Illumination

Starting in the 1970s, neurobiologist Otto-Joachim Grüsser spent 10 years collating the light sources in 2,124 paintings selected at random from Western art originating between the 14th and 20th centuries. He found that in most paintings considered Western works of art, especially those painted around the time of the Scientific Revolution, the light falls from the left.

“At the beginning of modern Western art during the early Gothic period, a preference for diffuse illumination or light sources distributed around the painted scene was found,” Grüsser noted. “In a minority of paintings from the fourteenth century that show a clear light direction, a bias to the left side is present. This left-sided preference increased at the expense of diffuse or middle light sources up to the sixteenth and seventeenth centuries and declined thereafter. In the twentieth century, the diffuse or middle type of light distribution again became dominant.”

It’s not clear what to make of this. It seems reasonable that a right-handed artist might favor light falling from the left, but why should this vary with time? Grüsser found that the left-handed Leonardo da Vinci applied light sources from varying angles, and Hans Holbein the Younger, also a dominant left-hander, favored light falling from the right.

“From such observations in the works of these two left-handed painters who painted, drew, and wrote with the left hand, one gains the impression that the distribution of left, middle, and right light direction in left-handed painters deviates significantly from the average distribution of light found in the paintings of other contemporary painters. It would be interesting to study the drawings and paintings of other confirmed left-handed artists, who worked exclusively with the left hand.”

(Otto-Joachim Grüsser, Thomas Selke, and Barbara Zynda, “Cerebral Lateralization and Some Implications for Art, Aesthetic Perception, and Artistic Creativity,” in Ingo Rentschler, Barbara Herzberger, and David Epstein, Beauty and the Brain, 1988.)

October 25, 2014October 24, 2014 | Art · Science & Math

Hot and Cold

Suppose you have three identical Dewar flasks labeled A, B, and C. (A Thermos is a Dewar flask.) You also have an empty container labeled D, which has thermally perfect conducting walls and which fits inside a Dewar flask.

Pour 1 liter of 80°C water into flask A and 1 liter of 20°C water into flask B. Now, using all four containers, is it possible to use the hot water to heat the cold water so that the final temperature of the cold water is higher than the final temperature of the hot water? How? (You can’t actually mix the hot water with the cold.)

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Pour half the cold water into D and put D into the hot flask (A). The final temperature in both A and D will be about 60°C. Empty D into C and the repeat the procedure, pouring the remaining half of the cold water in B into D and immersing it in the hot water. Now the final temperature of the water in A and D will be about 47°C. Empty D again into C; the final temperature of the mixture there will be about 53°C. So in two operations we’ve warmed the cold water to 53°C and cooled the hot water to 47°C. From Christopher Jargodzki’s books Science Brain-Twisters (1976) and Mad About Physics (2002).

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October 24, 2014November 2, 2014 | Puzzles · Science & Math

A Dice Puzzle

Timothy and Urban are playing a game with two six-sided dice. The dice are unusual: Rather than bearing a number, each face is painted either red or blue.

The two take turns throwing the dice. Timothy wins if the two top faces are the same color, and Urban wins if they’re different. Their chances of winning are equal.

The first die has 5 red faces and 1 blue face. What are the colors on the second die?

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Throwing two six-sided dice produces 36 possible outcomes. If Timothy and Urban have equal chances of winning, then there must be 18 outcomes in which both dice display the same color. If x is the number of red faces on the second die, then: 18 = 5x + 1(6 – x) x = 3 The second die must have 3 red faces and 3 blue faces. (From Pierre Berloquin, 100 Numerical Games, 1994.)

October 23, 2014November 3, 2014 | Puzzles · Science & Math

Presto

Discovered by R.V. Heath in 1950:

Think of two positive integers.

Add them to get a third number.

Add the second number and the third number to get a fourth number.

Continue in this way until you have 10 numbers.

The sum of the 10 numbers is 11 times the seventh number.

October 22, 2014October 21, 2014 | Science & Math

Return to Sender

Mathematician Yutaka Nishiyama of the Osaka University of Economics has designed a nifty paper boomerang that you can use indoors. A free PDF template (with instructions in 70 languages!) is here.

