@jsiehler Unfortunately I don't understand enough German to appreciate this.

0xDE boosted

A triangulated polygon, from Les Amusemens Mathématiques by André-Joseph Mancoucke, 1749. This is the earliest appearance I know of the theorem (stated without proof) that every simple polygon can be triangulated by diagonals. In particular, Mancoucke predates both Meister's (1770) and Poinsot's (1809) seminal treatises on polygons.

Numberphile 15-minute video on Dehn invariants: youtube.com/watch?v=eYfpSAxGak

The Dehn invariant is a value derived from a polyhedron that doesn't change if you cut up the polyhedron into smaller polyhedral pieces and rearrange them into a different polyhedron. It's 0 for the cube and nonzero for other Platonic solids, proving that they can't be cut and rearranged into a cube. See en.wikipedia.org/wiki/Dehn_inv for more, in somewhat more technical detail than the video.

Leiden wall formulas: muurformules.nl/

The last time I was in Leiden they were decorating the exterior walls of all their buildings with poems of many different languages. Now they've moved on to the language of mathematics.

@pschwede If all else fails you could encode whatever you're trying to represent as a binary string, and then encode that string as a long path with leaves attached to some path vertices to represent 1's and with path vertices that have no attached leaf representing 0's. That's a very special case of a bipartite graph. If you want your typed graph to look more like a graph you could use this kind of tree-binary representation just for the types, attached to each vertex of a larger graph.

@pschwede Closest I can think of is en.wikipedia.org/wiki/Gadget_(

Essentially, you want to use small recognizable parts of your bipartite graph to simulate parts of whatever other structure you're trying to simulate.

@pschwede "Typed" with what sort of type system? But in some sense the answer is always yes: bipartite graphs are complicated enough to encode whatever information you want. How direct the encoding might be is a different question.

New blog post "Connectivity and finiteness in modal graph logic", 11011110.github.io/blog/2019/0

Inspired by Joel David Hamkins’ recent blog post on modal model theory (jdh.hamkins.org/modal-model-th), I look at how one can express some basic graph properties in this kind of logic.

Trávník's smooth self-referential formula: jtra.cz/stuff/essays/math-self
via twitter.com/johncarlosbaez/sta

It is actually a linked set of formulas, described in a typeset image, that when plotted as described in the image produces the image itself. It follows the same ideas as earlier self-referential formulas like en.wikipedia.org/wiki/Tupper%2 but unlike them describes a smooth vector image based on splines instead of a pixelated bitmap.

Modal model theory: jdh.hamkins.org/modal-model-th

For graphs, this extends first order logic (where the only quantification is over vertices and the only predicate is adjacency) with operators \(\Box\) and \(\Diamond\): \(\Box(F)\) is true when all supergraphs model \(F\) and \(\Diamond(F)\) is true when at least one supergraph models \(F\).

This can express nontrivial graph properties like \(k\)-colorability, and comes in two variants depending on whether you can quantify outside the operators.

Evoboxx, a retro-styled portable device that does only two things: run Conway's Game of Life and generate sounds from it. Not very practical in these days of cell phones but then maybe that's what makes it a fun project. boingboing.net/2019/07/08/chec

This week I'm in Milan for the Symposium on Geometry Processing, sgp2019.di.unimi.it/
They have an official twitter stream, mostly consisting of event photos: twitter.com/geometryprocess

So far the sightseeing highlight of my trip has been seeing pages of Da Vinci's Codex Atlanticus at the Ambrosian Library. For online copies of some of it, see ambrosiana.it/en/discover/code

A formula for designing lenses with no spherical aberration: petapixel.com/2019/07/05/goodb

This seems to have little practical value as there was already a numerical solution, and I don't think it handles chromatic aberration, but it's interesting that there is an analytic formula for these shapes.

Via news.ycombinator.com/item?id=2

0xDE boosted

The other day I was thinking about graphs that have degree sequence 1,3,3,3,... such graphs can't be bipartite, so I was wondering whether it is possible for such a graph to have no even cycle. It turns out that graphs with no even cycle have at most (3n-3)/2 edges (I think), whereas my graphs have exactly (3n-2)/2 edges!

Chinese scientists guilty of ‘researching while Asian’ in Trump’s America: scmp.com/magazines/post-magazi
via news.ycombinator.com/item?id=2

The story focuses on star cancer researcher Xifeng Wu (en.wikipedia.org/wiki/Xifeng_W), forced to resign from the University of Texas, apparently because she fostered collaboration with Chinese cancer research institutions at the behest of her higher administration.

See also "The U.S. Is Purging Chinese Cancer Researchers From Top Institutions", bloomberg.com/news/features/20

@anne @christianp Wikipedia says the members of this field are called the constructible numbers: en.wikipedia.org/wiki/Construc

You can also call it the quadratic closure of the rationals.

The annual Great Internet Math-off is happening at @blog — view and vote on your favorites!

So far we've had commutativity of log-exponentiation vs weather infovis (aperiodical.com/2019/07/the-bi), the geometry of the Sydney Opera House vs straight lines on a donut (aperiodical.com/2019/07/the-bi), multiplication tables and muffins (aperiodical.com/2019/07/the-bi) and a video on shapes in La Sagrada Familia (shot on location?!) vs an intro to fractals (aperiodical.com/2019/07/the-bi). More daily for roughly a month.

U.Alaska budget gutted by 40%: insidehighered.com/quicktakes/
chronicle.com/article/Unpreced
npr.org/2019/07/03/738569508/u

From the Chronicle story, the total amount cut over the past five years (including this new biggest cut) is more like 63%, from $522M to $192M. And from the NPR story, the likely response is to close one of its three main campuses and all 13 smaller community campuses.

Ironically, the cause is right-wing insistence on a universal basic income of $3000/person from fuel extraction revenues.

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