Negative area/volume in a higher dimension?

Impossible!

                Well no. Not to the degree of logic. This paper is a logical exploration into the idea of negative volume and how the principle of it  might be able to exist in a higher dimension.

                We will first take a look at the idea of a negative area, and how the question of this whole topic arose. Let us be presented with a simple f(x) = x^2 - 1 graph. We can find the area of this graph between -1 and 1 by finding the intergral of the function between the two limits. If one takes a look at the graph it will be noticed that the area in question is actually below the f(x) = 0 line, in other words, below the x-axis. This will therefore give the area a negative value. Of course the area has an actual positive value, but it just happens to be below the x-axis. We can find the actual (positive) area by taking the modulus of the area. In this case, negative area is valueless, or nonsense. This is often said about any form of negative area, that it is meaning less. It is my personal belief that nothing is meaningless, even the concept of an actual negative area. This is the history of the topic, and is the seed of thought from which the logic grows.

                We have just looked at a two-dimensional case of 'negative' area. The negative in this case isn't quite what we mean, so we shall ponder into the relavent meaning now. In order to achieve this with clarity, we use  the three dimensions purely for ease (we live in a three-dimensional world so it is easier for us to understand concepts in this dimension).

                Before us stands a bowl. This bowl is full of water with a volume of 100 cm^3. We take our special cube which has a volume of 25 cm^3. The place the cube into the water and observe what happens. It is obvious that the water level will rise partially due to the volume taken up by the 25 cm^3. Our second cube has a volume of 25 cm^3 but is hollow, and has walls of infinite thinness. When this cube is placed in the bowl the water level remains the same so we can draw the conclusion that the cube actually has 0 volume. What happens when we put a cube of -25 cm^3 into the water? The only option left is that the level of the water actually decreases. For more clarity we can take a look at the mathematics of this situation.

                First we have

                                100 cm^3 + 1 x 25 cm^3 = 125 cm^3; the water level rises.

                Then

                                100 cm^3 + 0 x 25 cm^3 = 100 cm^3; the water level remains constant.

                Finally

                                100 cm^3 + -1 x 25 cm^3 = 75 cm^3; the water level decreases.

The question we have to ask ourselves is; what happened to the water? Some how 25 cm^3 has disappeared. By taking a closer look at the mechanics of the situation we might be able to understand things better. In the first case, the volume of the cube 'pushes' the water away and then occupies its space. The second cube doesn't interact with the water, so nothing happens. The third cube seems to 'suck up' the water. Somehow we have lost 50 cm^3 of volume, 25 cm^3 of which is water. The only real logical explanation to this would be to say that the density of water within the cube has doubled. So there is twice as much water in that area as there was before. We can quite easily rename water and call it space, forming the statement; "Negative volume is a volume of twice the density of space".

                With this we can return back to the two-dimensional example which makes life easier in explaining the idea of higher dimensions. We can imagine a piece of squared (mathematical) paper on a table infront of us. This is our two-dimensional world. On it we draw a sqaure 10 cm x 10 cm in area. If we imagine sliding another two-dimensional square, of an area of let us say 5 cm x 5 cm into the first sqaure, the total area is 100 + 25 = 125 cm^2. This is almost  the same as we did with the water. So when we put a negative area into the first square, the area decreases to 75 cm^2. Where did the 50 cm^2 go? 25 of this can be accounted for by the actual area of the second sqaure, but we are still missing the other 25 cm^2. Now comes the interesting part. All we have to do is introduce a new factor into this 'equation'. That new factor is a higher dimension. What dimension is higher then the second? The third, obviously. So we pull this 25 cm^2 sqaure up into the third-dimension giving it the principle of height. It still has 0 cm height, but is elevated into a height. Given this extra space, we can easily hide, and in this case find the missing 25 cm^2. Two dimensions in a three-dimensional world suddenly gain two sides. A top and a bottom. Before, this was impossible, but now we are given an extra degree of freedom. Having these two sides, in a two-dimensional sense, we have doubled the surface area, therefore the actual area. An analogy is a cube with a blue top and a red bottom. We squeeze the cube to give it a height of 0 (producing a square), but one side is still red and one is blue. We have managed to double the area of a two-dimensional square simply by bringing  it up into the third-dimension.

                Using this principle we can very easily explain the loss of 25 cm^3 of water using the idea of higher dimensions. This negative volumed cube gets its volume from the fith-dimension (taking time as the fourth) just as the negative area of the square comes from the third. We can conclude with the following statement; "Negative area/volume is a bi-product of a higher dimension.

Fraser Scott,
sometime at the beginning of this school year.