Double Maths First Thing is halfway through Ouch to 5k.
Hello! My name is Colin and I am a mathematician on a mission to spread the joy of doing maths. And this week, I’ve been doing some cool maths, looking at Oskar van Deventer’s Blocks puzzle and models of crowd movement.
I’m also half-following the British & Irish Lions rugby tour: I’m told by reliable sources that no pair of the 47 squad members share a birthday. What are the chances?!
Links
Since pi approximation day is coming up next week, let’s start by looking at pi’s evil twin, the lemniscate constant.
If you or someone you know has recently finished a great dissertation, and wants to write an article about it for Chalkdust (a magazine, I understand, for the mathematically curious), the IMA are offering a £100 prize to the best submission. Learn more here.
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
Double Maths First Thing couldn’t possibly comment.
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight in engaging with maths, for any reason or for none. I’ve made it back from rehearsal and am feeling a lot better (thanks for asking).
(By the way, if you know of any Colin-shaped work, please let me know about it — I’d love something stable, remote, part-time and reasonably paid but neither soul-crushing nor burnout-inducing. Plus the moon on a stick, obviously. I can write. I can code. I can solve weird problems. There must be something, right?)
Links
Let’s start with something on my to-read list: Tai-Danae Bradley’s guide to category theory and notes on Applied category theory. I feel like there’s something going on a level up from where I normally look at things and that category theory might be a lens to examine it through.
And following on from Minesweeper in issue 2B, we’ve now got Mark Round’s Primesweeper. Which is not in the slightest bit stressful.
Currently
I enjoyed the Finite Group livestream last week, especially the bit about bees’ ancestry following a Fibonacci sequence. You can join and, for a very reasonable monthly outlay, have access to some of the previous livestreams and all of the new ones. (You can even join for free and hang out in the Discord with awesome maths people.)
One small appeal from me: it’s striking, looking back through past DMFTs how heavily the links skew towards male (or male-presenting) mathematicians. I’m keen to amplify the voices of women and other members of under-represented groups. if you stumble on a link spoken in such a voice and think “I’m not sure Colin would be interested” or “I don’t want to bother him” — I’ll tell you right now, it’s no bother and I’m certainly interested.
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
Double Maths First Thing could really do with a hot lemon and honey
Hello! My name is Colin and I am a mathematician on a mission to shift this blooming cold before the weekend, when I’m meant to be in Peterborough to rehearse for the PRE show in August. Have I mentioned that previously? We’re playing at the Warwick Arts Centre as part of Talking Maths in Public.
Speaking of TMiP (and I know this belongs under “Currently”, but continuity > consistency), the animation challenge for July is live. Can you show that two reflections make a rotation?
As I say, I’m a bit under the weather this week (which might explain, if not excuse, my triple DNF at the cubing competition; I’ll be trying again in September. One of them was very close, but you don’t get anything for that) and I’ll be keeping DMFT correspondingly short.
The only horrifying thing about Simon Tatham’s Portable Puzzles Collection is the amount of time it has sucked out of the world over the last however-many years. His version of Minesweeper — which, unlike the version that comes with a popular OS, is always solvable, is twenty years old.
Another great name of British espionage and subterfuge was William Playfair, who — among other pieces of villainy, invented (or at least popularised) the pie chart.
Meanwhile, there’s been some commotion over the recently-constructed monostable tetrahedron, a shape that always lands on the same face. The paper claims that they built and lost a model in the 1980s. It’s not lost, it’s on Colin Wright’s desk, potato patato.
The deadline for the next Carnival is… not approaching as quickly as I thought, it’s a double-header and posts need to be in by August 1st.
What is approaching quickly is the next Finite Group livestream: all four of the generators will be trying to tell us something we don’t know on Friday July 4th (7pm UK time). Free membership grants you access to the Discord, where all the cool people hang out, and paid memberships allow you to watch the livestreams, as well as other goodies.
