## EUCLIDEAN SPACES OF 2, 3, 4, …, n DIMENSIONS The prosedure of representing (in a plane) the points defined in Euclidean spaces of one and two diensions, has been considered solved for a very long time. A two-dimensional E. space q1 q2 can be represented in a plane so that the very plane of the projection is defined by the coordinate axes q1 and q2, thus in the plane of the projection q1 q2.

Representing a point A1 (Q1, Q2) defined in the system q1 q2 is done by drawing a parallel from point Q1 on the axis q1, with the axis q2, and a parallel with axis q1 from point Q2 on the axis q2. The intersection of these two lines determines the projection (in a plane) of point A1 (Q1,Q2), defined in a two-dimensional E. space q1 q2 (Fig. 1)*. However, the problem of representing a three-dimensional E. space q1 q2 q3 in a plane is somewhat more complex. One of the basic traits of the two-dimensional E.space allows for the possibility of no more than two lines to intersect at a right angle. Which means that it isn’t possible to represent a three-dimensional E. space in a plane so that q1 q2 q3. As a solution to this problem the following procedure has been adopted: if the plane of the projection is determined by axes q1 and q2 then in that case in the plane of the projection q1 q2. The third axis q3 is represented so that it does not form right angles with q1 and q2 in point 0 with the condition that in the three-dimensional E. space which is represented q1 q2 q3. This means that in this procedure it is assumed that the q3 axis is perpendicular to q1 and q2 since this cannot be claimed for the two-dimensional E. space (Fig. 2a). One way to represent point A2 (Q1, Q2, Q3) in this case is to first locate A1 (Q1, Q2) on the plane q1 q2. Then to draw a parallel from Q3 with 0A1, and then from point A1 a parallel with q3. The intersection of these two lines determines the projection of point A2 (Q1, Q2, Q3) on the plane of the projection, defined by the axes q1 q2 (Fig. 2b). If we now suppose the possibility of a four-dimensional E. space q1 q2 q3 q4, there is the problem of its representation in a plane (in an Euclidean two-dimensional space). The solution is possible by applying an already explained procedure for the three-dimensional case. If we start with the three-dimensional system (q1 q2 q3) represented in the plane q1 q2, then the four-dimensional E. space q1 q2 q3 q4 can be represented on the plane q1 q2, by drawing an axes q4 from point 0, which (in the projection) in the general case does not need to be perpendicular to any of the projections of the other axes (the same goes for the q3 axis, in this case) (Fig. 3a). This means that a system represented this way is a projection of a four-dimensional E. space q1 q2 q3 q4 on the plane q1 q2. It is justifiable to assume such a projection plane in which non of the axes’ projections q1 q2 q3 q4 would be perpendicular on the projection plane to the other three axes (Fig. 3b). Let’s say that point A3 (Q1, Q2, Q3, Q4) is defined in a four-dimensional E. space and let’s say that the projection of this system in a plane is given like in image 3a, then the procedure of determining A3 in such a represented system is as follows: on the axes q1, q2, q3 and q4 we place the values of the coordinates of point A3 (Q1, Q2, Q3, Q4). From point Q1 we draw a parallel with the axis q2 and from Q2 we draw a parallel with axis q1. The intersection of these two lines determines the projection of point A1 on the plane q1 q2. Then from Q3 we draw a parallel with 0A1, and from A1 a parallel with q3. The intersection of these two lines determines the projection of point A3 on the plane q1 q2. If we now draw a parallel with 0A2 from point Q4 and from point A2 a parallel with the axis q4, then the intersection of these two lines represents the projection of point A3 (Q1, Q2, Q3, Q4) on the plane q1q2 (Fig. 3c). In the case of the four-dimensional E. space shown in image 3b, the procedure of determining point A3 (Q1, Q2, Q3, Q4) within it is shown in image 3d. It can be seen that the position of the projection of point A3 on the plane does not depend on the order of operations that determine it. Based on everything written here, the problem of representing in a plane (a two-dimensional E. space) some n-dimensional E. space (n ϵ N) would be solved by representing projections of all axes through a point 0 in a plane with arbitrary mutual angles, while in the n-dimensional E. space it would be q1 q2 q3 q4 … ┴ qn. So the projection of a six-dimensional E. space is q1 q2 q3 q4 q5 q6 on an arbitrary plane and the projection of some point A5 (Q1, Q2, Q3, Q4, Q5, Q6) defined in that space, is shown in Fig. 4. The analysis of thus defined multidimensional spaces leads to some more general conclusions. The axes of a two-dimensional E. space q1 q2 determine a plane. The axes of a three-dimensional E. space q1 q2 q3 determine three planes (q1,q2); (q1,q3); (q2,q3), while the axes of a four-dimensional E. space determine six planes: (q1 q2); (q1 q3); (q1 q4); (q2 q3); (q2 q4); (q3 q4). This means that the axes of an n-dimensional E. space would determine the planes whose number is given by the sum: So, it is possible to draw only two mutually perpendicular planes** through one line in a three-dimensional E. space. In a four-dimensional E. space it is possible to draw a maximum of three mutually perpendicular planes through one line (Fig. 5). In a five-dimensional E. space it is possible to draw a maximum of four mutually perpendicular planes through one line (Fig. 6). Based on all this we can conclude that in the n-dimensional E. space it is possible to draw an n-1 number of mutually perpendicular planes. *For easier deliberation in all the examples only positive aspects of the coordinate axes, since it is assumed that it does not greatly affect the heart of the problem.

** Two planes that have three or more common points (which do not belong to one line) are considered to be identical (we cannot distinguish them).

*** For simpler writing, q1 q2 q3 is used instead of q1 q2, q1 q3, q2 q3

Goran Đorđević, New Belgrade

This article titled On Representing a Point Defined in Euclidean Spaces of 2, 3, 4, …, n, Dimensions  was  first published in the  Journal for Mathematics and Phisics 4/XXV, 1974 – 75, Zagreb