Malfatti Circles

https://commons.wikimedia.org/wiki/File:Malfatti_circles_in_equilateral_triangle.svg

What’s the best way to squeeze three circles into a triangle so that the area of the circles is maximized? In 1803 Italian mathematician Gian Francesco Malfatti decided that the best course was to place each circle tangent to the other two and to two sides of the triangle (left) — he thought that some instance of this arrangement would give the best solution.

But that’s not actually so: In an equilateral triangle, Malfatti’s circles occupy less area than the solution on the right, found by Lob and Richmond in 1930 — their suggestion is to inscribe the largest possible circle in the triangle, then fit the second circle into one of the triangle’s three corners, and then fit the third circle into one of the five spaces now available, taking the largest available option in each case.

In the case of an equilateral triangle, Lob and Richmond’s solution is only about 1% larger than Malfatti’s. But in 1946 Howard Eves pointed out that for a long, narrow isosceles triangle (below), simply stacking three circles can cover nearly twice the area of the Malfatti circles.

Subsequent studies have borne this out — it turns out that Malfatti’s plan is never best. We now know that Lob and Richmond’s procedure will always find three area-maximizing circles — but whether their approach will work for more than three circles is an open question.

(Thanks, Larry.)

https://commons.wikimedia.org/wiki/File:Malfatti%27s_circles_in_sharp_isosceles_triangle.svg
Image: Wikimedia Commons
| Science & Math

“Let Liking Last”

Inscriptions found in 17th-century English wedding rings, from William Jones’ Finger-Ring Lore, 1898:

  • I LOVE AND LIKE MY CHOYSE.
  • I CHUSE NOT TO CHANGE.
  • Let reason rule affection.
  • A token of good-will.
  • Live in Loue.
  • As I expect so let me find, A faithfull ❤ and constant mind.
  • Time lesseneth not my love.
  • Love the truth.
  • In loving wife spend all thy life.

A diamond ring bore the inscription “This sparke will grow.”

| History · Society

A Mouthful

In 1854, a correspondent wrote to Notes and Queries asking about the origins of this couplet:

Perturbabantur Constantinopolitani
Innumerabilibus sollicitudinibus.

[“Constantinople is much perturbed.”]

He got this reply:

“When I first learned to scan verses, somewhere about thirty years ago, the lines produced by your correspondent P. were in every child’s mouth, with this story attached to them. It was said that Oxford had received from Cambridge the first line of the distich, with a challenge to produce a corresponding line consisting of two words only. To this challenge Oxford replied by sending back the second line, pointing out, at the same time, the false quantity in the word Constantinŏpolitani.”

| Language

Sound Sense

The opening of Chapter 5 in E.M. Forster’s 1910 novel Howards End describes a concert performance of the third movement of Beethoven’s Fifth Symphony — and succinctly illustrates six ways that people listen to instrumental music:

Whether you are like Mrs. Munt, and tap surreptitiously when the tunes come — of course, not so as to disturb the others –; or like Helen, who can see heroes and shipwrecks in the music’s flood; or like Margaret, who can only see the music; or like Tibby, who is profoundly versed in counterpoint, and holds the full score open on his knee; or like their cousin, Fräulein Mosebach, who remembers all the time that Beethoven is ‘echt Deutsch’; or like Fräulein Mosebach’s young man, who can remember nothing but Fräulein Mosebach: in any case, the passion of your life becomes more vivid, and you are bound to admit that such a noise is cheap at two shillings.

“All these responses are, of course, presented in highly ironical terms,” writes University of Graz literature professor Walter Bernhart, “but surely it is a very clever brief outline of a basic typology of music reception.”

(From Werner Wolf, Walter Bernhart, and Andreas Mahler, Immersion and Distance: Aesthetic Illusion in Literature and Other Media, 2013.)

| Art

Proteomics

Reader Eliot Morrison, a protein biochemist, has been looking for the longest English word found in the human proteome — the full set of proteins that can be expressed by the human body. Proteins are chains composed of amino acids, and the most common 20 are represented by the letters A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, and Y. “These amino acids have different chemical properties,” Eliot writes, “and the sequence influences how the whole chain folds in three dimensions, which in turn determines the structural and functional properties of the protein.”

