Tuesday, 25 December 2018

sequestering carbon, one Christmas at a time VI

His ’n’ hers Christmas presents (plus one still in transit..)

Monday, 17 December 2018

Hannover by night

I've arrived in Hanover for the conference on Cognitive Computing 2018: Merging concepts with hardware, starting tomorrow.  Here's a view from my hotel window around late morning, just before I went to suss out the tram system and find some lunch.

In the evening I went into the city centre, by tram, to meet up with some of the other conference delegates.  Between the tram stop and the Brauhaus, I took in the Christmas lights and markets.

all the trees are decked with lights
every open area has a little marketplace
just up the street from the Brauhaus
on my way back to the hotel
Looking forward to the conference starting tomorrow.

Sunday, 9 December 2018

Narrating Complexity

My complimentary editor copies of our latest book have recently arrived.

This is an outcome of a fascinating collaboration we started way back in 2012.

This book stages a dialogue between international researchers from the broad fields of complexity science and narrative studies. It presents an edited collection of chapters on aspects of how narrative theory from the humanities may be exploited to understand, explain, describe, and communicate aspects of complex systems, such as their emergent properties, feedbacks, and downwards causation; and how ideas from complexity science can inform narrative theory, and help explain, understand, and construct new, more complex models of narrative as a cognitive faculty and as a pervasive cultural form in new and old media.

The book is suitable for academics, practitioners, and professionals, and postgraduates in complex systems, narrative theory, literary and film studies, new media and game studies, and science communication.

See the Springer site for table of contents.

(This is the first time I have produced a book in anything other than LaTeX.  I won't be doing that again in a hurry.  But Springer made a great job of typeseting the ... shudder ... Word sources.)

Friday, 30 November 2018

Engineering Simulations as Scientific Instruments

My complimentary author copies of our latest book have just arrived.

This book describes CoSMoS (Complex Systems Modelling and Simulation), a pattern-based approach to engineering trustworthy simulations that are both scientifically useful to the researcher and scientifically credible to third parties. This approach emphasises three key aspects to this development of a simulation as a scientific instrument: the use of explicit models to capture the scientific domain, the engineered simulation platform, and the experimental results of running simulations; the use of arguments to provide evidence that the scientific instrument is fit for purpose; and the close co-working of domain scientists and simulation software engineers.

In Part I the authors provide a managerial overview: the rationale for and benefits of using the CoSMoS approach, and a small worked example to demonstrate it in action. Part II is a catalogue of the core patterns. Part III lists more specific “helper” patterns, showing possible routes to a simulation. Finally Part IV documents CellBranch, a substantial case study developed using the CoSMoS approach.

See the Springer site for table of contents.  The Springer official cover in not quite the same as the one I designed a while back.  That’s probably for the best, I suspect.

This is one of the outcomes of our not-that-recent EPSRC project, CoSMoS.

Saturday, 24 November 2018

These are the days of miracle and wonder

Amazing time-lapse video of a rocket launch… seen from space!

On November 16, 2018, a Soyuz-FG rocket launched from Kazakhstan, and perched atop it was a Progress resupply ship carrying two and a half tons of materials for the astronauts on ISS. The launch occurred at 18:14 UTC, or just after midnight local time. As it happens the ISS was over the Russian/Mongolian border at that time, far to the east of the launch but high enough above the Earth’s surface to still see it directly.

screen grab from video showing the second stage re-entry burnup (top right) – the video is even more impressive

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Friday, 16 November 2018

noise from a hotel

I’m in Swindon, at a two day meeting.  The view from the hotel window is surprisingly autumnal, given it is mid November.

This was the view just before I left for the meeting.  But I was awake much much earlier.

I was woken by a loud buzzing noise.  I thought it was the alarm clock in the next door room.  But it kept going.  I looked at my clcok.  5:45.  The noise continued intermittently.  I thought it might be a water hammer, or a heating rattle.  But it didn’t sound like that.  Of all things, it sounded like a drill: about the right duration of each burst of noise, with that little “chirp” at the end as the load is taken off.  No sleep was possible.  I eventually got up.

