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Chapter 10 How Round Is Your Circle? > 10.12 Modern and Accurate Roundness Meth... - Pg. 224

224 C H A P T E R 10 a triangle with some irrational angles cannot be accommodated within this scheme. Indeed, a remarkable theorem of Kamenet- ski (1947) applied to triangles proves that necessary and suf- i ficient conditions for the existence of a rotor of a triangle are that all angles are rational. Thus, a single irrational triangle has no rotors, and hence would provide a sufficient three-point summit test for roundness. While constructing an irrational- angled triangle might not be practical for the engineer, using fractional denominators that have a large lowest common mul- ¯ tiple would result in a much greater n and hence less deviation from roundness. It is precisely this approach that is rather cleverly adopted by Goho, Kimiyuki and Hayashi (1999). They propose an angle of 7 14 = 180 . In a symmetrical summit test the other two angles in 1 this triangle are 2 (180 - 14 ) = 83 . Since 83 is prime, 83/360 ¯ cannot be simplified and we would need to take n = 360, giving n = 359. With = 1, the convexity constraint of (10.5) gives