Archive for the mathematics Category

Coronavirus in Ireland – the Latest!

Posted in mathematics on March 28, 2020 by telescoper

Just a quick post to point out that I’ve set up a page on which I’m tracking the number of cases of Covid-19 in the Republic of Ireland.

I intend to keep the data on that page up to date as information is announced by the HSE and won’t do lots of updates as posts. I thought I’d show the latest figures here though.

The second (log-linear) plot is perhaps more informative. It shows some evidence of a flattening compared to an exponential curve. The plot in green is an exponential with a time constant of 3.5 days; it’s not a fit to the data, it’s just there to show what exponential growth would look like on such a plot (ie a straight line).

Clifford’s `Space-Theory of Matter’

Posted in Beards, History, mathematics, The Universe and Stuff with tags , , , on February 26, 2020 by telescoper

Well, here’s another thing I didn’t know until I was informed by Twitter.

Way back in 1876 –  forty years before Einstein presented his Theory of General Relativity – the mathematician W.K. Clifford (who is most famous nowadays for the Clifford Algebra) presented a short paper in the Proceedings of the Cambridge Philosophical Society in which he speculated that space might be described by Riemannian rather than Euclidean Geometry.

Here are a couple of excerpts:

and

The paper does not contain any actual equations, and his concentration on small scales rather than large was misguided, but it is quite remarkable that he was thinking about such matters such a long time ago!

Unfortunately Clifford died very young, in 1879, at the age of 33, tuberculosis. Had he lived longer he might have been able to develop these ideas a bit further.

As a postscript I should mention that Clifford had an impressive beard.

What’s the Vector, Victor?

Posted in Film, mathematics, The Universe and Stuff with tags , , , on February 18, 2020 by telescoper

Following on from Sunday’s post about the trials and tribulations caused by Storm Dennis, here is a clip of a plane (an Airbus 380)  landing at Heathrow airport on Saturday.

There are other clips of this same event on Youtube and some of them describe this landing as `dangerous’. Although it undoubtedly involved skill and concentration by the pilot it’s not actually dangerous. Aircrew are trained to land in windy weather like this, and it’s fairly routine. My plane to Dublin (an Airbus 320) landed like this on Saturday evening and, although the pilot got a well-deserved round of applause on landing, nobody was ever really at risk.

As it happens, this week I start teaching vector algebra to my first-year Engineering students, so the weekend’s weather events have  given me a good illustration of vector addition. The plane has to have a velocity vector relative to the air such that the sum of it and the wind vector adds to a resultant vector directed along the runway. Lots of people seem to think this is just guesswork but it isn’t. It’s applied mathematics.

This is in principle simple as long as the crosswind is steady, but obviously the pilot needs to be alert to gusting and make adjustments along the way. When the plane has slowed down enough to land in normal conditions, the wind over the wings is still causing a bit of extra lift. You can see that in the last moments before touchdown this aircraft is gliding because of this effect. I’m told that because of this, in windy conditions planes usually descend at a steeper angle than usual.

The interesting bit for me is that the plane touches down in such a way that its body is at an angle to the runway. As soon as it has landed it has to correct this and point along the runway. I think this is done with the rudder rather than the undercarriage, but I don’t know. Perhaps any experienced pilots that happen to be reading this could give more details through the comments box?

P.S. The title of this post is a reference to the film Airplane!

 

It’s all a question of angles.

Posted in History, mathematics on February 14, 2020 by telescoper

I couldn’t resist reblogging this fascinating post on the origins of trigonometry by the inestimable Thony Christie..

The Renaissance Mathematicus

Thomas Paine (1736–1809) was an eighteenth-century political radical famous, or perhaps that should be infamous, for two political pamphlets, Common Sense (1776) and Rights of Man (1791) (he also wrote many others) and for being hounded out of England for his political views and taking part in both the French and American Revolutions.

Portrait_of_Thomas_Paine Thomas Paine portrait of Laurent Dabos c. 1792 Source: Wikimedia Commons

So I was more than somewhat surprised when Michael Brooks, author of the excellent The Quantum Astrologer’s Handbook, posted the following excerpt from Paine’s The Age of Reason, praising trigonometry as the soul of science:

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My first reaction to this beautiful quote was that he could be describing this blog, as the activities he names, astronomy, navigation, geometry, land surveying make up the core of the writings on here. This is not surprising as Ivor Grattan-Guinness in his single volume survey of the history…

View original post 1,567 more words

Searching for Synge

Posted in History, mathematics, Television with tags , , on November 14, 2019 by telescoper

John Lighton Synge (above; 1897-1995), who was an expert on geometrical approaches to general relativity, was regarded by many as the most eminent Irish mathematician and physicist since Sir William Rowan Hamilton. Synge (whose uncle was the famous playwright John Millington Synge) was born in Dublin and had spells at Trinity College Dublin, the University of Toronto and various universities in the USA before taking up a position as Senior Professor at Dublin Institute for Advanced Studies (DIAS) in 1948 from which he retired in 1972.

I have been asked by a friend to find out if there are any video recordings of Synge talking or lecturing. A quick google search turns up nothing, so I thought I would put this request out into the blogosphere to see if anyone is aware of anything.

Given the dates it seems likely that any recordings of him would be originally on film (or perhaps television) which would have to be transferred to digital format. Perhaps there is archive material at Trinity College or DIAS that could be suitable?

Happy Birthday, Quaternions!

Posted in History, mathematics with tags , , , , on October 16, 2019 by telescoper

Sir William Rowan Hamilton (1805-1865)

Today, October 16th, is Hamilton day! It was on this day 176 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 episode  is commemorated by an annual Hamilton Walk. Sadly,  Broombridge (Droichead Broome) is near the bridge (Broom Bridge) where Hamilton had his Eureka moment and it is on the main commuter line from Maynooth into Dublin. This is ironic because Quaternion algebra does not commute. (Geddit?)

Although it is quite easy to reach Broombridge from Maynooth, I sadly can’t attend the walk this year because I’m teaching this afternoon.

P.S. Maynooth is also home to the Hamilton Institute which promotes and facilitates research links between mathematics and other fields.

Additional Mathematics O-level 1979

Posted in Biographical, Education, mathematics with tags , , , on July 1, 2019 by telescoper

Yesterday a comment appeared on an old post of mine about the O-level Examination I took in Mathematics when I was at School. With a shock that reminded me that it was FORTY years ago this summer that I was taking my O-levels at the Royal Grammar School in Newcastle. That’s a memory lane down which I wasn’t anxious to take a trip.

For any youngsters reading this, the GCE (General Certificate of Education) Ordinary Level Examinations O-levels were taken at age sixteen in the United Kingdom back in the day; they were replaced during the 1980s by the modern GCSE Examination. For readers in Ireland the O-levels were roughly equivalent to the Junior Certificate, just as A-levels are roughly equivalent to the Leaving Certificate.

Anyway, that also reminded me that I never got round to posting the other O-level I took in Mathematics that summer, in Additional Mathematics. I thought I’d remedy that failing now, so here are the two papers I took (on Tuesday 26 June 1979 and Thursday 5 July respectively.

I had forgotten that there was so much mechanics in this actually (Section C of each paper). Is that different from equivalent papers nowadays? In fact I’d be interested in comments about the content and level of difficulty of this compared to modern examinations in mathematics via the box below.

P.S. I did ten O-levels that summer of ’79: Mathematics; Additional Mathematics; Combined Science (2); English Language; English Literature; French; Latin; History; and Geography. I still have all the papers and have only posted a subset. If anyone has requests for any others please let me know and I’ll scan them.