Indira Lara Chatterji has some nice open-licensed animated gifs of concepts in low-dimensional geometry and topology at

This is one of them, unzipping a 2-torus.

Triangle dissection, no shared edges:
(the intended target of a recent broken link from

The question is how to divide a triangle into smaller triangles, no two sharing a whole edge, but the solutions shown also have no separating triangles and a straight angle at each interior vertex. Double-counting angles and combining with Euler shows that, with these conditions, \(F=V-1\). Do all internally-4-connected planar graphs with \(F=V-1\) work?

The plum trees (or cherry? I don't know) are in bloom outside the building I work in this week.

Square packing:

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 \((2n+1)^2\), for any \(n\). Via and

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 ( The flooring in is an example, with regular pentagons and thin rhombs as tiles. But it's not a Penrose tiling! It's one I analyzed in in connection with a Copenhagen plaza that uses the same tile shapes (but bigger) in a slightly different arrangement.

Computer scientists say they’ve solved the mystery of the orb in Leonardo da Vinci’s _Salvator Mundi_:

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 work shows that the model that Leonardo painted from was likely a hollow glass ball, not a solid crystal. Original paper:

How few \(k\)-gons can make a polyhedron, for different choices of \(k\)?

The answers include an amazing high-genus polyhedron with 12 faces, each of which is an 11-gon, posted Nov 2018 by Ivan Neretin (sadly, with multiple adjacencies for some pairs of faces, dubious by some definitions of polyhedra, rather than having one edge per face pair).

Via and indirectly via

Continuing the new articles related to my revision of the Wikipedia convex hull article, here's one on relative convex hulls:

These are the shapes you get by stretching a rubber band around one set (here, some blue points) while fencing it inside another set (the yellow polygon). As you can see, even for points inside a polygon, you don't always get a simple polygon as a result, but it's "weakly simple".

Convex hull of a simple polygon (new article on Wikipedia):

I've been working on cleaning up the main convex hull article (not yet finished) and added this one to fill in some details.

The problem has an interesting history: it's been known to be solvable in linear time since 1979, and many different algorithms for the same problem have been published before and since, but most of them are wrong.

Chalkdust magazine provides a compass-and-straightedge construction for the girih pattern on the cover of a recent issue:

This image by Adam Majewski from shows the osculating circles of an Archimedean spiral. The spiral itself is not shown, but you can see it anyway, where the circles become dense.

It is not unusual that the circles nest. By the Tait–Kneser theorem ( this happens whenever the curvature along a curve is monotonic. And on most smooth curves, the curvature is monotonic except at a small number of points called vertices.

Hoffman's packing puzzle, and its connection to the inequality of arithmetic and geometric means:

The one I have is not quite so colorful as this Wikipedia image. My father-in-law made it for me some 30 years ago; you can see it in a corner of the photo at

I don't unpack it very often, though, because I lost track of the handwritten table of solutions that I made when I first got it and it's quite difficult to re-pack.

Jessen's icosahedron ( a non-convex shape with right dihedrals but not axis-aligned faces. It's rigid but not infinitesimally, so paper models are hard to make accurately.

I drew the new lead image after the article had for years shown the wrong shape (with vertices of a regular icosahedron). Surely someone else can do better?

Also it can be cut into pieces and reassembled into cubes. Anyone know an explicit construction?

Pen refill mishap added a little extra challenge to my daily newspaper sudoku

Quasiperiodic bobbin lace patterns:, Veronika Irvine, Therese Biedl, and Craig S. Kaplan, via — aperiodic tilings in fiber arts.

The attached image is a photo of lace (not an illustration), braided into an Ammann–Beenker tiling pattern.

Kotzig's theorem ( Every convex polyhedron has an edge whose endpoints have total degree at most 13. (New article on Wikipedia.)

You might think that (because average vertex degree in a convex polyhedron is < 6) there will always be an edge whose endpoints have total degree at most 11, but it's not true. As Anton Kotzig proved in 1955, the answer is 13. A worst-case example is the triakis icosahedron, whose minimum-degree edges connect vertices of degrees 3 and 10.

My new dining room ceiling lamp is a trefoil knot! It's the "Vornado" LED lamp from WAC lighting (

We chose it to replace a halogen lamp that shorted out, burned through its power cable, fell onto the table below it, and shattered hot glass all over the room, fortunately without causing a fire or seriously damaging the table and while the room was unoccupied.

Full photo set at

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