From the Narrow Desert
by William James Tychonievich
Tuesday, February 25, 2020
Cat bitten by radioactive spider
This is Geronimo, one of my home's nine or ten resident felines. I haven't the slightest clue how he managed to get up on top of this tchotchke cabinet, which is a sheer vertical face with no protruding shelves or anything to use as stepping stones. I think he forgot how he did it, too, since I had to use a stepladder to get him back down.
Thursday, February 20, 2020
Captivity and power: Man, he never had a chance
Content warning: In this post I discuss (among other things) what I consider to be an artistically very effective use of what those parents' guides would call an "fword derivative" in a punk rock music video. If that's not the sort of thing you want to expose yourself to, skip this one.
Tuesday, February 18, 2020
The captivity and power of the devil
Here are some of the very naturalsounding phrases that come up if you search the Bible for the string "and power": strength and power, force and power, spirit and power, authority and power, faith and power, honour and power, dominion and power, glory and power.
And here's a line from the Book of Mormon which I've been brooding over recently.
In the past I always thought that, while the syntax may be a bit infelicitous, the meaning is clear enough. The "power of the devil" refers to the power which the devil himself possesses, while the "captivity of the devil" indicates other people's being in captivity to the devil. Only a tedious grammar pedant (something no one would ever accuse me of being!) would read it any other way.
After giving it some thought, though, I think that there are in fact good reasons, above and beyond overliteralism, for reading this passage as stating that both the devil and those who follow him are in the condition described by the seemingly oxymoronic conjunction "captivity and power."
The logic is simple enough. The devil tempts people to be evil and sinful, and is himself evil and sinful; therefore, whatever condition the devil himself is in, those who fall into his snare will tend toward that same condition. If Being Evil has given the devil great power, we can assume that people can also acquire great power by Being Evil. Likewise, if sin leads to captivity, we can assume that the devil, as the sinner par excellence, is also in a state of captivity. In other words, it doesn't make sense that the very same course of action should lead to power when pursued by the devil but only to captivity when pursued by anyone else.
Against this line of reasoning, there is the possibility that the archtempter is not also the archsinner  that, like any reasonably competent pusher, the devil knows better than to get high on his own supply  that he suckers people into committing sins that he himself isn't stupid enough to commit, thus bringing them into captivity while maintaining his own freedom.
There clearly has to be some truth to this. The devil can't possibly be the exemplar of every vice in the same way that God is arguably the exemplar of every virtue. A slothful devil couldn't be bothered to actually tempt people, for example, while a cowardly one would never have defied God in the first place. One of the litany of names applied to the devil in Revelation 12:9 is "the great," and I think we must concede that there is a sense in which he lives up to that title. If the devil were a mere nogoodnik, a congeries of vices, a contemptible sinridden fleabag of a spirit, he would be of no account, and there would be no need for us to so much as take notice of his existence. Fallen angels do not become vermin but dragons, roaring lions seeking whom they may devour.
But for all that, there is also a sense in which the devil is contemptible. As Lehi puts it in the Book of Mormon passage quoted above, "he seeketh that all men might be miserable like unto himself." I think in the end we must insist on the literal aptness of the phrase captivity and power.
To try to get a handle on this, I tried to think of other instances of "captivity and power" occurring together, and the first example that came to mind was the beast of burden. A draft ox is an immensely powerful animal, a ton and a half of pure muscle, but it nevertheless lives in captivity. Or, considering political power rather than muscular strength, we might think of a tyrant, reigning with blood and horror, hated by his people and obeyed out of fear alone. How much freedom does such a man, living under the constant threat of assassination or revolution, really have? What choice does he have to bar himself up in a virtual prison, surrounded by guards? What choice does he have but to rule with conspicuous brutality, lest any show of weakness embolden his enemies? Or, coming closer to our Satanic theme, we might consider anyone who has made a Faustian bargain of the kind reportedly offered to Jesus: "All these things will I give thee, if thou wilt fall down and worship me" (Matthew 4:9). "The captivity and power of the devil": The devil offers power, but only on condition of captivity.
What the ox, the tyrant, and the Faustian soulseller have in common is that, while they have the ability to do things that others cannot do (power), their ability to decide which things to do (liberty) is severely curtailed. Marlowe's Faustus at first wants the devil to reveal to him the secrets of the universe  only to find out that, sorry, that's not one of the things that sold souls are permitted to pursue. In the end, he is reduced to frittering away his Satanic powers on trifles and degradation  playing practical jokes, trying to sleep with Helen of Troy, that kind of thing. Captivity and power.