The longest English word he’s found is TARGETEER, at nine letters, in the uncharacterized protein C12orf42. The whole sequence of C12orf42 is:

MSTVICMKQR EEEFLLTIRP FANRMQKSPC YIPIVSSATL WDRSTPSAKH IPCYERTSVP 
CSRFINHMKN FSESPKFRSL HFLNFPVFPE RTQNSMACKR LLHTCQYIVP RCSVSTVSFD 
EESYEEFRSS PAPSSETDEA PLIFTARGET EERARGAPKQ AWNSSFLEQL VKKPNWAHSV 
NPVHLEAQGI HISRHTRPKG QPLSSPKKNS GSAARPSTAI GLCRRSQTPG ALQSTGPSNT 
ELEPEERMAV PAGAQAHPDD IQSRLLGASG NPVGKGAVAM APEMLPKHPH TPRDRRPQAD 
TSLHGNLAGA PLPLLAGAST HFPSKRLIKV CSSAPPRPTR RFHTVCSQAL SRPVVNAHLH

And there are more: “There are also a number of eight-letters words found: ASPARKLE (Uniprot code: Q86UW7), DATELESS (Q9ULP0-3), GALAGALA (Q86VD7), GRISETTE (Q969Y0), MISSPEAK (Q8WXH0), REELRALL (Q96FL8), RELASTER (Q8IVB5), REVERSAL (Q5TZA2), and SLAVERER (Q2TAC2).” I wonder if there’s a sentence in us somewhere.

(Thanks, Eliot.)

| Language · Science & Math

A Prose Maze

https://commons.wikimedia.org/wiki/File:Henry_James_by_John_Singer_Sargent_cleaned.jpg

Max Beerbohm wrote two parodies of Henry James’ impenetrable style. The first, “The Mote in the Middle Distance,” appeared in The Saturday Review in 1906:

It was with the sense of a, for him, very memorable something that he peered now into the immediate future, and tried, not without compunction, to take that period up where he had, prospectively, left it. But just where the deuce had he left it? The consciousness of dubiety was, for our friend, not, this morning, quite yet clean-cut enough to outline the figures on what she had called his ‘horizon,’ between which and himself the twilight was indeed of a quality somewhat intimidating. He had run up, in the course of time, against a good number of ‘teasers;’ and the function of teasing them back — of, as it were, giving them, every now and then, “what for” — was in him so much a habit that he would have been at a loss had there been, on the face of it, nothing to lose.

He wrote the second, “The Guerdon,” when he learned that James was about to receive the Order of Merit:

That it hardly was, that it all bleakly and unbeguilingly wasn’t for “the likes” of him — poor decent Stamfordham — to rap out queries about the owner of the to him unknown and unsuggestive name that had, in these days, been thrust on him with such a wealth of commendatory gesture, was precisely what now, as he took, with his prepared list of New Year colifichets and whatever, his way to the great gaudy palace, fairly flicked his cheek with the sense of his having never before so let himself in, as he ruefully phrased it, without letting anything, by the same token, out.

He wasn’t alone. “It’s not that he bites off more than he can chaw,” Clover Adams once wrote of James, “but he chaws more than he bites off.”

| Literature

The Bingo Paradox

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Image: Wikimedia Commons

Surprisingly, when a large number of people play bingo, it’s much more likely that the winning play occupies a row on its card rather than a column.

The standard bingo card is a 5 × 5 square in which the columns are headed B-I-N-G-O. The columns are filled successively with numbers drawn at random from the intervals 1-15, 16-30, 31-45, 46-60, and 61-75. And it turns out that, during play, it’s very likely that at least one number from each column group will be called (enabling a horizontal win) before some five numbers are called that occupy a single column (enabling a vertical win). In fact it’s more than three times as likely.

The math is laid out rigorously in the article below. If a free space appears in the middle of the board, as is common, the effect still obtains — Joseph Kisenwether and Dick Hess found that the chance of a horizontal win is still 73.73 percent.

(Arthur Benjamin, Joseph Kisenwether, and Ben Weiss, “The BINGO Paradox,” Math Horizons 25:1 [2017], 18-21.)

| Science & Math

Podcast Episode 268: The Great Impostor

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Ferdinand Demara earned his reputation as the Great Impostor: For over 22 years he criss-crossed the country, posing as everything from an auditor to a zoologist and stealing a succession of identities to fool his employers. In this week’s episode of the Futility Closet podcast we’ll review Demara’s motivation, morality, and techniques — and the charismatic spell he seemed to cast over others.

We’ll also make Big Ben strike 13 and puzzle over a movie watcher’s cat.

See full show notes …

| Crime · Podcast · Society