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En Garde, whole no. 16, January 1946
Page 14
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page 14. All this relies considerable on your intuition, but should demonstrate that there is nothing mysterious about higher dimensions if we assume them (as you will grant we may reasonably assume in the absence of any other hypothesis) to be simply spatial dimensions behaving exactly like our own basic 3. Much more can be done, of course. I venture to say that the number of different regular polyhedra possible in 4-space could be determined--and proved-- by the same method used in 3-space. Whether this has ever been done or not I don't know, but if Stanley or somebody wants to try it it might prove interesting; though you might quite likely find that the only regular polyhedron is the tesseract. II Now for the general exposition of the subject. the natural place to start is with the concept of a vector space. Those fans who were subjected at some time in their careers to college physics or analytic geometry will have some familiarity with the animal known as the vector. I hope they understood it at the time, but for those whose exposure was insufficient for the idea to take I'll try to explain it now. (The mathematically erudite should skip the next paragraph or two.) My freshman physics instructor opened one of his first few lectures with the statement, "A vector is something which has magnitude and direction." The students remarked to each other behind the backs of their hands that the definition didn't mean a damn thing; the instructor gave them all a look of tacit agreement and went on. Actually, the definition does put it pretty well; but I think it's better to say a vector is a magnitude and a direction. Visualize an arrow pointing in a given direction and with length equal to a given magnitude. Since position is not one of the things which determine a vector, the vector corresponding to this arrow is the same as that corresponding to a like arrow parallel to it at any point in space. It might represent, say, the velocity of the given particle at a given instant (its direction being that of the particle's motion, and its magnitude representing the particle's speed, in any appropriate units); and the vector's non-dependence on position is a reminder of the fact that 2 particles in widely separated positions may have the same velocity at the same instant. Or it might represent the "displacement" between 2 points, that is, the direction and distance of straight-line travel necessary to reach one from the other. Those two examples should be sufficient demonstration of the significance and importance of the concept of vector. Remember, you can move a vector parallel to itself all you want and it doesn't lose its identity; but if you change its direction or length it becomes a different vector altogether. Now suppose we have a system of vectors (with any physical meaning whatever, it makes no difference) which is restricted to a plane. If we set up a system of rectangular coordinates in the plane and then move one of the vectors so that its base (the tail end of the arrow) is at the origin of the coordinates, the setup is like so (Fig. 1).
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page 14. All this relies considerable on your intuition, but should demonstrate that there is nothing mysterious about higher dimensions if we assume them (as you will grant we may reasonably assume in the absence of any other hypothesis) to be simply spatial dimensions behaving exactly like our own basic 3. Much more can be done, of course. I venture to say that the number of different regular polyhedra possible in 4-space could be determined--and proved-- by the same method used in 3-space. Whether this has ever been done or not I don't know, but if Stanley or somebody wants to try it it might prove interesting; though you might quite likely find that the only regular polyhedron is the tesseract. II Now for the general exposition of the subject. the natural place to start is with the concept of a vector space. Those fans who were subjected at some time in their careers to college physics or analytic geometry will have some familiarity with the animal known as the vector. I hope they understood it at the time, but for those whose exposure was insufficient for the idea to take I'll try to explain it now. (The mathematically erudite should skip the next paragraph or two.) My freshman physics instructor opened one of his first few lectures with the statement, "A vector is something which has magnitude and direction." The students remarked to each other behind the backs of their hands that the definition didn't mean a damn thing; the instructor gave them all a look of tacit agreement and went on. Actually, the definition does put it pretty well; but I think it's better to say a vector is a magnitude and a direction. Visualize an arrow pointing in a given direction and with length equal to a given magnitude. Since position is not one of the things which determine a vector, the vector corresponding to this arrow is the same as that corresponding to a like arrow parallel to it at any point in space. It might represent, say, the velocity of the given particle at a given instant (its direction being that of the particle's motion, and its magnitude representing the particle's speed, in any appropriate units); and the vector's non-dependence on position is a reminder of the fact that 2 particles in widely separated positions may have the same velocity at the same instant. Or it might represent the "displacement" between 2 points, that is, the direction and distance of straight-line travel necessary to reach one from the other. Those two examples should be sufficient demonstration of the significance and importance of the concept of vector. Remember, you can move a vector parallel to itself all you want and it doesn't lose its identity; but if you change its direction or length it becomes a different vector altogether. Now suppose we have a system of vectors (with any physical meaning whatever, it makes no difference) which is restricted to a plane. If we set up a system of rectangular coordinates in the plane and then move one of the vectors so that its base (the tail end of the arrow) is at the origin of the coordinates, the setup is like so (Fig. 1).
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