along with the location of the centre of the circle and the positions of the 3 points around the circle. Details at gregegan.net/SCIENCE/Thre... BTW, if you want a laugh, here is one of the results offered up by the clown car that used to be Google Search. The MathOverflow post…

If you pick 3 points at random on a unit sphere, the average radius, R, of the circle that contains them is 7/6. For 3 points on a flat disk of radius ρ, mean(R) diverges. However, the curvature, κ = 1/R, has a finite mean which can be computed exactly: 16 (3 – 16/π^2) / (15ρ)

TIL about “cancellous bone”. I’d previously misunderstood what the solid sample taken in a bone marrow biopsy was composed of; I thought it was part of the outer, “cortical” bone. But today the doctor doing the biopsy showed me the sample and explained exactly what it was.

The image shows two cases, where the plane through 3 points intersects the hyperboloid either in a closed curve or an open one, with the latter corresponding to a hypercycle.

After recent observations with the VLT: “While there was evidence for the existence of this companion star, including a possible direct detection with the Gemini North Telescope in Hawaiʻi, USA, this is the strongest evidence for, and clearest image yet of, Betelgeuse B.” www.eso.org/public/news/...

The Burau representation of 𝐵ₙ associates a matrix with each element of the group, in such a way that multiplying these matrices is the same as joining braids together to make new ones.

An element of the Braid group 𝐵ₙ is a way of joining two sets of n points with strings, where two elements are the same if the strings can be moved around and/or stretched to follow the same paths in 3-dimensional space while keeping their endpoints fixed.

... “a 3D version of cryo-EM called cryo–electron tomography” that will reveal all kinds of cellular structures in situ with molecular resolution. The picture here shows two cryo–electron microscope images of the protein apoferritin, with and without dual “laser phase plates”.

How does this work? Light passing through the sample is both scattered slightly off-course, and shifted in phase by about 90 degrees. If the sample is illuminated by an annular light source, it is possible to use the angular separation of light scattered by the sample ...

Phase contrast microscopes were invented in the 1930s by the Dutch physicist Frits Zernike, and developed and manufactured by physicist and entrepreneur Caroline “Lili” Bleeker. Zernike won the Nobel Prize for this in 1953 ...

These are the tallies of cones for each dimension from 2 to 10. The first number in each pair is the number of vectors that generate the cone, the second is the count of such cones.

And this is what we get for tetrahedra.

… the set of linear combinations of these vectors with positive coefficients. Then these 2^n-1 cones will all fit together to form a half-space! This is what we get for triangles.

How can we generalise the fact that the angles at the vertices of a triangle add up to 180°, to apply to tetrahedra and higher-dimensional n-simplexes? Here’s a result discovered by Benoît Bertrand and Lucía López de Medrano that I learned from Omar Antolín.

Most likely, no human typed the word “permission” here. Rather, they misspelled “persimmon” in such a way that autocorrect considered “permission” the closest valid word. Pure spellchecking that *flagged* the typo would have alerted the user, who would have made the right choice.

By reflecting the solid triangle in the side AB, we can see that its height from BC to A is Q'A = QA = 1. Equating area from ½bh with area from vertex coords ½det(AB,AC) gives: ½√(x²+y²) = ½(x+y-xy) Solving this gives us coords for C', which then yields: AC·AC' = 0 So ∠CAC' = 90° Red ∠CAB = 45°

A new knot invariant that “uniquely identifies more than 97% of the knots with 18 crossings. By comparison, the Jones polynomial, one of the most widely used invariants for cataloging knots, identifies about 42%, and the Alexander polynomial only about 11%.”

A more objective answer is that the lightlike geodesics remain “equally spaced” … but the spacing itself has no observer-independent value! What does that mean? The cyan curves in this image join neighbouring geodesics with equal-length segments …

Here lightlike geodesics grow further apart, as measured along the grey circles. This is how the cosmological red shift works: wavefronts in an expanding universe move apart. But that’s a measurement by *particular observers*, at rest in galaxies that are themselves moving apart.

But the timelike geodesics behave like lines in the hyperbolic plane: if they start out parallel, they accelerate *away* from each other. The grey curves here, perpendicular to the geodesics and equally spaced along them, have a length that grows with the cosh of the distance.

The cyan curves here, perpendicular to the blue geodesics and equally spaced along them, vary in length like the circles of latitude on a sphere crossing the great circles of meridians, with the cosine of the latitude.

In 3D Euclidean space, the surface equidistant from a point is a sphere, with constant curvature. In (2+1)-dimensional Minkowski spacetime, the equidistant surface is a (1+1)-dimensional de Sitter spacetime, which also has constant curvature.

Those staff layoffs and vibe coding seem to be really working out for Amazon ...

This image (from www.gregegan.net/SCIENCE/Elli...) sums up for me (roughly, intuitively) why there is SO(4) symmetry for orbits: the two equal-sized vectors A and B can be rotated *independently* while preserving the semi-major axis, and hence the energy, of a Kepler orbit. And so(3)+so(3) = so(4).

A new collection of my stories in Korean 잠과 영혼, East Asia Publishing, Seoul, 2026. Translated by Kim Sang-hoon. ISBN13 979-1193078808 Contents: “Crystal Nights” / “크리스털의 밤” “This Is Not the Way Home” / “고향으로 돌아가는 길” “You and Whose Army?” / “너 혼자서?” “Dream Factory” / “꿈 공장”

In special relativity, observers in relative motion will disagree about various measurements. But if Alice draws any planar figure, and fires a simultaneous pulse of light from each point on it, perpendicular to the plane, any observer will agree on the figure’s shape and size.

These are the paths of the individual ants, from the starting point until they bump into each other.

Two pairs of ants set off from the same starting point, walking side-by-side. One pair takes the red path, the other pair takes the blue path. The red and blue paths are geodesics (the straightest possible paths, like great circles on a sphere) …

"We're trying to compete with the big boys, and part of that is you've got to keep your content refreshed and new all of the time," Mr Hennessy said. Refreshed, new and hallucinated: tourists book trips to nonexistent hot springs. www.abc.net.au/news/2026-01...

For crying out loud, ABC news, never send a line graph to do a bar chart’s job. Link: www.abc.net.au/news/2026-01...