Today, October 16th, is Hamilton day! It was on this day 183 years ago, in 1843, that William Rowan Hamilton first wrote down the fundamental result of quaternions. Apparently he was walking from his residence at Dunsink Observatory into Dublin when he had a sudden flash of inspiration, and wrote the result down on the spot, now marked by a plaque:
Picture Credit: Brian Dolan
This event is commemorated by an annual Hamilton Walk. Broombridge railway station (Droichead Broome), near the bridge (Broom Bridge), where Hamilton had his Eureka moment, is not far from Maynooth University the main commuter line from Maynooth into Dublin. This is ironic because Quaternion algebra does not commute. (Geddit?)
You may be interested that quaternions form the core mathematical framework used in the Euclid satellite’s telemetry system to represent and control its attitude (pointing direction and orientation). I also make extensive use of quaternions in preparing my tax return.
Although it is quite easy to reach Broombridge from Maynooth, I have never managed to attend the walk nor any of the other celebratory events because I’ve either been away or teaching. Perhaps one of these years, after I’ve retired, I’ll be able join the walk or attend one of the various ectures.
P.S. Maynooth University is also home to the Hamilton Institute which promotes and facilitates research links between mathematics and other fields.
It’s been a busy day and I haven’t the energy for anything strenuous so I thought I’d put up a cute problem I was thinking about (for some reason) on the train home from Friday’s concert.
You may have to refresh your memory about units as well as the geometry of ellipses. The following picture will help:
According to Kepler’s First Law of Cricket, the boundary of a cricket field is in the shape of an ellipse, with the two middle stumps each marking one of the foci. These are 22 yards apart, defining the length of the pitch P. The distance from one middle stump to the square leg or cover point boundary is 65 yards. This distance, called S, is the semi-latus rectum of the ellipse.
Derive a formula connecting the eccentricity e of the ellipse to P and S and calculate the value of e for the given values of S and P.
Calculate the major and minor axes of the ellipse for the given value of S.
Calculate the maximum distance from the stumps to the boundary for the given value of S.
Calculate the area of the playing surface in acres for the given value of S.
The value of S given above is the minimum allowed for First-class Cricket. The maximum is 90 yards. Repeat Qs 2-4 for this value of S. (P is always fixed at 22 yds.)
It’s been a lovely summery day in Maynooth so far, despite the summer having definitely ended and therefore having to return to teaching at the start of my penultimate academic year as a university teacher. The campus was very full this morning, and I was stopped many times on my walk into the office by new students trying to find their lecture theatres. A common (understandable) mistake for the uninitiated of Maynooth is to think that the Science Lecture Theatre is actually in the Science Building. To add to the excitement there was a fire alarm in Pugin Hall, where I sometimes go for lunch, so the building was evacuated. The kitchen staff and others were taking the opportunity to bask in the sunshine before the alarm ended. I think it was a false alarm. In any case lunch was available as soon as they opened up again, and no harm was done.
It’s hard to believe that it is two full years since I returned from sabbatical, but at least I’m teaching the same modules this semester as I did when I returned in the Autumn of 2024. I still have to get everything sorted out, though, including setting up my Moodle pages and preparing the materials, which is what I’ve been doing today, as well as giving my first lecture of the year, for MP469 Differential Equations and Complex Analysis, this afternoon at 2pm; my second, also MP469, will be at 11am on Tuesday (tomorrow).
The timetable for my Engineering Mathematics (EE206 Differential Equations and Transform Methods) module has not changed, so my first lectures on that (a double session) are not until tomorrow. That means I have three hours of teaching tomorrow and every Tuesday throughout the semester, which will make Tuesday my most tiring day.
Both the modules I am teaching this term are mathematical in nature. As last year, I am concerned about the integrity of the coursework element of these modules in the light of improvements in Generative AI. Only a couple of years ago GenAI could not solve the sort of problems I set for homework, but now it generally can – especially for EE206. I don’t altogether object to people applying artificial intelligence to solve mathematical problems, but the issue is that it does make mistakes. Moreover, instead of saying “sorry I can’t solve that problem” it will generally present a superficially plausible but incorrect solution. Although students will probably use GenAI for problem-solving, I think it is important that they learn to do such problems themselves, otherwise they won’t know whether the solution coughed up by the algorithm is correct or not. That way lies disaster.
