Scribal Waugh Fare

Because I thought I’d accidentally deleted it, for years I’ve been thinking fondly about a little essay I’d once written comparing errors by scribes in the ancient world with typos in printed copies of Evelyn Waugh’s books. Then I dug up an old CD with back-up copies of various files on it. And it turned out, first, that I hadn’t deleted the little essay and, second, that it wasn’t as good as I remembered it. Here it is anyway, following the essay that originally accompanied it.


The Purloined Letter

The writer and musician Alexander Waugh was once looking through a bound collection of Alastair Graham’s letters to his grandfather Evelyn Waugh.[1] Graham had been Evelyn’s first great love at Oxford, but the letters were not at all diverting and Waugh petit fils had reached a point of tedium at which any interruption was welcome when an interruption fortunately arrived: the collection fell apart revealing that the following words had been concealed along its spine:

RIEN N’EST VRAI QUE LE BEAU.

The words, which mean “Nothing is true but beauty”, were probably taken from the French romantic Alfred de Musset (1810-57),[2] and they are interesting not only for the light they shed on Waugh’s æsthetic attitudes in his youth but also for the way in which they were uncovered: the collection fell apart because a letter had been forcibly removed from it.

Why this should have been done is probably now an unanswerable question, but not beyond all conjecture. Graham was a central model for Sebastian Flyte in Brideshead Revisited (1945), and “Nothing is true but beauty” might be taken as the principle that guides Charles Ryder early in the novel. This is why he falls in love with Sebastian, who is “entrancing, with that epicene beauty which in extreme youth sings aloud for love and withers at the first cold wind.”[3] But the entrancing, epicene Sebastian hints at another famous French saying when he concludes one of his letters to Charles like this: “Love, or what you will. S.”[4]

The valediction may conflate the two great rules of the occultist Aleister Crowley (1875-1947), which appear in Waugh’s short-story “Out of Depth” (1930) when Dr Kakophilos,[5] a black magician based on Crowley, confronts the story’s lapsed Catholic protagonist, Rip Van Winkle:

“Do what thou wilt shall be the whole of the law,” said Dr Kakophilos, in a thin Cockney voice.

“Eh?”

“There is no need to reply. If you wish to, it is correct to say, ‘Love is the Law, Love under will.’”

“I see.”

The famous French saying underlying Crowley’s first law is Rabelais’s Fay Ce Que Vouldras, or “Do What Thou Wilt”, which was written over the entrance to the Abbey of Thélème in Rabelais’s novel Gargantua (1532).[6] It is not an invitation to unbridled hedonism in either Rabelais or Crowley,[7] but it might nevertheless be read as justifying the “silk shirts and liqueurs and cigars and […] naughtiness high in the catalogue of grave sins”[8] of Ryder’s first term at Oxford.

Some of this gravely sinful naughtiness was Waugh’s in reality as well as Ryder’s in fiction, and if Ryder’s naughtiness included dabbling in the occult, perhaps Waugh’s did too. If he was initiated at the country house of his lost novel The Temple at Thatch,[9] perhaps this explains why so many references to the occult are attached to the Flyte family who own the country house at Brideshead. They range from Lady Marchmain’s alleged sanguinivorous “witchcraft” in Book One[10] through Julia’s “magic ring” and “fawning monster” in Book Two[11] to the “wand” Julia wields against Charles on a night of “full and high” moon in Book Three.[12] But one of the references was cut from the revised edition of the novel published in 1960. In the older edition (which is still issued in the United States), Anthony Blanche, who has “practised black art” at Crowley’s Abbey of Thelema “in Cefalù”,[13] says of the Flytes that they are “a subject for the poet — for the poet of the future who is also a psycho-analyst — and perhaps a diabolist too.”[14]

Poet, psycho-analyst, and diabolist are all gone in the revised edition, but Blanche’s insistent warnings against the Flytes’ charm are left untouched: “I warned you expressly and in great detail of the Flyte family. Charm is the great English blight.”[15] Someone as interested in etymology as Waugh almost certainly knew that “charm” was once a supernatural term: it meant a spell cast to control or influence and came from the Latin carmen, meaning “song”. Such echoes of ancient meaning are also apparent in, for example, the names Cordelia and Julia. Cordelia is the exemplar of unselfish Christian love in Brideshead, and her name probably comes from the Latin cor, meaning “heart”; Julia is the exemplar of ultimately sterile beauty and sexual attractiveness, and her name comes from the Julius family of ancient Rome, who were said to be descended from Venus, the goddess of beauty and sex.