It may be readily observed that, judging by what economists would call "revealed preferences," most people obviously don't really want to be saints, either  but at least it is possible to want to be a saint, to want good and nothing but good, and a great many of us at least want to want to be saints. We can imagine gradually purifying our hearts and our desires until "we have no more disposition to do evil, but to do good continually" (Mosiah 5:2). The reverse  wanting evil and nothing but evil  is not really possible and cannot be coherently imagined. No one, not even the devil himself, can love sin as such completely and unreservedly; all we can do is love certain aspects of sin and try not to think too much about the others. The devil's power is captivity because, in the last analysis, it is only the power to obtain what no one could ever really want.
To close with another Book of Mormon quote, Samuel the Lamanite told the wicked Nephites, "your destruction is made sure; yea, for ye have sought all the days of your lives for that which ye could not obtain"  namely "ye have sought for happiness in doing iniquity" (Helaman 13:38). That's the bottom line. Those who follow the devil are seeking what they simply cannot, by the nature of things, ever obtain. It is in that sense that the devil is the "father of lies," and that his power  the power he offers, and the power he himself possesses  is really only captivity.
And here's a line from the Book of Mormon which I've been brooding over recently.
[Men] are free to choose liberty and eternal life, through the great Mediator of all men, or to choose captivity and death, according to the captivity and power of the devil; for he seeketh that all men might be miserable like unto himself (2 Nephi 2:27; emphasis added).Captivity and power. How does that work? Isn't captivity a lack of power?
In the past I always thought that, while the syntax may be a bit infelicitous, the meaning is clear enough. The "power of the devil" refers to the power which the devil himself possesses, while the "captivity of the devil" indicates other people's being in captivity to the devil. Only a tedious grammar pedant (something no one would ever accuse me of being!) would read it any other way.
⁂
After giving it some thought, though, I think that there are in fact good reasons, above and beyond overliteralism, for reading this passage as stating that both the devil and those who follow him are in the condition described by the seemingly oxymoronic conjunction "captivity and power."
The logic is simple enough. The devil tempts people to be evil and sinful, and is himself evil and sinful; therefore, whatever condition the devil himself is in, those who fall into his snare will tend toward that same condition. If Being Evil has given the devil great power, we can assume that people can also acquire great power by Being Evil. Likewise, if sin leads to captivity, we can assume that the devil, as the sinner par excellence, is also in a state of captivity. In other words, it doesn't make sense that the very same course of action should lead to power when pursued by the devil but only to captivity when pursued by anyone else.
⁂
Against this line of reasoning, there is the possibility that the archtempter is not also the archsinner  that, like any reasonably competent pusher, the devil knows better than to get high on his own supply  that he suckers people into committing sins that he himself isn't stupid enough to commit, thus bringing them into captivity while maintaining his own freedom.
There clearly has to be some truth to this. The devil can't possibly be the exemplar of every vice in the same way that God is arguably the exemplar of every virtue. A slothful devil couldn't be bothered to actually tempt people, for example, while a cowardly one would never have defied God in the first place. One of the litany of names applied to the devil in Revelation 12:9 is "the great," and I think we must concede that there is a sense in which he lives up to that title. If the devil were a mere nogoodnik, a congeries of vices, a contemptible sinridden fleabag of a spirit, he would be of no account, and there would be no need for us to so much as take notice of his existence. Fallen angels do not become vermin but dragons, roaring lions seeking whom they may devour.
But for all that, there is also a sense in which the devil is contemptible. As Lehi puts it in the Book of Mormon passage quoted above, "he seeketh that all men might be miserable like unto himself." I think in the end we must insist on the literal aptness of the phrase captivity and power.
⁂
To try to get a handle on this, I tried to think of other instances of "captivity and power" occurring together, and the first example that came to mind was the beast of burden. A draft ox is an immensely powerful animal, a ton and a half of pure muscle, but it nevertheless lives in captivity. Or, considering political power rather than muscular strength, we might think of a tyrant, reigning with blood and horror, hated by his people and obeyed out of fear alone. How much freedom does such a man, living under the constant threat of assassination or revolution, really have? What choice does he have to bar himself up in a virtual prison, surrounded by guards? What choice does he have but to rule with conspicuous brutality, lest any show of weakness embolden his enemies? Or, coming closer to our Satanic theme, we might consider anyone who has made a Faustian bargain of the kind reportedly offered to Jesus: "All these things will I give thee, if thou wilt fall down and worship me" (Matthew 4:9). "The captivity and power of the devil": The devil offers power, but only on condition of captivity.
