• Computer scientists say they’ve solved the mystery of the orb in Leonardo da Vinci’s Salvator Mundi ($$\mathbb{M}$$). My colleague and co-author Mike Goodrich has been working with computer graphics specialists to model the refraction in the clear ball (representing the universe) held by Jesus in Leonardo’s painting. Their preprint shows that the model that Leonardo painted from was likely a hollow glass ball, not a solid crystal.

• I asked on Mastodon how Semantic Scholar got their pdfs and whether they have a license to redistribute them ($$\mathbb{M}$$). The upshot was that they scrape pdfs from the web, and in doing so they sometimes accidentally re-host pirated material, but remove it when they learn about it. So if you are trying to avoid piracy (as I am in my Wikipedia editing if not necessarily in my own use) they may not be safe to link.

• The early-18th-century Pilgrimage Church of Saint John of Nepomuk in the Czech Republic is known for its repeated use of the number five in its layout and proportions, and its floor tiles are an example, with regular pentagons and thin rhombs as tiles ($$\mathbb{M}$$). But despite their five-way symmetry these tiles (also used in a more modern Copenhagen plaza) can’t form Penrose tilings.

Since the case analysis from my earlier post was a little undetailed, here’s an illustration with more. If you have a pentagon with three consecutive sides adjacent to other pentagons, then a wedge of the plane is forced to be filled by repeating tiles. And if not, then you get repeating patterns along parallel lines and the tiling has at least one direction of translational symmetry.

• The danger of hidden jargon ($$\mathbb{M}$$): if non-specialist readers don’t recognize that your terms have technical rather than colloquial meanings, they’re likely to think they understand more than they do. The issue was identified by Christine Anderson-Cook, in (ironically) a more technical paper on hidden jargon in statistics. Both links have useful advice for writing to make technical concepts more accessible.

• The English-language Wikipedia reached a new milestone of six million articles ($$\mathbb{M}$$). Counting these things is a little inexact but the officially-declared six-millionth article is on Maria Elise Turner Lauder, a 19th-century Canadian school teacher, writer and philanthropist, created by Rosie Stephenson-Goodknight.

• Open-access book on the theory of random graphs by Alan Frieze and Michal Karoński ($$\mathbb{M}$$). Some background in basic probability theory is probably required to get much out of this.

• Visual proof of Fermat’s two-square theorem ($$\mathbb{M}$$):

The theorem states that a prime number can be represented as the sum of two squares of integers iff it is congruent to 1 mod 4. In 1990, Don Zagier published a “one-line proof” involving a tricky algebraically-defined involution among triples of numbers. This video explains how to see the same involution more intuitively, as a dissection of polyominoes.

• I am sad that in 2020 I still don’t have a satisfactory workflow for LaTeX-formatted mathematics in vector graphics ($$\mathbb{M}$$). I need to be able to create PDF and SVG, and edit files in either format from other software; I’m using Illustrator, but no longer have a working equation plugin. In the Mastodon discussion, the most common alternative suggestion seems to be Inkscape; Ipe might also work, and TikZ can incorporate equations easily but can’t edit other files.

• The pigeonhole principle was not invented by Lagrange (1768) or Dirichlet (1834) ($$\mathbb{M}$$). Jean Leurechon briefly alluded to it in 1622 and it was spelled out in detail in 1624 in a book either by him or one of his students. Also there is a new history of science and mathematics stackexchange.

• Ringel’s conjecture solved (for sufficiently large $$n$$) ($$\mathbb{M}$$, via). This goes back to 1963 and states that the edges of the complete graph $$K_{2n+1}$$ can be partitioned into $$2n+1$$ copies of your favorite $$n$$-edge tree.

• Square packing ($$\mathbb{M}$$, via, via2). Adam Ponting found a way to cover arbitrarily large (e.g. as measured by inradius) contiguous patches of the plane by distinct squares of sizes from $$1$$ to $$2n+1$$, for any $$n$$.

• Triangles admitting periodic billiard orbits ($$\mathbb{M}$$). Cleaning out old bookmarks, I found a 9-month-old unanswered MathOverflow post. It’s a famous open problem which triangles have a closed billiard path (mirror the triangle edges and a beam of light in the right place will come back to itself) but it’s true if the maximum angle is at most $$100^\circ$$, or if two angles are rationally related or have a rational combination equal to $$\pi$$. Is it known for other classes of triangles?