The only way to learn mathematics is by doing it. If students get GenAI to do the mathematics for them, then they won’t learn it. In the past we have given marks for coursework (usually 20% of the module mark) mainly to encourage students to do them. Students who don’t bother to do these exercises generally do badly in the final exam (80%).
For these reasons I moved the assessment from weekly homework sheets – which could be tackled with AI – to supervised in-class tests for which students can use notes on paper, but not laptops or phones, just like they would in the final examination. I will of course give examples for the students to have a go at themselves, and I will give feedback on their attempts, but they will not contribute to the module score. Another advantage of this approach is that students won’t have to do so much work against deadlines outside of class. Some students will probably use AI to solve these non-contributory assignments, which is fine by me. It’s like having an extra tutorial. To learn they must actually try the problems, rather than absorb the methods by just watching CoPilot solve them.
I followed this approach last year and it worked quite well so I’m keeping the same basic idea, although with some tweaks. As I described here, this means I have taken a much harder line on assignments than other lecturers and indeed other departments. I allow no assignment to contribute to a module grade unless it is supervised in such a way as to prevent “AI” being used. I believe that this is the only way to maintain the integrity of these assessments and that to do anything else will result in a lowering of academic standards. I believe there should be university-wide regulations to manage it, but there are none…
When I moved to Ireland, almost nine years ago, one of the words I discovered had a usage with which I was unfamiliar was grind. My first encounter with this was after a lecture on vector calculus when a student asked if I knew of anyone who could offer him grinds. I didn’t know what he meant but was sure it wasn’t the meaning that sprang first into my mind so I just said no, I had just arrived in Ireland so didn’t know of anyone. I resisted the temptation to suggest he try finding an appropriate person via Grindr.
I only found out later that grinds are a form of private tuition and they are quite a big industry in Ireland, particularly at secondary school level. School students whose parents can afford it often take grinds in particular subjects to improve their performance on the Leaving Certificate. It seems to be less common for third level students to pay for grinds, but it does happen. More frequently, university students actually offer grinds to local schoolkids as a kind of part-time employment to help them through college.
The word grind can also refer to a private tutor, i.e. you can have a maths grind. It can also be used as a verb, in which sense it means `to instil or teach by persistent repetition’.
This sense of the word grind may be in use in the United Kingdom but I have never come across it before, and it seems to me to be specific to Ireland at least in current usage. A colleague sent me a link to an article about Carew Arthur Meredith, who worked at Trinity College Dublin from 1943 to 1964 but who was previously at Cambridge University and lived there until 1939:
After graduating, Meredith married Sybil Clark and settled down to what was to be for many years his occupation: that of university coach or “grinder.”
This establishes a possible conduit for the transfer of the word “grinder” from England to Ireland and it inspired me to look at the trusty online Oxford English dictionary which includes examples of usage. In fact the OED gives as examples of “grind” as a transitive or intransitive verbs and as a noun as well as the verbal noun “grinding” and the deverbal noun “grinder”. Here are some pertinent quotes. The first below, incidentally, is from a book of American college slang, demonstrating that the word had crossed the Atlantic by 1851.
Here are the most relevant examples, some of them hilariously susceptible to a double entendre.
n. Steady hard work; labour of a monotonous kind, esp. close and hard study; an instance of this, a dull and laborious task.
1851 Grind, an exaction; an oppressive action. Students speak of a very long lesson which they are required to learn, or of anything which is very unpleasant or difficult to perform as a grind. B. H. Hall, Collection of College Words
1857 A medical student would have told him, that a ‘Grind’ meant the reading up for an examination under the tuition of one who was familiarly termed ‘a Grinder’—a process which Mr. Verdant Green’s friends would phrase as ‘Coaching’ under ‘a Coach’. ‘C. Bede’, Mr. Verdant Green Married xi. 93
vi. To work hard at a subject of study under the direction of a tutor or ‘grinder’.
1835 I am obliged to ‘grind’..that is, undergo a private examination with an authorized teacher or tutor. E. Forbes in G. Wilson & A. Geikie, Memoir of Edward Forbes (1861) vi. 176
1870 After grinding with private-tutor Mylius the requisite time, Lessing entered the school of Camenz. J. R. Lowell, Among my Books (1873) 1st Series 308
vt. To teach (a subject) in a steady laborious manner; also, to prepare (a pupil) in a subject.