What the ox, the tyrant, and the Faustian soulseller have in common is that, while they have the ability to do things that others cannot do (power), their ability to decide which things to do (liberty) is severely curtailed. Marlowe's Faustus at first wants the devil to reveal to him the secrets of the universe  only to find out that, sorry, that's not one of the things that sold souls are permitted to pursue. In the end, he is reduced to frittering away his Satanic powers on trifles and degradation  playing practical jokes, trying to sleep with Helen of Troy, that kind of thing. Captivity and power.
⁂
But if the "captivity and power of the devil" simply refers to the fact that the devil offers power with strings attached, power that can only be used in certain ways and for certain ends  well, doesn't the "liberty and eternal life" offered by God also come with restrictions? Even more restrictions than the devil's offer, in fact, since there are many more ways of being evil than of being good, just as there are many more ways of dying than of staying alive. In the same vein, Tolstoy famously observed (in words that I dare not attempt to translate, for fear of angering the ghost of Nabokov, but will paraphrase) that happy families are all alike, while each unhappy family is unhappy in its own way. The devil's disciples ought logically, then, to have considerably more elbow room than God's  but the problem is that people don't want to die, or to live in unhappy families, so the "freedom" promised by those options is meaningless.
It may be readily observed that, judging by what economists would call "revealed preferences," most people obviously don't really want to be saints, either  but at least it is possible to want to be a saint, to want good and nothing but good, and a great many of us at least want to want to be saints. We can imagine gradually purifying our hearts and our desires until "we have no more disposition to do evil, but to do good continually" (Mosiah 5:2). The reverse  wanting evil and nothing but evil  is not really possible and cannot be coherently imagined. No one, not even the devil himself, can love sin as such completely and unreservedly; all we can do is love certain aspects of sin and try not to think too much about the others. The devil's power is captivity because, in the last analysis, it is only the power to obtain what no one could ever really want.
To close with another Book of Mormon quote, Samuel the Lamanite told the wicked Nephites, "your destruction is made sure; yea, for ye have sought all the days of your lives for that which ye could not obtain"  namely "ye have sought for happiness in doing iniquity" (Helaman 13:38). That's the bottom line. Those who follow the devil are seeking what they simply cannot, by the nature of things, ever obtain. It is in that sense that the devil is the "father of lies," and that his power  the power he offers, and the power he himself possesses  is really only captivity.
Monday, February 17, 2020
Applying Kevin McCall's logic to squares and other noncentered figurate numbers
Note: this post uses special terminology and notation introduced in the last post. You should read that first in order to understand what follows.
Kevin McCall's proof of the RPS theorem for reduced triangular numbers is based on the following observation:
At the end of this 20step cycle, we are back where we started, with a number that is congruent to 0 (mod 10), and the cycle starts again.
Note that we have to go through two cycles of adding and subtracting the numbers because +5 ≡ 5 (mod 10). We count it as +5 the first time around and 5 the second time, so that it cancels out. If the modulus is odd, only one cycle is necessary.
⁂
The series of triangular numbers is generated by starting with 0, then adding 1, then adding 2, then 3, and so on through the succession of natural numbers. Due to the fact that k  n ≡ n (mod k), one you've added numbers up to a certain point, you start doing the modular equivalent of subtracting those same numbers in reverse order, creating a palindrome.Thus, if we consider the sequence of triangular numbers reduced modulo 10:
 0
 +1
 +2
 +3
 +4
 +5
 +6 ≡ 4 (mod 10)
 +7 ≡ 3 (mod 10)
 +8 ≡ 2 (mod 10)
 +9 ≡ 1 (mod 10)
 +10 ≡ 0 (mod 10)
 +11 ≡ +1 (mod 10)
 +12 ≡ +2 (mod 10)
 +13 ≡ +3 (mod 10)
 +14 ≡ +4 (mod 10)
 +15 ≡ 5 (mod 10)
 +16 ≡ 4 (mod 10)
 +17 ≡ 3 (mod 10)
 +18 ≡ 2 (mod 10)
 +19 ≡ 1 (mod 10)
 +20 ≡ +0 (mod 10)
 etc.
