Triangular number congruence patterns proven

Kevin McCall, whose name you may recognize because his thoughts on dice and the Tarot (qv) have appeared on this blog in the past, has worked out a proof of the congruence patterns in the series of triangular numbers which I postulated here. He has proven that the series of triangular numbers reduced modulo k repeats itself, that the repetition has a period of k (if k is odd) or 2k (if k is even), and that the repeating series is always palindromic.

I have only seen his proof of the first of those statements; I am not going to look at the remainder of his proof until after I have proven it myself independently -- at which point I will post both his proof and mine.

A.E.'s "language of the gods"

Sunset, by A.E.

I first encountered the Irish mystic George William Russell, who wrote under the pseudonym A.E., in the pages of Ulysses, where Stephen Dedalus borrows money from him and then quips, "A.E.I.O.U." I thought that was a pretty good pun but, compared to some of the other work of the greatest punster ever to walk the face of the earth, nothing spectacular.

Only recently (as in this week) have I actually gotten around to reading any of A.E.'s work. I found that in The Candle of Vision he makes the letters of the alphabet an object of mystical contemplation, considers the vowels separately from the consonants, and takes pains to get the letters in what he considers to be the correct order. Given that the Candle was published just two years before Ulysses, and that the two authors knew each other, I can only conclude that the allusion to A.E.'s alphabetical mysticism was deliberate.

So, Joyce, I apologize for having underestimated you here. Despite everything (and there's a lot to forgive!), you were, in your own way, God.

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On pp. 116-118, A.E. explains his project of "brooding upon the significance of separate letters":

I was first led to brood upon the elements of human speech by that whisper of the word "Aeon" out of the darkness, for among many thoughts I had at the time came the thought that speech may originally have been intuitive. I discarded the idea with regard to that word, but the general speculation remained with me, and I recurred to it again and again, and began brooding upon the significance of separate letters, and had related many letters to abstractions or elements . . . . I then began to rearrange the roots of speech in their natural order from throat sounds, through dentals to labials, from A which begins to be recognisable in the throat to M in the utterance of which the lips are closed. An intellectual sequence of ideas became apparent. This encouraged me to try and complete the correspondences arrived at intuitively. I was never able to do this. Several sounds failed, however I brooded upon them, to suggest their intellectual affinities, and I can only detail my partial discoveries . . . .

In the following chapter, he lays out these partial discoveries, which may be summarized thus:

1. A: the self, God; a circle
2. R: motion; red; a vertical line
3. H: heat; orange; a triangle
4. L: fire, light, radiation; a shape like an upside-down Y
5. Y: binding, concentration, condensation, gravitation, the will; yellow; a triskelion
6. W: liquidity, water; green; the lower half of a circle, a smile-like shape
7. G: earth; a square
8. K: mineral, rock, crystal, hardness; a square crossed by a diagonal, so oriented that the diagonal is vertical
9. S: impregnation, inbreathing, insouling, the genesis of the cell; a circle with a horizontal line through the center, like the letter theta
10. Z: multiplication, division, reproduction; a circle with a cross in it, like the astronomical symbol for Earth
11. TH: growth, expansion, swelling
12. SH: scattering, dissolution, decay
13. T: individual action, movement, initiative, ego, extroversion; a symmetrical cross, like a plus sign
14. D: silence, sleep, immobility, abeyance, inwardness; the upper half of a circle, with a horizontal line joining its ends, like the letter D rotated 90 degrees counterclockwise
15. J: --
16. TCH: --
17. V: life in water, all that swims; blue
18. F: what lives in air and flies; blue
19. P: masculine life, paternity; indigo
20. B: feminine life, maternity; indigo
21. N: continuance of being, immortality; violet
22. M: close, limit, measure, end, death; violet

A.E. writes, "In all there are twenty-one consonants which with the vowels make up the divine roots of speech. . . . I despair of any attempt to differentiate from each other the seven states of consciousness represented by the vowels." I am not sure why A alone is discussed together with the consonants. At first I had thought it must, like the Hebrew aleph, represent a glottal stop, but A.E. clearly says that there are 21 consonants, which means A is a vowel.

The selection of sounds, and the order in which he puts them, are highly idiosyncratic. These 22 sounds correspond neither to any alphabet I know of nor to the phoneme inventory of any language with which I am familiar. It appears to be based on the sounds of English, as understood by someone deeply ignorant of phonetics. Many native speakers of English do not realize that there are two different "th" sounds (as in thy and thigh, respectively), that the "s" sound in vision is a distinct sound, or that "ng" is not just a combination of the "n" and "g" sounds.

