From Martin Gardner, via Michael Stueben: Obtain a slab of gold measuring 10″ x 11″ x 1″. Divide it diagonally and then cut a triangular notch in two corners as shown. Remove these notches as profit, and slide the remaining halves together to produce a new 10″ x 11″ x 1″ slab. The process can be repeated to yield any amount of money you like!
Kepler’s second law holds that a line segment connecting an orbiting planet to its sun sweeps out equal areas in equal periods of time: In the diagram above, if the time intervals t are equal, then so are the areas A.
If gravity were turned off, would this still be true?
Pretend that you’ve never seen this before and that it’s an actual living person whose personality you’re trying to read. If you look directly at her face, she seems to hesitate, but if you look near it, say beyond her at the landscape, and try to sense her mood, she smiles at you.
In studying this systematically, Harvard neurobiologist Margaret Livingstone found that “if you look at this painting so that your center of gaze falls on the background or her hands, Mona Lisa’s mouth — which is then seen by your peripheral, low-resolution, vision — appears much more cheerful than when you look directly at it, when it is seen by your fine-detail fovea.
“This explains its elusive quality — you literally can’t catch her smile by looking at it. Every time you look directly at her mouth, her smile disappears because your central vision does not perceive coarse image components very well. People don’t realize this because most of us are not aware of how we move our eyes around or that our peripheral vision is able to see some things better than our central vision. Mona Lisa smiles until you look at her mouth, and then her smile fades, like a dim star that disappears when you look directly at it.”
(From her book Vision and Art: The Biology of Seeing, 2002.)
William D. Harvey offered this in Omni in 1980 — a mnemonic for spelling mnemonics:
Mnemonics neatly eliminate man’s only nemesis: insufficient cerebral storage.
We [Einstein and Ernst Straus] had finished the preparation of a paper and were looking for a paper clip. After opening a lot of drawers we finally found one which turned out to be too badly bent for use. So we were looking for a tool to straighten it. Opening a lot more drawers we came upon a whole box of unused paper clips. Einstein immediately started to shape one of them into a tool to straighten the bent one. When asked what he was doing, he said, ‘Once I am set on a goal, it becomes difficult to deflect me.’
— Ernst Straus, “Memoir,” in A.P. French, ed., Einstein: A Centenary Volume, 1979
(Einstein said to an assistant at Princeton that this was the most characteristic anecdote that could be told of him.)
“[John] von Neumann gave me an interesting idea: that you don’t have to be responsible for the world that you’re in. So I have developed a very powerful sense of social irresponsibility as a result of von Neumann’s advice. It’s made me a very happy man ever since. But it was von Neumann who put the seed in that grew into my active irresponsibility!” — Richard Feynman
He expands on this in Christopher Sykes’ No Ordinary Genius (1994):
“I got the idea of ‘active irresponsibility’ in Los Alamos. We often went on walks, and one day I was with the great mathematician von Neumann and a few other people. I think Bethe and von Neumann were discussing some social problem that Bethe was very worried about. Von Neumann said, ‘I don’t feel any responsibility for all these social problems. Why should I? I’m born into the world, I didn’t make it.’ Something like that. Well, I’ve read von Neumann’s autobiography and it seems to me that he felt perpetually responsible, but at that moment this was a new idea to me, and I caught onto it. Around you all the time there are people telling you what your responsibilities are, and I thought it was kind of brave to be actively irresponsible. ‘Active’ because, like democracy, it takes eternal vigilance to maintain it — in a university you have to perpetually watch out, and be careful that you don’t do anything to help anybody!”
Hans Bethe:
“Feynman somehow was proud of being irresponsible. He concentrated on his science, and on enjoying life. There are some of us — including myself — who felt after the end of the Second World War that we had a great responsibility to explain atomic weapons, and to try and make the government do sensible things about atomic weapons. … Feynman didn’t want to have anything to do with it, and I think quite rightly. I think it would be quite wrong if all scientists worked on discharging their responsibility. You need some number of them, but it should only be a small fraction of the total number of scientists. Among the leading scientists, there should be some who do not feel responsible, and who only do what science is supposed to accomplish.”
