 Pythagoras of Samos |
Greek mathematician Pythagoras (~500 BC) is credited with discovery of the Pythagorean Theorem: The square of the hypotenuse is equal to the sum of the squares of the other two sides. His larger belief was that everything can be explained mathematically and that numbers are the ultimate reality. If Pythagoras were alive today, he might work at APC, where math and cable lubrication are the ultimate reality. APC touts its position as technical leader, but you may wonder how that comes about. It's the employees. Consider this actual email (edited for brevity) from Charles Cole, APC Int'l Sales Director to Sheri Dahlke, APC Lab Manager, as they noodle a mundane lubrication question: Sheri, I appreciate your "puzzle" of the cables in conduit. While you're correct that there is no solution in Euclidean space, I'm happy to inform you that I have found solutions in non-Euclidean space. Euclidean space is endowed with an intrinsic measure or metric resulting from the non-trivial specification of the origin as a special point in that space. Endowed with such a non-trivial point, the concepts of distance and orthogonality allow for a Euclidean distance measure based on the work done by Pythagoras and others. In other words, orthogonal transformation of Euclidean space implies conservation of distance. As such, limiting the cable in conduit problem to simple Euclidean spaces allows one to make a rather simplistic assertion that there is no solution. An affine space, though, is not endowed with such a metric, as no point is non-trivial. Without an intrinsic metric, only parallelism and rectilinearity are conserved in an affine space. (For proof, try commuting the momentum operator in A4.) The existence of spaces, where a distance measure is not intrinsic to any transformation group, suggests the possibility of the solubility of the cable/conduit problem using non-Euclidean spaces. Evariste Galois, before his death in 1837, developed the concept of group transformations. While Galois Theory is discrete in nature, further development of these group concepts by Sophus Lie allowed for continuous transformations to take center stage in the 1880's. The study of such continuous transformations led to the development of the study of Lie Groups and the symmetries they generate. It is here where the argument of non-solubility breaks down. After a few simple calculations using both Unitary and Symplectic groups, I was able to prove the existence of not just one but an infinite number of solutions to the puzzle. The symplectic group solutions are a bit more challenging because they allow not only anti-symmetrical operators but also non-commutativity. It is the latter condition that makes child's play of the solution to the puzzle. The non-commuting operators used in my symplectic group solution generate complex tensor spaces of degree six and higher. Recall that dimensions outside the domain of the four standard dimensions of space-time curl up into infinitesimal point spaces. This essentially shrinks any physical manifestation of the total cable/conduit configuration proportionally to Plank's constant. What this boils down to is that if your cable/conduit ensemble is embedded in a symplectic space of dimensionality 6 or higher, and if you apply a uniform translational force on that ensemble, the dialation of the cables themselves will be sufficient to allow the cables to pass through the conduit unimpeded. (To visualize, just think of Pauli's Exclusion principle applied to physical ensembles traveling close to the speed of light.) Your cable/conduit puzzle was fun to think about. I'm sorry I only had five minutes to consider it. With more time I'd have liked to analyze the implications of superluminary velocities of the cable/conduit ensemble. Thanks for the quick diversion. --Charlie |
