By Melvin Hausner

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During this booklet the main points of many calculations are supplied for entry to paintings in quantum teams, algebraic differential calculus, noncommutative geometry, fuzzy physics, discrete geometry, gauge conception, quantum integrable structures, braiding, finite topological areas, a few elements of geometry and quantum mechanics and gravity.

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Assign the mass 1 to each vertex and let M be the center of mass of the resulting configuration. There are quite a few ways to find M. As in Fig. 18, we may replace three mass-points 1A, 1B, 1C with 3M1, where M1 is the centroid of ABC. Then M is found along M1D by dividing M1D in the ratio 1/3. In Fig. 19, 1A and 1B are replaced by 2X1 (where X1 is the mid-point of AB). Similarly, X2 is found along CD, and M is then the mid-point of X1X2. 19 The lines connecting a vertex of a tetrahedron to the centroid of the opposite face all meet in a common point M, called the centroid of a tetrahedron.

But we may also very nicely explain Eq. (3) in terms of point of division. In general, was taken to mean that P is on segment QR and (See Fig. ) While m and n were understood positive in (4), we are nevertheless going to compare (3), (4), and (5). This inevitably leads to the equation for the points of Fig. 56. Does it make sense? * In Fig. 56, it is seen that the directions QP and PR are opposite. This accounts for the negative sign in Eq. (6). Thus we may extend the meaning of Eqs. (4) and (5) to include negative m or n.

This is justified, if justifications are indeed necessary, by the remark that one of the coordinates (say, C) is unnecessary; the equation a + b + c = 1 determines it from the other two. But retaining it has advantages, since the vertices are then all treated as equals and a certain symmetry in the notation is retained. The location of P = aA + bB + cC (a, b, c > 0) has already been found in the previous sections. In Fig. 52, P is found via the device of grouping A and B first: P = (aA + bB) + cC Thus, C′ is first constructed, and then P.