Posts Tagged mathematics

It’s Ada Lovelace Day! aimee whitcroft Oct 16


Today, October 16th, is Ada Lovelace Day.

Ada Lovelace, for those who didn’t already know (and you all do, right? * wink *), also called Augusta Ada King, Countess of Lovelace, is one of the shining stars in mathematics and computer history.

Ada Lovelace, a woodcut graphic by Colin Adams based on the original watercolor by Alfred Edward Chalon. Donated by the Ada Initiative to Wikimedia Commons.

Yep, you heard right – a _girl_ was incredibly good at maths :P

She was born and lived during the 1800s, and it known primarily for her work on Babbage’s analytical machine*.

Which means that, in some circles at least, she is also considered to have been the world’s first computer programmer.

Anyway, today is her day!

There’s a lovely page on the website Finding Ada devoted to the day, and I encourage you to go check it out. Christchurch City Libraries has also put out this great list of nonfiction books about female scientists.

And a fun challenge – what’s YOUR favourite story about women in STEM (science, technology, engineering. ,maths).  Who has inspired you?


Related posts

Greetings, 2012ers (in which I talk about the Ada Initiative, and going to a barcamp held by them)


* I’ve seen a giant replica of its predecessor, the Difference Engine, at the Computer History Museum in Mountain View, California. Huge, and amazing.**

** The Difference Engine was never actually built during Babbage’s lifetime, and the question remains whether it would have been possible to do so (even if anyone had tried), given milling technology at the time.  Now, anyone can make their own using 3d printing, and plans are underfoot to construct a wor

TOSP Episode 15: December 19th 2011 aimee whitcroft Dec 19

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The last TOSP before Christmas!

And, because we’re daring, it _isn’t_ Christmas-themed.  Just to give y’all a break.

Instead, Elf and aimee cover the winners of the Prime Minister’s Science Prize(s), how bees reach consensus (warning: headbutting), a very special new crab, Jupiter’s heart-cannibalisation, the race to create the bionic eye, and the effect that Foo Fighters concert has both on GeoNet (seismic sensors) and a human body :) And the Higgs boson results get a mention, too.


You can read the rest of this entry on the Sciblogs The Official Sciblogs Podcast site

Google launches STEM competition aimee whitcroft Nov 22

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I’ve just received a press release from Google, about a competition which looks kinda cool :)

google teslaOn January 11th, they’re going to be launching the (first ever, as is so common with the company) Google Online Science Fair.  Because of its online nature, it means it can be global pretty easily, and it’s open to anyone from the ages of 13-18.

And I can imagine that the prizes will be pretty cool :) (The release mentions internships, scholarships and other, even better, prizes).

The official announcement will be going out on December 1st (and no, I’m not breaking an embargo here).  Anyone who wants more info, and/or thinks they might be interested in telling others about it, go here and they’ll keep you updated!

Wish I was still young enough to enter…


Oh, and STEM stands for Science, Technology, Engineering and Mathematics

The mathematics of war aimee whitcroft Sep 13


War!  Hngh!

warWhat is it good for?  Well, the development of some interesting mathematics, if nothing else.  And raised eyebrows.  And scheming/strategising.

Last Monday morning (yes yes, I know – been busy, ‘k?!) I successfully managed to hie myself off to Dr Sean Gourley’s speech about, you guessed it, the mathematics of war.

Or, to be particular, the mathematics behind insurgencies’ ability to stave off defeat by much larger, more well-equipped forces.  Asymmetric warfare, in other words.

I am now going to attempt to share with you the (hopefully not too garbled) notes and learnings which I took therefrom.

Nash equilibria, open source intelligence, and oranges

Firstly, a brief explanation.  Up until relatively recently – the last coupla decades, really – warfare has been a more traditional affair.  Two sides, lined up against each other, having it out.  Clear knowledge of who one’s enemy is, where they are, and at least to some extent, what they’re up to.

