University of California psychologist Diana Deutsch discovered this phenomenon in 1972. When the tones of a familiar melody are distributed among three different octaves, people find it difficult to identify. But once the underlying melody has been revealed to them, they can hear it more readily in the distributed version. Knowing what to listen for makes the tune easier to follow.
The Final Frontier
The original Star Trek was presented in the rather boxy aspect ratio of 1960s television. Now San Francisco illustrator Nick Acosta has stitched together screenshots to see how it would have appeared in a widescreen format:
I created this project of what the show would have looked like in Cinerama widescreen. As a kid the show always felt bigger and more epic than it appears to me as an adult. I was able to create these shots by waiting for the camera to pan and then I stitched the separate shots together. The result is pretty epic. It reminds me of the classic science fiction movies of the 50’s and 60’s. Suddenly the show has a ‘Forbidden Planet’ vibe. Other shots remind me of how director Robert Wise would use a camera technique to keep the foreground and background elements in focus.
Arithmetic Billiards
To find the least common multiple and the greatest common divisor of two natural numbers, construct a billiard table whose side lengths correspond to the two numbers (here, 15 and 40). Set a ball in one corner, fire it out at a 45-degree angle, and let it bounce around the table until it stops in a corner.
Now the least common multiple is the total number of unit squares traversed by the ball (here, 120).
And the greatest common divisor is the number of unit squares traversed by the ball before it reaches the first intersection (here, 5).
Remembrance
The Shrine of Remembrance in Melbourne, built to honor the men and women of Victoria who served in World War I, contains a marble stone engraved with the words Greater love hath no man (from John 15:13, “Greater love hath no man than this, that a man lay down his life for his friends”).
The shrine is constructed so that once a year, at 11 a.m. on Nov. 11, a ray of sunlight will shine through an aperture in the roof to illuminate the word love.
Arizona’s Anthem Veterans Memorial has a related design.
11/14/2020 UPDATE: An interesting addendum: The introduction of daylight saving in 1971 led designers to introduce a system of mirrors to ensure the right timing. Thanks to everyone who wrote in about this.
A Prime Formula
A team of mathematicians in Canada and Japan discovered this remarkable polynomial in 1976 — let its 26 variables a, b, c, … z range over the non-negative integers and it will generate all prime numbers:
![\displaystyle (k+2)(1-\newline [wz+h+j-q]^{2}-\newline [(gk+2g+k+1)(h+j)+h-z]^{2}-\newline [16(k+1)^{3}(k+2)(n+1)^{2}+1-f^{2}]^{2}-\newline [2n+p+q+z-e]^{2}-\newline [e^{3}(e+2)(a+1)^{2}+1-o^{2}]^{2}-\newline [(a^{2}-1)y^{2}+1-x^{2}]^{2}-\newline [16r^{2}y^{4}(a^{2}-1)+1-u^{2}]^{2}-\newline [n+\ell +v-y]^{2}-\newline [(a^{2}-1)\ell ^{2}+1-m^{2}]^{2}-\newline [ai+k+1-\ell -i]^{2}-\newline [((a+u^{2}(u^{2}-a))^{2}-1)(n+4dy)^{2}+1-(x+cu)^{2}]^{2}-\newline [p+\ell (a-n-1)+b(2an+2a-n^{2}-2n-2)-m]^{2}-\newline [q+y(a-p-1)+s(2ap+2a-p^{2}-2p-2)-x]^{2}-\newline [z+p\ell (a-p)+t(2ap-p^{2}-1)-pm]^{2})\newline >0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+++%28k%2B2%29%281-%5Cnewline++%5Bwz%2Bh%2Bj-q%5D%5E%7B2%7D-%5Cnewline++%5B%28gk%2B2g%2Bk%2B1%29%28h%2Bj%29%2Bh-z%5D%5E%7B2%7D-%5Cnewline++%5B16%28k%2B1%29%5E%7B3%7D%28k%2B2%29%28n%2B1%29%5E%7B2%7D%2B1-f%5E%7B2%7D%5D%5E%7B2%7D-%5Cnewline++%5B2n%2Bp%2Bq%2Bz-e%5D%5E%7B2%7D-%5Cnewline++%5Be%5E%7B3%7D%28e%2B2%29%28a%2B1%29%5E%7B2%7D%2B1-o%5E%7B2%7D%5D%5E%7B2%7D-%5Cnewline++%5B%28a%5E%7B2%7D-1%29y%5E%7B2%7D%2B1-x%5E%7B2%7D%5D%5E%7B2%7D-%5Cnewline++%5B16r%5E%7B2%7Dy%5E%7B4%7D%28a%5E%7B2%7D-1%29%2B1-u%5E%7B2%7D%5D%5E%7B2%7D-%5Cnewline++%5Bn%2B%5Cell+%2Bv-y%5D%5E%7B2%7D-%5Cnewline++%5B%28a%5E%7B2%7D-1%29%5Cell+%5E%7B2%7D%2B1-m%5E%7B2%7D%5D%5E%7B2%7D-%5Cnewline++%5Bai%2Bk%2B1-%5Cell+-i%5D%5E%7B2%7D-%5Cnewline++%5B%28%28a%2Bu%5E%7B2%7D%28u%5E%7B2%7D-a%29%29%5E%7B2%7D-1%29%28n%2B4dy%29%5E%7B2%7D%2B1-%28x%2Bcu%29%5E%7B2%7D%5D%5E%7B2%7D-%5Cnewline++%5Bp%2B%5Cell+%28a-n-1%29%2Bb%282an%2B2a-n%5E%7B2%7D-2n-2%29-m%5D%5E%7B2%7D-%5Cnewline++%5Bq%2By%28a-p-1%29%2Bs%282ap%2B2a-p%5E%7B2%7D-2p-2%29-x%5D%5E%7B2%7D-%5Cnewline++%5Bz%2Bp%5Cell+%28a-p%29%2Bt%282ap-p%5E%7B2%7D-1%29-pm%5D%5E%7B2%7D%29%5Cnewline++%3E0++&bg=ffffff&fg=000&s=0&c=20201002)
The snag is that it will sometimes produce negative numbers, which must be ignored. But every positive result will be prime, and every prime can be generated by some set of 26 non-negative integers.
(James P. Jones et al., “Diophantine Representation of the Set of Prime Numbers,” American Mathematical Monthly 83:6 [1976], 449-464.)
Spreading Word
When a Western scrub jay discovers the body of a dead jay, it summons other birds to screech over the body for up to half an hour. It’s not clear why they do this — the birds are territorial and not normally social. Possibly it’s a way to share news of danger, concentrate attention to find a predator, or teach young about dangers in the environment.
The gatherings are sometimes called funerals, though we don’t know enough to understand the reasons behind them. But UC Davis student Teresa Iglesias said, “I think there’s a huge possibility that there is much more to learn about the social and emotional lives of birds.”
The Buchowski Paradox
Suppose you have two brothers, both of whom are older than you are.
Then it becomes perversely accurate to say, “My younger brother is older than I am.”
Something Else
During a visit to Oxford in May 1931, Albert Einstein gave a brief lecture on cosmology, and afterward the blackboard was preserved along with Einstein’s ephemeral writing. It now resides in the university’s Museum of the History of Science.
Harvard historian of science Jean-François Gauvin argues that this makes it a “mutant object”: It’s no longer fulfilling the essential function of a blackboard, to store information temporarily — it’s become something else, a socially created object linked to the great scientist. The board’s original essence could be restored by wiping it clean, but that would destroy its current identity.
“The sociological metamorphosis at the origin of this celebrated artifact has completely destroyed its intrinsic nature,” Gauvin writes. “Einstein’s blackboard has become an object of memory, an object of collection modified at the ontological level by a social desire to celebrate the achievement of a great man.”
The Kobon Triangle Problem
Three lines can be arranged to make one triangle. Four lines can make two, and five lines can make five.
But, generally, no one can say how many non-overlapping triangles can be formed by an arrangement of k lines — the problem remains unsolved.
Going Up
The world’s largest vertical maze is the Al Rostamani Maze Tower in Dubai. Designed by Adrian Fisher, it rises 57 stories from the entrance at the bottom to the goal at the top.
The facade of the 12-story car park presents a second maze.