Dinosaur fossils are often found in this dramatic posture, with the head thrown back, the tail extended, and the mouth wide open. The reason isn’t clear. It could be caused by muscle spams during the animal’s death throes, or the contraction of neck ligaments after death, perhaps facilitated by immersion in water. It’s called the opisthotonic death pose.
In 1970, Penn State meteorologist Hans Neuberger examined 12,000 European paintings to reconstruct historical climate data between 1400 and 1967. He found a marked increase in darkness and cloud cover between 1550 and 1849, corresponding roughly to the Little Ice Age. Painters in the Mediterranean School depicted bluer skies and clearer conditions than their British counterparts, whose paintings showed overcast skies and lower visibility. No English paintings depicted a completely clear sky.
In 2011, researchers Karen Aplin and Paul Williams quantified the frequency of musical allusions to weather in orchestral music over time. Storms, not surprisingly, were the weather type most frequently represented, followed by wind, cloud, rain, sun, and fog. Of the pieces they chose to study, all those depicting frontal storms were in minor keys, and all those depicting fair weather were in major keys. By a large margin, the most common nationality of weather-depicting composers was British. “This appears to support the stereotypical assumption that people from the UK are more enthusiastic about the weather than their colleagues overseas, although this effect could be due to sampling bias, given that the authors of this paper are both from the UK.”
(Hans Neuberger, “Climate in Art,” Weather 25:2 [February 1970], 46-56; Karen L. Aplin and Paul D. Williams, “Meteorological Phenomena in Western Classical Orchestral Music,” Weather 66:11 [November 2011], 300-306. Here’s a related study looking at popular music.)
A visual proof that the sum of all positive odd numbers up to 2n – 1 is a perfect square.
Hansel and Gretl have discovered a gingerbread cottage and are wondering whether to eat some of the tiles on its walls. A witch appears and tells them how they must go about it. “Each of you is to name a whole number between 0 and 100. Hansel’s must be odd and Gretl’s even. No conferring. Whoever chooses the lower number can eat twice that number of gingerbread tiles. Whoever chooses the higher number can eat the lower number.” So, for example, if Hansel chooses 57 and Gretl chooses 30, Hansel will get 30 tiles and Gretl will get 60.
This sounds fine, but the children have just had lessons in game theory and regard this as a non-cooperative game between rational utility maximizers. Gretl knows that Hansel will not choose 99, because 97 would leave him better off if she chose 98 and no worse off if she chose any other number. By the same reasoning, she will avoid 98 and choose 96. In her mind she can follow this train all the way to its end: Rationally, it seems, she must choose 2. Hansel, following it also, finds himself indifferent between 3 and 1. In the end he will receive a paltry two tiles and Gretl either one or four.
Is all of this sound? Gretl says, “There is something radically peculiar about trains of thought which proceed in the subjunctive. You are to work out what you would be rational to do, if I were to choose a number which I shall not choose. I am to do likewise, with each train of thought reproduced inside the other. What happens if either player derails a train by choosing in defiance of it? In that case it becomes radically unclear whether either player still has a rational choice.”
(Martin Hollis, “The Gingerbread Game,” Analysis 54:4 [October 1994], 196-200.)
In his 1908 autobiography, Francis Galton described a “beauty map” he’d compiled of the British Isles:
Whenever I have occasion to classify the persons I meet into three classes, ‘good, medium, and bad,’ I use a needle mounted as a pricker, wherewith to prick holes, unseen, in a piece of paper. … I used this plan for my beauty data, classifying the girls I passed in streets or elsewhere as attractive, indifferent, or repellent. … I found London to rank highest for beauty; Aberdeen lowest.
In 2008, psychologists Viren Swami and Eliana Hernandez set out to compile a beauty map of their own, this time focusing on London. They asked 461 residents to rate the physical attractiveness of men and women in the city’s 33 boroughs. For the record, the City of London, the City of Westminster, and Kensington and Chelsea were rated highest — which correlates with the affluence but not the health (life expectancy) of the residents in those boroughs.
Choose a number on this clock face and, starting from 12, spell out that number’s English name as you advance clockwise around the face, one letter per numeral. For example, if you’ve chosen 3, count out T-H-R-E-E and you’ll land on the numeral 5. Adopt this new position as your next chosen number and proceed as before (in this case, counting F-I-V-E and landing on 9). After three or more moves you’ll reliably land on 1.
This works because of a characteristic of Markov chains first observed by Russian mathematician Evgenii Borisovich Dynkin. Here’s a card trick that exploits the same principle.
This recording presents four pairs of tones, each pair separated by three whole tones, or half an octave. Curiously, some listeners hear the interval as ascending, others as descending. (In fact the tones used are ambiguous as to octave, so there’s no objectively right answer.)
Even more curiously, sometimes a listener’s perception reverses when an interval is transposed, say from C-F# to G#-D, even though nothing else has changed.
A pleasing detail from Built for Speed, University of Idaho zoologist John A. Byers’ 2003 account of a year studying pronghorn antelope in western Montana:
“It can be really bizarre if, when I’ve been alone for several hours, I spot a human standing or walking in the distance. For an instant, my reaction is, ‘What the hell is that?’ A large mammal that walks on its hind legs seems very strange.”
This street scene, taken in Cork in 1927, seems to give a 3-D effect, but it’s only alternating between two photos taken from slightly different perspectives. The sense of depth comes from parallax and the occlusion of distant objects.
Small animals bob their heads to produce this effect as they plan a jump; it helps them to estimate distance.
Albert Einstein and Kurt Gödel used to walk home together from the Institute for Advanced Study in Princeton. In Incompleteness (2005), Rebecca Goldstein gives a sample of their conversation, broached by Gödel:
All of his thinking is governed by an ‘interesting axiom,’ as Ernst Gabor Straus, Einstein’s assistant from 1944 to 1947, once characterized it. For every fact, there exists an explanation as to why that fact is a fact; why it has to be a fact. This conviction amounts to the assertion that there is no brute contingency in the world, no givens that need not have been given. In other words, the world will never, not even once, speak to us in the way that an exasperated parent will speak to her fractious adolescent: ‘Why? I’ll tell you why. Because I said so!’ The world always has an explanation for itself, or as Einstein’s walking partner puts it, Die Welt is vernunftig, the world is intelligible. The conclusions that emanate from the rigorously consistent application of this ‘interesting axiom’ to every subject that crosses the logician’s mind — from the relationship between the body and the soul to global politics to the very local politics of the Institute for Avanced Study itself — often and radically diverge from the opinions of common sense. Such divergence, however, counts as nothing for him. It is as if one of the unwritten laws of his thought processes is: If reasoning and common sense should diverge, then… so much the worse for common sense! What, in the long run, is common sense, other than common?
Somewhat related: Richard Feynman’s sense of “social irresponsibility.”
