The tangencies are exact: https://en.wikipedia.org/wiki/Golden_spiral#/media/File:FakeRealLogSpiral.sv...
My reason for doubting the tangency was based on the idea that EVERY point on the spiral can serve as a point of tangency (just perform an appropriate spiral similarity on the nest of rectangles while keeping the spiral fixed), and that if the tangency property held at every point, the "spiral" would have to be a circle. But I never thought it through clearly.
No, because the inward normal does not point to the centre of the spiral. Sincerely, Adam P. Goucher
Jim Propp
On Monday, June 22, 2015, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 22/06/2015 04:47, James Propp wrote:
What is the small but non-zero angle of tangency between the golden spiral
and the associated nest of golden rectangles? And does this angle occur elsewhere in geometry?
I assume "angle of tangency" means "angle cut off by that thing that looks like a tangent but isn't one". But there are two things that aren't obvious to me. (Spoiler alert: these are answered below.)
1. What exactly is the associated nest of golden rectangles? I know how to go from a nest of golden rectangles to a thing that looks like a golden spiral but isn't one (draw lots of quarter-circles) but that's no help.
Without thinking about it too much -- are you sure, in fact, that there isn't a way of associating the two that *does* give exact tangencies? At first glance it looks to me like there is, but first glances are notoriously unreliable and I have found my own especially so.
2. Is there really a single such angle? All the "tangencies" look the same because they're related by the rotation+scaling under which the spiral is invariant, but don't you get two slightly different angles at each "tangency"?
Actually, I'm having trouble working out exactly what the book thinks a "golden spiral" *is*. It describes a bunch of things that "approximate, but do not exactly equal, a golden spiral"; then it says "The golden spiral is essentially a logarithmic spiral" but what does "essentially" mean?; then, in discussing nautilus shells, it says "the curve tends to a logarithmic spiral as it expands" but it's not clear whether "the curve" is meant to be the golden spiral or the actual shape of a nautilus shell or what.
*
So, over to Wikipedia, which says a golden spiral *is* a log spiral (good!) r = a exp(2/pi log(phi) theta). So draw lines parallel to the axes where the spiral crosses them; I guess that gives our "associated nest of golden rectangles". E.g., one rectangle is defined by theta = 0, pi/2, pi, 3pi/2 giving (taking a=1 because why not?) r = 1, phi, phi^2, phi^3 so that the rectangle sides are 1+phi^2 and phi times that. And it does give a "nest of golden rectangles" cut off by squares, the relevant fact being that phi+phi^3 = phi^4-1.
OK, so now is the spiral tangent to the rectangle at those points? We have z = exp(2/pi log(phi) theta) exp(i theta) or z = exp(k theta) with k = 2/pi log phi + i. So dz/dtheta = k z and e.g. at theta=0 a point moving along the spiral is going in direction z, whose angle to the y-axis is arctan(2/pi log phi) ~= 0.2973 ~= 17 degrees. That's pretty big. And then for each of these there will be a crossing point slightly earlier, where the angle turns out to be (numerically, with apologies for laziness) ~= 0.2802 ~= 16 degrees.
But wait, this needn't be the nest of golden rectangles we use. What if instead we take the points where the spiral is parallel to the axes? Then we need real*{1,i,-1,-i} = dz/dtheta = kz = exp(k(theta+1)) = exp(real) exp(i(theta+1)) which means theta = pi/2*integer-1; if theta=pi/2*n-1 then we get z = (i phi)^n/k.
So, e.g., we have z moving right at 1/k, up at i phi/k, left at -phi^2/k, down at -i phi^3/k. And again we get golden rectangles and again they nest the way they need to. (Do the algebra if you don't believe me.)
So why isn't *this*, with its perfect tangencies, the "nest of golden rectangles associated with the golden spiral"?
*
So, my answers to my own two questions. 1. There are multiple ways to associate a nest of golden rectangles with a golden spiral. One of them, which seems like a pretty natural one, has exact tangencies. 2. For one "wrong" choice of rectangle-nest, which may or may not be the "right wrong choice", you get angles of about 17 degrees and about 16 degrees.
-- g
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