At the end of this 20step cycle, we are back where we started, with a number that is congruent to 0 (mod 10), and the cycle starts again.
Note that we have to go through two cycles of adding and subtracting the numbers because +5 ≡ 5 (mod 10). We count it as +5 the first time around and 5 the second time, so that it cancels out. If the modulus is odd, only one cycle is necessary.
⁂
Now let's consider the sequence of square numbers. We generate this series by starting with 0, then adding 1, then adding 3, then 5, and so on through the succession of odd natural numbers. This results in an RPS for much the same reason that the triangular series does: adding successive numbers is the modular equivalent of adding up to a certain point and then subtracting the same numbers in reverse order. Here's how it works modulo 10.
 0
 +1
 +3
 +5
 +7 ≡ 3 (mod 10)
 +9 ≡ 1 (mod 10)
 +11 ≡ +1 (mod 10)
 +13 ≡ +3 (mod 10)
 +15 ≡ 5 (mod 10)
 +17 ≡ 3 (mod 10)
 +19 ≡ 1 (mod 10)
 etc.
As with the triangular numbers, we have to go through two cycles so that the two 5s cancel each other out. Notice that, unlike the triangular numbers, this sequence never returns to adding 0 (mod 10). That is why the triangular numbers reduced mod 10 = RPS (0136051865), while the squares are RPS (0)1496(5)  the extra parentheses indicating that there are not two 5s in a row in the middle of the cycle, nor two 0s in a row at the end of one cycle and the beginning of the next.
⁂
Moving on to the pentagonal numbers, they are generated by starting with 0, then adding 1, then 4, then 7, then 10, and so on  every third natural number. The pattern should be obvious by now: The sequence of ngonal numbers is generated by starting with 0 and adding, successively, every (n  2)th natural number, beginning with 1. Here's the generation of the pentagonal sequence modulo 10.
 0
 +1
 +4
 +7
 +10 ≡ +0 (mod 10)
 +13 ≡ 7 (mod 10)
 +16 ≡ 4 (mod 10)
 +19 ≡ 1 (mod 10)
 +22 ≡ +2 (mod 10)
 +25 ≡ +5 (mod 10)
 +28 ≡ +8 (mod 10)
 +31 ≡ +1 (mod 10)
 +34 ≡ +4 (mod 10)
 +37 ≡ +7 (mod 10)
 +40 ≡ +0 (mod 10)
 +43 ≡ 7 (mod 10)
 +46 ≡ 4 (mod 10)
 +49 ≡ 1 (mod 10)
 +52 ≡ 8 (mod 10)
 +55 ≡ 5 (mod 10)
 +58 ≡ 2 (mod 10)
 +61 ≡ +1 (mod 10)
 etc.
The cycle here is more involved because we are adding every third natural number, which means that after we reach the 1 which cancels out the original +1, we do not go on to either  or +1 and the cycle does not yet begin anew.
⁂
Skipping hexagonal numbers for the time being, let's jump straight to what we're really interested in: the heptagonal numbers  the only figurate numbers yet examined which do not yield an RPS when reduced modulo 10. In keeping with the pattern, the heptagonal numbers are generated by adding, successively, every 5th natural number  yielding, modulo 10:
 0
 +1
 +6
 +11 ≡ +1 (mod 10)
 +16 ≡ +6 (mod 10)
 +21 ≡ +1 (mod 10)
 +26 ≡ +6 (mod 10)
 +31 ≡ +1 (mod 10)
 +36 ≡ +6 (mod 10)
 etc.
As can be seen, we just continue adding 1 and 6 (or subtracting 9 and 4) forever. This gives us a repeating cycle with a period of 20  because 10(1 + 6) ≡ 0 (mod 10)  but no palindrome is created because we never reach 1/+9 or 6/+4.
⁂
My tentative conclusion is that the sequence of (noncentered) ngonal numbers reduced modulo k will always be an RPS if n  2 and k are relatively prime. When that condition holds, adding every (n  2)th natural number in succession will (I think) mean in hitting all possible modular values, resulting in an RPS. The triangular numbers are a special case because for that sequence n  2 = 1, which is coprime to every integer.
Where n  2 and k are not coprime, an RPS may result, but not necessarily. I need to think a little more about what exactly determines which such sequences are RPSs and which are not.
What species was Bitter Green?
Is the title character in the Gordon Lightfoot song "Bitter Green" (1968) a dog, a horse, or a woman?