The placing of R in second place implies that A.E. used a uvular ("guttural") "r" sound, as in French. According to Wikipedia, "A guttural/uvular [ʁ] is found in north-east Leinster. Otherwise, the rhotic consonant of virtually all other Irish accents is the postalveolar approximant, [ɹ]." A.E. was from County Armagh, adjoining north-east Leinster, so it is possible that he spoke this way -- or it could be just another sign of his general confusion regarding where in the mouth different consonants are articulated.

The consonants from 7 to 22 appear in pairs, often but not always representing voiced and voiceless versions of the same sound. Sometimes the voiced sound comes first (G-K, J-TCH, V-F), and sometimes the voiceless (S-Z, T-D, P-B). TH and SH are paired, presumably because A.E. was not aware that each had its own voiced counterpart, and because these two "orphan" fricatives seemed vaguely similar. N and M are paired because they are the only nasal consonants of which he was aware. I surmise that A.E. had no knowledge of the voiced/voiceless distinction but simply put together those consonants that seemed intuitively to be similar in sound.

Despite his use of such terms as "dental" and "labial," and despite his account of how he brooded over these sounds, repeating them to himself again and again, A.E. seems not to have been very clear on what was going on in his mouth when he pronounced them. The sounds are supposed to be organized according to place of articulation, from back to front, but there are many puzzling exceptions. L is an alveolar sound, articulated in the same place as T and D, but it is placed far in the back. When he repeated the L sound to himself, he must have been saying "ull, ull, ull" -- giving the consonant its "dark," velarized pronunciation, rather than the "clear" pronunciation found in "luh, luh, luh." The "dark l" sound is unusual in Hiberno-English but apparently does occur in some Ulster dialects.

The placing of the palatal sound Y in the back, behind the velars, is incorrect but understandable. Since the tongue only comes close to the hard palate without actually touching it, the sound's palatal character is not easy to discover by self-observation. In fact, we can see that A.E. put all approximants in the back, regardless of place of articulation, perhaps because it was not easy to observe how they were pronounced.

The interdental TH and the post-alveolar SH, J, and TCH are also also misplaced, less understandably. (I assume this is tch as in Tchaikovsky, thus spelt to differentiate it from ch as in Bach.) If you pronounce TH, S, and SH in succession, I think it's pretty easy to observe the tongue moving from the front of the mouth toward the back.

But the most obvious exception is N, which is placed with the labial consonants apparently on the strength of its similarity to M.

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These amateurish errors, together with the incompleteness of the mappings, demonstrate the sincerity with which the project was carried out. I have not the slightest doubt that A.E. did just what he said he did: brooded over these sounds for a long time and wrote down only those correspondences which were confirmed by intuition. Sometimes he didn't get anything (as with TCH and J), and other times his results were only tentative. (Although my summary does not show it, many of the correspondences are qualified with "I think," or "it vaguely suggested itself to me.")

On one level, this whole list of correspondences seems to be obvious nonsense. Aside from the linguistic difficulties, what does it even mean to say that R corresponds to motion, the color red, and a vertical line? What would follow from such a statement's being true or false? It seems like a classic example of an assertion that is (as Wolfgang Pauli would put it) "not even false." I am reminded of Valentin Tomberg's statement that the traditional planet-metal mappings of astrology (Sun to gold, Moon to silver, Venus to copper, etc.) have been confirmed for him time and again by experience. What sort of experience, I wondered when I read that, could possibly confirm such a thing? What properties of tin unfolded themselves to his understanding when he reflected on the "fact" that tin corresponds to the planet Jupiter? Or what light did tin shed on the nature of Jupiter?