And game theory (more specifically, Nash equilibria) was able to adequately help model how such conflicts might go.

Things are now different.  The concept of ‘your enemy’ has become far more complicated.  There are many conflicts all over the world at the moment, each featuring its own cast of insurgents/guerillas/terrorists/organised criminals and, um, often at least one ‘major’ force.

Gourley was interested in collecting data on these sorts of things (attack size, when, which conflict, deaths, injuries etc), but of course didn’t have official access to such data.  Because govts like to keep them to themselves.  So, how to get the data he wanted?  The answer: open source intelligence.

Yes, once again, ‘open source’ raises its beautiful shiny head above the parapet and grins charmingly at us.

Open source intelligence is, simply, the information that one can gather from citizen reporting, the news, NGO stats and so forth.  Yes, there’s a lot of noise, but there’s also some signal in there.  Data, in other words.    Even looking for second order effects can help one determine if something’s going on.  Perhaps the classic story about open source intelligence and second order effects is about the Alliance. Unable to be directly sure whether they had successfully bombed bridges in Germany, they looked at the price of oranges in cities, which had to be imported.  Spikes in the price of this acidic, vitamin C-containing fruit corresponded to bridges going down.

We haz data – now what?

So yes.  Gourley and co. collected a bunch of data for different conflicts around the world (Iraq, Colombia, etc), and then set about analysing it.  What they found was interesting – when deaths were plotted against cumulative frequency for a number of conflicts on a log-log graph, the resulting line looked an awful lot like something adhering to the power law, with an alpha (slope) hovering around 2.5.

Or, to put it more simply: the data suggested that insurgent conflicts (fought in different places, for different reasons) around the world might cluster around this value.

The next step was to try to explain this phenomenon.

war equation

Gosh, I hope I copied this down correctly :)

[Where P (the probability of an attack killing x no. people in a time window t) = a constant multiplied by x (the size of the attack), raised to the power of negative alpha (which is, roughly speaking, the slope of the line when plotted on a log-log graph)]

Lost yet?

For those of you not terribly comfortable with the equation, not to worry.  Of more interest is what it means.

Basically, it looks like the number of people killed in an attack is correlated with the strength of the attacking group.  And it’s worth being clear here – that’s strength, not size. A smaller group of people with oodles of moolah and weaponry is going to do more damage than a larger group with less moolah and weapons.  So one could look at alpha as the distribution of attack strengths, which leads us towards an organisational structure.


Well, ask yourself: how does one organise one’s forces to best fight the opposition? Now, bearing in mind that insurgencies aren’t centrally controlled but rather self-organising, how does the insurgency organise itself to take on a much stronger, conventional armed force?

There are a couple of possibilities – one might be taking the whole force, and dividing it up equally.  But the attacks that come out of that look more like a Gaussian distribution, not a power law distribution.  Which means that’s not how insurgencies are organising.

Instead, the organisational structure which best fits what Gourley et al observed, was that each group would have a small number of groups which killed lots of people, lots of groups which killed very few people (per attack), and a bunch of groups in the middle.  Indeed, the maths suggests that it’s 316 times more difficult to kill ten people than it is one person, and 316 times more difficult again to kill 100 rather than 10 people.  And that multiplying exponent seems to stay as one looks at different conflicts.

Now, here’s where biology had some lessons for the mathematicians (hah!*).  Each group is subjected to forces of coalescence  and fragmentation.

With coalescence, there can be a formation bias towards the formation of large groups or towards the formation of small groups.  And the actions can be geographic (i.e. dominated by people one is near) in nature, or non-geographic. (hello mobile phones, internet etc).

Similarly, with fragmentation, groups can either split into two and factionalise, or they can split into many parts/shatter.

The interaction between these factors gives rise to different distributions – an understanding of how allowed Gourley et al to start looking at which structures best fit insurgencies.

How do insurgent organisational structures behave?

