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Linkage
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Linkage
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Linkage
- Brick territories experiment (\(\mathbb{M}\)). What shapes do you get when \(n\) simultaneous breakout games compete against each other for pixels?
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Periodic coloring of infinite planar graphs
The de Bruijn–Erdős theorem states that the number of colors needed to color an infinite graph is the same as the maximum number needed for its finite subgraphs. So for any reasonable definition of an infinite planar graph, the 4-color theorem for finite planar graphs implies that every infinite planar graph is also 4-colorable.
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Linkage with two research problems
In case you’ve been worrying that the recent publicity blitz of LLM solutions to open mathematics problems is causing us to run short, there are two more mixed in among my usual links here. Because I haven’t solved them, they are not very precisely formulated, and I don’t know how difficult or interesting they are nor even whether someone else might have already considered them; that’s often the way at the start of research.
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Fractional rings of tangent spheres
Soddy’s hexlet consists of a ring of six spheres, tangent to each other consecutively around the ring, and another ring of three consecutively-tangent spheres, so that all the spheres in the first ring are tangent to all the spheres in the second ring. If you keep one ring fixed, you can rotate the other ring continuously, possibly changing the sizes of some of the spheres as they rotate but keeping the pattern of tangencies unchanged. Here’s a nice animation I found on Wikipedia, where the ring of six spheres rotates continuously while the other ring of three spheres (the central blue one and the two green planes, considered as degenerate spheres tangent at infinity) stays fixed. The larger red sphere is not part of this configuration and I don’t know why the author of this animation included it.
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Linkage
- A permutation generation algorithm in the work of 13th-century Kabbalist Abraham Abulafia (\(\mathbb{M}\), via). The resulting permutation sequence is the one you get by reversing suffixes whose lengths form the sequence \((((2, 3)^2, 2, 4)^3, 2, 5)^4, \dots\) but that’s not the generation rule. Instead the rule is: to generate the permutations of \(1, 2, 3,\dots, n,\) form its \(n\) cyclically rotated permutations (starting with \(1, 2, 3,\dots, n,\)) and for each one in order, recursively generate the permutations of its length-\((n-1)\) suffix.
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Non-coplanar unit distances
Reports that LLMs have killed the Erdős unit distance problem turn out to be greatly exaggerated. There is still plenty not yet understood about the problem.
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Linkage
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Integer complexity and cographs
The integer complexity of a number \(n\) is the minimum number of ones needed to express \(n\) as a parenthesized combination of sums and products of ones. For instance, 10 has complexity 7 as it can be expressed using seven ones, but not fewer:
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