The name “Marchmain”, on the other hand, seems much harder to analyze, although it echoes mortmain, literally meaning “dead hand”,[16] and may hint at the impending loss of Brideshead by the Flytes, none of whom has any true heirs. However, its first syllable is also an anagram of “charm” — m-ar-ch <-> ch-ar-m — and “Charmmain” is very like the French charmant, or “charming”. So the Flytes are charming, and perhaps Waugh is hinting that the originals on whom he based them were charming in more senses than one. If so, perhaps that explains why a letter from the original Sebastian is missing from a collection of letters that occultly proclaimed “Nothing is true but beauty”.

However, the phrase also sheds light on Brideshead Revisited itself. Charles Ryder discovered first Sebastian’s beauty and then Julia’s and thought he had discovered truth too. In the end he, like Waugh, concluded that he was wrong. The “beaten-copper lamp” Ryder finds burning anew in the untouched art nouveau Catholic chapel of an otherwise ruined Brideshead is of “deplorable design”, but its pure light, “shining in darkness, uncomprehended”,[17] is beautiful because it is the light of truth.

2. Scribal Waugh Fare

The avant-garde self-publicist Will Self once described the Book of Revelation as “an insemination of older, more primal verities into an as yet fresh dough of syncretism”.[18] One can see what he means, but Evelyn Waugh’s pastiche of Revelation in Decline and Fall (1928) is still much funnier. The novel’s protagonist Paul Pennyfeather is in prison talking with a religious maniac, who describes a vision he has had:

No words can describe the splendour of it. It was all crimson and wet like blood. I saw the whole prison as if it were carved of ruby… And then as I watched all the ruby became soft and wet, like a great sponge soaked in wine, and it was dripping and melting into a great lake of scarlet. […] I sometimes dream of a great red tunnel like the throat of a beast and men running down it […] and the breath of the beast is like the blast of a furnace. D’you ever feel like that?”

I’m afraid not,” said Paul. “Have they given you an interesting library book?”

Lady Almina’s Secret,” said the lion of the Lord’s elect. “Pretty soft stuff, old-fashioned too. But I keep reading the Bible. There’s a lot of killing in that.”[19]

However, the Book of Revelation isn’t always as crazy as it seems:

4:2 And immediately I was in the spirit: and, behold, a throne was set in heaven, and one sat on the throne. 3 And he that sat was to look upon like a jasper and a sardine stone: and […] a rainbow round about the throne, in sight like unto an emerald.

Rainbows that look like emeralds are crazy but priests surrounding an emerald throne are not, and the traditional image may be nothing more than a mistake by a scribe taking dictation: Greek hiereis, “priests”, was pronounced much like Greek iris, “rainbow”.[20].

When scribes were copying texts by eye rather than ear, they made other kinds of mistake, as in Romans 6:5, where two ninth-century codexes[21] have , hama, “together” against a more general , alla, “but”: two lambdas, , are easily mistaken for a mu, M. And perhaps, in the words of Peter Simple, it is a triumph of the rich human past over the tinpot scientific present[22] that more than a thousand years later, despite all advances in the manufacture of books, one can find the same kind of mistake in the Penguin edition of Waugh’s Brideshead Revisited (1945):

We went across the hall to the small drawing-room where luncheon parties used to assemble, and sat on either side of the fireplace. Julia seemed to reflect some of the crimson and gold of the walls and lose some of her warmness.[23]

If the earth is struck by an asteroid and the few copies of Brideshead that survive are in the Penguin edition, the scholars of some future resurrected civilization should be able to reconstruct the “wanness” of the manuscript (even without the assistance of an earlier line that runs “in the gloom of that room she looked like a ghost”).

Those are what are technically known as errors of permutation; elsewhere in Waugh one can match the New Testament’s errors of omission. In 1 Thessalonians 2:7, for example, the egenêthemen nêpioi or “we were mild” of later manuscripts seems to be a haplography for the egenêthemen nêpioi or “we were children” of earlier ones.[24] Many centuries later, in the Penguin edition of Helena (1950), we can find this:

Carpicius looked at him without the least awe. Two forms of pride were here irreconcilably opposed; two pigs stood face to face.[25]

Those scholars of our putative post-apocalyptic future should be able to reconstruct the original “prigs”.

