iiinova — where your digital self keeps evolving

James Munro @jamesmunromaths.bsky.social · 13/05/2026

In #MathsToday I'm getting ready for Oxford Online Maths Club tomorrow, where DPhil student Alice will introduce General Relativity to an audience of mostly #ALevelMaths students. You / your students could join in too; it's free (no sign-up!) with over 100 other episodes at www.maths.ox.ac.uk/r/club

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James Munro @jamesmunromaths.bsky.social · 11/05/2026

Teachers with girls in Y13 or equiv. who love maths or CS or both: here's a free summer program where they can learn about applications of maths at Jane Street. The timeline is not great considering exams, but do pass it on if you know someone who might like this! www.janestreet.com/join-jane-st...

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James Munro @jamesmunromaths.bsky.social · 28/08/2025

Yesterday I saw a question here about mixed partial derivatives and it got me thinking about how a surface might or might not have d^2 f / dx dy = 0 if we rotate it. Today I saw these two ruled surfaces at #TMiP25 made by Laura Bradby. No such thing as coincidence in maths!

Two ruled surfaces (in this case, a hyperbolic paraboloid, like a Pringle crisp). Each is made of a collection of cross-sections in two colours, criss-crossing each other at right-angles. At first glance they're different; one has cross-sections aligned with the curvature and for the other they're at 45 degrees. But the surface is the same!

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James Munro @jamesmunromaths.bsky.social · 12/06/2025

In #MathsToday we've started the MAT Livestream again; free online support for people thinking of applying for Maths or CS at university (and especially Oxford). See www.maths.ox.ac.uk/r/matlive for more! ^James

Two computer screens, and a mess of tech peripherals like a tablet and microphone and a mini stream deck. One screen says MAT Livestream 2025 in large letters.

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James Munro @jamesmunromaths.bsky.social · 10/05/2025

Maths for everyone at the Oxford Maths Festival, even dragons! #MathsToday

A large red dragon mascot (wearing a T-shirt identifying him as the mascot for Templars Square shopping centre) attempts a maths activity with dice at the Oxford Maths Festival.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

And finally, in admissions, I wrote Q26 on MAT 2024, which I’m quite proud of because it involves this incredible plot. www.maths.ox.ac.uk/r/mat Thanks for following this self-indulgent birthday thread. 10/10

A square grid with dots. The dots have high density in the top-right of the picture, where they form a regular grid - every integer point over there gets a dot. But along the left side and lower side of the plot, there are dots missing from the grid, with lower density nearer the edges. There are patterns of diagonal lines in the dots, and overlapping triangular regions formed by these diagonal lines. It looks like lots things have accumulated together, with more and more dots as you move right and or up.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

And I taught on Opportunity Oxford, where I got to use this slide about the perils of natural language. www.ox.ac.uk/admissions/u... 9/10

A slide with title “Claim: In France, they only eat croissants at the weekend.” and then text
Which, if any, of the following would be a counterexample to the claim? Followed by a list of options;
A person in France not eating a croissant on a Saturday.
A person in France who never eats croissants.
A person in France who didn’t eat a croissant last Friday.
A person in France eating a croissant on a Friday.
A person in France doing a crossword on a Saturday.
A person in France reading a newspaper on a Tuesday.
A person in France eating a baguette on a Sunday.
A person in France looking at a croissant on a Sunday.
A person in Guadeloupe eating a croissant on a Monday.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

I visited the UKMT’s National Mathematics Summer School to talk about derangements (permutations where noting stays still; the image shows two examples). You can find out more about the summer schools at ukmt.org.uk/enrichment/s...

Two diagrams with seven numbered nodes in each, and arrows between some pairs of the nodes. In the first diagram, 1 goes to 3 goes to 5 goes back to 1, while 7 and 2 swap places, and while 4 and 6 swap places. In the second diagram, 1 goes to 2 goes to 3 goes to 5 goes back to 1, while 7 goes to 4 goes to 6. The node numbered 7 is highlighted in each case.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

For my second-year students, I wrote this introductory question for them to get a sense of how raising and lowering operators might work in Quantum Mechanics 5/10

Suppose that M is a matrix such that M M dagger minus M dagger M equals I. Let N equal M dagger M.
(i) Prove that N dagger = N and that the eigenvalues of N are real.
(ii) Suppose that x is an eigenvector of N with eigenvalue lambda. Prove that, if it’s non-zero, then M x is also an eigenvector of N, and find the corresponding eigenvalue in terms of lambda.
(iii) By considering x dagger N x, prove that the eigenvalues of N are positive or zero.
(iv) Hence show that N does not have any eigenvalues in the range 0 < lambda < 1.
For more quantisation effects, see A11 Quantum Mechanics.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