Marvin Minsky:
“I must say I have a little of this sense of social irresponsibility, and Feynman was a great inspiration to me — I have done a good deal of it since. There are several reasons for a scientist to be irresponsible, and one of them I take very seriously: people say, ‘Are you sure you should be working on this? Can’t it be used for bad?’ Well, I have a strong feeling that good and bad are things to be thought about by people who understand better than I do the interactions among people, and the causes of suffering. The worst thing I can imagine is for somebody to ask me to decide whether a certain innovation is good or bad.”
Set an ant down on a grid of squares and ask it to follow two rules:
That’s it. At first the ant will seem to mill around uncertainly, as above, producing an irregular jumble of black and white squares. But after about 10,000 steps it will start to build a “highway,” following a repeating loop of 104 steps that unfolds forever (below). Computer scientist Chris Langton discovered the phenomenon in 1986.
Will this happen even if some of the starting squares are black? So far the answer appears to be yes — in every initial configuration that’s been tested, the ant eventually produces a highway. If there’s an exception, no one has found it yet.
Points
Have no parts or joints.
How then can they combine
To form a line?
— J.A. Lindon
In 1960 Jane Goodall watched a chimpanzee repeatedly poking pieces of grass into a termite mound in order to “fish” for insects, the first observation of tool use among animals. When she notified anthropologist Louis Leakey of her discovery, he responded with a telegram:
NOW WE MUST REDEFINE TOOL, REDEFINE MAN, OR ACCEPT CHIMPANZEES AS HUMAN.
In late March 1938, Antonio Carrelli received a letter and a telegram in short succession. Both were from Ettore Majorana, the brilliant Italian physicist who had recently joined the faculty of the Naples Physics Institute, where Carrelli was director.
The letter read, “Dear Carrelli, I made a decision that has become unavoidable. There isn’t a bit of selfishness in it, but I realize what trouble my sudden disappearance will cause you and the students. For this as well, I beg your forgiveness, but especially for betraying the trust, the sincere friendship and the sympathy you gave me over the past months. I ask you to remind me to all those I learned to know and appreciate in your Institute, especially Sciuti: I will keep a fond memory of them all at least until 11 pm tonight, possibly later too. E. Majorana.”
The telegram had been sent immediately afterward: “Dear Carrelli, I hope you got my telegram and my letter at the same time. The sea rejected me and I’ll be back tomorrow at the Hotel Bologna traveling perhaps with this letter. However, I have the intention of giving up teaching. Don’t think I’m like an Ibsen heroine, because the case is different. I’m at your disposal for further details. E. Majorana.”
On investigation it was found that Majorana had withdrawn all the money from his bank account and taken the night boat from Naples to Palermo on March 23. He had sent both messages from Palermo and then boarded the steamer to return to Naples on the night of March 25.
But there the trail ended. On the return journey Majorana had shared a compartment with a local university professor, but beyond this point no trace of him could be found. His family offered a reward of 30,000 lire for his whereabouts, and Enrico Fermi implored Mussolini for aid, citing the “deep brilliance” of Majorana’s physics, which he compared to those of Galileo and Newton. A police search found no body but offered no clues.
What happened to him? Theories abound: The most natural explanation, that he committed suicide, is discounted by both his family and the bishop of Trapani, citing his strong Catholic faith. (Also, it doesn’t explain the withdrawal of the money.) Other theories contend that he was murdered, that he fled physics because he foresaw the advent of nuclear weapons, that he had a spiritual crisis and joined a monastery, that he became a beggar, and that he moved to South America. No one knows.
(Barry R. Holstein, “The Mysterious Disappearance of Ettore Majorana,” from the Carolina International Symposium on Neutrino Physics, 2008.)
