By Daniel J. Velleman
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Extra info for American Mathematical Monthly, volume 117, number 3, March 2010
An } is a sequence of distinct points on the plane). Then the following are called simple measurements: 1. distances Ai A j , i = j, 2. angles Ai A j Ak for i, j, k distinct, 3. ‘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.
11 (1983) 393– 402. 1214/aop/1176993604 12. H. M. Mahmoud, P´olya Urn Models, Texts in Statistical Science Series, CRC Press, Boca Raton, FL, 2009. 13. C. St. J. A. Nash-Williams, Random walk and electric currents in networks, Proc. Cambridge Philos. Soc. 55 (1959) 181–194. 1017/S0305004100033879 14. J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2, Cambridge University Press, Cambridge, 1998; reprint of 1997 original. 15. R. Pemantle, A survey of random processes with reinforcement, Probab.
We translate P if necessary to ensure that no plane that contains a face of P passes through the origin. This allows us to give an equation for the plane containing f j in the form a j (P)x + 236 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 b j (P)y + c j (P)z = 1 for some nonzero triple of real numbers (a j (P), b j (P), c j (P)); similarly, for any realization Q of that’s close enough to P the ith vertex of Q is a triple (xi (Q), yi (Q), z i (Q)) and the jth plane of Q can be described by an equation a j (Q)x + b j (Q)y + c j (Q)z = 1.