When I went down to the hotel lobby on my way to breakfast, I discovered it was a drill!  Christmas decorations were being attached to pillars.  With a drill.  Since 5:45am.

When I checked out, the following conversation ensued.
“I didn’t appreciate being woken at 5:45 by drilling.”
“Ohh. ... Yes. ... Sorreee.  We didn’t know they would be drilling.”
“When you found out, you could have stopped it.”
“Ohhh. ... Yes. ... We could have. ... Sorreeee.”

Customer service in action!

Wednesday, 14 November 2018

book review: Lean Out

Dawn Foster.
Lean Out.
Repeater. 2015

This slim book is a response to Sheryl Sandborg’s book Lean In. I had already seen Jennifer Dziura’s acid comment on Lean In:
Lean In is an absolutely joyless tutorial on how to give your entire life to a corporation while also having a husband and kids that are a lot of work. If you’re working that hard, there should at least be something rewarding about it.
which gave me a fair idea of its message. Dawn Foster’s response is political, taking apart the underlying assumptions and context.

That context is patriarchal capitalism, where most women are poor and exploited, and simply don’t have the opportunities to Lean In even if they wanted to. Just because a few women can manage to break through the glass ceiling and become rich and successful, doesn’t mean all women can, although it does provide a useful distraction. Any resulting “trickle-dowm feminism” is just as ineffective as “trickle-down economics”; highly successful women are more likely to stick with others of their class than of their gender. And, of course, those rich successful women are relying on the labour of poorer women to allow them to run their lives, by providing cleaning, child care, and so on.

Dawn Foster’s solution is to point to those who Lean Out: women who organise and lobby for decent wages and working conditions from the bottom up. And there are many examples.
A refreshing polemic.

For all my book reviews, see my main website.

Monday, 12 November 2018

book review: Predictably Irrational

Dan Ariely.
Predictably Irrational: the hidden forces that shape our decisions.
Harper. 2008

Classical economic theory is predicated on self-interested rational actors. But people aren’t rational. So much the worse for classical economics.

Although people aren’t rational, they aren’t randomly irrational, either. Instead, they are predictably irrational, in a way that can be studied and measured, and be built into a more realistic economic theory: behavioural economics.

Dan Ariely, psychologist and behavioural economist, engagingly describes a range of experiments he and his colleagues has performed (mostly on undergraduate students, in the time-honoured experimental psychology manner) to unpick a wide range patterns of irrationality. He looks at the over-strong lure of free items, how we overvalue our possessions, how keeping options open can be a mistake, why shops will often display an expensive option they don’t expect to sell, why we are happy to do things for free we wouldn’t do if paid for, how more expensive items are more effective than identical cheaper ones, how dishonesty varies when cash is involved, how some people choose second best, and more.

I found the chapter on free work versus paid work interesting, the difference being between social norms and market norms. The world is moving us towards the latter, seemingly to the detriment of enjoyment. Similarly the chapter on honesty highlights how people are more honest when cash is involved: while taking a pencil from work is barely noticed, taking the equivalent value in cash would be beyond the pale. Yet we are moving towards a cashless world, maybe to the detriment of honesty?

This is a good read, with the experiments clearly described, and the context and consequences well explained. I am not entirely convinced that the experimental situations, with their small values and low consequences, can be safely extrapolated to larger scale cases, but they are very illuminating. Several of the examples will be useful to help avoid faulty reasoning in certain cases. (Although I already order what I want from the menu, whether or not someone else in the party has previously ordered the same.)

For all my book reviews, see my main website.

Sunday, 11 November 2018

who does like the rain?

Trump cancelled a planned outing because it was raining.

amidst ceremonies in the French capital to mark the 100th anniversary of the end of the first world war, a planned presidential visit to the American cemetery at Belleau, a site of immense importance to the US military, was cancelled because it was raining.

C’mon, give the guy a break.  Who wants to go out in the rain, after all?

he never goes out in the rain

Wednesday, 31 October 2018

view from a hotel window

I'm in Cambridge, attending the two day Novel Computational Paradigms workshop at the Isaac Newton Institute.  Yesterday I learned about using magnonics in computational devices, about different kinds of DNA computing, and about some unconventional computational architectures.  Today I'm off to talks about industry's computational problems, and discussions sessions.  All very interesting.