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
We do these things, not because they are difficult, but because they are ridiculous – Matt Parker, probably
Matt Parker is going to the moon. I mean, not literally. Everyone’s favourite Stand-Up Mathematician is the sort of person who’s more likely to go to hyperspace than to space. However, when Matt was approached to do “something ridiculous” with spare computing power on a lunar rover, there was only ever going to be one outcome: an attempt to calculate \( \pi \) on the moon.
But… why?
That is an excellent question, beautifully presented. I compliment you for asking it. Next!
Moon-te Carlo
Because it’s important to do ridiculous things properly — there’s no point in going to the moon and doing calculations you could do on Earth — Matt made the decision to approximate \( \pi \) using the readings from the rover as a source of randomness for a Monte Carlo calculation.
Monte Carlo methods are typically used to work things out when it’s too difficult or too boring to do them analytically. While there were some earlier randomised calculation methods — Buffon’s needle, for example — the first real Monte Carlo experiment was done by Stanisław Ulam while recovering from an illness. Ulam wondered how likely it was that a game of solitaire would come out successfully and, rather than calculate it properly, decided to play a hundred games and count how many they won. It was a short step from there to the atomic bomb.
Matt showed the standard Monte Carlo approach to calculating \( \pi \) in the video announcing the moon \( \pi \) project — it’s often used as a simple example when introducing the idea. If you put a circular dartboard in a square box that just fits it, and threw darts at the box, assuming you were equally likely to hit any point in the box, each dart would have a probability of \( \frac{\pi}{4} \) of hitting the dartboard. If you threw 100 darts and 80 of them hit the board, you would conclude that \( \frac{\pi}{4} \approx 0.8 \) and that \( \pi \approx 3.2 \). Throwing more darts should get you a better estimate — although rather slowly. If you throw \( N \) darts, the standard error of your probability is proportional to \( N^{-\frac{1}{2}} \), which means becoming half as inaccurate requires four times as many darts.
An image from Christian’s Pi Day simulation, which does exactly this experiment. 10,233 out of 13,083 “darts”, coloured blue, have landed in the quadrant while the remainder landed outside, coloured red. That gives an estimate for \( pi \) of 3.129 or so, off by about 0.013. To improve that to “off by about 0.0065”, we’d need four times as many darts — 52,000.
Holding out for a Hero
Matt famously thinks Heron’s formula is one of trigonometry’s most remarkable results. It’s been known to make Matt extremely emotional. So, naturally, my first thought was “I bet an approach based on Heron’s formula could converge more quickly.”
And it could! The approach entailed starting with a right-angled triangle with legs of length one inside a unit circle in the first quadrant. It would then pick a random x-coordinate between 0 and 1, figure out the corresponding point on the arc, and add a triangle based on the two adjacent points. Here’s the code. It converges to several decimal places within 10,000 iterations.
But that’s not really Monte Carlo, now, is it?
That is an excellent question, beautifully presented. I complime… what do you mean, I need to answer it? Who’s writing… OK. Fine. Sheesh.
You’re right, this isn’t a traditional Monte Carlo method. While it uses random points, it doesn’t use them to generate a probability. I do still maintain that it’s technically a Monte Carlo method, using a very involved adaptive weighting function, but I take the point.
What about proper Monte Carlo methods?
A less sophisticated (but still significantly more efficient than the integration-by-darts method) approach is to use the fact that \( x^2 + y^2 = 1 \) on the unit circle. If you pick an \( x \) value at random, you can immediately calculate the probability of a random \( y \) value giving a point inside the circle — it’s \( \sqrt{1- x^2} \). Rather than sample and add 1 or 0 to your total to approximate the probability, why not just add the probability? This converges very nicely.
The mean distance of a point on the circle from the axis is \( \frac{\pi}{4} \) — a fact we’ll (likely) be using for the actual experiment on the moon.
