# Malfatti Circles

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.)

# 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.)

# Math Notes

The difference of any pair of the numbers {1873432, 2288168, 2399057, 6560657} is a perfect square (of 644, 725, 2165, 333, 2067, or 2040).

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.)

# The 36 Officers Problem

Suppose we have a group of officers in six regiments, each regiment consisting of the same six ranks (say, a colonel, a lieutenant colonel, a major, a captain, a first lieutenant, and a second lieutenant). Is it possible to arrange these 36 officers into a 6 × 6 square so that no rank or regiment is repeated in any row or column? That is, each row and column must contain an officer of each regiment and of each rank.

In 1782 Leonhard Euler wrote, “After we have put a lot of thought into finding a solution, we have to admit that such an arrangement is impossible, though we can’t give a rigorous demonstration of this.” He saw that the equivalent problem is impossible in a 2 × 2 square and surmised that it’s impossible in every case where the side of the square contains 4k + 2 cells.

It wasn’t until 1901 that French mathematician Gaston Terry proved that the 6 × 6 square has no solution, and it wasn’t until 1960 that Euler’s conjecture about the pattern of impossible squares was proven wrong: In fact, the task is impossible only in these two cases, 2 × 2 and 6 × 6.

# Actual Size

This is neat: Figure 1 in this physics paper is a 1:1 scale image of a primordial black hole whose mass is 5 times that of Earth.

A black hole of 10 Earth masses “is roughly the size of a ten pin bowling ball.”

(Jakub Scholtz and James Unwin, “What If Planet 9 Is a Primordial Black Hole?”, arXiv preprint arXiv:1909.11090 [2019].)

# The Speech-to-Song Illusion

In 1995 UCSD psychologist Diana Deutsch was fine-tuning the spoken commentary on a CD when she noticed something odd. When the phrase “sometimes behave so strangely” was repeated on a loop, it came to sound as though it were being sung rather than spoken. When the full surrounding passage was then played in its entirety, this phrase still sounded as though it were being sung (you can hear this here).

The phenomenon is not completely understood, but “the present experiments show that for a phrase to be heard as spoken or as sung, it does not need to have a set of physical properties that are unique to speech, or a different set of physical properties that are unique to song,” the researchers write. “Rather, we must conclude that, assuming the neural circuitries underlying speech and song are at some point distinct and separate, they can accept the same input, but process the information in different ways so as to produce different outputs.”

(Via MetaFilter.)

# Another One

Humans aren’t the only species that tend to move to a musical beat: Animals that mimic vocally (such as parrots and, here, a sulphur-crested cockatoo) bob their heads and move their feet. Animals that don’t mimic vocally don’t do this. So possibly our urge to move to music is a by-product of our tendency to mimic vocally — it’s a motor response to something we hear.

(Aniruddh D. Patel, et al., “Experimental Evidence for Synchronization to a Musical Beat in a Nonhuman Animal,” Current Biology 19:10 [2009], 827-830; Adena Schachner, et al., “Spontaneous Motor Entrainment to Music in Multiple Vocal Mimicking Species,” Current Biology 19:10 [2009], 831-836.)

# Math Notes

```                                       1
1 1
1 0 1
1 0 0 1
1 0 0 0 1
1 0 0 0 0 1
1 0 0 0 0 0 1
1 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 1 1 1 9 1 1 1 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 1 9 9 9 9 1 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 1 9 2 9 1 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 1 9 9 9 9 1 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 1 1 1 9 1 1 1 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 1
1 0 0 0 0 0 1
1 0 0 0 0 1
1 0 0 0 1
1 0 0 1
1 0 1
1 1
1
```

Amazingly, this is a prime number. When the digits are assembled into one long string, beginning at the top of the star and reading each row left to right, they form a 1093-digit number whose only factors are 1 and itself.

It was discovered by Australian mathematician Michael Hartley. See more of his prime curios.

# The Copenhagen Faustparodie

In The Roots of Coincidence, Arthur Koestler mentions that the participants at a 1932 conference on nuclear physics put on a parody of Goethe’s Faust in which Wolfgang Pauli played Mephistopheles. “His Gretchen was the neutrino, whose existence Pauli had predicted, but which had not yet been discovered.”

MEPHISTOPHELES (to Faust):

Beware, beware, of Reason and of Science
Man’s highest powers, unholy in alliance.
You’ll let yourself, through dazzling witchcraft yield
To weird temptations of the quantum field.

Enter Gretchen; she sings to Faust. Melody: ‘Gretchen at the Spinning Wheel’ by Schubert.

GRETCHEN:

My rest-mass is zero
My charge is the same
You are my hero
Neutrino’s my name.

There’s more here, including a link to the original script (in German).