Some of A.E.'s mappings did ring true, but it's hard to be sure why. For example, when I read about F ("what lives in air and flies"), I thought of English fly, flap, flutter, fowl, and feather; of Chinese fēi and Hebrew 'af, both meaning "fly"; and of T. H. White's geese singing "Free, free: far, far: and fair on wavering wings" -- but of course this is just picking cherries, and most flight-related words in most languages do not feature the "f" sound. Does the abundance of such words in English mean that Anglophones have historically been particularly sensitive to the "true" meaning of their consonants? Do Anglophones think "better" or more clearly about flight because they call it flight? Is that why it was the English-speaking Wright brothers who invented the airplane? (On the other hand, the Montgolfier and Breguet brothers were French, and the great American ornithologist Audubon was born Jean Rabin in French-speaking Saint-Domingue.) Or, more likely, are A.E.'s intuitions unconsciously influenced by the vocabulary of his native language? One feels that the influence of English is also at play in A.E.'s mapping of R to red and Y to yellow. (Both of those mappings were found to be particularly common in Sean Day's survey of 43 colored-letter synesthetes, qv; Day regrettably neglected to record the languages spoken by his subjects.)

Still, despite these very deep misgivings, both about the general meaningfulness of the questions A.E. was asking and about the validity of his specific answers, I found that I reacted to many of his mappings with delight, and with a sense that, somehow and for some reason, his project was after all worth doing. I suppose what I am reacting to, more than to anything specific, is the general attitude of taking things seriously, of refusing to take arbitrariness and meaninglessness as the null hypothesis.

⁂

Incidentally, the bit about the alphabet is only a few pages long; the bulk of The Candle of Vision is a serious and sustained meditation of the phenomena of imagination and clairvoyance and on what they mean. It's definitely something I'll be rereading.

Patterns in the digits of triangular numbers

Here's another way of saying what I said in my post on congruence patterns in the series of triangular numbers.

List the triangular numbers, starting with 0. The final digits of the numbers in the series will exhibit a repeating pattern with a period of 20, and the pattern will be palindromic. The penultimate (tens-place) digits will also exhibit a repeating palindromic pattern, with a period of 200. The repeating palindrome for the hundreds-place digits will have a period of 2000, and so on for any "place" you care to choose.

This is assuming you write the numbers in the familiar decimal system -- but any number base will yield corresponding results. If the triangular numbers are written in base b, the pth-to-last digit will exhibit a repeating palindromic pattern with a period of b^p (if b is odd) or 2b^p (if b is even).

(Oswald Wirth, whose writings first alerted me to the existence of such patterns, was doing what he called "Theosophic reduction" -- i.e., adding up the digits of a number until a single digit is arrived at. "Theosophically reducing" a number in base b corresponds to taking its final digit in base b - 1.)

The patterns are easiest to see in very small bases such as binary and ternary, shown below with color-coding to highlight the repeating palindromes.

I have still not figured out why these patterns exist. Even stating the pattern algebraically is proving somewhat difficult for someone whose mathematical training is as limited as mine.

Hungry Eyes

I've always like the Eric Carmen song "Hungry Eyes" (despite having once had a weird nightmare vision triggered by it, as I shall perhaps relate in another post) -- but the English teacher in me cringes every time I hear the repeated line, "I feel the magic between you and I."

Today it suddenly occurred to me that, since it obviously can't be "between you and I," the line in question must actually be "between U and I" -- i.e., between the 21st and 9th letters of the alphabet -- and the letter located precisely halfway between those two is the 15th letter, O.

The Latin letter O looks identical to, and is descended from, the Phoenician letter ʿayin -- whose name means "eye" (cf. the coincidentally similar archaic English plural eyen).

The Chinese word for "hungry" is 餓 -- which, in the Wade-Giles system of romanization, would be transcribed o.

I'm truly sorry

G. at he Junior Ganymede blog has been boasting about having originated "one of the worst puns ever written," so I think it's time to put him in his place.

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A long time ago in Russia there was a an ancient church famed far and wide for its beautiful art and architecture. One night a group of fanatical iconoclasts broke into the church and vandalized it, shocking all of Russia. While many parts of the church were damaged, the act that generated the greatest public outrage was the wanton destruction of the priceless and irreplaceable 10th century carvings of angels that had one adorned the large stone baptismal font. The font itself was still intact but was now bereft of all decoration.

The iconoclasts were apprehended, and such was the notoriety of the case that it came to the attention of the Tsar himself. He ordered that the perpetrators be temporarily locked up while he took his time thinking up a truly suitable punishment for such an enormous crime against art and religion. Finally, perhaps drawing inspiration from Dante, he decided that the vandals would be chained together, crammed into the very font they had vandalized, and "baptized" with boiling water. Despite protests from some that this punishment would be just as much a sacrilege as the vandalism itself, the Tsar stood firm.

