Space of n The conceptions introduced in the analysis situs of two-dimen sional surfaces in three-dimensional space per mit of generalization. The indictatnx of an m-dimensional surface will consist of in direc Lions in (i.e., tangent to) the surface perpen dicular to each other. There will be only two indicatrices (right and left), since we may bring about coincidence between a first pair of axes of different indicatrices, then between a second pair, etc. To do this, in the case of the kth pair, we have m— k-space at our dis posal. This becomes a common plane for the m — first pair. For the last pair there remains a line only, so that coincidence, if not existing, cannot be forced. As a consequence, unilateral and bilateral m-surfaces must be distinguished.
The indicatrix of a line is the line-element (or the tangent) taken in one of two possible directions. A closed curve with a cusp might be considered one-sided, as the di rection here changes abruptly as we make the circuit. A four-sided prism abcda'b'c'd' in R4 can be twisted like Moebius' sheet and its face abed joined to c'd'as b' . The resulting surface is bounded by one bilateral 2-surface from ab to cc-a'b' and back to and two unilateral ones. We may further join these two latter 2-surfaces, their juncture only form ing a unilateral surface such as we may imagine inside any solid. There will remain only one bounding bilateral 2-surface of spherical con nectivity, just as Moebius' sheet has one edge. This further shows that an incision is possible along a surface of spherical connectivity, which does not divide our 3-surface, but renders it bilateral (bilateralizing closed section).
We shall now consider bilateral m-surfaces. They may be given by making the co-ordinates xt. . xn of a point in n-space, functions of m parameters t, . . . tm: xi= xi(ti . . . tm). en planes will then be linear functions. Lines common planes, etc., are m-planes for m=1,2 ...
In n-space, we call surfaces complementary if their dimensions add up to n, dual if they add up to n-1. In R,, lines and 2-surfaces are complementary, while lines are dual to lines (self-dual).
Closed m-surfaces are boundless and contain no points with infinite co-ordinates. They separate the dual planes of n-space into interior and exterior ones. Taking any complementary plane (n—rn-plane) that does not intersect the closed m-surface, we can move into it any ex terior dual plane (n —1 — m-plane) without allowing it to intersect the surface on the way, can reverse it there by turning it through 180°, and bring it back to its original position along the path on which it was brought. An interior
n —1— m-plane, if we attempted to do the same, would describe an n — m-surface which must intersect the given rn-surface. Besides distinguishing between the interior and exte rior of our closed m-surface this also shows that the interior is bilateral, the exterior uni lateral, with regard to the dual planes.
(1) The limiting case of a closed figure without dimension is a couple of points. In 1-space (straight line) it bounds a segment. It separates the straights of 2-space (common plane) into those passing between it (interior) and the exterior ones, and does the same for the 2-planes of 3-space. (2) The in terior of a circular circumference is an area in 2-space; in 3-space it consists of the straight lines passing through it. Take a plane not in tersecting it: An exterior straight line may be moved into the same without coming in contact with the circle, may there be turned through 180° and brought back.—Although points can not be reversed, it is as natural in an analysis sites as in projective geometry to assume uni laterality for the infinite plane. This is merely to extend to a limiting case what is true generally.
Interiors of Different Orders and Degrees.
— In a plane a closed curve may have overlap ping portions so as to contain a certain area twice or r times. In 3-space it may, without having any double points, coil r times about certain straight lines of its interior, which thereby become an interior of the rth order. But if, as we follow the curve in a given di rection, it coils p times in an assumed positive rotation about this interior of rth order, and r — p times negatively, we may then call p (r — p) =2p the degree of the interior. The curve C (Fig. 13) has an interior of order 2 and degree 0, to which line L belongs. These considerations are applicable to n-space, where n —1-surfaces, however, cannot possess in teriors of higher orders without self-intersec tion (double points, etc.). Interiors of higher orders than the first can be removed by de formation, unless the curve is knotted (see below).