Category Archives: Mathematics

Record highs

As well as being a demonstration of nationalism and the fact that money can buy gold, the London Olympics gave us the usual advances in human performance. But is there a limit to how fast, high and strong we can get? Sounds like a question for science.

It’s a tricky one though: you’d expect there to be restrictions from physiology or mechanics, but we don’t necessarily understand those well enough to draw a line. Plus there’s always the chance of some innovation in technology or technique that pushes those limits further – for example, the Fosbury flop in the high jump.

But the other way to tackle the problem is to look at the statistics and see whether records are approaching a limit, which is just what Spanish physicist Filippo Radicchi from the Universitat Rovira i Virgili has done (Radicchi F 2012, “Universality, limits and predictability of gold-medal performances at the Olympic Games”, PLoS ONE, vol. 7, no. 7, e40335, doi:10.1371/journal.pone.0040335).

Graph of the probability of reaching particular time milestones as a function of the year, for five different athletic events - men's 100 m, men's 400 m, men's 10,000 m, men's marathon and women's 100 m. The probability of breaking these records increases over time.
The probability of reaching particular milestones – measured in seconds – in five different athletic events increases every year (image F Radicchi, PLoS ONE)

Radicchi looked at medal-winning performances from previous Olympics and showed that they followed a normal distribution relative to each other, which implies that they can be considered to be exponentially approaching a future limit or asymptote. Looking at the rate they’re approaching this limit also enables a prediction for each future Olympic Games.

As an example, we’ll look at the big one, the men’s 100 metre sprint. Radicchi’s calculations, published on 12 July 2012, predicted a time of 9.63±0.13 seconds for the London Olympics. On 5 August 2012, Usain Bolt won the gold medal in this event with a winning time of… 9.63 seconds.

Unfortunately, Radicchi says that some of the early data for these shorter events is unreliable, so calculating the ultimate limit is a bit tricky. The unadulterated calculations give an estimate of 8.28 seconds. But discarding some of the dodgier figures – blame them on less sophisticated timing techniques – a more conservative figure is 8.80 seconds.

Compare that with Usain Bolt’s own 2009 world record time of 9.58 seconds. Even with the conservative estimate, the limit is still a while off.

The table below shows a similar story in a number of other major events:

Event Predicted limit Current
world record
Gold medal
2012 London Olympics
Men’s 100 m (sec) 8.28 9.58 9.63
Men’s 110 m hurdles (sec) 11.76 12.87 12.92
Men’s 400 m (sec) 41.62 43.18 43.94
Men’s marathon (hr:min:sec) 1:36:11 2:03:38 2:08:01
Men’s pole vault (metres) 6.87 6.14 5.97
Women’s 100 m (sec) 9.72 10.49 10.75
Women’s long jump (metres) 8.12 7.52 7.12

At each successive Olympics the advances are expected to be smaller and smaller, meaning breaking records will get less and less likely. Eventually, we’ll reach the limit of not only human ability, but reasonable comparison between events, when you take into account things like weather conditions or even the accuracy of the length of the track.

But at least Radicchi’s predictions suggest we haven’t quite reached those limits. So if you’re an athlete with world record aspirations, don’t give up just yet.

Recently on the radio

We’ve been a bit quiet recently on the Lost in Science blog. But that doesn’t mean the team hasn’t been busy, oh no!

Here are some links to go with our recent radio broadcasts. Or, you can download the podcasts, for our shows from 3 November 2011 (25:54 min / 12 MB) and 10 November 2011 (28:09 min / 26 MB).

  • Analysis of corporate ownership networks shows that out of 43,060 transnational companies, only 147 of them – mostly banks – control 40% of the wealth. Read more in New Scientist, or see the entire paper in the arXiv database.
  • Protesting about this risks exposure to pepper spray, or Oleoresin Capsicum, which uses the chemical capsaicin ((CH3)2CHCH=CH(CH2)4CONHCH2C6H3-4-(OH)-3-(OCH3)), extracted from chilli peppers, to cause eye and skin irritation. Read about its health effects in Investigative Opthalmology and Visual Science and the North Carolina Medical Journal, or see treatment recommendations from Melbourne’s Royal Children’s Hospital.
  • The Berkeley Earth Surface Temperature study, partly composed of and funded by climate change sceptics, has performed a massive re-analysis of global land temperature records and verified that yes, the world really is warming.
  • Aside from being real, climate change seems to have caused Australian seaweed species to move between 50 and 200 km south, risking the habitat of many other species that depend on them. Read more at ABC Science, or see the paper in Current Biology.
  • In more extinction news, Tasmanian devils are currently threatened by a contagious cancer, which seems to spread due to their genetic similarity. Hope is held for a small, genetically different and mostly disease-free population in the northwest of the state, research into which has won a team of scientists the 2011 Eureka Prize for Environmental Research (also see their paper in Conservation Biology). Although the recent discovery of devils with facial tumour disease in even that remote area has increased concern for this unique species.
  • (A good friend of ours, John Cook of Skeptical Science, was also awarded the 2011 Eureka Prize for Advancement of Climate Change Knowledge. Congratulations John!)
  • Speaking of genetic diversity, research on the Sandy Island mouse has shown that polygamous females produce more viable embryos. See the paper in Ecology Letters, or read more at the University of Western Australia.
  • Finally, to space. Three recent discoveries have shed new light on how solar systems like ours form: there’s a planet called LkCa 15b, 473 light years away, which has been discovered in the process of forming; water seen in the planet-forming disk around the young star TW Hydrae (175 light years away) supports the theory that it collects around grains of dust to make comets, which then deposit the water on planets like Earth; and photos of the asteroid Lutetia, taken by the European Space Agency’s Rosetta probe, suggest that, at around 3.6 billion years old it’s a relic of the early Solar System, and have given clues to its formation.

