La Formule de François

Here is a beautiful and astonishingly simple formula for π created by the French mathematician François Viète (1540-1603):

• 2 / π = √2/2 * √(2 + √2)/2 * √(2 + √(2 + √2))/2…

I can remember testing the formula on a scientific calculator that allowed simple programming. As I pressed the = key and the results began to home in on π, I felt as though I was watching a tall and elegant temple emerge through swirling mist.


At last… it’s OFFICIAL!



Cult pop singer Morrisey — hailed as hero by his fanatical fans — is a twat, according to experts. And that will come as bad news to his many admirers who have worshipped the pop idol since he came to fame as lead singer of The Smiths.


Professor Ivan Sogorski of Barrow-in-Furness University’s Department of Advanced Human Behavioural Studies came to his dramatic conclusion about the star after listening to many of his records and watching video footage of his TV appearances. And he summed up his professional opinion in a few short words.


“The man is an absolute twat,” he told us.


Professor Sogorski cited examples of behaviour which had lead him to his controversial conclusion. “Take for example Mr Morrisey’s appearance on Top Of The Pops in the early eighties when he wore oversized shirts, National Health glasses, a hearing aid, and flailed about the stage with daffodils sticking out of his back pocket. Clearly, even the most casual analysis could only conclude this to be the behaviour of an arsehole,” said the Professor.


As a part of his painstaking research, Professor Sogorski consulted a colleague to obtain a second independent opinion. “I submitted manuscipts and recordings of many Morrissey songs to a leading Professor of Composition at the Royal College of Music, and he says they are crap.”


The Professor quoted examples of Morrisey’s song titles as further evidence to support his views. “Girl In A Coma. Big Mouth Strikes Again. Heaven Knows I’m Miserable Now. These are all bullshit,” said Professor Sogorski.

During his career Morrisey has endeared himself to a huge cult following of pop fans, among them many students, and has also won artistic acclaim for his work.


But Professor Sogorski’s comments are bound to fuel speculation that whilst some of his songs might be quite good, the man is, quite frankly, a bit of an arsehole. “I am convinced Morrisey is a twat, and anyone who says otherwise is a wanker,” said the Professor yesterday.

Professor Sogorski last hit the headlines in 1988 when he claimed that page three model Samantha Fox was a “boiler”.

• From Viz

Morrissey ‘Still a Twat in Parallel Universe’ — Hawking

PHYSICS boffin Professor STEPHEN HAWKING has confirmed that pop singer MORRISSEY would remain a bell-end in every conceivable alternate universe.

Hawking, 73, was delivering a lecture at the Sydney Opera House last night when he made the startling announcement regarding the ex-Smiths frontman.


The Cambridge egghead told attendees: “Theoretical physics may one day be able to prove the existence of multiple universes outside our own. We can predict very little about what these parallel universes would be like, but we do know one thing: Morrissey would still be a twat in them.”

Hawking went on to explain his revelation with a series of complex equations.


He said: “Multiple universes would probably differ from our own in almost every way. They would be made up of different chemical elements which themselves would be made of fundamental particles different from the ones we have identified. They may even be governed by completely differnt laws of physics. The only constant would be Morrissey behaving like an arse and saying twattish things.”

the chas

The Brief History of Time author continued: “The possibilities in a parallel universe are genuinely limitless: the sky could be purple, the moon could be made of Styrofoam, cats could talk. Absolutely anything is feasible — except Morrissey not being a dick.”

“He still would be one, I’m afraid,” he added. “Nothing so sure.” Hawking has announced plans for a follow-up lecture next week at the Royal Albert Hall, in which he will hypothesise that Sting could still get on everybody’s tits in a black hole.

• From Viz

Mötley Vüe

Here’s the Fibonacci sequence, where each term (after the first two) is created by adding the two previous numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765...

In “Fib and Let Tri”, I described how my eye was caught by 55, which is a palindrome, reading the same backwards and forwards. “Were there any other Fibonacci palindromes?” I wondered. So I looked to see. Now my eye has been caught by 55 again, but for another reason. It should be easy to spot another interesting aspect to 55 when the Fibonacci numbers are set out like this:

fib(1) = 1
fib(2) = 1
fib(3) = 2
fib(4) = 3
fib(5) = 5
fib(6) = 8
fib(7) = 13
fib(8) = 21
fib(9) = 34
fib(10) = 55
fib(11) = 89
fib(12) = 144
fib(13) = 233
fib(14) = 377
fib(15) = 610
fib(16) = 987
fib(17) = 1597
fib(18) = 2584
fib(19) = 4181
fib(20) = 6765

