The Vista Paradox

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Image: Wikimedia Commons

In Bologna, the former convent of San Michele in Bosco contains a 162-meter hallway that’s “aimed” at the Asinelli tower 1,407 meters from the window. This produces an odd effect: As you move north along the hallway toward the window, you’re approaching the tower, yet it seems to shrink. This is because the retinal size of the window’s aperture increases enormously as you approach it, while the retinal size of the distant tower remains relatively unchanged.

Similarly, as you back away from the window the tower seems to grow and draw closer, because the “shrinking” window shuts out the panorama, leaving only the tower in view. The illusion was first reported on a 1714 map by Paolo Battista Baldi of the University of Bologna.

The city contains a second “vista paradox” in the hermitage of Ronzano, where another long hallway is oriented toward the Sanctuary of San Luca 1,970 meters away. The Sanctuary seems to shrink as one approaches the frame and to grow as one retreats.

| Oddities

Memoranda

Excerpts from the literary notebooks of Thomas Hardy:

  • “Loughborough used to say, ‘Do what you think right, & never think of what you are going to say to excuse it beforehand.’ — a good maxim.”
  • “Bonaparte had not the patience requisite for defensive operations, said Wellington.”
  • “Miracles, scriptural & ecclesiastical — how make a difference?”
  • “Brahms – The individual character of his ideas. … With him beauty seems to hold a place subordinate to expression.” [Grove Dictionary of Music]
  • “‘Be it so; then minimize pain.’ Words of Jeremy Bentham when his physician told him he was about to die.” [F.R.E. Dowden]
  • “Epicurean philosophy – always in vogue in declining & sickly states.” [Life of Virgil]
  • “Indirect road to honour. Virgil’s introduction to Octavius was because of his reputation as a horse doctor.”
  • “Caesar, & Brutus, tampered with the muses. Poems curiously bound, & lodged in the most famous libraries; but neither the sacredness of those places, nor the greatness of their names, cd. preserve ill poetry.”
  • “You will find, perhaps to your surprise, that nine-tenths of all human suffering endured by men is useless.” [Tolstoy]
  • “Swift says some men know books as others do lords: learn their titles & then boast of their acquaintance with them.”
  • “Like all persons who have looked a great deal at human life, Balzac had been greatly struck by most people’s selfishness.” [Henry James]
  • “A fair test of the value of an institution is this — Supposing it did not exist, should we set about to establish it?” [Montague Cookson]
  • “It is the customary fate of new truths to begin as heresies & to end as superstitions.” [Thomas Huxley]
  • “The scientific spirit is of more value than its products; & irrationally held truths may be more harmful than reasoned errors” [Huxley]
  • “The artist may be known rather by what he omits.” [Schiller]
  • “Nothing is so great as it seems beforehand.” [George Eliot]

“‘He who has to act on his own responsibility is a slave if he does not act on his own judgment.’ Saying of Sir H. Edwardes — highly valued by Livingstone.”

| Literature

Alternating Tread Stairs

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Image: Wikimedia Commons

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.)

https://commons.wikimedia.org/wiki/File:Escalier.a.45.degres.png

| Technology

Secondhand

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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

| Science & Math

A Hazy Mate

smullyan undetermined mate

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.)

kruglov improvement

| Oddities