Have you missed any other shows? Catch up on our old episodes!

Vehicle traffic and fluid flow

Traffic can be difficult to model mathematically, comprising as it does thousands of drivers in metal boxes making their own decisions and moving in a coordinated – or uncoordinated – fashion. But at the risk of over-simplifying things, it can be instructive to treat certain road conditions as a fluid.

Consider bottlenecks, where a blockage reduces multiple lanes of traffic down to one.

Diagram comparing free flowing traffic in one direction to a bottleneck in the other, which forces all the cars into one lane
Diagram of a road bottleneck: the rate of cars entering the section is the same in both directions, but the reduced flow around the roadworks forces the traffic into one lane, slowing cars to a crawl (Image by Smurrayinchester, via Wikimedia Commons)
In the diagram above, roadworks have forced all the cars from three lanes into one. The restriction of the single lane determines how many cars can pass through the entire section of road in any period of time.

Since cars can’t magically appear or disappear, they must be going into the blockage at the same rate – if you like, we can call this the conservation of cars. But because the three lanes mean there are three times as many cars going in, to satisfy the law of conservation of cars, they must be moving at one third the speed.

This may seem counter-intuitive, after all we normally assume that wider roads make traffic move faster. But you’ve probably observed the effect yourself when you encounter roadworks. It implies that widening roads won’t help if there’s some form of limiting factor, like an exit to a freeway. Adding more lanes to the diagram above will only cut the speed further.

This same principle applies to fluids, except it’s not conservation of cars, it’s more like conservation of mass. And the overall rule is called the continuity principle.

You can easily see the continuity principle in action in your home: for instance, water coming out of a tap accelerates under gravity, so when it’s moving faster the cross-sectional area has to reduce to keep the rate of flow the same. Which is why the stream narrows further from the tap.

Or for another example, when you put your thumb over the end of a hose you restrict the area it can flow through, and so the water moves faster – and you can squirt the water further.

But as a final interesting twist, there is a maximum speed you can reach by reducing the area like this: thanks to Bernouilli’s principle, increasing the speed of flow also reduces the pressure. And eventually the pressure gets so low that the water can actually boil at room temperature.

This is what sets the maximum speed: it’s known as choked flow. And this room temperature boiling, with bubbles of vapour forming and collapsing, is the cause of the characteristic hissing noise of taps.

And that’s what freeway construction and roadblocks have to do with noisy plumbing!

Shaken by a moment of great magnitude

What’s all this talk about the “magnitude” of an earthquake? Whatever happened to the Richter scale?

I’m glad you asked. The moment magnitude scale has gradually taken the place of the Richter scale since 1979, when it was developed by Hiroo Kanamori and Tom Hanks (no, not that Tom Hanks). It’s designed to be more reliable for larger events, being based on the energy released by the quake rather than how much the ground has moved.

Actually, it’s based on the logarithm of the energy, meaning that the scale goes up by one when the energy goes up by a factor of 10. There are many measurement scales based on logarithms, from pH to decibels, stellar magnitude to information, even the octave scale. In fact there is evidence to suggest that human beings find it more natural to estimate quantities based on logarithmic scales.

In the 19th century, Ernst Heinrich Weber and Gustave Theodor Fechner performed experiments like gradually increasing weights held by blindfolded people. Small increases were barely perceptible, but when the weight was increased by an amount comparable to what it started with, it could easily be detected – regardless of the starting weight.

As is often the case, the maths may sound complicated, but we’re awfully good at doing it subconsciously.

Victims of interpretation

Following the release of the most recent Victoria Police crime statistics, there’s been a lot of attention on doing something about the state’s “culture of violence”. Overall crime has dropped by 6.4% from 2008/09 to 2009/10, but in the same period the number of assaults has risen by 3.8%.

Naturally this has caused a lot of concern, with calls in the media for a reexamination of liquor licensing. And the still-new Baillieu government has been swift to respond with a crackdown on drunken louts:

“We promised to send a strong signal that drunken, loutish and threatening behaviour on our streets would not be tolerated and that people who engaged in such behaviour could expect a punishment that would hurt.”

But is this what the numbers really say? Are beer barns and young hooligans the real problem, or is something else going on?

Graph of percentage of assaults from family incidents in Victoria, rising from 11% in 2000/01 to 25% in 2009/10
Percentage of assaults arising from family incidents, 2000/2001 to 2009/2010 (Victoria Police)

Well, it’s pretty clear from the reports that the big rise in assaults is due to family violence, in particular violence against women.

Continue reading Victims of interpretation