55 is fib(10), the 10th Fibonacci number, and 5+5 = 10. That is, digsum(fib(10)) = 10. What other Fibonacci numbers work like that? I soon found some and confirmed my answer at the Online Encyclopedia of Integer Sequences:

1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222 — A020995 at OEIS

And that seems to be the lot, according to the OEIS. In base 10, at least, but why stop at base 10? When I looked at base 11, the numbers of digsum(fib(k)) = k didn’t stop coming, because I couldn’t take the Fibonacci numbers very high on my computer. But the OEIS gives a much longer list, starting like this:

1, 5, 13, 41, 53, 55, 60, 61, 90, 97, 169, 185, 193, 215, 265, 269, 353, 355, 385, 397, 437, 481, 493, 617, 629, 630, 653, 713, 750, 769, 780, 889, 905, 960, 1013, 1025, 1045, 1205, 1320, 1405, 1435, 1501, 1620, 1650, 1657, 1705, 1735, 1769, 1793, 1913, 1981, 2125, 2153, 2280, 2297, 2389, 2413, 2460, 2465, 2509, 2533, 2549, 2609, 2610, 2633, 2730, 2749, 2845, 2893, 2915, 3041, 3055, 3155, 3209, 3360, 3475, 3485, 3521, 3641, 3721, 3749, 3757, 3761, 3840, 3865, 3929, 3941, 4075, 4273, 4301, 4650, 4937, 5195, 5209, 5435, 5489, 5490, 5700, 5917, 6169, 6253, 6335, 6361, 6373, 6401, 6581, 6593, 6701, 6750, 6941, 7021, 7349, 7577, 7595, 7693, 7740, 7805, 7873, 8009, 8017, 8215, 8341, 8495, 8737, 8861, 8970, 8995, 9120, 9133, 9181, 9269, 9277, 9535, 9541, 9737, 9935, 9953, 10297, 10609, 10789, 10855, 11317, 11809, 12029, 12175... — A020995 at OEIS

The list ends with 1636597 = A18666[b11] and the OEIS says that 1636597 almost certainly completes the list. According to David C. Terr’s paper “On the Sums of Fibonacci Numbers” (pdf), published in the Fibonacci Quarterly in 1996, the estimated digit-sum for the k-th Fibonacci number in base b is given by the formula (b-1)/2 * k * log(b,φ), where log(b,φ) is the logarithm in base b of the golden ratio, 1·61803398874… Terr then notes that the simplified formula (b-1)/2 * log(b,φ) gives the estimated average ratio digsum(fib(k)) / k in base b. Here are the estimates for bases 2 to 20:

b02 = 0.3471209568153086...
b03 = 0.4380178794859424...
b04 = 0.5206814352229629...
b05 = 0.5979874356654401...
b06 = 0.6714235829697111...
b07 = 0.7418818776805580...
b08 = 0.8099488992357201...
b09 = 0.8760357589718848...
b10 = 0.9404443811249043...
b11 = 1.0034045909311624...
b12 = 1.0650963641043091...
b13 = 1.1256639207937723...
b14 = 1.1852250528196852...
b15 = 1.2438775226715552...
b16 = 1.3017035880574074...
b17 = 1.3587732842474014...
b18 = 1.4151468584732730...
b19 = 1.4708766105122322...
b20 = 1.5260083080264088...

In base 2, you can expect digsum(fib(k)) to be much smaller than k; in base 20, you can expect digsum(fib(k)) to be much larger. But as you can see, the estimate for base 11, 1.0034045909311624…, is very nearly 1. That’s why base 11 produces so many results for digsum(fib(k)) = k, because only a slight deviation from the estimate might create a perfect ratio of 1 for digsum(fib(k)) / k, i.e. digsum(fib(k)) = k. But in the end the results run out in base 11 too, because as k gets higher and fib(k) gets bigger, the estimate becomes more and more accurate and digsum(fib(k)) > k. With lower k, digsum(fib(k)) can easily fall below k or match k. That happens in other bases, but because their estimates are further from 1, results for digsum(fib(k)) = k run out much more quickly.

