The 20th International Congress on Modelling and Simulation (MODSIM2013), in conjunction with the 22nd National Conference of the Australian Operations Research Society (ASOR2013), begins tomorrow in Adelaide, South Australia.
AntarcticArctic is a blog by a “Waste Management Specialist” at the Amundsen–Scott South Pole Station over the dark Antarctic winter. It has covered several interesting topics, such as the Aurora Australis and Aurora Borealis (see satellite image below).
I’m not sure if the blog will survive its author’s return to warmer climes – but, through both words and photographs, it has succeeded in sharing a perspective on our planet that most of us will never experience.
The vessels of plant xylem consist of tubular cells which undergo programmed cell death, leaving long thin tubes which conduct water to the top of even the tallest tree, through capillary action. As illustrated below, the individual xylem vessel elements include perforation plates at their ends; it is these which can act as water filters.
Lee, Boutilier, Chambers, Venkatesh, and Karnik found that the water filter illustrated above could remove particles larger than 0.1 µm – enough to filter out bacteria, though not viruses. Epoxy glue is needed to seal the filtering wood into tubing, however, and the flow rate is a low 180 ml per hour, even under pressure. Still, this is a very interesting low-cost water-purification technique.
This visualisation (made using R) shows the sizes of major Australian plant families, compared to world totals (based on slightly old data from here).
Australia’s 836 orchid species are only 3% of the world total, for example, but Australia has 89% of the Goodeniaceae. Plants illustrated in the chart include the Golden Wattle and Sturt’s Desert Pea.
I have been experimenting recently with R, my favourite statistical toolkit. The plot above (of the gender balance in Australian states and territories) uses a palette extracted from Vermeer’s famous Girl with a Pearl Earring (below) – a brown-and-blue diverging scale with some pink highlights. Interestingly, this is a palette which still works well under simulated colour-blindness. It looks good too.
Oh, and those numbers on the map? Partly related to an excess of males in the mining industry, and partly to an excess of females among retired people.
One of the more famous unsolved questions of mathematics is the twin prime conjecture. There are many “twin prime” pairs of the form (p, p+2), where both p and p+2 are prime numbers. The pair (3, 5), for example. Or (5,7), (11,13), (17,19), (29,31), (10007,10009), or (10000139,10000141). The question is whether there are infinitely many such pairs.
A recent burst of work has seen progress towards answering this question, and a paper on arXiv.org by James Maynard (Centre de recherches mathématiques, Université de Montréal) reports that there are infinitely many pairs of primes separated by at most 600. See also the story in Wired. The underlying unsolved question might be cracked soon!
Data visualisation is one of my favourite topics. We can examine data visualisation in terms of three “lenses.” The first lens is that of mathematical mapping. Typically we have a visual element that represents one or more real-world variables, and we want a clear mapping from one to the other – that is, a one-to-one mapping without too much distortion.
For example, Fisher’s Iris flower data set gives 4 measurements for each of 150 flowers of three species of Iris. Multidimensional scaling can be used to map distances in the 4-dimensional data to distances in a 2-dimensional image (below). Flowers with similar measurements map to dots close together in the image, and flowers with wildly different measurements map to dots far apart. Some distortion is inevitable here, but the diagram clearly shows that Iris setosa flowers (se) are quite different from Iris virginica (vi) and Iris versicolor (ve), which sometimes have similar measurements:
In many visualisations, the mathematical mapping is defined by a colour scale, as in this map of fine particulate air pollution (credit: Aaron van Donkelaar, Dalhousie University):
The second lens is that of cognitive psychology. We build visualisations for people, not Martians, and people see some things more easily than others. A great deal has been written on this topic. To pick just one example, rainbow scales like the one above have been strongly criticised. They fail when turned into grey-scale, they may over-emphasize the transition at whichever value is coded yellow, and they can also confuse people with colour-blindness. It must be said, however, that the rainbow scale above (an improved version) combines hue and brightness in a way that actually makes good sense to people with most forms of colour-blindness (see the simulation below of what a person with red/green colour-blindness would see). Consequently, the rainbow scale above probably works better for this dataset than many of the alternative colour scales would. On the other hand, the nonlinearity in the colour scale is rather confusing.
The final lens is that of design. Visualisations that are mathematically and cognitively satisfactory may still be boring or ugly. Although the design lens is a little more subjective, there are many examples of good visualisation design out there. The tide prediction infographic by Kelvin Tow below (submitted to the 2013 Information is Beautiful Awards) is just one. The classic books by Edward Tufte also have a lot of good advice in this area.
