-space and a plane in a right line, a 3-space and a right line in a point, and two planes in a point. If the intersections are at an infinite distance the intersecting elements are said to be parallel, and if two 3-spaces are parallel all figures or volumes in one 3-space are at equal distances from the other 3-space. In the case of planes there are two kinds of parallelism, and parallel planes are either completely or incompletely parallel according as they are in the same or different 3-spaces, or as their intersection at infinity is a right line or a point.
To a given right line at a given point one can erect in a plane but one perpendicular, while in a 3-space one can erect an infinite number of perpendiculars, forming together a perpendicular plane, and in hyperspace an infinite number of perpendicular planes forming together a 3-space perpendicular to the given right line. A 3-space can also be perpendicular to a plane or to another 3-space. Planes may be perpendicular in two ways, incompletely or completely perpendicular, according as they are in the same 3-space or not; in the latter case every right line of either plane is perpendicular to every right line of the other.
The position of a point in a plane may be determined by its distance from each of 2 perpendicular right lines; in our space, by its distance from each of 3 mutually perpendicular planes; and in hyperspace, by its distance from each of 4 mutually perpendicular 3-spaces. In hyperspace these distances are accordingly measured along 4 mutually perpendicular right lines, which, taken by twos, determine 6 mutually perpendicular planes; and, taken by threes, determine the above-mentioned 4 mutually perpendicular 3-spaces. Just as in our space it requires at least 3 points to determine a plane, so in hyperspace it requires at least 4 points to determine a 3-space. A 3-space may also be determined by 2 non-intersecting right lines or by a plane and one point not in it.
Just as portions of our space are bounded by surfaces, plane or curved, so portions of hyperspace are bounded by hypersurfaces (three-dimensional), i. e., flat or curved 3-spaces. Hyperspace contains not only an infinite number of flat 3-spaces like ours but also an infinite number of curved 3-spaces or hypersurfaces of different types. A hypersphere, for instance, is a closed hypersurface all the points of which are equally distant from its center. Five points not in the same 3-space determine it, just as 4 points not in the same plane determine a sphere, and 3 points not in the same straight line a circle. All of its plane intersections are circles, all of its space intersections are spheres. A hypersphere of radius R passing through our space would appear as a sphere with a radius gradually increasing from zero to R and then gradually decreasing from R to zero.
While in our space there are but 5 regular polyhedrons (solids bounded by equal regular polygons), namely, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron; in hyperspace there are 6 regular hyper-solids (cells), bounded by equal regular polyhedrons. These are C5 (bounded by 5 tetrahedrons), C8 (by 8 cubes), C16(by 16 tetrahedrons), C24 (by 24 octahedrons), C120 (by 120 dodecahedrons), and C600 (by 600 tetrahedrons). All of them have been exhaustively studied by mathematicians, and models of their projections on our space have been constructed. Of these, C8 (or the hyper-cube) is the simplest, because, though with more bounding solids than C5, it is right-angled throughout, and therefore the standard form for measuring hyperspace. It is generated by a cube moving in the direction perpendicular to our space for a distance equal to one of its sides. In Fig. 1 where all dotted lines are supposed to be in hyperspace the initial cube is symbolically represented by A B C D E F G H and the final cube by A' B' C' D' E' F' G' H', the direction AA' being supposed perpendicular to our space. Projecting the edges of a hypercube on our space we get a network model of which Fig. 2 is a plane projection. The eight bounding cubes are represented in the model by the following projections: (1, 2, 3, 4, 5, 6, 7, 8), (5, 6, 7, 8, 9, 10, 11, 12), (9, 10, 11, 12, 13, 14, 15, 16), (13, 14, 15, 16, 1, 2, 3, 4), (1, 5, 9, 13, 2, 6, 10, 14), (2, 6, 10, 14, 3, 7. 11, 15), (3, 7, 11, 15, 4, 8, 12, 16), (4, 8, 12, 16, 5, 9, 13, 1). The form of the hypercube is conditioned by the mutual relations of these cubes that form its boundaries merely, as it contains an infinite number of cubes, just as a cube contains an infinite number of squares. In generating a hypercube by the motion of a cube, the latter's corners generate edges, its edges generate faces (squares) and its faces generate cubes. The resulting number of elements of the hypercube are therefore:
In Final In Hypercube.
Cube. Generated. Cube.
Faces (squares). 6 12
Each corner is common to 4 mutually perpendicular edges, to 6 faces and to 4 cubes; each edge is common to 3 faces and 3 cubes; and each face is common to 2 cubes. Every cube therefore has one face in common with 6 of the 7 others. We must conceive of the hypercube as composed of cubes starting from squares parallel to the faces of the cube and of these cubes all that exist in our space are the parallel squares from which they start.
In a plane the only kind of rotation possible is that about a point, in 3-space rotation can take place about an axis line, and in hyperspace about an axis plane.
Two symmetrical plane figures such as the triangles A and B (Fig. 3) cannot be made to coincide by any movements in their plane, but by rotating one of them 180 deg. in the third dimension, it can be made to coincide with the other. Similarly, two symmetrical volumes (with faces equal but in reverse order) such as the hollow pyramids C and D (Fig. 4) cannot be made to coincide by any movements in our space, but by rotating one of them 180 deg. in hyperspace this can be done. The rotating pyramid disappears from our space, and upon its return after rotating 180 deg. it can be slipped into the other. In our space two movements of rotation will combine into a single resultant rotation, similar to its component rotations except that the direction of the axis is different. In hyperspace, however, there is in general no resultant for two rotations. Hence there are two different types of rotation in hyperspace and a body subject to two rotations is in a totally different condition from that which it is in when subject to one only. When subject to one rotation a whole plane of the body is stationary. When subject to the double rotation no part of the body is stationary except the point containing the two planes of rotation; and if the two rotations are equal, every point in the body, except that one, describes a circle.
