Supratridimensional Spatial Graphs

For some reason the thought struck me today that a spatial graph does not need to be confined to three dimensions. In other words, the fourth dimension and beyond need not be some mysterious and barely comprehensible phenomena. Each variable in an equation can be interpreted in a variety of ways, and spatial coordinates happens to be one of the more well-known of those ways. But traditionally the fourth dimension has been interpreted graphically as some sort of weirdness resulting in such items as the enigmatic “hypersphere” and others.

The first thought that I had on this subject was simply that our standard graphing coordinates rely on the position of axes such that the same scalar value on each axis (a single point) is equidistant from the others, i.e. x = y = z = K where K is any number. Connecting these points results in the plane x + y + z = K. My feeling was that by simply adding another axis and repositioning the four of them so that w = x = y = z = K for any K, one could then graph any given object in four dimensions. Once this is proven, one could likely demonstrate that any number of axes could be arranged so that any object could be graphed therein. This would simply be another interpretation of multiple dimensions—one where the traditional mystique of the fourth and fifth dimensions, etc., is discarded in favor of a system whereby any number of dimensions could be represented spatially. I have made several observations already about the possibilities of this system, but I can already see that it is fraught with difficulties that only mathematicians far more accomplished than I could fully confront. In fact, I believe someone has very likely thought of this theory before me, but I am not aware of that being the case.

One observation is that this system, particularly the arrangement of axes, can be constructed from the primary “nonexistent” 2D elements: points, lines, and planes. One end result of this is that the 0, first, and second dimensions remain theoretical and non-spatial, while every dimension from the third onward is spatial. I wondered for a while how I could determine the position of the axes for the fourth dimension, then I finally had a thought. The 3D coordinate system has three axes, which can be considered as six “sticks” protruding from the origin. You can create symmetry between three of these sticks and the other three by separating them with a plane. If you were to view one side of this plane from the origin point, you would see three lines extending outward, each of which is equidistant from the next and previous line. I wish I could provide a picture, but I am a bit too lazy to draw one up right now—however, I plan to do so in the future. However if you can see these lines extending from the origin point as I am describing, you will notice that connecting the tips of the lines as they extend out results in an equilateral triangle. And this is precisely the graph resulting from x + y + z = K where the value of K is the same on each of the three axes and the graph is limited to one of the eight octants of the coordinate system. So now what would happen if we added a fourth axis? My thought is that if we divided the resulting eight “sticks” with a plane, as we did with three axes, and positioned ourselves at the origin, we would see four lines extending, each of which is equidistant from the previous and next. So in this case a square is projected.

The major problem that arises from this is that for any value K, the value of K on each axis is not equidistant from the other three, but only from the next and previous axis. I am not sure at this juncture whether the possibility exists for arranging four points such that each one is equidistant from all the rest. My thinking is that it is not possible, and I am uncertain whether this would ultimately matter—I do not think it nullifies the plausibility of my system, but it will result in variant graphs for equations that may have the same form. In other words, z = y + x may look different from z = w + x even though the forms are the same. This does not come as a major surprise, since the 3D coordinate system is arranged such that the angle between each “stick” and its neighbors is 90 degrees. This right angle property of the 3D coordinate plane facilitates a level of simplicity and intuitiveness that additional dimensions would lack in my system, but only for the majority of the basic shapes to which we are accustomed. Certain crystalline structures may be much simpler to graph in a supratridimensional coordinate system.

Inconsistency from one dimension to the next is expected. For instance, the graph x² + y² = K in two dimensions results in a circle, but in three dimensions it is a cylinder. A circle in three dimensions is an intersection of two graphs: for example x² + y² = K and z = 0. Similarly a 3D sphere x² + y² + z² = K would look much different in the 4D coordinate system. My initial thought was that a typical sphere could be graphed in four dimensions as w² + x² + y² + z² = K, but the aforementioned lack of equidistance between each axis from the other would preclude this. However my only goal is demonstrating that any number of dimensions can produce purely spatial graphs.

Returning to the projected shapes I mentioned earlier, you can see how this can easily extend to more dimensions. Five dimensions would result in a projected pentagon, six dimensions in a hexagon, and so on. The angle between each of these axes and the dividing plane would seemingly be equal, but the angle of each stick in relation to the others would differ by varying amounts (to be determined later by me or someone else perhaps). This projected shape must appear on any given arrangement of sticks that are all on one side of the dividing plane. As an aside, I made sure this theory was “backwards compatible” with two dimensions, which it is: in this case a line is projected, but it is separate from the lines that comprise the axes. In one dimension a point is projected.

Another pattern I noticed is the number of projected figures that can appear on the coordinate systems. A 1D graph has two sides, a 2D graph has four quadrants, and a 3D graph has eight octants. The computer programmer in me immediately notices a binary squares pattern. 1D = 2¹ projections, 2D = 2² projections, 3D = 2³ projections. I am hypothesizing that a 4D coordinate system must contain 2⁴ projections, or 16 squares. (Coincidentally I have for a long time been fascinated with the number 16 and its frequent appearance in nature and society. This is merely another example of its ubiquity.) Now looking at a 2D graph of its projected lines (|x| + |y| = K) from its normal perspective results in a tilted square (or a 2D diamond). Similarly a 3D graph of all eight of its projected triangles pieced together results in an eight-sided diamond. So I might expect that the 4D shape formed from its projections would be a 16-sided figure where each side is a square—or perhaps that would be the case only if the aforementioned “equidistance problem” did not exist. As of now, I can only envision a typical cube as the result of connecting the dots of four intersecting axes, which is a six-sided figure, but I feel like I am missing something. Are the rules different for even-numbered dimensions? Or should I be trying to envision “2D diamonds” rather than “2D squares” even though they are essentially the same? Or is the occurrence of triangles in this figure impossible to escape? Right now I am not sure about these questions. I only know that my basic concept of a dividing plane and projected 2D shapes (where the number of sides is equal to the number of dimensions) seems to make sense and may be a potentially useful method for constructing supratridimensional coordinate systems that can render entirely spatial graphs.