Dutch author Leo Lionni devoted most of his career to children’s books, but in 1977 he undertook a weird experiment. Parallel Botany is a catalog of made-up plants, whose made-up features are described by made-up botanists and illustrated by Lionni’s pencil drawings. Sigurya barbulata, at left, is distinguished by its crowning “cephalocarpus”; a specimen discovered in a Mexican pyramid was found to have been metallized into an organic mace, but how this had come about is the subject of “furious debates.”
“The difficulties of applying traditional methods of research to the study of parallel botany stem chiefly from the matterlessness of the plants,” Lionni wrote. “Deprived as they are of any real organs or tissues, their character would be completely indefinable if it were not for the fact that parallel botany is nonetheless botany, and as such it reflects, even if somewhat distantly, many of the most evident features of normal plants.”
Why do all this? Lionni closes with a quote by the made-up Swedish philosopher Erud Kronengaard: “There are two kinds of men, those who are capable of wonder and those who are not. I hope to God that it is the first who will forge our destiny.”
I’m not sure who came up with this — this simple diagram reflects all possible true trigonometric identities of the form x ÷ y = z or x × y = z, where x, y, and z are the basic trigonometric functions of the same angle t.
For any three neighboring functions on the perimeter of the star, the product of the ends always equals the middle (e.g., tan t × cos t = sin t) and the middle function divided by one of the end functions is equal to the other end function (e.g., sin t ÷ tan t = cos t and sin t ÷ cos t = tan t). If you memorize the diagram you can reel off a list of 18 simple relations.
I found it in Michael Stueben’s Twenty Years Before the Blackboard, 1998.
The index to the fourth edition of George Thomas’ Calculus and Analytic Geometry contains an entry for “Whales” on page 188. That page contains no reference to whales, but it does include the figure above.
German mathematician Erich Bessel-Hagen was often teased for his protruding ears.
In 1923 his colleague Béla Kerékjártó published a book, Vorlesungen Über Topologie, whose index lists a reference to Bessel-Hagen on page 151.
That page makes no mention of Bessel-Hagen, but it does contain this figure:
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?
Yes. The planet would pass into space in a straight line, and each time interval would define a triangle with height h and base t. Since a triangle’s area is half the product of its height and base, these are all equal.
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.)
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:
If you find yourself on a white square, turn 90° right, change the color of the square to black, and move forward one unit.
If you find yourself on a black square, turn 90° left, change the color of the square to white, and move forward one unit.
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.