They found that the formation bias was towards large groups , with connections that aren’t geographical in nature.  Of course, the stronger they get, the more liable they’d be to find themselves on the radar of whichever the larger, conventional force is.  Who would then attack them.

Classic tall poppy syndrome stuff.

Said insurgent group would proceed to shatter (rather than factionalise).  However, it wouldn’t shatter randomly – instead, the next most successful group starts to accrete members.  So there’s this fluid system which allows a great deal of learning and innovation, as opposed to the conventional forces which have static, rigidly defined operating procedures.

All nicely explained in the picture below.

Figure 4 | Model framework for insurgency. The insurgent population comprises an overall strength N, distributed into groups with diverse strengths at each time-step t. This distribution changes over time as groups join and break up. Dark shadows indicate strength, and hence casualties that can be inflicted in an event involving that group. Figures 1 and 2 are derived from the number of events of size x aggregated over time. Figure 3 is derived from the number of events at a given time-step aggregated over size.

Model framework for insurgency. The insurgent population comprises an overall strength N, distributed into groups with diverse strengths at each time-step t. This distribution changes over time as groups join and break up. Dark shadows indicate strength, and hence casualties that can be inflicted in an event involving that group. Figures 1 and 2 are derived from the number of events of size x aggregated over time. Figure 3 is derived from the number of events at a given time-step aggregated over size. Credit: Nature, doi:10.1038/nature08631

Alpha – higher or lower?

So interesting patterns around alpha – 2.5 appears to be the value at which an insurgency is stably/sustainably fighting against the larger/stronger force.  That is, the conflict won’t end with an alpha of 2.5.  But what happens when alpha is higher or lower?

If one can drive alpha higher, then one drives the insurgency towards fragmented, fluid groups, and more groups (basically, more towards the guerilla feel).  These tend to peter out eventually.

If one can drive alpha lower, one has an insurgency made up of stronger, more robust groups, but fewer of them (more like a conventional war). There is the possibility of an actual win/defeat here.

How does one decide whether to drive alpha higher or lower?  Well, look at where alpha is currently, and  has been over time, and from there make a decision about what’s achievable.

Fascinatingly, the strategy that the maths suggests is counterintuitive – attack the weak groups, not the strong ones.


Now, what are the applications of this?  Well, it’s certainly got the US NSA and other such organisations interested, because of what it might be able to teach them about how to defeat the insurgencies with which they’re involved.

The model’s powerful – it can suggest how many insurgent groups are active, and how to deal with them.  Because insurgent wars which drag on kill an awful lot of people.

Which helps answer the question: why did Gourley start this research?  Well, he’s of the belief that the more we understand war, and how people die, the more we can stop it happening.  Hear hear :)

Oh yes, and there are other applications – the model can probably be drawn out to look more generally at how small groups successfully fight large groups.  As with companies (how small tech startups beat large tech giants), or even ,medicine (how drugs attack tumours in the body).

Brilliant stuff!

I’m hoping to be able to podcast Gourley’s talk (and the questions afterwards!), but while I wait for permission, here’s a brief TED talk he gave.

UPDATE: Sean’s given me permission to podcast his talk.  /celebrates

You can find it here**.


Oh yeah: and I highly recommend the BBC’s “The Story of Maths”, hosted by Oxford professor Marcus du Sautoy.

* You see?  Biology’s not just, ahem, “stamp collecting”…

** Full props to the Internet Archive for letting people upload files, for free :)



Bohorquez, J., Gourley, S., Dixon, A., Spagat, M., & Johnson, N. (2009). Common ecology quantifies human insurgency Nature, 462 (7275), 911-914 DOI: 10.1038/nature08631

Reprise #3: the value of memory aimee whitcroft Aug 16


Last reprise for the moment!

According to this marvellous post, a British (yes, the Brits feature again) brain has actually worked out a formula able to place a precise, sterling value on one’s memories.