Upon the bitter green she walked the hills above the townThis certainly sounds like a horse. A grazing animal would naturally spend its time "upon the . . . green," and only the footsteps of a large hoofed animal would echo through the hills. A woman's footsteps might echo on pavement, but not on a green. It's hard to know what to make of the last phrase, since "footsteps as soft as eiderdown" surely means silent footsteps, the kind that don't echo. Anyway, on balance these lines support the horse theory.
Echo to her footsteps as soft as eider down
Waiting for her master to kiss away her tears"Master" normally refers to the human owner of a domestic animal, but it might be used poetically of a woman's husband or lover. Tears of sorrow, on the other hand, are shed only by human beings, but again could be ascribed poetically to other species. Waiting through the years for one's master to return is a behavior most stereotypically associated with dogs, but horses and women have also been known to do it.
Waiting through the years
Bitter Green they called her"Loving everyone that she met" sounds like an animal, and specifically like a dog. Applied to a woman, the phrase is rather scandalous and also seems inconsistent with the idea of a Penelope patiently waiting for her true love to return. "Bitter Green" itself also seems like a name that would be more naturally given to an animal than to a person. A woman would be known by that rather strange nickname only if no one knew who she was or what her real name was, which is inconsistent with "loving everyone that she met."
Walking in the sun
Loving everyone that she met
Bitter Green they called herThis also sounds like an animal. A woman whose husband or lover had disappeared would still have a home and would naturally wait there (perhaps by the window, wearing a face that she keeps in a jar by the door) rather than in the sun.
Waiting in the sun
Waiting for someone to take her home
Some say he was a sailor who died away at seaHorses don't kiss people, but dogs and women do.
Some say he was a prisoner who never was set free
Lost upon the ocean he died there in the mist
Dreaming of her kiss
But now the bitter green is gone, the hills have turned to rustIs this just a seasonal change? But Bitter Green waited "through the years."
There comes a weary stranger, his tears fall in the dustThis strongly supports the woman theory. The most natural interpretation is that the longawaited "master" finally returns, but too late, and kneels in tears at Bitter Green's grave. It seems unlikely that a dog or horse would have been buried in a churchyard, especially in her owner's absence. (Against this, note that this line is "dreaming of a kiss"  not "her kiss" as in the chorus. Perhaps the wife whose grave the stranger is visiting is distinct from Bitter Green.)
Kneeling by the churchyard in the autumn mist
Dreaming of a kiss
⁂
In one of the scifi stories I wrote as a very young child, the astronaut protagonist had, among other items of spacefaring equipment, a "space shovel" which he used for digging on the surface of distant planets. I had no very clear idea of how a space shovel might differ from a commonorgarden shovel, but the phrase "space shovel" just sounded right, and it seemed that an astronaut ought to have one. It was not until years later that it dawned on me that "space shovel" sounded an awful lot like the thencommon phrase "space shuttle," and that it was almost certainly this unconscious echo that made me think the former phrase "sounded right." I think some similar unconscious association was likely behind Gordon Lightfoot's choice of the phrase "Bitter Green."
Sunday, February 16, 2020
Are all reduced sequences of figurate numbers repeating palindromes?
Time for another mathematical interlude.
Preliminaries: Terminology and notation
A palindrome is any series of elements that is the same forwards and backwards. Each palindrome thus consists of two parts, which we shall call the head and the tail. The tail consists of the same series of elements as the head, but in reverse order. For example, in the palindromic word “noon,” the string “no” is the head, and “on” is the tail. In “noon,” the head and tail are entirely separate, but in a palindrome with an odd number of elements, the end of the head will overlap with the beginning of the tail. For example, the head of the palindromic word “level” is “lev,” and the tail is “vel”; a single “v” does double duty as the last element of the head and the first element of the tail.
No special notation is required to write a simple palindrome such as “noon” or “level,” but we are concerned in this post with repeating palindromic series (RPSs). If a palindrome consists of a head followed by a tail (possibly overlapping), an RPS is a head followed by a tail, then the head again, then the tail again, and so on to infinity. In the notation we will be using here, an RPS is represented by "RPS" followed by its head enclosed in parens (round brackets), thus:
If the end of the head overlaps with the beginning of the tail, an additional openparen is placed before the first element of the tail, thus:
In an RPS, unlike a simple palindrome, overlap at the other end is also possible. For example, the word “grammar” is not itself a palindrome, but “grammar” endlessly repeated is an RPS. The letter “g” is both the beginning of the head and the end of the tail. This sort of overlap is indicated by placing a closeparen after the last element of the tail, thus:
It is possible to have overlap at both ends, as in this example.