Hold it vertically, like a paper airplane, and throw it straight ahead at eye level, snapping your wrist as you release it. The greater the spin, the better the performance. It should travel 3-4 meters in a circle and return in 1-2 seconds. Catch it between your palms.

October 21, 2014January 20, 2018 | Entertainment · Science & Math · Technology

The Modern Prometheus

jacobson railroad

By 1958 many of the attributes of living things could be found in our technology: locomotion (cars), metabolism (steam engines), energy storage (batteries), perception of stimuli (iconoscopes), and nervous or cerebral activity (computers). The missing element was reproduction: We hadn’t yet created a nonliving artifact that could make copies of itself.

So Brooklyn College chemistry professor Homer Jacobson built one. Using an HO gauge model railroad, he designed an “organism” made of boxcars that could use sensors to select other cars on the track and assemble them on a siding into models of itself. “Head” cars have “brains,” and “tail” cars have “muscles” and “eyes”; together, a head and a tail make an organism in which the head directs the tail to watch for available cars elsewhere on the track and shunt them appropriately onto a siding to create a new organism.

“Any new ‘organisms’ formed continue the propagation in a linear fashion,” Jacobson wrote, “until the environment runs out of parts, or there are no more sidings available, or a mistake is made somewhere in the operation of a cycle, i.e., a ‘mutation.’ Such an effect, like that with living beings, is usually fatal.”

(Homer Jacobson, “On Models of Reproduction,” American Scientist, September 1958.)

October 19, 2014October 18, 2014 | Science & Math · Technology

The Vacuum Airship

A conventional balloon rises because its airbag displaces a large volume of air. But the gas that fills the bag has some weight; it, along with the weight of the gondola, reduces the balloon’s total lift.

Realizing this, Italian monk Francesco Lana de Terzi in 1670 proposed a “vacuum airship,” a balloon whose airbag was filled with nothing at all. Since a vacuum weighs nothing, this should maximize the vehicle’s lift — the vacuum could displace a large volume of air without itself adding any weight.

In principle this might work; the problem is that the vacuum would tend to collapse its container, and building a shell sturdy enough to withstand it would leave us with a ship too heavy to lift. It’s not clear whether any material or structure could overcome this problem.

October 17, 2014October 16, 2014 | Science & Math · Technology

Stamps and Math

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Lee Sallows tells me that the postal system of Macau is releasing a new series of stamps based on magic squares. The full set will touch on everything from the Roman SATOR square to Dürer’s Melencolia. Details are here.

Charmingly, the values of the stamps will be 1, 2, …, 9 Macau patacas, so that the sheet of the nine stamps will itself form a classic Lo Shu magic square. Lee’s contribution, above, is a Nasik 2D geomagic square of order 3 — not only are all the rows and columns magic, but so are all six diagonals, including the four “broken” diagonals.

Somewhat related: In 2000 Finland issued seven stamps in classic tangram shapes, featuring images of science and education. (One of the small triangles, barely visible here, is a Sierpinski gasket.) Only three of the seven shapes are denominated postage, but I should think the temptation is overwhelming to arrange all seven on an envelope in the shape of a little man or a fish or something. I wonder what the post office makes of that.

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October 16, 2014October 16, 2014 | Science & Math

Stretch Goals

stretch goals puzzle

Two circles intersect. A line AC is drawn through one of the intersection points, B. AC can pivot around point B — what position will maximize its length?

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Call the other intersection point D and draw AD and DC. All angles inscribed in a circle and subtended by the same chord are equal, so angle BAD retains the same measure as A travels around its circle. Similarly, angle BCD remains the same as C travels around its circle. This means that triangle ADC will always have the same shape: As line AC pivots around B, triangle ADC will vary in size but remain self-similar. So which position will maximize its size? AD and DC are chords of their respective circles, and the longest chord is a diameter. So turn the triangle until both of these lines are diameters (this will happen simultaneously); at that point triangle ADC will reach its maximum size and line AC its maximum length. From Mogens Larsen in Richard Guy and Robert Woodrow, eds., The Lighter Side of Mathematics, 1994.

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