The Plimpton 322 tablet is a Babylonian clay tablet, written in cuneiform, from around 1,800 BC (now held at Columbia University). The tablet contains four columns of numbers, written in base 60 (a system that survives in our hours, minutes, and seconds):
|A||B (“width”)||C (“diagonal”)||D|
|1.59:00:15 = 1 + 59/60 + 15/216000 = 1.983402777777778||1:59 = 60 + 59 = 119||2:49 = 2×60 + 49 = 169||#1|
|1.56:56:58:14:50:06:15 = 1.949158552088692||56:07 = 3367||1:20:25 = 4825||#2|
|1.55:07:41:15:33:45 = 1.918802126736111||1:16:41 = 4601||1:50:49 = 6649||#3|
|1.53:10:29:32:52:16 = 1.886247906721536||3:31:49 = 12709||5:09:01 = 18541||#4|
|1.48:54:01:40 = 1.815007716049383||1:05 = 65||1:37 = 97||#5|
|1.47:06:41:40 = 1.785192901234568||5:19 = 319||8:01 = 481||#6|
|1.43:11:56:28:26:40 = 1.719983676268861||38:11 = 2291||59:01 = 3541||#7|
|1.41:33:45:14:03:45 = 1.692709418402778||13:19 = 799||20:49 = 1249||#8|
|1.38:33:36:36 = 1.642669444444444||8:01 = 481||12:49 = 769||#9|
|1.35:10:02:28:27:24:26:40 = 1.586122566110349||1:22:41 = 4961||2:16:01 = 8161||#10|
|1.33:45 = 1.5625||45||1:15 = 75||#11|
|1.29:21:54:02:15 = 1.489416840277778||27:59 = 1679||48:49 = 2929||#12|
|1.27:00:03:45 = 1.450017361111111||2:41 = 161||4:49 = 289||#13|
|1.25:48:51:35:06:40 = 1.430238820301783||29:31 = 1771||53:49 = 3229||#14|
|1.23:13:46:40 = 1.38716049382716||28||53||#15|
Column B (with a label on the tablet containing the word “width”) is one of the sides of a Pythagorean triangle, and C (with a label on the tablet containing the word “diagonal”) is the hypotenuse. Column A is the ratio of the square on the hypotenuse to the square on the third side. The tablet does contain a small number of errors, as well as some numbers missing through damage, but correcting them gives the very accurate results below: C2 − B2 is always a perfect square (yellow in the diagram), and C2 / (C2 − B2), the ratio of blue to yellow, is always equal to A.
|C2 − B2 is a perfect square||C2 / (C2 − B2) = A||D|
|1692 − 1192 = 14400 = 1202||1692 / 14400 = 1.983402777777778||#1|
|48252 − 33672 = 11943936 = 34562||48252 / 11943936 = 1.949158552088692||#2|
|66492 − 46012 = 23040000 = 48002||66492 / 23040000 = 1.918802126736111||#3|
|185412 − 127092 = 182250000 = 135002||185412 / 182250000 = 1.886247906721536||#4|
|972 − 652 = 5184 = 722||972 / 5184 = 1.815007716049383||#5|
|4812 − 3192 = 129600 = 3602||4812 / 129600 = 1.785192901234568||#6|
|35412 − 22912 = 7290000 = 27002||35412 / 7290000 = 1.719983676268861||#7|
|12492 − 7992 = 921600 = 9602||12492 / 921600 = 1.692709418402778||#8|
|7692 − 4812 = 360000 = 6002||7692 / 360000 = 1.642669444444444||#9|
|81612 − 49612 = 41990400 = 64802||81612 / 41990400 = 1.586122566110349||#10|
|752 − 452 = 3600 = 602||752 / 3600 = 1.5625||#11|
|29292 − 16792 = 5760000 = 24002||29292 / 5760000 = 1.489416840277778||#12|
|2892 − 1612 = 57600 = 2402||2892 / 57600 = 1.450017361111111||#13|
|32292 − 17712 = 7290000 = 27002||32292 / 7290000 = 1.430238820301783||#14|
|532 − 282 = 2025 = 452||532 / 2025 = 1.38716049382716||#15|
The obvious question here is: why on earth were they doing this? There have been quite a few theories. A good recent discussion is by Eleanor Robson [“Words and pictures: new light on Plimpton 322,” American Mathematical Monthly 109 (2): 105–120]. We don’t really know enough about how Babylonian mathematicians thought to ever be totally certain why they were doing this, but they were clearly very good at this kind of geometric calculation. Which makes this 3,800-year-old clay tablet really cool.
Florence Nightingale was passionate about collecting and visualising health statistics – not just her famous wartime visualisation (below), but also national health statistics. The Institute for Health Metrics and Evaluation at the University of Washington has realised this dream with a visualisation tool showing the impact of injury and disease globally (see also the report in Wired).
The image above shows Years of Life Lost (a statistic giving greater weight to deaths among the young) for women aged 15–49 in the USA. Colours mark injuries (greens), infectious diseases (reds), and other diseases, such as cancers (blues). Darker colours highlight growing problems. On the right, we can see the large (but shrinking) problem of death by 4-wheel vehicle accident, and the growing problem of death by poisoning. In the USA, 42,917 people died from poisoning in 2010 (almost double the figure of 22,242 for 2001). Most of these (77.0%) were unintentional, the result of ingesting drugs (pain relievers, sedatives, heart medication, etc.) or cleaning substances. But, as Florence Nightingale understood, if you can see the problem, you can deal with the problem.