Overlap need not be limited to a single element; it can be a series of elements, provided that series is itself a palindrome. For example, consider the RPS created by endlessly repeating the word “sestet.”
I should also mention that, much like a repeating decimal, an RPS has a beginning but not an end. As with a decimal like 0.16666666..., there may be a nonrepeating segment at the beginning, before the RPS proper starts. Our notation can deal with this by putting this nonrepeating segment before the first paren, as in this example.
One final note about the notation: If each element in the RPS can be represented by a single character (a letter, a onedigit number, etc.), it can be written as in the examples above, without commas or spaces. If the elements of an RPS are words, multidigit numbers, etc., the elements should be separated by commas and spaces.
RPSs in the sequence of triangular numbers
Several posts on this blog have dealt with the fact that, for any modulus k, the sequence of triangular numbers reduced modulo k will be an RPS. Two different proofs of this have been given (here and here). The pattern is easiest to see when k = 10, since (in the decimal system) any number reduced modulo 10 is equal to the final digit of that number.
The first several triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...
If we reduce that sequence modulo 10 (by replacing each number in the sequence with its final digit), we get: 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6...
This reduced sequence is RPS (0136051865). Any other modulus will also yield an RPS.
Square numbers
Only some three months after proving the RPS theorem for triangular numbers did I notice that the sequence of square numbers shows a similar pattern. Below are the square numbers from 0^{2} to 50^{2}. Start at the upper left, follow the zigzag down to the bottom, and then come back up the zigzag on the right.
Numbers in the same column have the same last digit (i.e., are congruent modulo 10). Numbers in the same row have the same last two digits (i.e., are congruent modulo 100).
Square numbers reduced modulo 10 = RPS (0)1496(5).
Square numbers reduced modulo 100 = RPS (0), 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 21, 44, 69, 96, 25, 56, 89, 24, 61, 0, 41, 84, 29, 76, (25).
I haven't checked if other moduli also yield RPSs, but, based on my experience with triangular numbers, and on the general principle that there is nothing mathematically special about powers of 10, I feel quite certain that they do.
Octagonal numbers: 0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720, 4961, 5208, 5461
Octagonal numbers reduced modulo 10 = RPS (01810)56(3)
Enneagonal numbers: 0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364
Enneagonal numbers reduced modulo 10 = RPS (0194651441564910)69)
Decagonal numbers: 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326
Decagonal numbers reduced modulo 10 = RPS (010)725(6)
Centered figurate numbers
Centered triangular numbers: 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529
Centered triangular numbers reduced modulo 10 = RPS (1409164596)
Centered square numbers: 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, 4513
Centered square numbers reduced modulo 10 = RPS (15(3)
Centered pentagonal numbers: 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976, 3151, 3331, 3516, 3706, 3901, 4101, 4306, 4516, 4731, 4951, 5176, 5406
Centered pentagonal numbers reduced modulo 10 = RPS (16)
Hex numbers: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487
Hex numbers reduced modulo 10 = RPS (17(9)
Centered heptagonal numbers: 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246
Centered heptagonal numbers reduced modulo 10 = RPS (1823168736)
Centered octagonal numbers (i.e., odd squares): 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569
Centered octagonal numbers reduced modulo 10 = RPS (19(5)
Centered enneagonal numbers (i.e., every third triangular number): 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316
Centered enneagonal numbers reduced modulo 10 = RPS (1085160356)
Centered decagonal numbers: 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901
Centered decagonal numbers reduced modulo 10 = RPS (1)
Centered decagonal numbers reduced modulo 100 = RPS (1, 11, 31, 61, 1, 51, 11, 81, 61, 51)
Star numbers (i.e., centered dodecagonal numbers): 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837
Star numbers reduced modulo 10 = RPS (13(7)
It certainly looks as if all such sequences reduce to RPSs, but the unexpected exception of the (noncentered) heptagonal numbers makes me hesitant to jump to that conclusion.
