The Shepard tone is an auditory illusion: A succession of overlapping scales are played, each ascending, and each scale fades out as its successor fades in an octave lower. The resulting impression is of a climbing pitch that never “arrives” anywhere, a rising note that never gets higher.

Among many other applications, this sound was used for the Batpod in Christopher Nolan’s films The Dark Knight and The Dark Knight Rises — the vehicle seems constantly to accelerate without ever changing gear. “When played on a keyboard,” wrote sound designer Richard King, “it gives the illusion of greater and greater speed; the pod appears unstoppable.”

(Thanks, Nick.)

01/31/2022 UPDATE: Similarly, the Risset Rhythm seems to speed up:

(Thanks, Chris.)

Here’s something remarkable: This formula, discovered in 1995 by David Bailey, Peter Borwein, and Simon Plouffe of the University of Quebec at Montreal, permits the calculation of isolated digits of π — it’s possible to calculate, say, the trillionth digit of π without working out all the preceding digits.

The catch is that it works only in base 2 (binary) and base 16 (hexadecimal), but not in base 10. So it’s possible to know, say, that the five trillionth binary digit of π is 0, but there’s no way to convert the result into its decimal equivalent without working out all the intervening binary digits.

“The new formula allows the calculation of the nth base 2 or base 16 digit of π in a time that is essentially linear in n, with memory requirements that grow logarithmically (very slowly) in n,” writes David Darling in The Universal Book of Mathematics. “One possible use of the Bailey-Borwein-Plouffe formula is to help shed light on whether the distribution of π’s digits are truly random, as most mathematicians suppose.”

08/14/2022 UPDATE: A new formula permits the extraction of decimal digits. (Thanks, Edward.)

A puzzle from reader Steven Moore:

Find A, B, and C as distinct integers. There is only one solution.

From Lee Sallows:

The traditional magic square is a square array of n×n distinct numbers, their magical property being that the sum of the n numbers occupying each row, column, and diagonal is the same. A variation on this theme that I introduced in 2011 is the geometric magic square in which distinct geometrical figures (usually planar shapes) occupy the cells of the array rather than numbers. The magical property enjoyed by such an array is then that the n shapes making up each row, column, and diagonal can be fitted together as in a jigsaw puzzle so as to yield (i.e. tile) a new compound shape that is the same in each case.

Beyond ‘ordinary’ geometric magic squares, it turns out that the combinative properties of shapes are such as to enable ‘magical’ constructions that are denied to analogous structures using numbers. For example, at left in the figure above is seen a 3×3 square of a kind that cannot be realized using distinct numbers rather than shapes. Note first that the square is to be understood as ‘toroidally-connected’, which is to say, as if inscribed on a torus. Its left-hand edge is then to be interpreted as adjacent to its right-hand edge and its top edge adjacent to its bottom edge. Its magical property is then that the four pieces contained within any 2×2 subsquare can be assembled to produce an identical shape, in this case a rectangle of size 4×5. In all there are nine such subsquares to be found in the square, as seen (again in a square) at right. Note that three of the pieces are disjoint, my attempts to produce a similar solution using nine unbroken pieces having failed. So whereas a 3×3 magic square, numerical or geometric, satisfies at least 8 separate conditions ( 3 rows + 3 columns + 2 diagonals), the square here shown satisfies one more.

(Thanks, Lee!)

Just a little detail that I thought was interesting: Famously the moon appears larger when it’s near the horizon, but you can defeat this illusion by bending over and viewing it between your legs.

Why this works isn’t clear — possibly it’s because the image is inverted on the retina, or it may be an effect of inverting the body’s orientation.

(Stanley Coren, “The Moon Illusion: A Different View Through the Legs,” Perceptual and Motor Skills 75:3 [1992], 827-831; Atsuki Higashiyama and Kohei Adachi, “Perceived Size and Perceived Distance of Targets Viewed From Between the Legs: Evidence for Proprioceptive Theory,” Vision Research 46:23 [2006], 3961-3976.)

In desert conditions, helicopter rotors are sometimes surrounded by sparkling rings. When flying sand strikes the abrasion strips on the leading edges of the blades, clouds of eroded titanium particles ignite as they’re exposed to oxygen.

The effect is most visible at night when the aircraft is near the ground, but it’s been observed as high as 1700 feet. It’s named after Benjamin Kopp and Joseph Etchells, two soldiers killed in combat in Afghanistan in July 2009.

Psychologist Abraham S. Luchins discovered a discouraging phenomenon in 1942: When people find a problem-solving strategy that works successfully in multiple trials, they’ll tend to adopt the same strategy even in situations when more efficient solutions are available.

Suppose you’re given water jugs with capacities of 21 units (A), 127 units (B), and 3 units (C) of water. Then you’re asked to use these to measure out 100 units of water. You find that you can do this by filling jug B and using that to fill jug A once and jug C twice, leaving 100 units in jug B. In several similar problems you find that this strategy, B – A – 2C, works.

But then you’re given an “extinction problem” in which this strategy doesn’t work. And in the next problem you’re given jugs of capacity 15, 39, and 3 and asked to measure 18 units of water. The old strategy, B – A – 2C, works here, but there’s a simpler solution: A + C.

In Luchins’ experiments, a control group that had not been primed with the early B – A – 2C successes hit immediately on the A + C solution in the last test. But subjects who did have those early successes tended to revert to the original strategy, reflecting a “mechanized” state of mind that prevented them from seeking more efficient solutions. When Luchins told them “Don’t be blind,” more than half of them found the A + C solution.

The reasons for this are still being studied in different populations and with different types of tasks. Luchins acknowledges that habit can be useful, but notes that when, “instead of the individual mastering the habit, the habit masters the individual — then mechanization is indeed a dangerous thing.”

(Abraham S. Luchins, “Mechanization in Problem Solving: The Effect of Einstellung,” Psychological Monographs 54:6 [1942]: i.)

Lime juice is phototoxic — on contact it sensitizes the skin to ultraviolet light, so that exposure to sunlight can then produce redness, itching, burning, and even blisters.

The reaction is called phytophotodermatitis, or margarita photodermatitis. It can also be caused by celery, parsnips, parsley, and figs.