# A lecture on 5-fold symmetry and tilings of the plane by Penrose R.

By Penrose R.

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I(T ) → 0 as t − T → ∞. To prove Thus we have shown that δq m (t) δI(T ) the corresponding relation for T − t → ∞ one may calculate δq m (t) with t < T from (17), and proceed in exactly the same manner. In this way we can establish the required relation (18). This then shows that I(T ) is an important quantity which is conserved. A particularly important example is, of course, the energy expression. This is got by the transformation of displacing the time, as has 8 In fact, for all practical cases which come to mind (energy momentum, angular moδy (σ) mentum, corresponding to time displacement, translation, and rotation), δq n (t) is acm tually zero if σ = t.

Conservation of Energy. Constants of the Motion Because of the importance in ordinary quantum mechanics of operators which correspond to classical constants of motion, we shall mention brieﬂy the analogue of these operators in our generalized formulation. Since these are not needed for the remainder of the paper, they have not been studied in detail. The notation will be as in the classical case described in section 3, of the ﬁrst part of the paper. The general discussion given there applies equally well in this case, so that we shall not repeat it.

Be written m 2 tk+1 −tk · tk −tk−1 2 tk+1 −tk The latter is inﬁnite. If, in (54), we had chosen for F the expression G 1 xk G2 , where G1 is any function of the coordinates, x j , belonging to times tj later than tk (tj > tk ), and G2 is any function of the coordinates belonging to times earlier than tk , we would have found, in place of equation (55), the relation, G1 = m i xk+1 − xk tk+1 − tk |G1 G2 | . · xk − xk · m xk − xk−1 tk − tk−1 G2 (58) The Principle of Least Action in Quantum Mechanics 37 This is equivalent to the usual relation among averages, |G1 (pq − qp)G2 | = i |G1 G2 | .