I’m not officially allowed to reveal that the reason for my interest in calculating \( \pi \) on the moon is that I’m helping to design Matt’s experiment, or that my codename is FizzBuzz Aldrin. (Is that lede sufficiently buried? Excellent.) And I’m definitely not officially allowed to say what we’re actually doing, because I imagine Matt will want to do a video about it.
However, I can say that the method above is equivalent to the fact that the expected distance between a point on the unit circle and an axis — any axis — is \( \frac{\pi}{4} \). By extension, using the magical incantation “SYMMETRY!” and a magisterial wave of the hand, it turns out that any point on a unit sphere is — on average — \( \frac{\pi}{4} \) from any axis of the sphere. That’s a fact that could be exploited, just to pick a random example, by a simulated rover making random moves on the surface in a sort of random moonwalk.
It would take small steps for a rover, and giant LOOPs for Matt-kind.
You can donate to Matt’s kickstarter here, if you’re so inclined. At the time of writing, they’ve raised well over a quarter of a million pounds towards their £75k target, so I’m looking forward to them launching their own space mission before long.
If you’re a teacher who wants to be involved, you can sign up here. Get the kiddos to estimate \( \pi \) by hand and they’ll get (I understand) a certificate, possibly their own value of lunar \( \pi \), and their name in a text file that goes to the moon. [Edited 2025-06-26 for formatting and to clarify that the personal \( \pi \) value is not guaranteed.]
Earlier this year, Brady Haran visited Newcastle to record a video with some Leverhulme scholars. Luckily for me he had a bit of spare time to record a video with me, so we did one about the Herschel enneahedron, which I first looked at back in 2013.
There were a few common questions among the comments on YouTube. I thought I’d quickly respond to them here.
Hello! My name is Colin and I am a mathematician on a mission to the moon! But I’m also on a mission to spread joy and delight in doing maths anywhere in the cosmos.
Talking of doing maths in strange and hostile environments, I’m going to Bristol this weekend for my Rubik’s cube competition. Going by my current success rate at blindfold, it’s about 60-40 whether I’ll manage to complete a solve.
This week I’ve also been working on a nice combinatorics problem with Zoe Griffiths for her AEOUD talk about finding sets of children’s names with no common letters. I learned that in 2023, at least three families decided to name a child “C”. I don’t like to judge, but that feels even worse than calling them JavaScript.
Links
There’s a breathless, magaziney furore over a link between primes and integer partitions (ArXiv paper here) — it’s not my field, and part of me doesn’t feel very surprised that there’s a link, but having used the method of “Leonhard Euler would have thought of that already” to rebut a “proof” this week, I feel like there’s a corresponding method of “if it was trivial, Ken Ono wouldn’t be publishing it.”
Here’s an old post from David R Hagen about an XKCD cartoon. He wonders “how come the 11th of the month doesn’t show up as often as it should?”
I enjoyed Vanessa Madu’s talk at Big MathsJam about rubber ducks and ocean currents, but I also enjoyed her article about consecutive odd semi-primes, another of those questions that you didn’t know you needed to wonder about, and which then drops out quite neatly. Quack!
Currently
It’s a big couple of weeks for the Finite Group, with a livestream today (June 25th, 2pm UK time) and another in a couple of weeks (Friday July 4th, 7pm UK time) — meaning that joining today for £4 will get you access to both. Bargain!
You’ve got a week or so to submit entries for this month’s Carnival — and I imagine Katie would love it if you volunteered to host a future event. (It’s not that much work if you’ve already got a blog.)
That’s all I’ve got for this week. If you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
Meanwhile, if there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something you want to tell me.
In my childhood memories, the lanterns in my hometown always fascinated me – circular palace lanterns, polyhedral colorful lanterns. How did my ancestors achieve the magical transformation from flat to three-dimensional through simple bamboo strips and paper? As a child, I was often confused: was there anything hidden behind these beautiful shapes?
