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Last week, when I was wrapping strings of beads around my Christmas tree, I initially had the following design in mind: I wanted each pass around the tree to be evenly spaced (e.g. exactly one foot below each part of the string would be another part of the string), and I wanted the slope of the string to be constant (with the exception of the vertex of the tree, since slope isn't defined there).

I quickly realized that this was mathematically impossible, since a constant slope along the entire string of beads would imply increased spacing between each pass down the tree due to the tree's increasing diameter at lower heights.

That said, if we only look at the points at which the string of beads intersects a given ray down the side of the tree, then it is possible for the string to be at the same spacing and slope for each point on that ray by flattening or steepening the string in that neighborhood. My intuition says that this cannot apply to all rays at once, no matter how varied or strange we make the path, but I'm struggling to prove why that is.

My thought is that flattening or steepening the string around whatever point we're looking at has to be compensated for in other neighborhoods, but I'm not sure how to formalize this. Any ideas?

Matthew Milone
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Put the tree looking downwards with the vertex at the origin. Its surface $S$ then is given by $$S:\quad (s,\phi)\mapsto\left\{\eqalign{x&:=s\cos\phi\cos\alpha\cr y&:=s\sin\phi \cos\alpha\cr z&:=s\sin\alpha\cr}\right.\qquad(s>0,\ -\infty<\phi<\infty)\ ,$$ where $\alpha\in\>]0,{\pi\over2}[\>$ is a certain constant. Your string then is defined by $$t\mapsto\left\{\eqalign{s&:=s(t)\cr \phi&:=t\cr}\right.\qquad(-\infty<t<\infty)$$ with an unknown function $t\mapsto s(t)$, and leads to the space curve $$\gamma:\quad t\mapsto\left\{\eqalign{x&:=s(t)\cos t\cos \alpha\cr y&:=s(t)\sin t\cos\alpha\cr z&:=s(t)\sin\alpha\cr}\right.\qquad(s>0,\ -\infty<\phi<\infty)\ .$$ The slope $\theta$ of $\gamma$ with respect to the $(x,y)$-plane computes to $$\tan\theta={\dot z\over\sqrt{\dot x^2+\dot y^2}}={\dot s\over\sqrt{s^2+\dot s^2}}\>\tan\alpha\ .$$ If we want $\theta$ to be constant we therefore need $\dot s=\lambda s$ for a certain constant $\lambda>0$. This means that the wrapped string is a logarithmic spiral on the surface $S$. In fact we have an infinite family of such spirals, all of them intersecting all generatrices of $S$ under the same angle. But they of course do not have the other desired property $s(t+2\pi)-s(t)\equiv c$.

Christian Blatter
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