1815 implied in: Perhaps when Tom leaves Oxford..we may contrive some gainful grinding [i.e. tutorial] scheme between us. [at grinding n.] J. Keble, Letter to Coleridge in Mem. (1869) iv. 63
1848 A pack of humbugs, and quacks, that weren’t fit to get their living but by grinding Latin and Greek. W. M. Thackeray, Vanity Fair: A Novel Without a Hero lvi. 502
1859 [Dr. Bennett] undertook to grind him in anatomy and physiology. G. Wilson & A. Geikie, Memoir of Edward Forbes vi. 180
In Hard Times by Charles Dickens there is a character by the name of Mr Thomas Gradgrind, a grimly utilitarain school superintendent who insisted on teaching only facts. Hard Times was published in 1854 which is after some of the quotations above so Dickens did not originate the use of “grind” in teaching. He may have amplified it, of course.
This demonstrates that these terms were established in 19th Century England, but does not explain how their usage died out there but still flourishes in Ireland. Can anyone shed light on this? First recorded usages of grind, etc in Irish literature would be most helpful!
With thoughts turning to the new academic year I noticed an Open Letter to Universities in the Netherlands, which I have signed. You can read the letter yourself but here is a summary of the main calls to action:
• Resist the introduction of AI in our own software systems, from Microsoft to OpenAI to Apple. It is not in our interests to let our processes be corrupted and give away our data to be used to train models that are not only useless to us, but also harmful.
• Ban AI use in the classroom for student assignments, in the same way we ban essay mills and other forms of plagiarism. Students must be protected from de-skilling and allowed space and time to perform their assignments themselves.
• Cease normalising the AI hype and the lies which are prevalent in the technology industry’s framing of these technologies. The technologies do not have the advertised capacities and their adoption puts students and academics at risk of violating ethical, legal, scholarly, and scientific standards of reliability, sustainability, and safety.
• Fortify our academic freedom as university staff to enforce these principles and standards in our classrooms and our research as well as on the computer systems we are obliged to use as part of our work. We as academics have the right to our own spaces.
• Sustain critical thinking on AI and promote critical engagement with technology on a firm academic footing. Scholarly discussion must be free from the conflicts of interest caused by industry funding, and reasoned resistance must always be an option.
I agree with all of these. I don’t think The Management does though.
In Maynooth University there has been a total absence of leadership from the Management about the academic threats posed by the uncritical adoption of “AI” . I’ve put “AI” in quotes because most of what is being described as AI nowadays (e.g. LLMs) is not is in no way Artifical Intelligence. There is a place for such things as Machine Learning. Indeed, much of the data-driven research in my own field of cosmology would be impossible without them. The key, however, is not whether you insist on calling it AI or not, but the word uncritical. Much of what I have heard from our “Leadership” is simple-minded boosterism. And yes, that word is in quotes for the same reason as “AI”.
In planning my teaching for next term, which starts in two weeks, I have taken a hard line on assignments. I allow no assignment to contribute to a module grade unless it is supervised in such a way as to prevent “AI” being used. Both modules I will be teaching next semester are mathematical in nature, and the summative assessments will consist of four (unseen) in-class tests and a written exam. Phones, laptops and tablets are banned from all these.
I believe that this is the only way to maintain the integrity of these assessments and that to do anything else will result in a lowering of academic standards. I believe this is the case across a great many disciplines, so there should be university-wide regulations to manage it. I find the absence of any “Leadership” on this scandalous (but I’m afraid not surprising).
This is not to say that I will band the use of LLMs etc entirely. Students will be free to use ChatGPT or CoPilot or whatever to check their answers to the formative assessments I hand out, if they think that will help, but these will not contribute to their module mark.
Today, Monday 3rd August 2026, being the first Monday in August, is a Bank Holiday in Ireland. This holiday was created by the Bank Holiday Act of 1871 when Ireland was under British rule. While the August Bank Holiday was subsequently moved to the end of August in England and Wales, it has remained at the start of August in Ireland. It has been a nice morning weather-wise, but in true Bank Holiday fashion we’re promised heavy rain later on, with a chance of thunderstorms.
As I mentioned on Saturday, the first day of August marks the old pagan festival of Lughnasadh, named after the God Lugh, on which is celebrated the beginning of the harvest season. This coincides with the English Lammas Day, one of many Christian festivals with pagan origins. Traditionally this is the start of the harvest season and is celebrated accordingly, with rites involving the first fruit and bread baked from flour obtained from the first corn. It is also one of the cross-quarter days, lying roughly half-way between the Summer Solstice and the Autumnal Equinox (in the Northern Hemisphere).
