Linkage

Sariel Har-Peled demonstrates how to make animated pdfs in beamer talk slides and present them on Linux using okular (\(\mathbb{M}\)): YouTube demo; beamer source code. On OS X, the same technique works when presenting using Acrobat, but not Preview.

Evidence of the Elements: Finding Euclid on scattered pot shards (\(\mathbb{M}\), via). Excerpt from Benjamin Wardhaugh’s new book Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World.

Yoshimura Crush Patterns (\(\mathbb{M}\)). See also Yoshimura buckling on Wikipedia, which repeats Robert Lang’s observation that these patterns can be seen on Mona Lisa’s sleeves.

Two more new Wikipedia Good Articles (\(\mathbb{M}\)):

• 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, 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.

Building a trivalent graph of harmonic relations among major and minor triads (\(\mathbb{M}\)). The graph has a vertex for each major or minor chord and an edge when you can get from one chord to another by changing a single note. It has 24 vertices (12 bottom notes of chords in 12-TET tuning, and two chords per note). It is vertex-transitive: shifting major or minor chords up or down the scale doesn’t change their adjacency patterns, and reversing the scale swaps major for minor. But it is not the Nauru graph because it is not edge-transitive: you can walk up or down the scale by removing the bottom note from a chord and adding a new top note, and this walk forms a 24-cycle, but these edges are different from the ones where you change a major chord into a minor or vice versa by changing the middle note. The resulting graph is the one with LCF notation \([7,-7]^{12}\), shown below.

My 5-year-old Macbook Pro decided to start shutting down the screen if you open it too wide (\(\mathbb{M}\)), so I had to replace it. New hardware is nice but I had been holding off on several versions of OS updates and the disruption in my software setup was a bit of a pain. The biggest issue: losing all my old paid-for non-subscription copies of Adobe software, because they won’t run on the new OS. Fortunately my campus has a subscription, for now.

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, … (\(\mathbb{M}\)). 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.

Henry Segerman on rolling acrobatic apparatus in the Notices (\(\mathbb{M}\)). Unfortunately he missed MOMIX dancer Alan Boeding’s work in this area from the late 1970s and early 1980s.

Hamming cube of primes (\(\mathbb{M}\)). 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, and a new Matt Parker video on the same concept in decimal.

A dead mathematician co-authored a paper after appearing in a dream (\(\mathbb{M}\)). The paper is “Higher algebraic \(K\)-theory of schemes and of derived categories” by Robert Wayne Thomason and Thomas Trobaugh (2007), doi:10.1007/978-0-8176-4576-2_10, MR1106918. 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.

Imperfect comb construction reveals the architectural abilities of honeybees (\(\mathbb{M}\), via). 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.

Solid Objects: 16th-Century Geometric and Perspective Drawings from the Herzog August Bibliothek in Wolfenbüttel (\(\mathbb{M}\), via).

You remember that simple fruit pictogram equation with the ridiculously complicated answer (\(\mathbb{M}\))? David Roberts has another one in the same style for which we don’t even know the answer.

Scott Aaronson takes a break from quantum supremacy to tell us about busy beavers (\(\mathbb{M}\)). 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.

Antoine Chambert-Loir posts a nice photo of multicolored foam on the surface of a cup of coffee.

A new paper by Asperó and Schindler (\(\mathbb{M}\)) 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, but I think the introduction of the paper is quite readable. (The rest is not.)