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.”
A set of 4 elements can be partitioned in 15 ways.
Pleasingly, this is also the number of rhyme schemes that a 4-line poem can take: AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, ABCD.
A poem with 5 lines has 52 possible schemes, corresponding to the partitions of a 5-element set, and so on. These are called Bell numbers.
A chestnut from physics:
Two cannons are aimed directly at one another. One is on the floor of a valley, and the other is on a promontory. Neglecting air resistance, if the two fire simultaneously, what will happen?
A puzzle from the site Riddle of the Day:
A warden offers a challenge to two prisoners. The first prisoner will enter a room that contains a chessboard. On each of the board’s 64 squares is a coin that’s either heads up or tails up. The guard will identify one square as the “target.”
The first prisoner must turn over exactly one coin and then leave the room. The second prisoner must then enter and, solely by viewing the board, determine which square is the target.
If they succeed, both prisoners will go free. They can confer beforehand on a strategy, but they may not communicate after that. Can they establish a plan that will always work?
In Pascal’s triangle, each number is the sum of the two above it. Obviously, the infinite pyramid contains an infinite number of 1s, but most numbers appear surprisingly seldom:
In 1971, Berkeley mathematician David Singmaster suggested that there may be a finite upper bound on the number of times that any number can appear (apart from 1). But that remains an unsolved problem.
I’d missed this: In 2006 a geneticist, a philosopher, and a chicken farmer all agreed that the egg came before the chicken.
Nottingham University geneticist John Brookfield pointed out that the first chicken (the first creature bearing chicken DNA) must have begun as an embryo in an egg. “The first living thing which we could say unequivocally was a member of the species would be this first egg, so I would conclude that the egg came first.”
David Papineau, philosopher of science at King’s College, London, agreed. “I would argue it is a chicken egg if it has a chicken in it. … If a kangaroo laid an egg from which an ostrich hatched, that would surely be an ostrich egg, not a kangaroo egg.”
And Charles Bournes, chair of trade body Great British Chicken, said, “Eggs were around long before the first chicken arrived. Of course they may not have been chicken eggs as we see them today, but they were eggs.”
According to the BBC, “Professor Brooke added the debate could finally be laid to rest.”
The second star in the Big Dipper’s handle is actually two stars, Mizar and Alcor. Distinguishing the two with the naked eye has been used as a test of vision for hundreds of years. Arabic tradition held that only those with the sharpest eyesight could see Mizar’s companion, and the 13th-century Persian astronomy writer Zakariya al-Qazwini wrote that “people tested their eyesight by this star.” In Japan, it was said that being unable to see Alcor with the naked eye foretold an impending death of old age, and Alexander von Humboldt and François Arago both noted that Alcor can be seen only with difficulty.
In The Herschels and Modern Astronomy (1901), Irish astronomer Agnes Mary Clerke wrote, “The Arabs in the desert regarded it as a test of penetrating vision; … Vidit Alcor, at non lunam plenam (Latin for ‘he saw Alcor, but not the full moon’), came to be a proverbial description of one keenly alive to trifles, but dull of apprehension for broad facts.”
(Thanks, Sharon.)
In 1739, Leonhard Euler devised a “tone net” to represent graphically the traditional harmonic relationships in European classical music. In the version above, the dark blue triangle at the center is an A minor triad; the red triangle to its right is the relative major, C; and the red triangle below it is the parallel major, A. Every such pattern is mirrored in all the other keys. The tones are numbered from 0 (A♭) to 11 (G). Major triads are red, and minor are blue. The thinnest lines denote minor thirds, thicker are major thirds, and the thickest horizontal lines are fifths. See this page for further relationships.
The heart of the figure, shown in somewhat darker colors in the diagram, is a parallelogram composed of 24 triangles. The top edge of this parallelogram can be joined to the bottom, which lists the same notes in their enharmonic equivalents, and if the resulting cylinder is twisted slightly then its ends can be joined in the same way, forming a torus.
Multiply 92 by 8, then multiply that product by 8, and so on. List the products, indenting them successively as shown and continuing indefinitely:
92 736 5888 47104 376832 3014656 24117248 192937984 9999999...
The sum converges to a string of 9s.
From Archimedes’ Lab’s Zoo of Numbers.