To see this base behavior represented visually, I’ve created Ulam-like spirals for k using three colors: blue for digsum(fib(k)) < k, yellow for digsum(fib(k)) > k, and red for digsum(fib(k)) = k (with the green square at the center representing fib(1) = 1). As you can see below, the spiral for base 11 immediately stands out. It’s motley, not dominated by blue or yellow like the other spirals:

Spiral for digsum(fib(k)) in base 9
(blue for digsum(fib(k)) < k, yellow for digsum(fib(k)) > k, red for digsum(fib(k)) = k, green for fib(1))

Spiral for digsum(fib(k)) in base 10

Spiral for digsum(fib(k)) in base 11 — a motley view of blue, yellow and red

Spiral for digsum(fib(k)) in base 12

Spiral for digsum(fib(k)) in base 13

Finally, here are spirals at higher and higher resolution for digsum(fib(k)) = k in base 11:

digsum(fib(k)) = k in base 11 (low resolution)
(green square is fib(1))

digsum(fib(k)) = k in base 11 (x2 resolution)

digsum(fib(k)) = k in base 11 (x4)

digsum(fib(k)) = k in base 11 (x8)

digsum(fib(k)) = k in base 11 (x16)

digsum(fib(k)) = k in base 11 (x32)

digsum(fib(k)) = k in base 11 (x64)

digsum(fib(k)) = k in base 11 (x128)

digsum(fib(k)) = k in base 11 (animated)


• მელიამ მგელს შესძახა: შე უმი ხორცის ჭამიაო!
•• Meliam mgels šesdzakha: še umi khortsis ch’amiao!
••• FOX-agentive WOLF-dative called: thou raw MEAT-genitive EATER-vocative
•••• The fox called to the wolf: “Thou eater of raw meat!”
••••• The pot called the kettle black.

Toxic Turntable #24

Currently listening…

• We Worship Silence, Pass the Gates (2011)
• House of Pyromania, Many Seek (Few Find) (1987)
• X-Newly Inc, Oz Wuwu 9 (2003)
• Iujisba, Abominable Abdominal (Killer Bees EP) (1986)
• Danny Yaup, Vision Ov (1969)
• Fizzy Glamsters, Keict (1991)
• Roxane Redmoor, Voxational DJ (2008)
• Kogar Fjö, Capnotic Micrographs (1993)
• Dynamic and the Zone, Cocodrilo Rock (1977)
• იჰვიუხე, პეპლები მთვარის (2002)
• Quickfinger, Ship on a Painted Ocean (1980)
• Earl Vanburgh, Glad (but) Sad (1965)
• Aquilæ ζ, Songs of Seventeen Stars (1979)
• Kozmik Krusaders, Hexen Zoo (1998)
• Quanta Thalassia, This Is Si Siht (2010)
• Thallium Addicts, Thanatographic (1999)
• X Xepj Xo, On an Ebb (2014)
• Orion’s Cradle, Live in Oslo (1987)
• Gazing on Bifrost, By the Swords (1976)
• Ausna, Z.M.E. (1977)
• Obelisk Pact, Long You’ll Slide (2003)
• Um Nuhotóbareac, L’Xac Rey (2011)

Previously pre-posted:

Toxic Turntable #1#2#3#4#5#6#7#8#9#10#11#12#13#14#15#16#17#18#19#20#21#22#23

Discharming Manc

A passionately socialist Anglican priest and proud member of the LGBTQ+ Community no longer approves of Moz:

The song I can no longer listen to

“This Charming Man”. Much as I like the song, Morrissey has ceased to be charming for me.

‘No Jacket Required would be the soundtrack of hell’: the Rev Richard Coles’s honest playlist, The Guardian, 10i22

Core War…

In terms of my core ambitions for 2022, I hope to continue the fight against such things as the reprehensible and repulsive phrase “in terms of”, the pretentious and throbbingly urgent adjective “core”, and the cheap trick of trailing dots… I know that I won’t win and that the Hive-Mind will continue to buzz deafeningly at core venues like The Guardian, The London Review of Books and The Shropshire Advertiser, but so what? In the core words of Samuel in terms of Johnson:

[I]t remains that we retard what we cannot repel, that we palliate what we cannot cure. Life may be lengthened by care, though death cannot be ultimately defeated: tongues, like governments, have a natural tendency to degeneration; we have long preserved our constitution, let us make some struggles for our language. — Samuel Johnson, Preface to a Dictionary of the English Language (1755)

Elsewhere Other-Accessible

Ex-term-in-ate! — core interrogation of why “in terms of” is so despicable, deplorable and downright disgusting…
Don’t Do Dot — core interrogation of why “…” is so despicable, deplorable and downright disgusting dot dot dot

Post-Performative Post-Scriptum

How should the first line of this incendiary intervention begin? I suggest: “In terms of my core ambitions for 2022…” → “Among my main ambitions…”

Triangular Squares

The numbers that are both square and triangular are beautifully related to the best approximations to √2:


Square Root

Factors of root

1 1 1
36 6 2 * 3
1225 35 5 * 7
41616 204 12 * 17

and so on.

In each case the factors of the root are the numerator and denominator of the next approximation to √2. — David Wells, The Penguin Dictionary of Curious and Interesting Mathematics (1986), entry for “36”.

Elsewhere other-accessible

A001110 — Square triangular numbers: numbers that are both triangular and square