It factors in elements such as how vividly you recall it, its perceived importance, and a host of other interesting factoids. It’s also available here, if you’d like to try it out…

It’s all part of his research into how, essentially, to use neuroscience to enable companies to tailor their marketing even more. And this is where it gets sticky, I think.

I will, quite freely, admit to not being a mathematician, and so will refrain from any pithy comments related to the formula.

As a former market researcher/analyst (amongst other things), I am generally not overly concerned by companies’ efforts to tailor their efforts to us. I would probably rather have my time wasted by ads I might be slightly interested in, than not. Probably. Certainly, I can understand the companies’ point of view.

Then again, people have varied levels of resistance to marketing messages, which is where some of my friends’ misgivings come in. They worry that such tailoring makes it more and more difficult for people to say ‘no’ to marketing messages. Particularly, well, the so-called ‘mass market’ (an ever more inaccurate phrase, frankly). No, this does not, sadly, paint a picture of humanity as reasonably able to make decisions for themselves. Nonetheless.

Personally, I find myself sitting uncomfortably on the proverbial fence with mixed feelings about this development and what it heralds. I’m amused, yes. It’s funny, after all.

I’m also slightly worried by it – it does seem that companies’ are increasingly looking for the edge in their messages, and are quite happy to manipulate us at levels where, frankly, the ability to filter messages is not longer an option.

Mostly, though, my overriding emotion is this: fantastic, we’ve managed to attach a monetary value to yet another integral part of the human experience…


In other news, I shall be attending Semi-Permanent this week in my ‘i heart design’ hat*, as one of Idealog’s roving writer-type people.  Hooray!  Look out for the tweets and the posts on Idealog‘s site :)


No, it’s not an _actual_ hat.

The first mathematical model for cow behaviour (I kid you not) aimee whitcroft May 25


Apologies for the slightly dodge agrarian pun in the subject line.

Coupled cows not displaying synchrony (perhaps due to their youth?)

Coupled cows not displaying synchrony (perhaps due to their youth?)

Reading the headline for this, however, has had me hanging onto my chair in hysterics for the last 5 minutes (a long time, believe me).  And, given the extent to which kiwis care about cows (and their climbing numbers here), it was, I thought, something to be shared immediately.

So, on to details!

The paper in question aims to explain, and predict, how it is that cows lie down/stand up* in synchrony.  Something, apparently, that they do, space and resources allowing**. Now, synchronous behaviour amongst beasties (including bacteria) is hardly unusual, but no one had attempted it with beasties of the bovine persuasion before, and so this behaviour wasn’t well understood.

And how have they achieved it?  Simple.  They treated cows like oscillators***.  Oscillators, as the name implies, oscillate between two states. On an ongoing basis (think sine wave).  If applying this logic to our dairylicious friends, it means treating cows as either standing up, or lying down, and doing this in cycles.  And they are then coupled, which doesn’t necessarily involve watching them do, um, documentary type things, but instead means that they are more likely to stand up or lie down depending on the behaviour of the cows around them. Or, to put it another way, the authors took their single cow equations, turned them into coupled cow equations, and then used those to construct networks of interacting cows (herd equations).

And they put in place some assumptions.  For example, they posited that a cow watching others around it standing and eating, might feel peckish too (in the same way that, no matter how unhungry you are, you will still nick someone’s chips).  Conversely, it might feel compelled to have a lie down when its herd-buddies do.  Certainly, it assumes that space is unlimited – i.e the cows are living a kiwi lifestyle, not an intensive-farming lifestyle.  The authors were at pains to point out, however, that these constraints are not necessary, and that it would be interesting to consider other options, as well as comparing the model’s predictions to real behaviour (apparently such observations are under way).

Using the constraints mentioned, however, they found that high degrees of herd synchrony don’t necessarily accompany strong coupling.  Certainly one can see the benefits inherent in an entire herd not, for example lying down (the better to see predators, my dear).

