Afghan Bands

afghan bands

Most people know that if you cut a Möbius band in two lengthwise you’ll produce one band rather than two. But splitting the ends and joining them also creates some surprising effects.

Starting with the figure above, join ends A and D directly, then pass B under A and join it to E. Now pass C over B and under A; pass F over D and under E; and join C and F. Extend the slits along the length of the band and you’ll have three linked rings.

Now compare this variant, suggested by Ellis Stanyon in 1930: Starting again from the diagram, give E a half-twist to the right and join it to C; give F a half-twist to the right and join it to B; then pass A under B and join it to D (without turning it over). Cut along the two slits and you’ll produce a small ring linked to a large one. What became of the third ring?

There’s a surprisingly simple way to produce a similar effect: Draw a line along the length of a Möbius band, one-third of the way across the strip. Cutting along this line will produce a large band linked to a small one — and this time the small band is itself a Möbius band, on which you can repeat the feat.

You Answer Quite Slowly

What key is “Lucy in the Sky With Diamonds” written in? It’s not easy to say; the harmony is strangely ambiguous. Musicologist Naphtali Wagner found that the song is notated differently in two reasonably authoritative sources, Wise Publications’ The Beatles Complete (1983) and Hal Leonard’s The Beatles: Complete Scores (1993):

"lucy in the sky with diamonds" key signatures

And he found that scholars disagree as well:

  • Steven Porter believes that it’s in A major, “surrounded by tonicized structural neighbour tones (B♭ major and G major), as a sort of substitute for the absence of a structural dominant.”
  • Walter Everett believes it’s in G major, citing a voice-leading graph that opens with the hypothetical notes G and B in the outer voices.
  • Allan Moore is ambivalent: The scale steps in the upper voice suggest G major, but the bass line contradicts this, and the key signatures suggest D major.

Wagner shows that a case can be made for three rival interpretations: A major, D major, and G major. “Each is consistent with the Beatles’ harmonic style and has precedents in many other songs.” But he adds that one solution might be to abandon the idea of monotonality and see the song as oscillating between two keys: A in the chorus and pre-chorus and G in the chorus. “This version could be defended with the argument that oscillation between tonal centres separated by a major second is found in other Beatles songs, such as ‘Doctor Robert,’ ‘Good Day Sunshine’ and ‘Penny Lane.’”

(Naphtali Wagner, “The Beatles’ Psycheclassical Synthesis: Psychedelic Classicism and Classical Psychedelia in Sgt. Pepper,” in Oliver Julien, ed., Sgt. Pepper and the Beatles, 2008)

Unquote

http://commons.wikimedia.org/wiki/File:Lemon_with_white_background.jpg

“My living in Yorkshire was so far out of the way, that it was actually twelve miles from a lemon.” — Sydney Smith

Stickeen

http://commons.wikimedia.org/wiki/File:Glacier_-_Stickeen_Valley_-_Alaska_Days_with_John_Muir.jpg

One day in 1880 John Muir set out to explore a glacier in southeastern Alaska, accompanied by Stickeen, the dog belonging to his traveling companion. The day went well, but on their way back to camp they found their way blocked by an immense 50-foot crevasse crossed diagonally by a narrow fin of ice. After long deliberation Muir cut his way down to the fin, straddled it and worked his way perilously across, but Stickeen, who had shown dauntless courage throughout the day, could not be convinced to follow. He sought desperately for some other route, gazing fearfully into the gulf and “moaning and wailing as if in the bitterness of death.” Muir called to him, pretended to march off, and finally ordered him sternly to cross the bridge. Miserably the dog inched down to the farther end and, “lifting his feet with the regularity and slowness of the vibrations of a seconds pendulum,” crept across the abyss and scrambled up to Muir’s side.

And now came a scene! ‘Well done, well done, little boy! Brave boy!’ I cried, trying to catch and caress him; but he would not be caught. Never before or since have I seen anything like so passionate a revulsion from the depths of despair to exultant, triumphant, uncontrollable joy. He flashed and darted hither and thither as if fairly demented, screaming and shouting, swirling round and round in giddy loops and circles like a leaf in a whirlwind, lying down, and rolling over and over, sidewise and heels over head, and pouring forth a tumultuous flood of hysterical cries and sobs and gasping mutterings. When I ran up to him to shake him, fearing he might die of joy, he flashed off two or three hundred yards, his feet in a mist of motion; then, turning suddenly, came back in a wild rush and launched himself at my face, almost knocking me down, all the while screeching and screaming and shouting as if saying, ‘Saved! saved! saved!’ Then away again, dropping suddenly at times with his feet in the air, trembling and fairly sobbing. Such passionate emotion was enough to kill him. Moses’ stately song of triumph after escaping the Egyptians and the Red Sea was nothing to it. Who could have guessed the capacity of the dull, enduring little fellow for all that most stirs this mortal frame? Nobody could have helped crying with him!