Three decades slipped by, and as I returned to the art of paper folding, that long-forgotten question suddenly became clear. It turns out that the structural principles of those traditional lanterns align with geometric calculations—how a flat sheet of paper can be perfectly transformed into a three-dimensional structure with specific curvature. This process contains profound mathematical essence: straight creases correspond to developable surfaces, while curved folds achieve their shapes. From rectangular paper-based to porcelain like smooth curved surfaces, every fold undergoes intricate geometric transformations.
This perfect fusion of craft and mathematics not only showcases the ingenuity of folk artisans but also remarkably anticipated the evolution of modern computational origami techniques. The craftsmanship behind traditional lantern-making constitutes humanity’s earliest exploration and practice of spatial geometry—a profound legacy that has continuously inspired the development of structural design approaches.
Historical photo: crafting circular lanternsAncient lantern festival: children carrying lanterns
Reflecting…
A question lingered unresolved for over three decades—until I encountered modern origami theory and finally grasped its profound mathematical wisdom. This long-delayed revelation compels me to ponder: In our relentless pursuit of so-called progress, have we overlooked something profoundly valuable? As masters worldwide have long pioneered innovations in paper folding, shouldn’t we pause and reflect? This isn’t merely an academic gap; it speaks to the neglect of foundational mathematical research. It’s time we reignite our appreciation for the enduring brilliance of mathematics!
As an archetype of curved-crease origami art, Origami Blue-and-White Porcelain transforms flat paper into exquisite three-dimensional forms through precise calculated creases.
Crease algorithm design
In the art of origami creation, I have always adhered to the concept of “subtractive aesthetics” – shaping forms by accurately hiding excess parts.
This principle is particularly typical in the structure of traditional Chinese circular lanterns: When the lantern ’ s uniformly arranged petal-like segments are unfolded into a planar sheet, the absent portions represent exactly those strategically eliminated excess regions.
This foundational geometric principle can be extended to designing curved vessels (e.g., circular lanterns, porcelain), with its computational workflow comprising three key steps:
Unit Segment Width Determination → Based on the target model’s diameter.
Structural Length Calculation → Computed using the Pythagorean theorem.
Curved Petal Optimization → Derives optimal curvature for each unit segment.
Ultimately, the algorithm generates precise crease distribution diagrams, providing scientific mathematical guidelines for vessel fabrication.
Using a bowl shape as a modelling caseCrease patternFolding form
Constructing universal geometric features
Applicable to rotatable vessels, papercraft lamps/lanterns, circular heritage architectures, and pagoda-style structures.
Engineering mathematics
This method overcomes the planar limitations of conventional origami techniques, offering an alternative computational approach for developable surface design with potential applications in industrial design and architectural mechanics.
Example: Bulbous porcelain vessels (Dulu)
We observe that exquisite ceramics and traditional circular lanterns alike are formed by rotational assembly of curved crease units. This exemplifies the core principle of origami engineering: a material optimization process where precise folding conceals redundant sections to achieve visually streamlined forms.
This artifact demonstrates uniform curved crease unit distribution in its structure. The waist contour is formed through rotational constriction, showcasing exceptional craftsmanship in curvature control. Notably, its spiral patterning precision surpasses comparable paper-folding implementations.
Wrong fold, it’s in another direction
These gear-like structures represent unanticipated structural outcomes during folding operations. The creative process requires continuous cognitive switching between 2D crease patterns and 3D topological realization, presenting significant neural adaptation challenges. Particularly without physical reference models, the spatial imagination ability of the brain often goes through a debugging stage, and these “wrong” works are considered a good thing from a certain perspective.
This is the mark of my encounter with mathematics. The folding and unfolding of paper is like a pair of hands revealing the science behind it. I returned to the old me, learning to discover the beauty of mathematics, exploring the mysteries of origami, and leaving my own thinking trajectory in this interlaced field.