Happily, the model could also be used to understand synchrony in other ruminants.  Hooray :)

And, as a final note****, the authors end with:

Milking these ideas as much as possible should prove to be very insightful from both theoretical and practical perspectives.


Anyone have any good cow herd stories to share? Or want to go play in fields with herds of them testing the maths?


As usual, there’s a fun arxiv article on the subject, and the paper can be read here.  The article does contain the fascinating tidbit that happy cows are more likely to act synchronously.  Hmmmm.  It’s certainly a very clear way of measuring their quality of life, albeit an amusing one.

* I blame this hyperlink on a friend to whom I am introducing Radiohead (yes, gasp).  It means I have ol’ Thom stuck on the brain at the moment.

** Not being agrarian, this was news to me.  Imagine my embarrassment to find I was not in possession of a well-known fact.  They stand to feed, and lie to ruminate (which latter phrase sounds suspiciously philosophical).  Both stages are necessary.

*** A piecewise affine dynamical system.  If you know what that is, I am most impressed.  I have no idea.

**** For another fine example of people being humorous, see my post on teapots and fluidic dynamics.

UPDATE: I just noticed another fine piece of sciency humour (in the arxiv article):

On the other hand, cows are so highly bred that it would hardly be a surprise if they had lost the ability to protect themselves from natural predators. That’s a topic ripe for rumination by a suitably interested PhD student.

* head smack *

All passengers: please remember your space insurance aimee whitcroft Apr 29

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Excellent stuff, this, and a fitting way of both apologising for my recent absence, and attempting to be sufficiently interesting to lure my readers (who have no doubt wandered) back to the fold.

virgin galactic

Aah, look at them playing so nicely together! Although this can't possible be safe...

The topic of today’s post?  Why space tourism and insurance, of course!

Perhaps it’s just me, but space insurance wasn’t something to which I’d ever given any thought.  Upon finding out that other people have, and serious thought at that, I thought I’d share.

To begin, then: why was this paper written at all?  Well, it says (fairly, I think) that space travel and tourism are both quite risky for humans and tech alike, particularly at these early stages.  And risk = insurance.  Of course, no one’s really looked into the issue of space insurance per se, (silly us, we’ve been getting stuck up in the tech and politics of it), and hence said paper.  Indeed, the opening line reads:

‘Amateurs talk propellant, professionals talk insurance’. These were the words of Pete Bahn, Founder of TGV rockets at the Space Access ‘07 Conference in Phoenix.

Further, it explains that 2009 and 2010 are, apparently, being seen as bridge years towards the upcoming commercial space flights, with Virgin Galactic‘s* first flight scheduled for sometime in the next year or so.  As a result, this is of course the perfect time for anyone interested in making huge amounts of moolah (while of course providing a valuable service) to get involved and to start figuring out how best to do so.  The author maintains that:

The creation of a viable and affordable insurance  regime for future space tourists would be a critical element in the development of the overall space tourism market.

So, what’s the next step?  To identify the risks, of course, some of which are summarised below.

Risk identification

Spaceflight is risky.  Calls to Houston regarding problems are made.  Craft explode.  People expire.  This will likely continue to happen.

The death rate of spaceflight is, according to NASA, some 4%.  A 1 in 25 chance.  That’s actually quite alarmingly high, considering heart disease kills 1 in 6 or so**, and strokes 1 in 28.  Of course, I’d like to suggest that to extrapolate this figure would be absurd – the sample size is tiny, and we haven’t exactly done a lot of it yet.  Nonetheless, entry/re-entry, radiation, tiny bits of hurtlingly-fast rock etc DO make it a bit more dangerous than, say, your average pastoral landscape (except for the rabid sheep, of course).

So, it’s going to be necessary to find insurance methods both for damaged craft, and for damaged people.  Damaged craft first:

One of the first things to work out is the official definition of space, in terms of defining where is atmosphere, and where is space.  I had assumed there would be consensus on this.  There is not.  The most generally used definition, though, is that space begins at the Karman line, some 100km or so above the earth’s surface.

virgin galactic 2

The VG spaceplanes. Like business jets, except not.