NOTES

1. The collection is called Litteræ Wellensis.

2. De Musset continued “rien n’est vrai sans beauté”, “[and] nothing is true without beauty”, but the original phrase seems to have been used first by the classicist Nicolas Boileau (1636-1711), who continued “le vrai seul est aimable”, “[and] truth alone is lovable”.

3. Op. cit., Book I, “Et In Arcadia Ego”, ch. 1, pg. 33 of the 1984 Penguin paperback.

4. Op. cit., Book I, “Et In Arcadia Ego”, ch. 3, pg. 71 of the 1984 Penguin paperback.

5. Greek for “Lover of Evil”.

6. Book I, ch. LVII. Thélème is from the Greek thelema, meaning “will”.

7. Rabelais amplified it thus: parce que gens liberes, bien nez, bien instruictz, conversans en compaignies honnestes, ont par nature un instinct et aguillon, qui tousjours les poulse à faictz vertueux et retire de vice, lequel ilz nommoient honneur: “because free people, well-born, well-taught, living in honest company, have by nature a sharp instinct and spur, which prompts them always towards virtue and away from vice, and which they name honor.”

8. Op. cit., Book I, “Et In Arcadia Ego”, ch. 2, pg. 46 of the 1984 Penguin paperback.

9. See ‘Adam and Evelyn: “The Balance”, The Temple at Thatch, and 666 at http://www.lhup.edu/~jwilson3/Newsletter_33.2.htm.

10. Ch. 1, pg. 56 of the 1984 Penguin paperback.

11. Ch. 2, pg. 56 of the 1984 Penguin paperback.

12. Ch. 3, pg. 277 of the 1984 Penguin paperback.

13. Book I, “Et In Arcadia Ego”, ch. 2, pg. 47 of the 1984 Penguin paperback.

14. Book I, “Et In Arcadia Ego”, ch. 2

15. Book III, “A Twitch Upon The Thread”, ch. 2, pg. 260 of the 1984 Penguin paperback.

16. Referring to land held under impersonal or institutional control by the Church.

17. “In fragments and whispers we get news of other saints in the prison camps of Eastern and South-eastern Europe, of cruelty and degradation more savage than anything in Tudor England, of the same, pure light shining in darkness, uncomprehended”. Introduction to Edmund Campion (1935): the reference is to John v,1: And the light shineth in darkness: and the darkness did not comprehend it. (Authorized Version and Douay).

18. From Self’s introduction, pg. xii, to Revelation, Authorized Version, published in a single book by Canongate, Edinburgh, 1998.

19. Part three, chapter iii.

20. Compare the initial vowels of the English derivates “hierarchy” and “iris”.

21. Augiensis and Boernerianus: see http://www.earlham.edu/~seidti/iam/permutation.html

22. The Stretchford Chronicles: 25 Years of Peter Simple, The Daily Telegraph, Purnell & Sons, Briston, 1980, “1962: Glory”, pg. 60.

23. Op. cit., Book II, “Brideshead Deserted”, ch. 3, pg. 200 of the 1984 Penguin paperback.

24. Haplography is writing once what should be written twice: an original  (egenêthemen nêpioi), meaning “we were children”, may have lost a nu, N, and became  (egenêthemen êpioi), “we were mild”. Alternatively, it may have gained a nu in an error known as dittography, or writing twice what should be written once.

25. Op. cit., ch. 8, “Constantine’s Great Treat”, pp. 107-8 of the 1963 Penguin paperback.

Russell in Your Head-Roe (Re-Visited)

“Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.” — Bertrand Russell, The Scientific Outlook (1931)


Previously pre-posted

Russell in Your Head-Roe — Bertrand Russell on mathematics
A Ladd Inane — Bertrand Russell on solipsism
Math Matters — Bertrand Russell on math and physics
Whip Poor Wilhelm — Bertrand Russell on Friedrich Nietzsche

Farnsicht

Photo of developing ferns by the German nature photographer Karl Blossfeldt (1866-1932)
(open in new window for full image)


Post-Performative Post-Scriptum

“Farnsicht” is a pun on German Farn, meaning “fern”, and Fernsicht, meaning “view” or “visibility” (literally fern, “far”, + Sicht, “visibility”).

