One of the most famous mathematical problems is the traveling salesman problem: Given a set of cities, how do you determine the shortest route that visits each of the cities exactly once? Of course, you can simply go through all possible routes, but this approach is feasible only for a very limited number of cities. There are more clever ways to solve the problem, but the time needed to find a solution always increases exponentially with the number of cities. In other words, there is no known efficient way to solve the traveling salesman problem.

Photo from Wikimedia Commons

It is quite cool, therefore, that bees appear to be able to solve the traveling salesman problem, as reported by Lihoreau, Chittka and Raine in a forthcoming paper in The American Naturalist. The authors made an artificial flower-arrangement. Recordings of the flight paths showed that the bees chose the most efficient route to visit all flowers. Quite a feat for such a tiny-brained animal!

I haven't read the entire paper (which is due in December, for now there is only a summary on the website of The American Naturalist), but I do wonder about a few things. One of the characteristics of the traveling salesman problem is that there are a finite number of connections and you're allowed to visit each city only once. In contrast, the bees can fly however they like and I would assume that they don't care if they have to fly over a flower that they …

In the image below (which I have adapted from an illusion by Kitoaka) there are three greenish “spokes” rotating counterclockwise. (You can download a script to generate this type of images here.)

Or are there? Actually, the greenish and blueish areas are the same “aqua” colour. The only difference is that in the greenish areas, aqua is interspersed with orange, whereas in the blueish areas, aqua is interspersed with pink. This becomes clear if you look at the image below, in which all pink has been replaced with orange. By the way, the illusion works equally well in a static image. I just added the rotation to make it look extra cool.

Initially I thought the illusion had to do with colour opponency. There are visual neurons which respond to yellow and are inhibited by blue, and vice versa. Because pink is essentially orange with some added blue, you might think that if aqua is interspersed with pink, it becomes more “yellowish” and, conversely, if aqua is interspersed with orange it becomes more “blueish.” However, on closer inspection the illusion works the other way around: Aqua actually looks more blueish next to pink, than next to orange!

Monnier and Shevell describe an illusion which is similar to the one presented here and offer a tentative explanation. Essentially they say that the colours appear to “spill over.” That is, if aqua is next to pink it simply “borrows” some of the blue that is present in the pink. However, this …

Random House, 2^nd edition, 2010

Conclusion Taleb argues that we grossly underestimate the unpredictability of the world around us. Unfortunately, his potentially interesting message suffers from a chaotic presentation and a pretentious style.

One of Holland's silly traditions is that around election time an agency called the CPB (het centraal planbureau) takes the programs of the larger political parties and outputs a bunch of numbers: How many jobs will be gained/ lost over the next 40 (!) years if we follow a particular program? What will happen to the energy consumption? And what will be the effect on the level of education? These numbers are taken quite seriously, even though no attempt has ever been made to validate the predictions of the CPB. And I dare say that if any such attempt were ever made the conclusion would be that the CPB can hardly predict what will happen tomorrow, let alone in 40 years from now.

This summer (it was election time) I walked into the university bookstore and saw “The Black Swan”. According to the cover, the book was about our belief that we can predict random events, when, in fact, we can't. At all. How delicious! Needless to say I left the bookstore with “The Black Swan”, by Nassim Nicholas Taleb, under my arm.

In a nutshell, Taleb argues that we can only say something about a probability if we have some knowledge about the underlying distribution. If we roll a die we know that the probability of …

One of the great creators of optical illusions is Akiyoshi Kitaoka and the following cool illusion is by his hand (he is also the creator of the awesome “rotating snakes” illusion). As you can see in the image below, the checkerboard gives a strong sensation of “bulging out the screen”. Of course, it's just a regular checkerboard. I made the picture fuzzy, so that (to me) it appears like a mysterious sphere hovering in a dark space, but that's really just being fancy. All that is needed for the illusion to work are the little dots (a script to generate this type of images can be found here).

How does this illusion work? According Michael Bach (who's website, incidentally, contains an impressive collection of optical illusions) this is unclear. Kitoaka has a book on optical illusions, which might contain the secret, but I don't have this book in my possession and, at any rate, it's Japanese. So I will venture a guess. I think it has to do with the fact that we tend to see lines in everything. If we consider, for example, the squares in the top-right part of the checkerboard, the pattern looks like (a) in the figure below. Because we tend to interpret the margin around the small white spots as lines (b) and because these perceived lines are not perfectly aligned, we perceive a slight tilt (c). By cleverly positioning the small dots differently in different quadrants of the checkerboard, the whole checkerboard appears to …

Uitgeverij Bert Bakker/ Prometheus, 2010

Conclusion An accessible and entertaining book, in which Victor Lamme argues that we have far less insight into our own motivations than we like to believe.

Have you ever wondered who's really in charge in your brain? Victor Lamme, professor of cognitive neuroscience at the University of Amsterdam, certainly has. In fact, he has devoted an entire book to it. “De vrije wil bestaat niet” (“Free will does not exist”) is a popular science book, which provides an accessible and amusing account of a large number of psychological studies. In the course of describing these studies, Lamme converges on the conclusion that our behaviour is largely determined by factors into which we have little insight: e.g., our tendency not to disagree with group consensus, our fear of other ethnicities, our tendency to select right-most socks (!), etc. Consciousness is simply a spectator which, after th4 Star Rating; Recommendede fact, interprets our behaviour in the best way it can. In this sense free will does not exist, according to Lamme.

Before I bought the book, I had read some mixed reviews. Some people feel that Lamme's claims are overstated, especially when he uses “case studies” like Winston Churchill's military tactics during WWII to drive his point home. However, these case studies (a murder case in which the assailant was aquitted after pleading “sleepwalking” is another one) are backed up by solid research. Lamme makes a compelling argument: We have very little insight into our motivations and …