And all set off by a glorious view of a frosty autumn morning:

Sunday, 28 October 2018

book review: Lost in Math

Sabine Hossenfelder.
Lost in Math: how beauty leads physics astray.
Basic Books. 2018

I have been reading Sabine Hossenfelder’s Backreaction blog for a while now, enjoying her “take no prisoners” style of explaining physics: clear and insightful on the science, blunt about the process. One of the things she is particularly blunt about is the lack of progress in fundamental particle physics, with its plethora of predictions over the last decades, accompanied by a paucity of experimental verification. So I was excited to hear she was writing a book about this aspect. And what a great book it is, both from a physics and a process perspective.

Particle physics has gone nowhere in the last 40-odd years. The Standard Model of the 1970s is still the best available, despite having known shortcomings. The most recent prediction to have been experimentally verified (the existence of the top and bottom quarks) is from 1973. (The Higgs boson was proposed in the previous decade.) Since the 1970s, experimental results have succeeded only in disconfirming attempts to improve on the Standard Model. Why this stunning lack of progress?

Hossenfelder, herself an accomplished theoretical physicist, argues it is because physicists are beguiled by the idea of (mathematical) “beauty”, and are being led astray in their work. Rather than being steered by real-world data, they are following beautiful theories, such as string theory, down mathematical rabbit holes, further and further from experimental checks, ending up in the bizarre place of having to redefine science itself to allow for the absence of even the possibility of experimental validation. The book weaves clear explanations of the problem with snippets of interviews with several of the physicists involved.

Her argument has two main components. Firstly, theoretical beauty is not a reliable guide. There are past examples of a beautiful theory having to give way to a less beautiful one when confronted with disconfirming experimental data. And many times the “beauty” of a theory becomes appreciated, or possibly learned, only after the theory has been accepted and established. Secondly, physics absolutely has to be grounded in experimental validation. Mathematics is the language used to express the theories, and like any language, you can say many different things in it. The only way to discover which, if any, of these is right, is to look to the real world.

This is a great book that will win Hossenfelder few friends in the subject. It cuts deep to the heart of the problem, in a very accessible manner, and exposes the groupthink lying at the heart of today’s fundamental physics.

For all my book reviews, see my main website.

Saturday, 27 October 2018

sequestering carbon, several books at a time XC

The latest batch, including a few venerable old titles purchased from a retiring colleague.

Thursday, 25 October 2018

using domain knowledge to evolve graphs

We are investigating using evolutionary algorithms to evolve graph structures directly, and are getting some rather nice results.

One thing that’s made a lot easier by using an explicit graph representation is encoding domain knowledge about “semantics preserving mutations”.  This allows domain specific neutral mutations to be added easily.  Since neutral mutations are supposed to help evolution, this should be good?  And yes, it is!

Our latest paper, just up on the arXiv, shows how including certain propositional logic tautologies, such as de Morgan’s laws, as neutral mutations, makes for imporved performance when evolving benchmark circuits.

Timothy Atkinson, Detlef Plump, Susan Stepney
Semantic Neutral Drift
arXiv:1810.10453 [cs.NE]

We introduce the concept of Semantic Neutral Drift (SND) for evolutionary algorithms, where we exploit equivalence laws to design semantics preserving mutations guaranteed to preserve individuals’ fitness scores. A number of digital circuit benchmark problems have been implemented with rule-based graph programs and empirically evaluated, demonstrating quantitative improvements in evolutionary performance. Analysis reveals that the benefits of the designed SND reside in more complex processes than simple growth of individuals, and that there are circumstances where it is beneficial to choose otherwise detrimental parameters for an evolutionary algorithm if that facilitates the inclusion of SND.

Wednesday, 17 October 2018

can I make this blob of goo compute?

Unconventional Computing manages to make the weirdest materials compute to a greater or lesser degree: slime moulds, chemicals, black holes, gold nanoparticles, swarms of crabs, carbon nanotubes.  But if I’m given a blob of goo, can I work out how well it can compute?