Z-Fall

Do you want a haunting literary image? You’ll find one of the strangest and strongest in Borges’ “La Biblioteca de Babel” (1941), which is narrated by a librarian in an infinite library. The librarian anticipates the end of his life:

Muerto, no faltarán manos piadosas que me tiren por la baranda; mi sepultura será el aire insondable; mi cuerpo se hundirá largamente y se corromperá y disolverá en el viento engenerado por la caída, que es infinita. — “La Biblioteca de Babel

When I am dead, compassionate hands will throw me over the railing; my tomb will be the unfathomable air, my body will sink for ages, and will decay and dissolve in the wind engendered by my fall, which shall be infinite. — “The Library of Babel” (translation by Andrew Hurley)

The infinite fall is the haunting image. Falling is powerful; falling for ever is more powerful still. But it can’t happen in reality: soon or later a fall has to end. Objects crash to earth or splash into the ocean. Of course, you could call being in orbit a kind of infinite fall, but it doesn’t have the same power.

However, there’s more kinds of falling than one and I think the arithmophile Borges would have liked one of the other kinds a lot. Numbers can fall — you sum their digits, take the sum from the original number, and repeat. That is, n = n – digsum(n). Here are some examples:


10 → 9 → 0
100 → 99 → 81 → 72 → 63 → 54 → 45 → 36 → 27 → 18 → 9 → 0
1000 → 999 → 972 → 954 → 936 → 918 → 900 → 891 → 873 → 855 → 837 → 819 → 801 → 792 → 774 → 756 → 738 → 720 → 711 → 702 → 693 → 675 → 657 → 639 → 621 → 612 → 603 → 594 → 576 → 558 → 540 → 531 → 522 → 513 → 504 → 495 → 477 → 459 → 441 → 432 → 423 → 414 → 405 → 396 → 378 → 360 → 351 → 342 → 333 → 324 → 315 → 306 → 297 → 279 → 261 → 252 → 243 → 234 → 225 → 216 → 207 → 198 → 180 → 171 → 162 → 153 → 144 → 135 → 126 → 117 → 108 → 99 → 81 → 72 → 63 → 54 → 45 → 36 → 27 → 18 → 9 → 0

The details are different in other bases, like 2 or 16, but the destination is the same. The number falls to zero and the fall stops, because digsum(0) = 0:


102 → 1 → 0 (n=2)
100 → 11 → 1 → 0 (n=4)
1000 → 111 → 100 → 11 → 1 → 0 (n=8)
10000 → 1111 → 1011 → 1000 → 111 → 100 → 11 → 1 → 0 (n=16)
100000 → 11111 → 11010 → 10111 → 10011 → 10000 → 1111 → 1011 → 1000 → 111 → 100 → 11 → 1 → 0 (n=32)
1000000 → 111111 → 111001 → 110101 → 110001 → 101110 → 101010 → 100111 → 100011 → 100000 → 11111 → 11010 → 10111 → 10011 → 10000 → 1111 → 1011 → 1000 → 111 → 100 → 11 → 1 → 0 (n=64)


1013 → C → 0 (n=13)
100 → CC → B1 → A2 → 93 → 84 → 75 → 66 → 57 → 48 → 39 → 2A → 1B → C → 0 (n=169)
1000 → CCC → CA2 → C84 → C66 → C48 → C2A → C0C → BC1 → BA3 → B85 → B67 → B49 → B2B → B10 → B01 → AC2 → AA4 → A86 → A68 → A4A → A2C → A11 → A02 → 9C3 → 9A5 → 987 → 969 → 94B → 930 → 921 → 912 → 903 → 8C4 → 8A6 → 888 → 86A → 84C → 831 → 822 → 813 → 804 → 7C5 → 7A7 → 789 → 76B → 750 → 741 → 732 → 723 → 714 → 705 → 6C6 → 6A8 → 68A → 66C → 651 → 642 → 633 → 624 → 615 → 606 → 5C7 → 5A9 → 58B → 570 → 561 → 552 → 543 → 534 → 525 → 516 → 507 → 4C8 → 4AA → 48C → 471 → 462 → 453 → 444 → 435 → 426 → 417 → 408 → 3C9 → 3AB → 390 → 381 → 372 → 363 → 354 → 345 → 336 → 327 → 318 → 309 → 2CA → 2AC → 291 → 282 → 273 → 264 → 255 → 246 → 237 → 228 → 219 → 20A → 1CB → 1B0 → 1A1 → 192 → 183 → 174 → 165 → 156 → 147 → 138 → 129 → 11A → 10B → CC → B1 → A2 → 93 → 84 → 75 → 66 → 57 → 48 → 39 → 2A → 1B → C → 0 (n=2197)

