You have to remember that Professor Koch's diagram shows a projection of four dimensions onto three dimensions and that's why it might have looked like two 3D objects tangled together. This is analogous to seeing a picture of something on a flat (2D) page or TV screen. These images are really projections of 3D objects onto a 2D object.
The best way I have ever understood the higher dimensions, and let's just stick with the fourth dimension, although you can go right out to an infinite number of dimensions using the same technique, is to use the process of dimensional analogy first explained to me by the famous geometer H.M.S. Coxeter in Toronto. You can see this explained graphically at www.science.ca/scientists/Coxeter/coxeter.html. The key thing to remember is that just because space is in 3 dimensions, it does not mean we have to stop there. Adding time to the 3 dimensions of space is typical, and so time is often thought of as the 4th dimension, but one can add an infinite number of dimensions. For instance, we could keep track of the where you are in 3D space, the time (the fourth dimension) you do things, and we could track your body temperature in the 5th dimension, how much money you had in the 6th dimension, your emotional state in the 7th dimension and on and on. Another good use of the higher dimensions is in telecommunications networks. Many many mathematics and physics applications require more than 3 or 4 dimensions to express relationships, and concepts.
Here is how Dimensional Analogy works. Think of each successive dimension to be a space containing multiple spaces of the dimension prior. 1-dimension consists of a straight line (the x-axis). 2-dimensions consists of the straight line in 1-D along with another line 90-degrees to it (the y-axis). The addition of the y-axis allows multiple x values to exist; in essence you can have multiple horizontal lines as well as lines that exist in both dimensions. 3-dimensions consists of the x-axis and the y-axis along with yet another line 90-degrees to both of them (z-axis). The addition of the z-axis allows multiple x-y planes (2-D spaces) to exist as well as surfaces that exist in all three dimensions. 4-dimensions consists of a 4th axis perpendicular to each of the x, y, and z axii. This is hard to visualize, but if we go along with our previous line of thinking, the addition of the 4th axis would allow multiple x-y-z spaces (3-D) to exist as well as spaces that exist in all four dimensions. A Moebius strip is a clever demonstration of 2-D existing in 3-D. To make a Moebius strip, take a thin strip of paper, flip one end upside-down, and tape the two ends together. To demonstrate the effect, take a pen and draw down the centre of the strip, following the strip in any direction and draw until you come back where you started. Notice how you have traversed both sides of the piece of paper without ever leaving the plane of the paper. The Klein bottle is a 4-D shape that exhibits the same analogous paradox in 3-D. A search on it may return several interesting results.
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