Chronological Order

By Lee Sallows: If the letters BJFGSDNRMLATPHOCIYVEU are assigned to the integers -10 to 10, then:

J+A+N+U+A+R+Y     = -9+0-4+10+0-3+7     =  1
F+E+B+R+U+A+R+Y   = -8+9-10-3+10+0-3+7  =  2
M+A+R+C+H         = -2+0-3+5+3          =  3
A+P+R+I+L         = 0+2-3+6-1           =  4
M+A+Y             = -2+0+7              =  5
J+U+N+E           = -9+10-4+9           =  6
J+U+L+Y           = -9+10-1+7           =  7
A+U+G+U+S+T       = 0+10-7+10-6+1       =  8
S+E+P+T+E+M+B+E+R = -6+9+2+1+9-2-10+9-3 =  9
O+C+T+O+B+ER      = 4+5+1+4-10+9-3      = 10
N+O+V+E+M+B+E+R   = -4+4+8+9-2-10+9-3   = 11
D+E+C+E+M+B+E+R   = -5+9+5+9-2-10+9-3   = 12

Similarly, if -7 to 7 are assigned SROEMUNFIDYHTAW, then SUNDAY to SATURDAY take on ordinal values. See Alignment.

(David Morice, “Kickshaws,” Word Ways 24:2 [May 1991], 105-116.)

Kürschák’s Tile

Hungarian mathematician József Kürschák offered this “proof without words” that a regular dodecagon inscribed in a unit circle has area 3. If the circle is inscribed in a square, the resulting figure can be tiled by triangles of two families — 16 equilateral triangles whose sides are equal to those of the dodecagon and 32 isosceles triangles with angles 15°-15°-150° and longest side 1. The area of the large square is 4, and the triangles that make up the dodecagon can be rearranged to fill 3 of its quadrants (see the video below). So the area of the dodecagon is 3/4 of 4, or 3.

(To see that the square and the dodecagon can be tiled as claimed, see Alexander Bogomolny’s discussion here.)

(Gerald L. Alexanderson and Kenneth Seydel, “Kürschak’s Tile,” Mathematical Gazette 62:421 [October 1978], 192-196.)


[Wittgenstein] once greeted me with the question: ‘Why do people say that it was natural to think that the sun went round the earth rather than that the earth turned on its axis?’ I replied: ‘I suppose, because it looked as if the sun went round the earth.’ ‘Well,’ he asked, ‘what would it have looked like if it had looked as if the earth turned on its axis?’

— G.E.M. Anscombe, An Introduction to Wittgenstein’s Tractatus, 1959

A Fuss Budget

Francis Galton quantified boredom. At a tedious meeting in 1885, he observed that the distance between neighboring heads in the listless crowd began to vary:

When the audience is intent each person forgets his muscular weariness and skin discomfort, and he holds himself rigidly in the best position for seeing and hearing. As this is practically identical for persons who sit side by side, their bodies are parallel, and again, as they sit at much the same distances apart, their heads are correspondingly equidistant. But when the audience is bored the several individuals cease to forget themselves and they begin to pay much attention to the discomforts attendant on sitting long in the same position. They sway from side to side, each in his own way, and the intervals between their faces, which lie at the free end of the radius formed by their bodies, with their seat as the centre of rotation varies greatly.

He wasn’t able to estimate this numerically, but he did find another measure: He counted about 50 fidgets per minute in each section of 50 people. “The audience was mostly elderly; the young would have been more mobile.” He urged “observant philosophers” at dull meetings to estimate “the frequency, amplitude, and duration of the fidgets of their fellow-sufferers” in hopes that “they may acquire the new art of giving numerical expression to the amount of boredom expressed by the audience generally during the reading of any particular memoir.”

(Francis Galton, “The Measure of Fidget,” Nature 32:817 [June 25, 1885], 174–175.)

Strähle’s Construction

In 1743, Swedish organ maker Daniel Stråhle published this method to calculate the sounding lengths of strings in a musical tuning with 12 pitches per octave that’s close to equal temperament. Draw segment QR 12 units long and establish it as the base of an isosceles triangle with sides of length 24. Find point P on OQ seven units above Q and draw a line through it from R to M such that PM = RP. Now MR is the string length of the lowest sounding pitch, MP is the pitch one octave higher, and the points labeled 2 through 12 give the endpoints for successive semitones within the octave.

Stråhle, who had no mathematical training, said he’d established the method after “some thought and a great number of attempts.” Exactly how he came up with it is not known.

Getting There

English mapmaker John Ogilby completed a startling project in 1675: a road atlas of 17th-century Britain, offering strip maps of most of the major routes in England and Wales. He wrote to Charles II:

I have Attempted to Improve Our Commerce and Correspondency at Home, by Registring and Illustrating Your Majesty’s High-Ways, Directly and Transversely, as from Shoare to Shoare, so to the Prescrib’d Limts of the Circumambient Ocean, from the Great Emporium and Prime Center of the Kingdom, Your Royal Metropolis.

It used a consistent scale of one inch per mile, with each mile comprising 1760 yards, a standard that later mapmakers would follow. You can see the whole atlas here.

A Boomerang Sequence

From reader Éric Angelini:

What distinguishes this sequence of integers?

10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18 …

Adding 9 to each successive digit and inserting a comma after the result reproduces the original sequence:

1 + 9 = 10
0 + 9 = 9
9 + 9 = 18
1 + 9 = 10
8 + 9 = 17, etc.

This example (A369603 in OEIS) is lexicographically the earliest such sequence beginning with 10.

(Thanks, Éric.)


  • Fletcher Christian’s first son was named Thursday October Christian.
  • 16384 = 163 × (8 – 4)
  • Of the 46 U.S. presidents to date, 16 have had no middle name.
  • “It is ill arguing against the use of anything from its abuse.” — Elizabeth I, in Walter Scott’s Kenilworth

Star Trek costume designer William Ware Theiss offered the Theiss Theory of Titillation: “The degree to which a costume is considered sexy is directly proportional to how accident-prone it appears to be.”

(Thanks, Michael.)

Einstein’s Sink
Image: Wikimedia Commons

This antique sink has been in use by the physics faculty of Leiden University since 1920, the year that Albert Einstein was made professor by special appointment.

It stood originally in the large lecture room of the old Kamerlingh Onnes Laboratory, and it accompanied the department to the Leiden Bioscience park in 1977.

In more than a century of use, it’s earned its renown: Its users also include Paul Ehrenfest, Hendrik Antoon Lorentz, Heike Kamerlingh Onnes, Albert Fert, and Brian Schmidt.

In 2015, when it became clear that sink would not accompany the department to a new science campus in 2025, a petition to “save the sink” received 197 signatures in a month. The faculty board agreed to move it to a lecture room in the new Oort building.

See Something Else.