⁂
Preliminaries: Terminology and notation
A palindrome is any series of elements that is the same forwards and backwards. Each palindrome thus consists of two parts, which we shall call the head and the tail. The tail consists of the same series of elements as the head, but in reverse order. For example, in the palindromic word “noon,” the string “no” is the head, and “on” is the tail. In “noon,” the head and tail are entirely separate, but in a palindrome with an odd number of elements, the end of the head will overlap with the beginning of the tail. For example, the head of the palindromic word “level” is “lev,” and the tail is “vel”; a single “v” does double duty as the last element of the head and the first element of the tail.
No special notation is required to write a simple palindrome such as “noon” or “level,” but we are concerned in this post with repeating palindromic series (RPSs). If a palindrome consists of a head followed by a tail (possibly overlapping), an RPS is a head followed by a tail, then the head again, then the tail again, and so on to infinity. In the notation we will be using here, an RPS is represented by "RPS" followed by its head enclosed in parens (round brackets), thus:
 RPS (no) = noonnoonnoonnoonnoon...
If the end of the head overlaps with the beginning of the tail, an additional openparen is placed before the first element of the tail, thus:
 RPS (le(v) = levellevellevellevellevel...
In an RPS, unlike a simple palindrome, overlap at the other end is also possible. For example, the word “grammar” is not itself a palindrome, but “grammar” endlessly repeated is an RPS. The letter “g” is both the beginning of the head and the end of the tail. This sort of overlap is indicated by placing a closeparen after the last element of the tail, thus:
 RPS (g)ram) = grammargrammargrammar...
It is possible to have overlap at both ends, as in this example.
 RPS (v)oo(d) = voodoovoodoovoodoo...
Overlap need not be limited to a single element; it can be a series of elements, provided that series is itself a palindrome. For example, consider the RPS created by endlessly repeating the word “sestet.”
 RPS (ses)t(e) = sestetsestetsestetsestet...
I should also mention that, much like a repeating decimal, an RPS has a beginning but not an end. As with a decimal like 0.16666666..., there may be a nonrepeating segment at the beginning, before the RPS proper starts. Our notation can deal with this by putting this nonrepeating segment before the first paren, as in this example.
 RPS n(eve)(r) = neverevereverevereverever...
One final note about the notation: If each element in the RPS can be represented by a single character (a letter, a onedigit number, etc.), it can be written as in the examples above, without commas or spaces. If the elements of an RPS are words, multidigit numbers, etc., the elements should be separated by commas and spaces.
⁂
Several posts on this blog have dealt with the fact that, for any modulus k, the sequence of triangular numbers reduced modulo k will be an RPS. Two different proofs of this have been given (here and here). The pattern is easiest to see when k = 10, since (in the decimal system) any number reduced modulo 10 is equal to the final digit of that number.
The first several triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666...
If we reduce that sequence modulo 10 (by replacing each number in the sequence with its final digit), we get: 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6...
This reduced sequence is RPS (0136051865). Any other modulus will also yield an RPS.
⁂
Square numbers
Only some three months after proving the RPS theorem for triangular numbers did I notice that the sequence of square numbers shows a similar pattern. Below are the square numbers from 0^{2} to 50^{2}. Start at the upper left, follow the zigzag down to the bottom, and then come back up the zigzag on the right.
Numbers in the same column have the same last digit (i.e., are congruent modulo 10). Numbers in the same row have the same last two digits (i.e., are congruent modulo 100).
Square numbers reduced modulo 10 = RPS (0)1496(5).
Square numbers reduced modulo 100 = RPS (0), 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 21, 44, 69, 96, 25, 56, 89, 24, 61, 0, 41, 84, 29, 76, (25).
I haven't checked if other moduli also yield RPSs, but, based on my experience with triangular numbers, and on the general principle that there is nothing mathematically special about powers of 10, I feel quite certain that they do.
⁂
Other noncentered figurate numbers
Pentagonal numbers: 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187...
Pentagonal numbers reduced modulo 10 = RPS (01522510)275607)
Hexagonal numbers: 0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, 3003, 3160, 3321, 3486, 3655, 3828, 4005, 4186, 4371, 4560
Hexagonal numbers reduced modulo 10 = RPS (016585610(3)
Heptagonal numbers: 0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688
Heptagonal numbers reduced modulo 10 = 01784512895623906734 endlessly repeated  not a palindrome!