Freedom of movement is greater in hyperspace than in our space. The degrees of freedom of a rigid body in our space are 6, namely, 3 translations along and 3 rotations about 3 axes, while the fixing of 3 of its points can prevent all movement. In hyperspace, however, with 3 of its points fixed it could still rotate about the plane passing through those points. A rigid body has 10 possible different movements in hyperspace, namely, 4 translations along 4 axes, and 6 rotations about 6 planes, while at least 4 of its points must be fixed to prevent all movement.
In hyperspace, a sphere if flexible could without stretching or tearing be turned inside out. Two rings of a chain could be separated without breakage. Our knots would be useless. Thus the knot shown in Fig. 5 could be unknotted without removing the fastened ends. Just as in our space a point can pass in and out of a circle without touching its circumference, so in hyperspace a body could pass in and out of a sphere (or other in closed space) without going through the surface surrounding it. In short, all of our space including the interior of the densest solids is open to inspection and manipulation from the fourth dimension, which extends in an unimaginable direction from every point of space.
Has hyperspace a real physical existence? If so, our universe must have a small thickness in the fourth dimension, otherwise like the geometrical plane assumed to be without thickness, our world too would be a mere abstraction (as indeed some idealistic philosophers have maintained), that is, nothing but "a shadow cast by a more real four-dimensional world." The real existence of a slight extension in the fourth dimension would, moreover, simplify certain scientific theories. For instance, in our space 4 is the greatest number of points whose mutual distances, 6 in number, are all independent of each other; but in hyperspace the 10 distances between any 2 of 5 points are geometrically independent. If this greater freedom of position were permissible to atoms, it would help to explain such chemical phenomena as isomerism, where molecules of identical composition have different properties. Again, rotation in hyperspace would explain the change of a body producing a right-handed into one producing a left-handed polarization of light. Further, Prof. McKendrick said before the British Association: "It is conceivable that life may be the transmission to dead matter ... of a form of motion sui generis." Hyperspace has been brought somewhat into disrepute because spiritualists have assumed its existence in order to give "a local habitation" to their vagaries. Nevertheless, the possibility of its existence has not yet been shown to be inconsistent with any scientific fact, and the limitation of space to three dimensions, though probably correct, is therefore purely empirical.
Of what use then is the conception of hyperspace? For one thing, it gives a deeper insight into geometry. Thus, a circle considered merely as a one-dimensional aggregate of points has very few properties, while in a plane it has a center, radii, tangents, etc., and in 3-space has further numerous geometrical relations with the sphere, cone, etc. Similarly, the properties of any given line or surface increase in number when investigated in hyperspace. Also, just as it requires a 3-space to include certain one-dimensional aggregates (the helix, for instance), so in hyperspace hitherto unknown lines and surfaces become mathematically possible. Lower spaces are contained in higher (if curved, not necessarily the next higher); and just as the comprehension of plane geometry is enlarged by viewing plane figures in 3-space, so solid geometry is much illuminated by the geometry of hyperspace. Fields of mathematics hitherto inaccessible to geometry are now elucidated by geometrical representations. Finally, this conception effects a complete divorce between geometric space and real space, no longer considered necessarily identical, and in other ways also enlarges our mental horizon.CHAPTER 2
NON-EUCLIDEAN GEOMETRY OF THE FOURTH DIMENSION.
BY LIEUT.-COL. GRAHAM DENBY FITCH, CORPS OF ENGINEERS, U. S. A.
The Fourth Dimension is an offshoot of the so-called "non-Euclidean" geometry, which has thrown so much light on the foundations of mathematics and on the nature of space.
For over 2,000 years Euclid was considered unassailable. His axioms were regarded as indisputable laws of real space, and his theorems as rigidly logical deductions therefrom. Neither view is correct. His axioms are metaphysical assumptions, and his theorems do not follow from them alone. The foundation of his method consists in establishing by superposition the congruence of lines, angles, plane figures, etc., and proof with him is thus merely a matter of constructive intuition. The axiom of "free mobility" (i. e., the possibility of moving figures in space without change of size or shape), which for instance does not hold on an egg-shaped surface but is essential to any system of geometrical measurement, is assumed by Euclid without being stated. (Hilbert discards proof by superposition, for motion itself needs a geometric foundation, and so cannot be a foundation for geometry.) Another of Euclid's tacit assumptions is that the straight line can be infinitely extended, which, true of Euclidean, is false of some non-Euclidean geometries (e. g. Riemann's).
Euclid proves that "if alternate angles are equal, then the lines are parallel," but of the converse propositions.
"If alternate angles are unequal, the lines meet."
"If the lines are parallel, alternate angles are equal" (either of which implies the other) he could prove neither, and hence assumed the first, his celebrated fifth postulate, without which he could not proceed, as it was needed to prove the early important theorem that the sum of the angles of a triangle is not less than two right angles. This postulate of parallels appeared to later mathematicians neither self-evident nor independent of the other axioms. Considered a flaw, fruitless efforts were made for centuries to prove it. Yet here Euclid is right; this axiom or some equivalent (e. g. two intersecting lines cannot both be parallel to the same line) is necessary to Euclidean geometry.
It was from endeavors to improve upon Euclid's theory of parallels that non-Euclidean geometry arose. If the fifth postulate is really involved in Euclid's other assumptions, its denial must lead to contradictions; but about 1830 the Russian Lobachevsky and the Hungarian Bolyai, independently of each other, showed that its denial led to a system of two-dimensional geometry as self-consistent as Euclid's. This new geometry is based on the assumption that through a given point a number of straight lines can be drawn parallel to a given straight line.
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