That’s what our latest paper, just up on the arXiv, sets out to do: it provides a framework for evaluating how well some arbitrary substrate can be configured to be a Reservoir Computer:

Matthew Dale, Julian F. Miller, Susan Stepney, Martin A. Trefzer
A Substrate-Independent Framework to Characterise Reservoir Computers
arXiv:1810.07135 [cs.ET]

The Reservoir Computing (RC) framework states that any non-linear, input-driven dynamical system (the reservoir) exhibiting properties such as a fading memory and input separability can be trained to perform computational tasks. This broad inclusion of systems has led to many new physical substrates for RC. Properties essential for reservoirs to compute are tuned through reconfiguration of the substrate, such as change in virtual topology or physical morphology. As a result, each substrate possesses a unique “quality” – obtained through reconfiguration – to realise different reservoirs for different tasks.

Here we describe an experimental framework that can be used to characterise the quality of any substrate for RC. Our framework reveals that a definition of quality is not only useful to compare substrates, but can also help map the non-trivial relationship between properties and task performance. And through quality, we may even be able to predict the performance of similarly behaved substrates. Applying the framework, we can explain why a previously investigated carbon nanotube/polymer composite performs modestly on tasks, due to a poor quality. In the wider context, the framework offers a greater understanding to what makes a dynamical system compute, helping improve the design of future substrates for RC.

Saturday, 13 October 2018

book review: Artemis

Andy Weir.
Del Rey. 2017

The moon has a small town, Artemis, population 2000. Most of the inhabitants are either super-wealthy, or the ordinary blue-collar people keeping the town running, and working as tourist guides. Some are criminals. Jazz is a bit of both: a legitimate porter, and a smuggler to keep the wolf from the door. But she’s making no headway in saving to pay off a big debt, so when one of her super-wealthy smuggling customers offers her what looks like a great deal for some sabotage, she decides to take it. But the job doesn’t go as planned, and soon Jazz is on the run, and the worst is, she doesn’t even know who’s after her! She will need all her moon-smarts just to survive, let alone win out.

This is a fast paced romp, essentially based on a huge McGuffin, but with plenty of engineering know-how needed to solve all the problems. Jazz is a drop-out, which is why she wants money; she also has a well-developed business ethic, which is why she gets into trouble in the first place; she also happens to be an engineering genius, which is how she gets even deeper into trouble.

The book superficially has a YA vibe, but Jazz is not a teenager, she’s in her mid-20s. So occasionally the story and the vibe clash somewhat. However, this is an interestingly-drawn world, both the politics of Artemis, and its Earth-based back story. I enjoyed the grunge-tech feel to life on the moon (although I’m sure there are holes in the engineering, and in the chemistry). And I particularly appreciated how the ending panned out.

For all my book reviews, see my main website.

Thursday, 11 October 2018

book review: The Nightmare Stacks

Charles Stross.
The Nightmare Stacks.
Orbit. 2016

Alex Schwartz, recently a highly paid bank employee, currently working for the Laundry on a much reduced civil service salary, and a reluctant vampire to boot, has been posted to Leeds to help set up a secondary HQ. Or at least, that’s why he thinks he’s there. Higher powers have sent him there for a rather different reason.

Cassie was a student, until she got her brain sucked out and replaced by the consciousness of the head spy of an invading Elven army. Alex has to save the world from Cassie’s warlord father. But all he thinks he has to do is convince his parents that Cassie is his new girlfriend. Good luck with that.

We first met Alex in The Rhesus Chart, where Bob Howard helped flush out the vampires. He’s now a Laundry employee (it was either that, or something more terminal), and we get to see how he is faring as the new boy.

Each of the books in the series has a schitck: here it’s elves. Usually in fantasy we get to see an army casually plough through the peasantry in the countryside before the big showdown. Here we get to see them plough through the ordinary people in the suburbs of Leeds: it cleverly demonstrates how utterly horrifying those fantasy battles should be, but somehow never are.

Although there are still the absurdities, of the clash of dark horrors and government bureaucracy, and we care about the various characters, the series is definitely getting darker, as it inexorably moves towards the End of Days.

For all my book reviews, see my main website.

