Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

38 marcus giaquintorecall the puzzlement one feels when first introduced to vector multiplicationin terms of pairs-of-reals. Given that i 2 =−1, it is clear that(x + iy)(u + iv) = (xu − yv) + i(xv + yu)But why does the term on the right denote the vector whose length is theproduct of the lengths of the multiplied vectors and whose angle is the sum ofthe angles of the multiplied vectors? Given the law for multiplication by addingexponents, the answer is immediate using the Euler notation for vectors:re iθ se iη = rse i(θ+η)For further light on Euler’s formula I recommend Needham (1997), 10–14.Another way in which visualization can be useful is as an aid to discoveryor proof. We have already seen an example in the use of the diagram inFig. 1.5 in getting to the theorem that, if x.y = k, the values of x and y forwhich x + y is smallest are x = √ k = y. The visual thinking does not get usall the way to the theorem, but it reduced the problem to simple algebraicmanipulations. Often a visual representation will seem to show properties thatmight be useful in solving a problem, a maximum here or a symmetry there;thus the investigators are supplied with plausible hypotheses which they canthen attempt to prove. An excellent example is the successful attempt to showthat the Costa minimal surface (Fig. 1.8) is embeddable¹⁰ by David Hoffman,William Meeks and James Hoffman.Fig. 1.8.¹⁰ A surface is embeddable if and only if it can be placed in R 3 without self-intersections. Moregenerally, a surface is n-embeddable if and only if it can be placed in R n without self-intersecting, butfor all m < n cannot be placed in R m without self-intersecting.