Then there are the craft themselves.  Currently built to look quite a lot like business jets, a lot of thought and design needs to go into how to design them so as to survive multiple entry/re-entry events, the various other vagaries of space, NOT explode due to their sitting on large amounts of extremely explosive fuel, and how best to certify and license them.  The author suggests a rigid certification scheme would likely be more appropriate.  I agree.

And then, damaged people.  There’s the effect of space flight on people themselves.  Psychological.  Emotional.  Physiological. Etc.  And what if someone does actually get hurt?  Currently, the only people who can afford this sort of fun own some of the most valuable arses on earth, so (while the paper doesn’t mention it) I imagine the insurance premiums would be astronomical.***  There would be the potential human capital loss as well…

We’ve identified the risks, what now?

What do they suggest be done?  Due to the very new, and complicated nature of space insurance, the author suggests that each case be evaluated separately, rather than using the aviation insurance industry as a model.  Of course, once things become a little more routine in terms of craft reliability and design, flight frequency etc, then perhaps an overarching framework could be considered.  He reckons some 5 – 15 accident-free flights would be necessary for underwriters to begin working out the crafts’ reliability and coming up with pricing models.

There’s an awful lot more in the paper itself, in which detail-oriented people can frolick.  But to end off, I’m going to quote the author:

It is a matter of when,not if!

Also, I think the author deserves either a smack, or a thumbs-up, for the use of the phrase “The final underwriting frontier’ as a subheading.

The jury’s still out on commercial spaceflight – is it a silly waste of money for the very bored and wealthy, or does it herald the beginning of humanity’s move off earth?  Time will tell.  And apparently, so will insurance.


Bensoussan, D. (2010). Space tourism risks: A space insurance perspective Acta Astronautica, 66 (11-12), 1633-1638 DOI: 10.1016/j.actaastro.2010.01.009


*For anyone who may have spent time in a cave the last few years, yes, commercial space tourism is here.  And you too could be a part of it!  Simply gather together 200,000 of your favourite US dollars, give them to VG, and see the earth as only, well, many of us have seen it, except in real life.  Note: at this point, all flights are suborbital.

** If you’re interested in comparing, this is fun.  It’s a death calculator!  It only uses publicly available data from Europe and the US, of course, but still.  Fun!  Although they could definitely work on the presentation of the site/data a bit…

*** Yes, you read that right.  And it was intentional.

A little bit of fun: how to (mathematically) park your car aimee whitcroft Jan 26


One wonders if this doesn’t have IgNobel potential.  Of course, it’s not particularly useful, so I doubt it, but it does have that slightly silly appeal :)

Vauxhall Motors commissioned a University of London researchers by the name of Simon Blackburn to figure out how much space any given car needs to parallel park without ‘see-sawing’.

As he says in his paper:

“I want to parallel park, and I’ve found a space. The road is wide, but the space looks narrow. I’m not interested in shuffling back and forth to get into the space: I want to reverse into the space at full lock, and then drive straight forward into the middle of the space to park. How narrow can the space be so that I can do this? This report uses some straightforward geometry to compute the smallest length that the space can be.”

And the final equation, which he came to using some circles and a bit of Pythagorean goodness, looks like this…

parallel parking 2Where:

  • r is the radius of my car’s turning circle (curb to curb)
  • l is my car’s wheelbase, defined as the distance between the centres of the front and rear wheels
  • k is the distance from the centre of my front wheel to the front of my car, and
  • w is the width of the car in front of mine once I’m done parking

parallel parking 3

(npr’s got some good graphics here)

So yes, folks, there you have it!  Of course, much the same effect can be achieved by practice, and various other techniques, but it’s good to know that, given the time, equipment and facility for maths almost no one has while attempting to parallel park, you could be absolutely certain whether that gap was big enough…Happy parking!

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