Tuesday, 9 October 2018

no longer an anti-social networker...

Farewell Google+.  Your interface was always clunky and annoying, but you had an interesting set of users.

For my 3.5 followers, you can still watch my eccentricities at my blog site (where I was always “archiving” my G+ posts anyway, in preparation for this day…)

For all my social networking posts, see my Google+ page

Sunday, 7 October 2018

It's About Time

O brave new world, That has such people in ’t!

Friday, 5 October 2018

It’s your job

an excellent post about honesty in science

No, it’s not The Incentives—it’s you

... a good reason why you should avoid hanging bad behavior on The Incentives is that you’re a scientist, and trying to get closer to the truth, and not just to tenure, is in your fucking job description. Taxpayers don’t fund you because they care about your career; they fund you to learn shit, cure shit, and build shit. If you can’t do your job without having to regularly excuse sloppiness on the grounds that you have no incentive to be less sloppy, at least have the decency not to say that out loud in a crowded room or Twitter feed full of people who indirectly pay your salary. Complaining that you would surely do the right thing if only these terrible Incentives didn’t exist doesn’t make you the noble martyr you think it does; to almost anybody outside your field who has a modicum of integrity, it just makes you sound like you’re looking for an easy out. It’s not sophisticated or worldly or politically astute, it’s just dishonest and lazy. If you find yourself unable to do your job without regularly engaging in practices that clearly devalue the very science you claim to care about, and this doesn’t bother you deeply, then maybe the problem is not actually The Incentives—or at least, not The Incentives alone. Maybe the problem is You.

For all my social networking posts, see my Google+ page

Friday, 28 September 2018

Wednesday, 26 September 2018

full moon at sunrise

The sun has just risen; the full moon is just setting.

7:07 BST, looking west

Friday, 21 September 2018

bits of ancient history

We drove from Westonbirt to Avebury, to see the the largest megalithic stone circle in the world.
some of the stones are big, taller than a person, ...
... some are smaller, and some are missing, marked by little concrete obelisks (like the one nearest the camera)
it’s possible to walk all the way round the circle (with some small detours), and to touch the stones; some people were clearly deeply communing with them (maybe because it is the autumn equinox today?)
here we can see the ditch and the bank (in the left foreground; I’m standing on the bank to take the photo)

Having walked round the circle, we followed the "Avenue" of stones (by car) down to Silbury Hill, the tallest prehistoric man-made mound in Europe.

this is as close as we were allowed to get
Over the road is the West Kennet Long Barrow, a Neolithic tomb.

view from the road: the barrow is that little line on the horizon
wow, we’re allowed just to walk into it
it’s dark in there (although there are a couple of skylights recently drilled through the rock ceiling)
view of Silbury Hill from the barrow
That’s enough tramping round the past: we’re off to our next B&B for some modern comforts.

Westonbirt, The National Arboretum

We poodled down from Hay-on-Wye to the Westonbirt Arboretum, arriving in time for an early lunch in the visitors’ centre cafe.  Then we set out on the “Autumn trail” (backwards, as we started from the cafe end, not the main entrance), which is a mile long.  The map says “allow 1–1.5hrs”; that’s because you will keep stopping to commune with so many marvellous trees.  Add yes, we took an hour to walk a mile.

the weather is considerably better than yesterday’s storm
gorgeous autumn colours
unretouched photos taken from roughly the same spot with two different cameras:
left, a Canon EOS 20D, right, a Samsung Galaxy S7
I love the fine tracery that the branches make
a beautiful pairing of colours and shapes
a magnificent specimen; it is hard to get a scale, but the point where the trunk narrows and the first branch comes outwards the camera is over 6 feet high, and something weird is happening there ...
... inosculation 
plane tree weirdness: this isn’t two branches growing up out from the ground; it is a single branch where the left part is growing down into the ground
another weirdly shaped tree
an absolutely magnificent birch

Highly recommended.  There’s a second, longer trail, which we did not tackle, since we were next off to Avebury for a spot of ancient history.

post storms Ali and Bronagh

Yesterday was very wet and windy: storm Ali hit the UK.  We spent some time driving around the Brecon Beacons, and some in Hay-on-Wye's famous bookshops.  (We bought only eight books.  The days of filling the back of the car with second-hand purchases are over, due to already having a lot of books, having already got them via Amazon.)

