With respect to the Golden Ratio, check out Andrew Miller's week 11 page and my response thereto.
. I think that the "golden" epithet just ref erred to the fact that this ratio was supposed to be beautiful. I'm fairly certain that the ancients didn't make the connection between the "postcard" proportions they used in their temple designs and the numbers that arise from patterns on yellow flower s. But I might be wrong. By the way, it wasn't made clear in the book, but the sunflower patterns are connected with the fairly familiar Fibonacci sequence: 1,1,2,3,5,8,13,21,34, . . . where each number is the sum of the preceding two. The ratios of successive Fibonacci numbers then magically approach the golden ratio. In formal terms, if we let Fn denote the nth Fibonacci number, then the defining property is that F(n+1) = Fn + F(n-1). Dividing both sides by Fn, we get F(n+1)/Fn = 1 + F(n-1)/Fn . As n gets large, the ratio F(n+1)/Fn approaches some number L, and Fn/F(n-1) approaches the same number so F(n-1)/Fn approaches 1/L. Thus the left side of the equation approaches L and the right side approaches 1 + 1/L. This leads to the same expressi on L = 1 + 1/L that we saw in class on Friday when we discussed pentagons and Brown cards, so L^2 = L + 1 and L^2 - L - 1 = 0, and there is comes!Your comments on the therapeutic value of formal mathematics might be very helpful to the folks in psych services who have to cope with people trying to write theses, et cetera. I agree that there is a certain predictability in this chapter, and it ev en manages to make a few things predictable that were mysterious before. By the time we introduce coordinates, I am more than ready to use them rather than having to writ things out in paragraph form. Still, I am frustrated that there is not yet adequat e html support for mathematical expressions, although that should come soon.
The complex function graphs do indeed suggest some of the same shapes that come up in cosmological models, and in particular they are negatively curved. The "parking ram p" form of the cube root is a bit confusing, I admit, but I also find parking garages confusing, as if there is some parallel universe that is not quite accessible where all the open parking spaces are hidden. There is probably a story there.