Yes – nice!

]]>If you apply any of the four matrices to a vector , you obtain a vector such that

and

Hence if we apply any sequence of matrices, repeated times, these two invariants are bounded by the th entry of a Fibonacci-like sequence. So they are bounded by a constant times the th power of the golden ratio. If the starting vector is an eigenvector, both invariants grow like the th power of the eigenvalue, so the eigenvalue is at most the th power of the golden ratio.

]]>Don’t get me wrong, I also love pure maths (even if my understanding is very limited as this point), but it’s a rather surprising, unexpected use case, so that’s lovely in its own right.

Also, watching that taffy puller is mesmerizing. ]]>

I tend to think of this result as more about beautiful pure math than anything practical, but there are related papers that focus on the practical issue of efficient mixing. Since real-world mixing is about *3-dimensional* fluids, it’s a lot more complicated. It’s very interesting, but I don’t know much about it.

On the lighter side, see:

• Jean-Luc Thiffeault, A mathematical history of taffy pullers.

Abstract.We describe a number of devices for pulling candy, called taffy pullers, that are related to pseudo-Anosov mapping classes. Though the mathematical connection has long been known for the two most common taffy puller designs, we unearth a rich variety of early designs from the patent literature.

This paper has 56 pictures and is definitely worth a look! I never knew the mathematics of taffy was so interesting.

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