• Neil Sloane has a new blog subtitled “interesting sequences I need help with”. His first post concerns the two-up sequence, formed in steps where the $k$th step adds two numbers that are not already in the sequence and are relatively prime to the preceding $k$. Most of the terms appear to be the primes (in order). The remaining terms appear to be prime powers or semiprimes but this has not been proven.
• ## The motion of bend lines on smooth surfaces

You may have played with a paper yoyo, a strip of paper wrapped around a stick so that when you flick it with your wrist, it extends outward into a long tube. Here’s one, the only example I could find on Wikimedia Commons:

• ## Reflections in an octagonal mirror maze

The second preprint from my CCCG papers is now online. It is “Reflections in an octagonal mirror maze”, arXiv:2206.11413. The title is quite literal: suppose you find yourself in a mirror maze, where the mirrors are aligned with the sides of an octagon, and have integer coordinates (meaning that, on a floorplan of the maze, the mirrors become line segments between points of an integer grid). What would you see if you looked in any given direction? It might be many reflections eventually leading to the back of your own head, to the exit, or some other non-reflective part of the maze. The example below, from the Museum of Science & Industry in Chicago, is hexagonal rather than octagonal, but otherwise has much the same effect:

• ## Dehn rank revisited

In a recent post, I discussed dissection of orthogonal polygons into each other by axis-parallel cuts, translation, and gluing. Each polygon has a value associated with it, called its Dehn invariant, that cannot be changed by dissection, so two polygons that can be dissected into each other must have equal invariants. And for past usage of Dehn invariants, that was pretty much all we looked at: are they equal or not? But my post pointed out that these invariants actually have a lot of structure (you can think of them as matrices, after an arbitrary choice of basis) and this structure is geometrically meaningful. Matrices (or tensors) have a rank, and the rank of the Dehn invariant is a lower bound on the number of rectangles into which a polygon can be dissected. This in turn has implications on the ability of a polygon or its dissections to tile the plane.

• ## The shapes of triangular pencils

The Institute of Mathematics & its Applications tells us that applications of the Reuleaux triangle include “the cross-section of some pencils that are thought to be more ergonomic than traditional hexagonal ones”. It it true?

• Claas Voelcker on academic work-life balance via). I think we all know that many academics (myself included!) struggle to keep our weekend and evening time free of work-related distractions. Voelcker investigates where this pressure to work comes from (often internally) and suggests that overwork may block creativity; taking time off can make you more productive.
• ## The analyst's minimum spanning tree

Infinite sets of points in the Euclidean plane, even discrete sets, do not always have Euclidean minimum spanning trees. For instance, consider the points with coordinates

• ## Maybe powers of π don't have unexpectedly good approximations?

After I wrote recently about Ramanujan’s approximation $$\pi^4\approx 2143/22$$, writing “why do powers of $$\pi$$ seem to have unusually good rational approximations?”, Timothy Chow emailed to challenge my assumption, asking what evidence I had that their approximations were unusually good. So that led me to do a little statistical experiment to test that hypothesis, and the experiment showed…that the approximations seem to be about as good as we would expect, no more, no less. Not unusually good. Chow was correct, and my earlier statement was overstated. So if Ramanujan’s approximation is not just random fluctuation (which for all I know it could be), it at least does not seem to be part of a pattern of many good rational approximations for small powers of $$\pi$$. Below are some details of how I came to this conclusion.