This morning, we saw some of the effect of the storm: the river Wye was somewhat higher than when we arrived, having flooded "the lower field".

The B&B proprietor said the river sometimes flooded that field, in winter, or in spring, but they had never seen it flood in September.

Wednesday, 19 September 2018

view from a car park

We have just arrived in Hay-on-Wye, for a short holiday.  Here is the evening view across the Wye, from the B&B car park.

Tuesday, 18 September 2018

view from a hotel window

I'm at a two day EPSRC meeting in Birmingham, near the NEC.  Standard view of a car park, although not the one I’m parked in:

Tuesday, 11 September 2018

view from a hotel window

I’ve just arrived in Glasgow, for a small workshop tomorrow.  The hotel is on a bit of a slope, and I’m on the ground floor, explaining the nice view up the rockery:

Monday, 10 September 2018

book review: Better by Mistake

Alina Tugend.
Better by Mistake: the unexpected benefits of being wrong.
Riverhead Books. 2011

We often hear that we should learn from our mistakes (or, preferably, from the mistakes of others). Here Tugend digs down into this aphorism, and shows how we are actually given mixed messages: if mistakes are such a wonderful learning opportunity, why are we actually ashamed of them, punished for them, and why do we try so hard to avoid them to the extend of over-constraining our ambitions?

Tugend recounts some work showing that mistakes while learning can be valuable, for some people at least, and should be allowed, even encouraged, to enable the deeper learning that results.
[p93] Gully and his colleagues found that not everyone learned better by making mistakes, but those with certain types of personality traits—who are good at processing information, open to learning, and not overly conscientious—were more effectively trained by being encouraged to make mistakes rather than avoid them. In other research, Gully and colleagues found that telling people to perform well during training resulted in higher immediate performance; but it also resulted in shallower processing of information, more superficial learning, and less confidence. In contrast, those people who were told to learn—and nor worry about mistakes—during training did more poorly initially but ended up with deeper processing of information, more complex learning, and more confidence about performance. Those in the latter group also showed higher performance when faced with a really challenging version of the task they were trained to perform.
Of course, mistakes while learning are somewhat different from mistakes that happen during operational use. But we are actually learning all the time, and so these other mistakes should also be exploited as learning opportunities, both to educate the one who errs, and to make the overall system more robust to inevitable mistakes.
[p112] “when an adverse event occurs, the important issue is not who blundered, but how and why the defenses failed,” […] In life-or-death situations, it is important to set up a system in which, to whatever degree possible, one person’s error cannot sink the ship. […] focussing on active errors lets the latent errors remain in the system, and “their accumulation actually makes the system more prone to future failure.”
The key point here is that a systems approach to error does not look just at the “active error”, the immediate surface error that seems the cause of the problem, but also digs down to find the “latent errors”, the systemic issues that allowed the active error both to occur and to have such poor consequences. Just fixing active errors results in a patchwork of arbitrary rules and regulations that can make the system more complex and fragile.

How to use errors during learning requires a supportive culture. Tugend gives an example of the culture in Japanese schooling.
[p191] Making a mistake, therefore, isn’t a reflection of your lack of ability or intelligence, but simply that you haven’t learned something yet. “You have to show you’re trying hard—they have this expression for ‘facing the desk,’ ” […]
So there are two aspects. First, the learner has to be trying: their mistake is a good-faith error, not the result of carelessness, or even deliberate sabotage; the learner genuinely thought that their mistake was the right thing to do. Second, the mistake can then be recognised as a signal that the learner needs to learn something, maybe a new fact (surface error), or a revised mental model of the problem space (latent error). Occasionally, it is the teacher who has to change: the learner’s mental model might be more advanced than the teacher had realised, and their “mistake” is due to tackling an over-simplistic problem in their more sophisticated manner. A systems approach to diagnosing the latent error should be able to distinguish these cases.

