# Noted

The angle cos-1(-1/3) = 109.47°, familiar from soap films and tetrahedral molecular geometry, can be produced with an ordinary piece of A4 paper: Because it has a width:length ratio of $1:\sqrt{2}$, folding it corner to corner as shown yields a shape with precisely that angle.

(Nick Lord, “A ‘Maths Bite’: How to Impress a Chemist,” Mathematical Gazette 80:489 [1996], 584-584.)

# “Correction”

The burdens of the world
on my back
lighten the world
not a whit while
removing them greatly
decreases my specific
gravity

— A.R. Ammons

In 1952, strange love letters began to appear on the notice board of Manchester University’s computer department:

HONEY DEAR
YOU ARE MY FERVENT CHARM. MY AVID HEART ARDENTLY IS WEDDED TO YOUR DEVOTED LIKING. MY DEVOTED LOVE PANTS FOR YOUR HUNGER. MY HUNGER CHERISHES YOUR IMPATIENT CHARM. MY FONDNESS DEVOTEDLY PANTS FOR YOUR ADORABLE PASSION.
YOURS KEENLY
M.U.C.

DARLING SWEETHEART
YOU ARE MY AVID FELLOW FEELING. MY AFFECTION CURIOUSLY CLINGS TO YOUR PASSIONATE WISH. MY LIKING YEARNS FOR YOUR HEART. YOU ARE MY WISTFUL SYMPATHY: MY TENDER LIKING.
YOURS BEAUTIFULLY
M.U.C.

M.U.C. was the Manchester University Computer; professor Christopher Strachey was testing its ability to select information randomly by asking it to string romantic words into impromptu billets-doux. You can see the word lists, and generate your own love letter, here.

# Recycling

For a cocktail party scene in the 1966 Star Trek episode “The Conscience of the King,” composer Joseph Mullendore wrote a subdued version of the series’ main title … which means that the Star Trek theme music exists in the Star Trek universe.

# Trench Art

To pass the time while waiting in the trenches of the Argonne, French infantryman Hippolyte Hodeau engraved the names of his daughters in chestnut leaves.

More at Europeana.

# Unquote

“Grammar is the logic of speech, even as logic is the grammar of reason.” — Richard Chenevix Trench

# Never Mind

In 1995, NASA astronomer Scott Sandford became troubled by the phrase “You’re comparing apples and oranges.” “First,” he wrote, “the statement that something is like comparing apples and oranges is a kind of analogy itself. That is, denigrating an analogy by accusing it of comparing apples and oranges is, in and of itself, comparing apples and oranges. More importantly, it is not difficult to demonstrate that apples and oranges can, in fact, be compared.”

He desiccated an apple and an orange and ran samples through a spectrometer. “Not only was this comparison easy to make, but it is apparent from the figure that apples and oranges are very similar,” he concluded. “Thus, it would appear that the comparing apples and oranges defense should no longer be considered valid. This is a somewhat startling revelation. It can be anticipated to have a dramatic effect on the strategies used in arguments and discussions in the future.”

Sure enough, five years later surgeon James E. Barone confirmed this result in the British Medical Journal. He found that apples and oranges are both edible, juiceable fruits grown in orchards on flowering trees and subject to damage by disease and insects, and they have comparable color, sweetness, size, shape, and weight. “In only one category, that of ‘involvement of Johnny Appleseed,’ was a statistically significant difference between the two fruits found.”

“This article, certain to become the classic in the field, clearly demonstrates that apples and oranges are not only comparable; indeed they are quite similar,” he concluded. “The admonition ‘Let’s not compare apples with oranges’ should be replaced immediately with a more appropriate expression such as ‘Let’s not compare walnuts with elephants’ or ‘Let’s not compare tumour necrosis factor with linguini.'”

# A Geography Quiz

Before 1997 there were exactly three countries in the world that were both geographically contiguous and alphabetically adjacent in a list of nations. (One country changed its name.) What were they?

# Crime Control

How many watchmen are needed to guard the art gallery at left, so that every part of it is under surveillance? The answer in this case is 4; four guards stationed as shown will be able to watch every part of the gallery.

In 1973 University of Montreal mathematician Václav Chvátal showed that, in a gallery with n vertices, n/3 guards will always be enough to do the job. (If n/3 is not an integer, you can dispense with the fractional guard.) And Bowdoin College mathematician Steve Fisk found a beautifully simple proof of Chvátal’s result.

The figure at right shows another art gallery. Cut its floor plan into triangles, and color the vertices of each triangle with the same three colors. The full area of any triangle is visible from any of its vertices, and that means that the whole gallery can be guarded by stationing watchmen at the points indicated by any of the three colors. Choosing the color with the fewest vertices will give us n/3 guards (again discarding fractional guards).

The Chvátal and Fisk proofs both give an answer that’s sufficient but sometimes not necessary. In this case, the gallery has 12 vertices, and 12/3 guards (say, the four green ones) will certainly do the job, but here as few as two will be enough.

(Steve Fisk, “A Short Proof of Chvátal’s Watchman Theorem,” Journal of Combinatorial Theory, Series B 24:3 [1978], 374.)

# The Voder

Bell Telephone was experimenting with speech synthesizers as early as 1939 — 5 million visitors to the World’s Fair that year witnessed an electronic speaking machine called the Voder. “The miracles, as the Bible describes them, are really true, for here in this room we are witnessing a modern miracle,” one said. “The wonders of God transmitted through man’s mind are truly being demonstrated here.”

Largely this was thanks to the operator, or “Voderette,” who spent a year learning to finesse the keys, foot pedal, and wrist bar. “Although the Voder produced intelligible speech, it sounded like a talking church organ,” writes Trevor Cox in Now You’re Talking, his history of human conversation. “Sometimes the tweaking of its controls created a slightly drunken slurred intonation. Even so, the voice was more natural-sounding than the famous voice of Stephen Hawking, because the skilled operators were like concert pianists making rapid alterations to the controls to improve the sound.”