Pat Ashforth and Steve Plummer make knitted illusions. When it’s viewed from the front, each piece presents only a series of unremarkable stripes, but when it’s viewed obliquely, an image emerges.
Here’s an introduction to the technique, and here are some patterns.
Conventional stairs are somewhat extravagant: Because users alternate their steps (1), half of each tread goes unused. In close quarters, floor space can be conserved by omitting these unused portions (3), permitting a slope as high as 65 degrees without sacrificing the depth of the treads (2).
Because each tread “overlaps” those that precede and follow it, an alternating staircase might require only half the horizontal space of conventional stairs, and users can face forward when descending, where a ladder would require them to turn. The disadvantage is that they’re steep, and users must take care to begin each traverse with the correct foot. For that reason these stairs may not be safe for children or the elderly.
Below: In the Orange Tower, built in Carpentras at the start of the 14th century, builders set alternate risers at a diagonal to achieve an ascending slope of 45 degrees. “We will recognize that it is never subtlety that our medieval architects lack. But these latter examples only provide service stairs.”
(Eugène Viollet-le-Duc, Dictionary of French Architecture from 11th to 16th Century, 1856.)
If nature be regarded as the teacher and we poor human beings as her pupils, the human race presents a very curious picture. We all sit together at a lecture and possess the necessary principles for understanding it, yet we always pay more attention to the chatter of our fellow students than to the lecturer’s discourse. Or, if our neighbor copies something down, we sneak it from him, stealing what he himself may have heard imperfectly, and add to it our own errors of spelling and opinion.
— G.C. Lichtenberg, quoted in W.H. Auden’s A Certain World, 1970
Raymond Smullyan presented this oddity in his Chess Mysteries of Sherlock Holmes in 1980. Suppose we come upon this abandoned chess game. Can White mate in two moves? The answer seems to be yes. If Black can’t castle, then White can play 1. Ke6 and then promote his g-pawn, giving mate. If Black can castle, that means that neither his king nor his rook has yet moved, and hence he must just have moved his pawn to e5. That permits White to capture the pawn en passant. Now if Black castles then 2. b7 is mate, and if he plays any other move then the g-pawn promotes. Either way, White mates Black on his second move.
But that’s odd. We’ve decided that a mate in two exists, but we can’t show it — and we don’t even know how White commences!
(F. Alexander Norman, “Classicists and Constructivists: A Dilemma,” Mathematics Magazine 62:5 [December 1989], 340-342. See Donkey Sentences.)
02/17/2025 UPDATE: Reader Eugene Kruglov points out that 1. Ke6 works in Smullyan’s position whether or not Black castles — if he does, then 2. a8=Q is mate. The position below works as Smullyan intended — when Black castles, only the en passant capture leads to mate in two. (Thanks, Eugene.)
Hungarian mathematician Paul Erdős’ epitaph reads “Végre nem butulok tovább” — “I’ve finally stopped getting dumber.”
This is beautifully well done — a humiliating list of all the ways your brain can deceive you (click to enlarge).
Designed by John Manoogian III.
Another geometric magic square from Lee Sallows:
(Thanks, Lee!)
This is a picture of a cow. If you can’t see it (I couldn’t), there’s an explanatory image on page 66 of John McCrone’s 1991 book The Ape That Spoke.
The experience illustrates how the brain is able to construct a picture from limited data by relying on patterns and experience.
Thomas Jefferson to the Rev. Isaac Story, Dec. 5, 1801, on the afterlife:
When I was young I was fond of the speculations which seemed to promise some insight into that hidden country, but observing at length that they left me in the same ignorance in which they had found me, I have for very many years ceased to read or to think concerning them, and have reposed my head on that pillow of ignorance which a benevolent Creator has made so soft for us, knowing how much we should be forced to use it.
“I have thought it better, by nourishing the good passions & controlling the bad, to merit an inheritance in a state of being of which I can know so little, and to trust for the future to him who has been so good for the past.”