Thereafter, Muir wrote, “Stickeen was a changed dog. During the rest of the trip, instead of holding aloof, he always lay by my side, tried to keep me constantly in sight, and would hardly accept a morsel of food, however tempting, from any hand but mine. At night, when all was quiet about the camp-fire, he would come to me and rest his head on my knee with a look of devotion as if I were his god. And often as he caught my eye he seemed to be trying to say, ‘Wasn’t that an awful time we had together on the glacier?’”

The Pill Scale (Part 2)

A variation on yesterday’s puzzle:

Suppose there are six bottles of pills, and more than one of them may contain defective pills that weigh 6 grams instead of 5. How can we identify the bad bottles with a single weighing?

Click for Answer

Delayed

http://commons.wikimedia.org/wiki/File:Charles_Clifford_Ogle.jpg

On the evening of Aug. 12, 1964, 41-year-old California businessman Charles Ogle climbed into a four-seat Cessna 210 and took off from Oakland International Airport without filing a flight plan. He was never seen again.

Rescuers searched for Ogle for 60 hours before giving up.

“This has hung over me my whole life,” Ogle’s son William told the San Francisco Chronicle in 2007. “I don’t remember the emotional impact because I was too young, but my teachers would complain to my mother because I would look out the windows all the time looking for his plane. I just thought he didn’t come down yet.”

The Pill Scale (Part 1)

An efficiency-minded pharmacist has just received a shipment of 10 bottles of pills when the manufacturer calls to say that there’s been an error — nine of the bottles contain pills that weigh 5 grams apiece, which is correct, but the pills in the remaining bottle weigh 6 grams apiece. The pharmacist could find the bad batch by simply weighing one pill from each bottle, but he hits on a way to accomplish this with a single weighing. What does he do?

Click for Answer

Dupli-Cat

http://www.flickr.com/photos/cloudsoup/494542407/

Image: Flickr

Two things cannot occupy the same place at the same time. That seems reasonable. But:

A cat called Tibbles loses his tail at time t2. But before t2 somebody had picked out, identified, and distinguished from Tibbles a different and rather peculiar animate entity — namely, Tibbles minus Tibbles’ tail. Let us suppose that he decided to call this entity ‘Tib.’ Suppose Tibbles was on the mat at time t1. Then both Tib and Tibbles were on the mat at t1. … But consider the position from t3 onward when, something the worse for wear, the cat is sitting on the mat without a tail. Is there one cat or are there two cats there? Tib is certainly sitting there. In a way nothing happened to him at all. But so is Tibbles. For Tibbles lost his tail, survived this experience, and then at t3 was sitting on the mat. And we agreed that Tib ≠ Tibbles. We can uphold the transitivity of identity, it seems, only if we stick by that decision at t3 and allow that at t3 there are two cats on the mat in exactly the same place at exactly the same time.

Tibbles and Tib were distinct material objects, but after the amputation they appear to occupy exactly the same space. Were we mistaken?

(From David Wiggins, “On Being in the Same Place at the Same Time,” The Philosophical Review, 77:1 [January 1968], 90-95.)

Further trouble:

No cat has two tails.

Every cat has one tail more than no cat.

Therefore every cat has three tails.

Cardiology

In other words, this is the day on which those charming little missives, ycleped Valentines, cross and intercross each other at every street and turning. The weary and all for-spent twopenny postman sinks beneath a load of delicate embarrassments, not his own. It is scarcely credible to what an extent this ephemeral courtship is carried on in this loving town, to the great enrichment of porters, and detriment of knockers and bell-wires. In these little visual interpretations, no emblem is so common as the heart,–that little three-cornered exponent of all our hopes and fears,–the bestuck and bleeding heart; it is twisted and tortured into more allegories and affectations than an opera-hat. What authority we have in history or mythology for placing the head-quarters and metropolis of god Cupid in this anatomical seat rather than in any other, is not very clear; but we have got it, and it will serve as well as any other thing. Else we might easily imagine, upon some other system which might have prevailed for any thing which our pathology knows to the contrary, a lover addressing his mistress, in perfect simplicity of feeling, ‘Madam, my liver and fortune are entirely at your disposal;’ or putting a delicate question, ‘Amanda, have you a midriff to bestow?’ But custom has settled these things, and awarded the seat of sentiment to the aforesaid triangle, while its less fortunate neighbours wait at animal and anatomical distance.

– Charles Lamb, Essays of Elia, 1823

A mathematical valentine:

http://commons.wikimedia.org/wiki/File:Heart3D.png

Black and White

kuznecov and plaskin chess problem

This week’s puzzle has a twist: Imagine that the board has been rolled into a cylinder so that the a- and h-files are joined and pieces can move across the boundary. How can White mate in two moves?

Click for Answer
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