Of course, not all mistakes occur when we are trying our hardest: we might be tired, or distracted, or careless, or selfish. A systems approach can help diagnose these problems as well: we don’t need to learn new procedures, rather a new attitude. Although Tugend doesn’t explore this aspect (other than in the chapter on apologising), this links to the rationale behind punishment of errors: the assumption that the error is in some sense “deliberate”, and the way to learn the required new attitude is through some sort of pain. But a culture change is needed, to distinguish good-faith learning opportunity errors, from sloppy poor attitude unnecessary mistakes – and to treat them differently.

There is more in the book, including how the air transport industry learns from mistakes and its use of checklists, how men and women react differently to their mistakes, and apologising for mistakes. There is a lot of food for thought here. I would have liked more guidance on the productive exploitation of mistakes, but maybe that is for another book.

For all my book reviews, see my main website.

Sunday, 9 September 2018

book review: The Irrationals

Julian Havil.
The Irrationals: a story of the numbers you can’t count on.
Princeton University Press. 2012

The Irrationals – real numbers that aren’t rational – have many fascinating properties. Havil takes us on an historical journey, from the Ancient Greek mathematicians’ geometrical investigations, to present day deep algebraic concepts. The earlier material is, unsurprisingly, less technically challenging, but no less interesting, than later results.

I thought I knew a reasonable amount about irrational numbers before I started (possibly more than the target audience), but I still learned many interesting new facts. Some of these are quite elementary (presumably I missed these back at school due to having studied a “modern maths” curriculum), and some are more advanced.

So I learned the rational roots theorem (see p.131), which (in its simplest form) states that, given a polynomial equation xn + an−1xn−1 +…+ a1x + a0 = 0, where all the coefficients ai are integers, then any rational root must be an integer that divides a0. That’s extremely useful, as it gives a very quick constructive way to test for the existence of rational roots. The theorem is presumably known to all children studying a more traditional maths curriculum. This result, along with the majority of the others, is accompanied by a proof; I confess I skimmed most of the proofs, but they are a useful resource.

We get discussions of constructing geometric figures, algebraic numbers, continued fractions, approximating irrationals with rationals, transcendental numbers, a “positive” definition of the irrationals (that is, not one like “all reals except for the rationals”), a way to quantify how irrational a number is, randomness, and much more.

One result I particularly liked is the following (see p.262): Let α be a positive irrational. Define β = α / (α−1). Define the sets A = {floor(nα) : n=1,2,3,…} and B = {floor(nβ) : n=1,2,3,…}. A and B partition the natural numbers; that is they have no elements in common, and together contain all the natural numbers. Different choices of α yield different partitions. Havil provides examples, for α = √2, π, and e. For e, the sets are: A = {2,5,8,10,13,16,19,21,…} and B = {1,3,4,6,7,9,11,12,14,15,17,18,20,…}. That would be fun enough but itself, but it leads on to even more fascinating results. It is obvious we can write a formula to generate the multiples of three: f(n) = 3n. But we can also write a formula that generates the non-multiples of three: g(n) = floor(n/2 − ¼)+n. The approach is general, and can be used to write formulae to generate the non-squares, the non-triangular numbers, etc. (I just love this kind of stuff.)

From the playful subtitle, to the Appendix on how to find the tomb of Roger Apéry in Paris, the whole book is full of gems, and is written in a very accessible manner, provided you know a little bit of maths to start with. Highly recommended if you like reading about maths.

For all my book reviews, see my main website.

Friday, 7 September 2018

not suspicious at all

I've just received an email.

It contains the following 4 images:

Yes, an email that looks like a load of text is actually four images.  The first bullet in this pseudo-text has the excellent advice "Please do not click on any links you do not recognise."

The middle two images (the last three bullets, with the bold telephone number and the bold "here") not only are links, but you are explicitly invited to click on one.  These links have extremely unrecognisable URLs like "http://links.mkt529.com/servlet/M= ailView?ms=3DMjE4MTcxMTAS1&r=3DMjExMTk3OTQyNDQwS0&j=3DMTMwMzMzNDgwNwS2&mt= =3D1&rt=3D0"

So, an email warning me about a potential data breach that is itself either (i) a phishing attack; or (ii) from someone who does not understand their own security advice!