By Daniel J. Velleman
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Extra resources for American Mathematical Monthly, volume 117, number 3, March 2010
Example text
Related Work. The results and ideas of Section 2 are, for the most part, not new. The idea of an abstract polyhedron represented by an incidence structure, and its realizations in R3 , appears in Section 2 of [14]. In Corollary 15 of that paper, Whiteley proves that the space of realizations of a ‘spherical’ incidence structure (equivalent to an incidence structure arising from a convex polyhedron) has dimension E. 8 is often referred to as ‘Steinitz’s lemma,’ and a variety of proofs appear in the literature ([10], [8]); we believe that the proof above is new.
Diagonal angles’ between Ai A j and Ak Al . Other quantities might also be considered, like the distance from Ai to A j Ak , but in practice these would require several measurements. It is natural to ask how many simple measurements are needed to determine a polygon up to isometry. In the case of a triangle the answer is 3. As mentioned earlier, there are a few ways to do it. 4). It is easy to see that for any n-gon P = A1 · · · An the following (2n − 3) measurements suffice: the n − 1 distances |A1 A2 |, |A2 A3 |, .
Probab. 26 (1998) 1212–1231. 1214/aop/1022855750 7. N. L. Johnson and S. Kotz, Urn Models and Their Application. An Approach to Modern Discrete Probability Theory, John Wiley, New York, 1977. 8. M. Krishnapur and Y. Peres, Recurrent graphs where two independent random walks collide finitely often, Electron. Comm. Probab. 9 (2004) 72–81. 9. D. A. Levin, Y. Peres, and E. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, Providence, RI, 2008. 10. R. Lyons and Y. Peres, Probability on Trees and Networks (in progress); available at http://php.