This is also from The Irrationals – though I had to ask for assistance over at Stack Exchange to get the right answer (as is so often the case the solution is reasonably obvious when you are presented with it).
Anyway, the question is: given a regular pentagon (of sides with length of 10 units) which is inscribed by a circle, what is the diameter of the circle?
This figure helps illustrate the problem:
We are trying to find and we know that .
If we knew then we could answer as . We do not know that but we do know that and hence, from the identity: .
From our knowledge of the pentagon with sides of unit length (you’ll have to trust me on this or look it up – it’s too much extra to fit in here) we also know that , where is the golden ratio.
Hence … well, the rest is left as an exercise for the reader 🙂
English: John Duns Scotus (c. 1266 – November 8, 1308) was a theologian and philosopher. Some think that during his tenure at Oxford, the notion of what differentiates theology from philosophy and science began in earnest. (Photo credit: Wikipedia)
His views annoyed the humanists (the humanists being of the view that humans needed to re-read the classics, the scholastics being content with the scholars’ existing interpretations) and hence his name became a term of abuse.
But Duns Scotus was actually quite perceptive in at least one way – he argued that the patterns of the celestial wanderers (i.e., the Sun, Moon and planets) could not repeat in a grand cycle because they were incommensurate – in other words some bodies moved at speeds that were irrational compared to others.
This is, in fact, the same point I made last September about the Spirograph tracing out cusps for inner wheels with incommensurate radii – so I am just eight centuries behind him.
The advantage of waiting a little longer to take some more shots is that the sky gets darker and you can drop the exposure time to 20 seconds. The disadvantage is that clouds appear…
It’s a clear night here in Norfolk, so here’s a shot – 30 seconds at 1600 ISO – of Vega, the rest of the Lyra constellation and the Milky Way (bottom of frame).
It’s quite eccentric – publishing passages in Ancient Greek alongside maths – but quite fun. Expect some entries based on the things it makes me think about.
One of the paper’s authors is Flavio D. Garcia, who is based in the UK (the University of Birmingham) and so would have to respect the injunction or face contempt proceedings (and the possibility of prison) – the two other authors Roel Verdult and Baris Ege, both of Radboud University Nijmegen are not in or from the UK so it’s not clear to me how effective the injunction would be against them if they opted to defy it.
It’s difficult to see this as an open-and-shut case about academic freedom – as Volkswagen were not trying to supress the fact of the vulnerability, merely the details of how to conduct the crack – but it does concern me. Many security researchers make it their normal practice to publish the full details of zero day exploits – some arguing that the pressure thus applied is the only way to ensure the problems get fixed. It’s a legitimate argument to make and it should not, in my view, be a matter for the courts to judge on as a matter of routine.
I am not a lawyer but my understanding is that temporary injunctions are usually granted pending a full hearing – to preserve the status quo ante while the full legal arguments are heard: so this may have some time to run yet.
Today is the twentieth anniversary of the launch of Windows NT. In fact I have been using it (I still have to – in the sense that Windows XP/7/8 are NT – on occasion) for a bit longer than that as I was a member of the NT beta programme – I even managed to write some software for it using the 32 bit SDK before it was officially released (a not very good version of exie-ohsies/naughts-and-crosses/tic-tac-toe – so poor was the coding that you could beat the computer every time if you knew its weakness).
NT would not run on my Amstrad 386 and in the end I bought a cheap 486 machine to match the software – it was a lot of fun in the early days – though I shudder to think of trying to get by, day by day, on a machine with such a weak command line.
One thing I remember was the late, great, Byte magazine running an issue in the summer of 1992 predicting that NT would be the death of Unix. I even thought that was right – Unix was for expensive workstations and now us users of commodity hardware were to get a proper multi-tasking 32 bit operating system – who needed time-sharing when we could all have PCs of our own?
Servers designed for Linux (Photo credit: Wikipedia)
Two big thoughts strike me as a result of the literature review I have just completed for my PhD:
Linux is not the centre of the universe, in fact it is a bit of an intellectual backwater;
The UK may have played as big a role in the invention of the electronic computer as the US, but these days it is hardly even in the game in many areas of computing research.
On the first point I am in danger of sounding like Andy “Linux is obsolete” Tanenbaum – but it is certainly the case that Linux is far from the cutting edge in operating system research. If massively parallel systems do break through to the desktop it is difficult to imagine they will be running Linux (or any monolithic operating system).