Octagonal numbers reduced modulo 10 = RPS (01810)56(3)
Enneagonal numbers reduced modulo 10 = RPS (0194651441564910)69)
Decagonal numbers: 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326
Decagonal numbers reduced modulo 10 = RPS (010)725(6)
This is so bizarre that I almost think I must have made some mistake, but I'm pretty sure I haven't. Inductively, it looks like virtually all noncentered figurate numbers reduce to RPSs, but the heptagonal numbers are an exception! Why? Are there other exceptions?
⁂
Centered figurate numbers
Centered triangular numbers: 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529
Centered triangular numbers reduced modulo 10 = RPS (1409164596)
Centered square numbers: 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, 4513
Centered square numbers reduced modulo 10 = RPS (15(3)
Centered pentagonal numbers: 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976, 3151, 3331, 3516, 3706, 3901, 4101, 4306, 4516, 4731, 4951, 5176, 5406
Centered pentagonal numbers reduced modulo 10 = RPS (16)
Hex numbers: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487
Hex numbers reduced modulo 10 = RPS (17(9)
Centered heptagonal numbers: 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246
Centered heptagonal numbers reduced modulo 10 = RPS (1823168736)
Centered octagonal numbers (i.e., odd squares): 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569
Centered octagonal numbers reduced modulo 10 = RPS (19(5)
Centered enneagonal numbers (i.e., every third triangular number): 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316
Centered enneagonal numbers reduced modulo 10 = RPS (1085160356)
Centered decagonal numbers: 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901
Centered decagonal numbers reduced modulo 10 = RPS (1)
Centered decagonal numbers reduced modulo 100 = RPS (1, 11, 31, 61, 1, 51, 11, 81, 61, 51)
Star numbers (i.e., centered dodecagonal numbers): 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837
Star numbers reduced modulo 10 = RPS (13(7)
It certainly looks as if all such sequences reduce to RPSs, but the unexpected exception of the (noncentered) heptagonal numbers makes me hesitant to jump to that conclusion.
⁂
So the new mission (paging Kevin McCall!) is to come up with a general proof that almost all figurate number sequences reduce to RPSs  a proof that makes it clear what the exceptions are and why. There's obviously a pattern here that goes beyond the triangular numbers, and it should be possible to express that pattern mathematically.
Thursday, February 13, 2020
Hatelove
George Orwell's Newspeak was in many ways prophetic of modern politically correct language, but he made two important errors. First, Orwell's Newspeak used crime in compound words (crimethink, crimestop, etc.) to indicate anything that was beyond the pale, whereas Nowspeak has found hate to be more effective for that purpose. Second, Orwell defined goodsex as "sexual intercourse only for procreation" and sexcrime as "sexual intercourse for pleasure"  implying, to put it mildly, values somewhat different from those currently endorsed by realworld totalitarianism. So, in the spirit of updating and correcting Orwell, I offer this addition to the PC lexicon:
hatelove: biologically natural relations between a man and a woman within the bounds of marriage
⁂
It's a felicitous enough coinage, I think you'll agree, capturing something of the spirit of blackwhite ("the ability to believe that black is white, to know that black is white, and to forget that one ever believed the contrary")  but, you might ask, what is actually so hateful about hatelove?
Glad you asked.
1. Being disproportionately practiced by privileged white people, hatelove is inherently racist and elitist. Furthermore, its procreative aspect (see 3 below) means that it leads to the production of more people of the hatelovers' own race and class, something only a white supremacist would want to do. In essence, hatelove boils down to a deliberate act of genocide directed against the Other.
2. Hatelovers may use the "consenting adults" excuse to argue that their predilections are their own business, but in fact each and every hatelove relationship contributes to the cancer of toxic heteronormativity. Heteronormativity is (as Studies Have Shown) one of the leading causes of suicide, so in a way hatelove is a kind of murder.
3. Worst of all, hatelove is known to produce fetuses  parasitic and often dangerous growths which, conveniently enough, never affect white cisgender men. This makes hatelove a form of misogynistic and transphobic violence. Moreover, each fetus thus engendered will, if (foolishly and irresponsibly) allowed to develop to maturity, go on to release as much as 30 tons of deadly carbon dioxide gas into the atmosphere, thus directly contributing to the destruction of the planet.
If that's not hate, what is?
⁂
This is, of course, satire, but we live in a world where satire dates very quickly, as it never takes long for reality to catch up with it. The SPLC already classifies promarriage organizations as hategroups (I recommend writing such terms as single words to emphasize their Newspeakiness), and I venture to predict that it won't be very long before we start hearing rhetoric very close to that used in this post.
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