But the fall to 0 made me think of another kind of number-fall. What if you count the 0s in a number, take that count away from the original number, and repeat? You could call this a z-fall (pronounced zee-fall). But unlike free-fall, z-fall doesn’t last long:


10 → 9
100 → 98
1000 → 997
10000 → 9996

And the number always comes to rest far above the ground, as it were. In a fall using digsum(n), the number descends to 0. In a fall using zerocount(n), the number never even reaches 1. At least, never in any base higher than 2. But in base-2, you get this:


10 → 1 (n=2)
100 → 10 → 1 (n=4)
1000 → 101 → 100 → 10 → 1 (n=8)
10000 → 1100 → 1010 → 1000 → 101 → 100 → 10 → 1 (n=16)
100000 → 11011 → 11010 → 11000 → 10101 → 10011 → 10001 → 1110 → 1101 → 1100 → 1010 → 1000 → 101 → 100 → 10 → 1 (n=32)
1000000 → 111010 → 111000 → 110101 → 110011 → 110001 → 101110 → 101100 → 101001 → 100110 → 100011 → 100000 → 11011 → 11010 → 11000 → 10101 → 10011 → 10001 → 1110 → 1101 → 1100 → 1010 → 1000 → 101 → 100 → 10 → 1 (n=64)

When I saw that, I had a wonderful vision of how even the biggest numbers in base 2 could z-fall all the way to 1. Almost all binary numbers contain 0, after all. So the z-falls would get longer and longer, paying tribute to la caída infinita, the infinite fall, of the librarian in Borges’ Library of Babel. Alas, binary numbers don’t behave like that. The highest number in base 2 that z-falls to 1 is this:


1010001 → 1001101 → 1001010 → 1000110 → 1000010 → 111101 → 111100 → 111010 → 111000 → 110101 → 110011 → 110001 → 101110 → 101100 → 101001 → 100110 → 100011 → 100000 → 11011 → 11010 → 11000 → 10101 → 10011 → 10001 → 1110 → 1101 → 1100 → 1010 → 1000 → 101 → 100 → 10 → 1 (n=81)

Above that, binary numbers land on what you might call a shelf:


1010010=82 → 1001110=78 → 1001011=75 → 1001000=72 → 1000011=67 → 111111=63 (n=82)

If binary numbers are an infinite tall mountain, 1 is at the foot of the mountain. 111111 = 63 is like a shelf a little way above the foot. But I conjecture that arbitrarily large binary numbers will z-fall to 63. For example, no matter how large the power of 2, I conjecture that it will z-fall to 63:


10 → 1 : 2 → 1 (count of steps=2)
100 ... → 1 : 4 ... → 1 (c=3)
1000 ... → 1 : 8 ... → 1 (c=5)
10000 ... → 1 : 16 ... → 1 (c=8)
100000 ... → 1 : 32 ... → 1 (c=16)
1000000 ... → 1 : 64 ... → 1 (c=27)
10000000 ... → 111111 : 128 ... → 63 (c=21)
100000000 ... → 111111 : 256 ... → 63 (c=60)
1000000000 ... → 111111 : 512 ... → 63 (c=130)
10000000000 ... → 111111 : 1024 ... → 63 (c=253)
100000000000 ... → 111111 : 2048 ... → 63 (c=473)
1000000000000 ... → 111111 : 4096 ... → 63 (c=869)
10000000000000 ... → 111111 : 8192 ... → 63 (c=1586)
100000000000000 ... → 111111 : 16384 ... → 63 (c=2899)
1000000000000000 ... → 111111 : 32768 ... → 63 (c=5327)
10000000000000000 ... → 111111 : 65536 ... → 63 (c=9851)
100000000000000000 ... → 111111 : 131072 ... → 63 (c=18340)
1000000000000000000 ... → 111111 : 262144 ... → 63 (c=34331)
10000000000000000000 ... → 111111 : 524288 ... → 63 (c=64559)
100000000000000000000 ... → 111111 : 1048576 ... → 63 (c=121831)
1000000000000000000000 ... → 111111 : 2097152 ... → 63 (c=230573)
10000000000000000000000 ... → 111111 : 4194304 ... → 63 (c=437435)
100000000000000000000000 ... → 111111 : 8388608 ... → 63 (c=831722)
1000000000000000000000000 ... → 111111 : 16777216 ... → 63 (c=1584701)
10000000000000000000000000 ... → 111111 : 33554432 ... → 63 (c=3025405)
100000000000000000000000000 ... → 111111 : 67108864 ... → 63 (c=5787008)
1000000000000000000000000000 ... → 111111 : 134217728 ... → 63 (c=11089958)
10000000000000000000000000000 ... → 111111 : 268435456 ... → 63 (c=21290279)
100000000000000000000000000000 ... → 111111 : 536870912 ... → 63 (c=40942711)
1000000000000000000000000000000 ... → 111111 : 1073741824 ... → 63 (c=78864154)

So the z-falls get longer and longer. But z-falling to 63 doesn’t have the power of z-falling to 1.