However, it happened that one of the iconoclasts came from a wealthy and influential family, and his parents came before the Tsar and begged that their wayward son be shown mercy. The Tsar was a hard man, but such was their persuasiveness that in the end he agreed to modify his original sentence: All the other perpetrators would be tortured as planned, but their son would instead be exiled to Georgia for life, never to return to Russia on pain of death. The tearful parents thanked him for his kindness, and the Tsar sent the modified sentence to the local magistrate to be executed.

But imagine the parents' horror when, on the day of the punishment, their son was chained up in the baptistry and tortured together with his partners in crime! When they sent to the Tsar for an explanation, they found that he was just as astonished as they were. He was adamant that he had sentenced their son to exile only, but the magistrate was just as adamant that he had received orders to punish all the malefactors in the same way. Obviously the sentence had been changed by some unknown hand on its way to the magistrate. The Tsar ordered an investigation into the matter, and it was discovered that the Tsar's own copy editor was to blame. The copy editor confessed but insisted that it had all been a misunderstanding.

"A misunderstanding?" bellowed the Tsar. "How could you, a mere copy editor, possibly think you had the right to change a punishment decreed by the Tsar himself?"

"Begging Your Imperial Majesty's pardon," said the copy editor, "I naturally thought it was just a mistake when I saw that that one sentence was in Georgia while all the other sentences were in the same sans-seraph font."

Texaco's Fire Chief gasoline

As so often, reading about strange coincidences seems to attract them. As I mentioned in my Copper State post, I have been reading the conspiracy classic "King Kill 33" by James Shelby Downard, which ostensibly argues that the Freemasons killed JFK but is in fact an extremely thoroughgoing catalogue of impressive and not-so-impressive coincidences, only a small fraction of which could be explained by a Masonic conspiracy even in principle. Today, for example, I read this.

[Elm Street in Dallas, Texas,] was also the home of the Blue-Front tavern, a Masonic hangout in the grand tradition of "tavern-Masonry." . . . The Blue-Front was the site of the "broken-man" ritual in which various members of the "Brotherhood of the Broom" swept the floor and tended some fierce javelino pigs. The Blue-Front was once a firehouse and was still sporting the pole in the late '20s. This is extremely germane symbolism. The national offices of the Texaco oil corporation are located on Elm Street, Dallas. Its chief products are "Haviland (javelino) oil" and "Fire Chief" gasoline.

As you can see, this is all pretty bizarre even by conspiracy-theory standards. I mean, suppose you were a Masonic Grand Master responsible for orchestrating a presidential assassination. Obviously, the first order of business would be to make sure the president's motorcade passed through a street that was home to a company whose chief products had names reminiscent of random facts about a Masonic hangout that once existed on the same street! Apparently, among Masons this is what passes for leaving one's calling card at the crime scene. What Downard is doing isn't a conspiracy theory; it's just random coincidence-noticing of the sort to which I am myself addicted.

Anyway, when I read this, I thought, "I know Havoline motor oil, which must be what he means, but Fire Chief? I've been to a few Texaco gas stations but never noticed that name."

While I was reading Downard, my wife was watching TV -- one of those reality shows about very fat people. Just minutes after reading about Fire Chief gasoline, I happened to glance up at the TV and saw that the very fat woman who is the star of the show had just ordered a genuine vintage Texaco Fire Chief gas pump and had it delivered to her house.

Those Masons!

A rationale for Waite's orientation of ROTA

Yes, yet another post (see the others) on this very minor bit of Tarot iconography.

In a past post (qv), I wrote that "of the four possible orientations of the ROTA, the only one I can't think of any good reason for is the one actually used by Lévi and Waite, with T at the top!" Now a possible rationale for it has occurred to me.

Notice that the wheel is apparently turning counterclockwise: the snake on the left is descending, and the cynocephalus on the right is ascending. So the letters ROTA correspond, respectively, to the low point, the descent, the high point, and the ascent. Lévi had already identified the A and O of ROTA as corresponding to alpha and omega -- representing the beginning and the end. It seemed to me that the T and R ought to correspond to some similarly fundamental opposition, but nothing came to mind until today. T and R correspond to the Hebrew words tov (טוֹב, "good") and ra (רַע "bad/evil"). Obviously, the top of the wheel represents good fortune; and the bottom, bad fortune. One starts at the bottom, rises (alpha, the beginning), reaches the top, and then falls (omega, the end).