Two years ago at this time I was still on sabbatical. That seems like ages ago! When I resumed teaching, in September 2024, I had to teach two modules I’d never taught before: a fourth-year Mathematical physics course on Differential Equations and Complex Analysis and a second-year Engineering Mathematics course. I am doing both of these again so resumption of teaching in September won’t be as challenging, although I will be changing a few things in the light of last year’s experience. Lectures start on Monday 22nd September.
After I return to work tomorrow the next big item on the agenda is the repeat examination period, which starts on Wednesday August 5th. The first paper for which I have responsibility is on Friday 7th, but I won’t be able to get the scripts until Monday 10th so I have a bit of time to do other stuff before I start marking. I am not expecting many scripts, actually, so it shouldn’t be too onerous a task to get them done promptly.
Next week also sees the deadline for the dissertations for our MSc course. I have been supervising a student this year and will probably be asked to give comments on the final draft of his report. Then the MSc students have their presentations on Friday 14th August. They have all completed the taught component of the course so the only thing left to assess are these projects. Students will get their final results later this month, along with the resitting undergraduate students.
In between all this we have our Eclipse Event on August 12th. This sold out very quickly so we expect a good turnout. The event was advertised quite a bit in the local media, e.g. here and here:
The viewing of the eclipse will of course depend on the weather, but we have talks lined up indoors and a pop-up planetarium so I hope people will turn up even if the weather isn’t great. Although it is officially full, there may be a few returns, so it is worth checking on the day to see if we can fit you in.
Anyway, after the repeat examinations and project assessments are done, the marks uploaded, and the Undergraduate and Postgraduate Examination Boards have done their work, the next job will be to prepare for the new intake of students. This year’s Leaving Certificate results will be announced on Friday 21st August, at which point we’ll see how many students (if any) we have studying Physics next academic year (which, if all goes to plan, will be my penultimate…)
Composition Derived from the Equationwith Green, Orange, Violet (Black) by Georges Vantongerloo (1930, Oil on Canvas, 119.4 x 68.2 cm, Guggenheim Museum, New York)
I leave it as an exercise to the reader to determine how this painting is derived from the stated equation.
In a post last week I hinted that I hoped soon to be able to provide an update on progress with the European Space Agency’s Euclid Mission. People within the Euclid Consortium have known for some time, but it has now been officially announced by ESA, that the timeline for the first full Data Release (DR1; originally scheduled for October 2026) has now been revised.
The plan now is for DR1 to happen in stages, with a first tranche to occur in November 2026 (precise date yet to be announced). The complete data release will take place in mid-2027 (probably in June). The sky area covered by DR1 will be about covering a large sky area of about 1900 deg².
To give a little more background, the data products from the Euclid survey are divided into three levels of data processing function:
LE1 – the “raw” data frames, prior to calibration, generated at the Science Operations Centre from the time-ordered data and telemetry received from the spacecraft
LE2 – the calibrated and corrected data for the two instruments (VIS and NISP) – images, spectra, catalogues of point sources, etc
LE3 – the high-level data products (galaxy catalogues, cosmic shear maps, etc) designed for cosmological analysis
The first release will comprise LE1 and LE2 only. This will be called DR-Foundation. The LE3 data will be added next year to make the full DR1. Since LE3 is required for the cosmological analysis that is the prime motivation for the Euclid mission, it follows that there will be no official cosmology results from Euclid DR1 until mid-2027 at the earliest. Other results based on “Foundation” data may of course emerge before then.
I hope this clarifies the situation.
P.S. Irish physicists and astronomers will be particularly interested to know that the LE1 data includes information about the spacecraft and instrument pointing orientations, which is stored in the form of quaternions…
As I mentioned a few days ago, examinations for the 2026 school Leaving Certificate are under way. One of the interesting things about the Irish system is that the examination papers are put up online immediately after the examinations. Students took their first paper in Mathematics (either Ordinary or Higher level) on Friday and the second was this morning. There has been some reaction in the news here and here.
Anyway, I thought I’d share the Mathematics papers here so you can see what you think of them.
People keep telling me that Newtonian Mechanics is simple. Here’s a demonstration of the apparently simple Double Pendulum that demonstrates very complicated behaviour.
P.S. I did not make the video – it can be found all over the interrnet. I am sharing it here for the education and entertainment of my readers, both of them.
P.P.S. The music is a track called “Swing Theory”…
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