Period Panes

In The Penguin Dictionary of Curious and Interesting Numbers (1987), David Wells remarks that 142857 is “a number beloved of all recreational mathematicians”. He then explains that it’s “the decimal period of 1/7: 1/7 = 0·142857142857142…” and “the first decimal reciprocal to have maximum period, that is, the length of its period is only one less than the number itself.”

Why does this happen? Because when you’re calculating 1/n, the remainders can only be less than n. In the case of 1/7, you get remainders for all integers less than 7, i.e. there are 6 distinct remainders and 6 = 7-1:

(1*10) / 7 = 1 remainder 3, therefore 1/7 = 0·1...
(3*10) / 7 = 4 remainder 2, therefore 1/7 = 0·14...
(2*10) / 7 = 2 remainder 6, therefore 1/7 = 0·142...
(6*10) / 7 = 8 remainder 4, therefore 1/7 = 0·1428...
(4*10) / 7 = 5 remainder 5, therefore 1/7 = 0·14285...
(5*10) / 7 = 7 remainder 1, therefore 1/7 = 0·142857...
(1*10) / 7 = 1 remainder 3, therefore 1/7 = 0·1428571...
(3*10) / 7 = 4 remainder 2, therefore 1/7 = 0·14285714...
(2*10) / 7 = 2 remainder 6, therefore 1/7 = 0·142857142...

Mathematicians know that reciprocals with maximum period can only be prime reciprocals and with a little effort you can work out whether a prime will yield a maximum period in a particular base. For example, 1/7 has maximum period in bases 3, 5, 10, 12 and 17:

1/21 = 0·010212010212010212... in base 3
1/12 = 0·032412032412032412... in base 5
1/7 =  0·142857142857142857... in base 10
1/7 =  0·186A35186A35186A35... in base 12
1/7 =  0·274E9C274E9C274E9C... in base 17

To see where else 1/7 has maximum period, have a look at this graph:

Period pane for primes 3..251 and bases 2..39


I call it a “period pane”, because it’s a kind of window into the behavior of prime reciprocals. But what is it, exactly? It’s a graph where the x-axis represents primes from 3 upward and the y-axis represents bases from 2 upward. The red squares along the bottom aren’t part of the graph proper, but indicate primes that first occur after a power of two: 5 after 4=2^2; 11 after 8=2^3; 17 after 16=2^4; 37 after 32=2^5; 67 after 64=2^6; and so on.

If a prime reciprocal has maximum period in a particular base, the graph has a solid colored square. Accordingly, the purple square at the bottom left represents 1/7 in base 10. And as though to signal the approval of the goddess of mathematics, the graph contains a lower-case b-for-base, which I’ve marked in green. Here are more period panes in higher resolution (open the images in a new window to see them more clearly):

Period pane for primes 3..587 and bases 2..77


Period pane for primes 3..1303 and bases 2..152


An interesting pattern has begun to appear: note the empty lanes, free of reciprocals with maximum period, that stretch horizontally across the period panes. These lanes are empty because there are no prime reciprocals with maximum period in square bases, that is, bases like 4, 9, 25 and 36, where 4 = 2*2, 9 = 3*3, 25 = 5*5 and 36 = 6*6. I don’t know why square bases don’t have max-period prime reciprocals, but it’s probably obvious to anyone with more mathematical nous than me.

Period pane for primes 3..2939 and bases 2..302


Period pane for primes 3..6553 and bases 2..602


Like the Ulam spiral, other and more mysterious patterns appear in the period panes, hinting at the hidden regularities in the primes.

Parlez-vous franchat?

French novelist Colette was a firm cat-lover. When she was in the U.S. she saw a cat sitting in the street. She went over to talk to it and the two of them mewed at each other for a friendly minute. Colette turned to her companion and exclaimed, “Enfin! Quelqu’un qui parle français.” (At last! Someone who speaks French!) — viâ Cat Ladies and a book whose title I forget