Mozilla weighs in on why Facebook's side of the story doesn't hold water: https://blog.mozilla.org/en/mozilla/news/why-facebooks-claims-about-the-ad-observer-are-wrong/

A new paper by Asperó and Schindler (https://annals.math.princeton.edu/2021/193-3/p03) argues that principles of maximal forcing, unified in their paper, provide natural models for set theory in which many natural questions that are independent of ZF have clear answers. For instance, in these models, there are \(\aleph_2\) real numbers, not \(\aleph_1\).

I got to this via a popularized treatment in Quanta (https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/), but I think the introduction of the paper is quite readable. (The rest is not.)

Scott Aaronson takes a break from quantum supremacy to tell us about busy beavers: https://www.scottaaronson.com/blog/?p=5661

These are Turing Machines that take as long as possible to do stuff. "As long as possible" is an explosively-quickly growing function of the number of states, but the gist of the post is that the "do stuff" part can be defined in various ways, some of which make the explosion happen earlier than others.

Imperfect comb construction reveals the architectural abilities of honeybees: https://doi.org/10.1073/pnas.2103605118, via https://arstechnica.com/science/2021/07/mergers-twists-and-pentagons-the-architecture-of-honeycombs/

How do bees cope with making hexagonal honeycombs when some kinds of cells have different sizes and some patches of honeycomb don't align when they come close to first meeting up? Answer appears to be: they see the problems coming and accommodate them gradually by intermediate variations in size and degree of cells.

A dead mathematician co-authored a paper after appearing in a dream: https://boingboing.net/2021/07/20/a-person-in-a-dream-co-authored-a-math-paper.html

The paper is "Higher Algebraic \(K\)-Theory of Schemes and of Derived Categories" by Robert Wayne Thomason and Thomas Trobaugh (2007), https://doi.org/10.1007/978-0-8176-4576-2_10, https://www.ams.org/mathscinet-getitem?mr=1106918

It appears to have been quite an influential one; the MR review calls it a landmark, and it has over 1000 citations on Google Scholar.

Hamming cube of primes: https://cp4space.hatsya.com/2021/07/20/hamming-cube-of-primes/

Make an infinite graph whose vertices are the binary representations of prime numbers and whose edges represent flipping a single bit of this representation. (For instance, 2 and 3 are neighbors.) Surprisingly, it is not connected! 2131099 has no neighbors.

See also the same question on MathOverflow, a year ago: https://mathoverflow.net/q/363083/440

A paper almost a decade in the making, now in the Notices of the American Mathematical Society. In 2012, Marco Paoletti asked me a question about rolling acrobatic apparatus. Answering this led to new designs, one of which was fabricated by Lee Brasuell and is now being performed with.

https://www.ams.org/journals/notices/202107/rnoti-p1106.pdf

If you modify the sieve of Eratosthenes so that each generated number \(p\) knocks out the numbers \(pn+2\) instead of the usual \(pn\), you get the prime-like sequence 2, 3, 7, 13, 19, 25, 31, 39, 43, 49, 55, 61, 69, ... (https://oeis.org/A076974).

Although many non-primes are in this sequence, and many primes are not, Bill McEachen has observed that with one exception the larger prime in every twin prime pair is part of this sequence! The proof is not difficult; see the OEIS link for spoilers.

Building a trivalent graph of harmonic relations among major and minor triads.

https://www.youtube.com/watch?v=O4UpNSlzKAM

Two more new Wikipedia Good Articles:

Cairo pentagonal tiling (https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling), a tiling of the plane by congruent but irregular pentagons, formed by overlaying two hexagonal tilings. It appears in street pavings, crystal structures, and the art of M. C. Escher.

Halin graphs (https://en.wikipedia.org/wiki/Halin_graph), the planar graphs formed from trees by connecting their leaves into a cycle. Studied by Kirkman long before Halin and significant in graph algorithms because of their low treewidth.

Yoshimura Crush Patterns: https://blogs.ams.org/beyondreviews/2021/07/18/yoshimura-crush-patterns/

See also https://en.wikipedia.org/wiki/Yoshimura_buckling, which repeats Robert Lang's observation that these patterns can be seen on Mona Lisa's sleeves.

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- https://www.ics.uci.edu/~eppstein/

I'm a computer scientist at the University of California, Irvine, interested in algorithms, data structures, discrete geometry, and graph theory.

Joined Apr 2017