… and we recently worked together to make a Desmos plot of Ford circles (a circle for each fraction p/q, with diameter 1/q^2) www.desmos.com/calculator/n... 4/10

Lots of colourful circles, each tangent to the x-axis. Two large red circles loom into view on each side of the picture. There’s an orange circle in the middle tangent to each of the red circles, then a whole rainbow of tangent circles disappearing off in each direction from there in either direction, each tangent to the previous, getting smaller as they get towards the points where the red circles meet the x-axis. Underneath each of those circles is another infinite series of rainbow-coloured circles tangent to each other, and so on.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

On my Chill Maths livestream we hit 196 followers and made a Cayley table in Minecraft to celebrate. We’re live most Sundays at 14:00 UK time www.twitch.tv/jamesmunro 3/10

Screenshot of a livestream. On the left, a large wooden structure in Minecraft; wooden blocks are stacked into a 14-by-14 square, with seven colours in diagonal stripes from lower right to upper left, with each colour appearing in 2-by-2 squares. On closer inspection, each 2-by-2 square is made of two log blocks and two plank blocks, with logs in lower-left and upper-right positions within each 2-by-2 square. In the top left, chat messages with hearts from erbs5621 and mariaeterna, and a message from spookyy6_ reads “C2 is the best group anyway”. Beneath chat, the name of the stream is Chill Maths, a URL reads twitch.tv/jamesmunro and under that is James Munro in the middle of a sentence, waving his arms in the air.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

Next, here’s a comparison of the harmonic series from music with a log plot www.desmos.com/calculator/c.... I thought this would work, but I didn’t think it would work this well! I made both of these for an Oxford Online Maths Club livestream www.maths.ox.ac.uk/outreach/oxf... 2/10

Musical staff notation for the harmonic series, which is a rising series of notes. A red line has been superimposed, going through all of the notes.

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James Munro @jamesmunromaths.bsky.social · 28/02/2025

It’s my birthday, so to celebrate here are ten things that I’ve made or worked on in the last year that I’m proud of, starting with this Desmos interactive about sine series that simultaneously varies colours and sounds and shapes, all at the same time www.desmos.com/calculator/q... 1/10

A purple curve is drawn on graph paper, with two distinct peaks between 0 and pi. It’s an odd function, and it’s 2 pi periodic. Red vertical lines mark 0 and pi.

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James Munro @jamesmunromaths.bsky.social · 29/01/2025

Related to other facts you've already got... you can tile the surface of a doughnut with hexagons! (no pentagons needed here) mathematica.stackexchange.com/questions/39...

A torus tiled with hexagons, with three hexagons meeting at each point. There are about 15 hexagons around the short direction, and perhaps 30 around the circumference. The colors vary smoothly through pinks and purples, according to the orientation of the hexagons. This image copied from Stack Exchange

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James Munro @jamesmunromaths.bsky.social · 10/01/2025

One of my students saw this pattern on a bus seat. What do you notice? What do you wonder? #MathsToday (yes my reaction to their photo was to look up the exact fabric online, and yes I am fun at parties)

A fabric pattern with a dark blue background, some white spots, and a design in orange made of a cluster of short intersecting orange lines, arranged in a mysterious way. Are the lines tangent to some other imagined shape? Are they the chords of something else? Sorry that this alt text is so mysterious; explaining what we're looking at is sort of the whole activity here.

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James Munro @jamesmunromaths.bsky.social · 08/01/2025

Perhaps this; find any vector a in one plane, and any vector b in the other plane. Find the unit normal n to the planes. Then take dot product (a-b).n to project the difference onto the unit normal. This works because |a-b| |n| cos(theta) is the distance you want, if |n|=1.

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James Munro @jamesmunromaths.bsky.social · 08/01/2025

The Oxford Online Maths Club starts again this week! It's a free livestream maths show from Oxford, aimed at prospective uni students, but open to everyone. Here's a clip from last year with Alice and topological data analysis. New episodes every Thursday 5pm at www.maths.ox.ac.uk/r/club #MathsToday

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James Munro @jamesmunromaths.bsky.social · 04/01/2025

One of my resolutions this year is to have another go at Hatcher's Topology of Numbers, so that's what I'll be working on Sunday 2pm over at www.twitch.tv/jamesmunro, for this week's Chill Maths livestream!

Farey diagram, from the cover of A. Hatcher's Topology of Numbers. A circle with blue and purple regions, each of which is bounded by three arcs. The arcs join points on the boundary of the circle, and for each region, the arcs join three points on the perimeter of the circle that are equally spaced. The arcs are nested within each other so that regions nearer the edge of the circle are much smaller than the regions at the centre.

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