In fact the first generation may do – because nobody has anything else right now – but Linux will be a kludge in that case.
Doing my MSc which did focus on a Linux related problem, it seemed to me that we had come close to “the end of history” in operating system research – ie the issue now was fine tuning the models we had settled on. The big issues had been dealt with in the late 60s, the 70s and the 80s.
Now I know different. Operating systems research is very vibrant and there are lots of new models competing for attention.
But along the way the UK dropped out of the picture. Read some papers on the development of virtual memory and it will not be long before you come across the seminal experiment conducted on EMAS – the Edinburgh University Multiple Access System – which was still in use when I was there in 1984. Now you will struggle to find any UK university – with the limited exceptions of Cambridge and York (for real-time) – making any impact (at least that’s how it seems to me).
It’s not that the Americans run the show either – German, Swiss and Italian universities are leading centres of research into systems software.
I am not sure how or why the UK slipped behind, but it feels like a mistake to me – especially as I think new hardware models are going to drive a lot of software innovation in the next decade (well, I would say that, wouldn’t I?)
I have just submitted my “qualifying dissertation” to the University of York.
Latterly I have been thinking of my PhD studies in terms of the film Dougal and the Blue Cat – I grew up at a bad time (the virtual collapse of cinema going in the 60s and 70s) and a bad place (in a civil war of varying intensity) and this was one of the few films I saw in my childhood (pre-13) in the cinema and it did have a profound impact!
In the film the Blue Cat has to undergo several trials on his route to kingship, rising in rank as he goes. I have just made my bid for the MSc stage (if the University fail me they may recommend I write up the literature review for a standard MSc). Next would come an MSc by Research, then an MPhil, then a PhD.
I don’t want any of them – except the last one of course – but it’s a decent guide to the amount of effort required, so this feels like a real milestone.
But it’s also the easiest part – I have to (assuming my dissertation is passed – my viva is in September) actually do some research and experimentation now!
My issue with the book is not atheism but the essential claim of the author – Alex Rosenberg – that human beings cannot reason about anything, can exercise no choice and have no free will and live a completely determined life.
Rosenberg grounds this in the claim that humans cannot have thoughts “about” anything – how can, he asks, your neurons be “about Paris” (or anything else) when they are merely electrical connections? And, he adds, our sense of free will, of conscious decision, is an illusion as demonstrated by multiple experiments that show we have “taken” any decision before we consciously “decide” to take it.
English: Plaque marking Alan Turing’s former home in Wilmslow, Cheshire. Español: Placa conmemorativa en la antigua casa de Turing (Photo credit: Wikipedia)
In the end I just think this is a tautology. How can the words on a page be “about Paris” either when they are just black ink? We end up abolishing the category of “about” if we follow this argument. Nothing is about anything else.
And how do humans advance their knowledge and understanding if they cannot reason, cannot decide? Knowledge cannot be immanent in experience, surely? Newton did not formulate gravity because being hit on the head by the mythical apple was a form of “percussive engineering” on his neural circuits – he reasoned about the question and yes, that reasoning helped reshape the neural connections, but it was not pre-destined.
And anyone who has read Godel, Escher, Bach will surely see conscious and unconscious decision making closely linked in any case – this is what a “strange loop” is all about.
Ultimately I find myself thinking of Turing’s idea of the “imitation game” and the more general idea that intelligence is what looks like intelligence. Computers have no free will, but they are not necessarily fully deterministic either – we can build a random number generator which is powered by nuclear decay events which, we must believe, are fully stochastic. Such a system could be made to appear as exercising choice in a completely non-deterministic way and look fully human within the bounds of Turing’s game. And when I say it is being “made to appear” to be exercising choice, I think it will be exercising choice in just the same way as we do – because there is no way that we could tell it apart from a human.
Or to take another example – if we build a genetic algorithm to find a heuristic solution to the travelling salesman problem in what sense has the computer not thought “about” the problem in developing its solution?
and I discover that the term “
cusps for inner wheels with incommensurate radii – so I am just eight centuries behind him.
– I grew up at a bad time (the virtual collapse of cinema going in the 60s and 70s) and a bad place (in a civil war of varying intensity) and this was one of the few films I saw in my childhood (pre-13) in the cinema and it did have a profound impact!
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will surely see conscious and unconscious decision making closely linked in any case – this is what a “strange loop” is all about.
cartesian product 