This Means RaWaR

The Overlord of the Über-Feral says: Welcome to my bijou bloguette. You can scroll down to sample more or simply:

• Read a Writerization at Random: RaWaR


• ¿And What Doth It Mean To Be Flesh?

მათემატიკა მსოფლიოს მეფე


Gweel & Other Alterities – Incunabula’s new edition


Tales of Silence & Sortilege – Incunabula’s new edition



If you’d like to donate to O.o.t.Ü.-F., please click here.

Maugham Muses Maupassant

An intelligent critic, who combines wide reading and a sensitive taste with a knowledge of the world rare among those who follow his calling, has found in my stories the influence of Guy de Maupassant. That is not strange. When I was a boy he was considered the best short story writer in France and I read his works with avidity. From the age of fifteen whenever I went to Paris I spent most of my afternoons poring over the books in the galleries of the Odéon. I have never passed more enchanted hours. The attendants in their long smocks were indifferent to the people who sauntered about looking at the books and they would let you read for hours without bothering. There was a shelf filled with the works of Guy de Maupassant, but they cost three francs fifty a volume and that was not a sum I was prepared to spend. I had to read as best I could standing up and peering between the uncut pages. Sometimes when no attendant was looking I would hastily cut a page and thus read more conveniently. Fortunately some of them were issued in a cheap edition at seventy-five centimes and I seldom came away without one of these. In this manner, before I was eighteen, I had read all the best stories. It is natural enough that when at that age I began writing stories myself I should unconsciously have chosen those little masterpieces as a model. I might very well have hit upon a worse.

Maupassant’s reputation does not stand as high as it did, and it is evident now that there is much in his work to repel. He was a Frenchman of his period in violent reaction against the romantic age which was finishing in the saccharine sentimentality of Octave Feuillet (admired by Matthew Arnold) and in the impetuous slop of George Sand. He was a naturalist, aiming at truth at all costs, but the truth he achieved looks to us now a trifle superficial. He does not analyse his characters. He takes little interest in the reason why. They act, but wherefore he does not know. “For me,” he says, “psychology in a novel or in a story consists in this: to show the inner man by his life.” That is very well, that is what we all try to do, but the gesture will not by itself always indicate the motive. The result with Maupassant was a simplification of character which is effective enough in a short story, but on reflection leaves you unconvinced. There is more in men than that, you say. Again, he was obsessed by the tiresome notion, common then to his countrymen, that it was a duty a man owed himself to hop into bed with every woman under forty that he met. His characters indulge their sexual desire to gratify their self-esteem. They are like the people who eat caviare when they are not hungry because it is expensive. Perhaps the only human emotion that affects his characters with passion is avarice. This he can understand; it fills him with horror, but notwithstanding he has a sneaking sympathy with it. He was slightly common. But for all this it would be foolish to deny his excellence. An author has the right to be judged by his best work. No author is perfect. You must accept his defects; they are often the necessary complement of his merits; and this may be said in gratitude to posterity that it is very willing to do this. It takes what is good in a writer and is not troubled by what is bad. It goes so far sometimes, to the confusion of the candid reader, as to claim a profound significance for obvious faults. So you will see the critics (the awe-inspiring voice of posterity) find subtle reasons to explain to his credit something in a play of Shakespeare’s that any dramatist could tell them needed no other explanation than haste, indifference or wilfulness. Maupassant’s stories are good stories. The anecdote is interesting apart from the narration so that it would secure attention if it were told over the dinner-table; and that seems to me a very great merit indeed. However halting your words and insipid your rendering, you could not fail to interest your listeners if you told them the bare story of Boule de Suif, L’Héritage or La Parure. These stories have a beginning, a middle and an end. They do not wander along an uncertain line so that you cannot see whither they are leading, but follow without hesitation, from exposition to climax, a bold and vigorous curve. It may be that they have no great spiritual significance. Maupassant did not aim at that. He looked upon himself as a plain man; no good writer was ever less a man of letters. He did not pretend to be a philosopher, and here he was well-advised, for when he indulges in reflection he is commonplace. But within his limits he is admirable. He has an astonishing capacity for creating living people. He can afford little space, but in a few pages can set before you half a dozen persons so sharply seen and vividly described that you know all about them that you need. Their outline is clear; they are distinguishable from one another; and they breathe the breath of life. They have no complexity, they lack strangely the indecision, the unexpectedness, the mystery that we see in human beings; they are in fact simplified for the purposes of the story. But they are not deliberately simplified: those keen eyes of his saw clearly, but they did not see profoundly; it is a happy chance that they saw all that was necessary for him to achieve the aim he had in view. He treats the surroundings in the same way, he sets his scene accurately, briefly and effectively; but whether he is describing the charming landscape of Normandy or the stuffy, overcrowded drawing-rooms of the eighties his object is the same, to get on with the story. On his own lines I do not think that Maupassant is likely to be surpassed. If his excellence is not at the moment so apparent it is because what he wrote must now stand comparison with the very different, more subtle and moving work of Chekov.

• From Maugham’s introduction to his Collected Short Stories (1951).

Mouche Appreciated

“Don’t ever think that magic is simply somebody taking a rabbit out of a hat. Our ancestors believed in magic and were right for the wrong reasons — for the most part they believed that magic was evil, not good. But the magic that lies all about you, from your own body to that of an elephant, to a fly’s wing as intricate as anything that lets the sunlight into Chartres Cathedral, to the great surging sea itself — that is magic. Anyone who goes through life unastounded by everything he sees is not alive.” — Gerald Durrell, Myself and Other Animals (2024), “Fragments from unpublished autobiography”

A FracTeasel on a Fract-L

Here are two new fractals, both of which remind me of the seedheads of the wildflower known as a teasel, Dipsacus fullonum:

A FracTeasel fractal


Dried seedheads of teasel, Dipsacus fullonum (Wikipedia)


Another FracTeasel fractal (embedded in the first)


Flowering seedhead of teasel, Dipsacus fullonum (Wikipedia)


How do you create the two FracTeasels? Let’s look first at the fractal they’re inspired by. In “Back to Frac’” I talked about this fractional fractal, a variant of what I call the limestone fractal:

Variant of a limestone fractal or gryke fractal


It’s a fractal on a fract-L, that is, the x and y co-ordinates of the red L represent pairs of fractions generating decimals between 0 and 1. The x represents the fractions a1/b1 = 1/n to (n-1)/n in simplest form: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, 5/8, 7/8,…

And what about the y? It represents the fraction found by taking the continued fraction of a1/b1, reversing it, and generating a new fraction, a2/b2, from the reversal. For example, here’s the continued fraction of a1/b1 = 3/23 = 0.1304347826…:

contfrac(3/23) = 7,1,2

The continued fraction of a1/b1 = 3/23 is used like this to reconstruct a1/b1:

7,1,2

0 → 1 / (0 + 2) = 1/2 → 1 / (1/2 + 1) = 2/3 → 1 / (7 + 2/3) = 3/23

Now reverse the continued fraction, 7,1,2 → 2,1,7, and generate a2/b2:

2,1,7

0 → 1 / (0 + 7) = 1/7 → 1 / (1/7 + 1) = 7/8 → 1 / (2 + 7/8) = 8/23 = 0.3478260869565…

The limestone fractal above appears when a1/b1 → a2/b2 for a1/b1 = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, 5/8, 7/8,… But you can do other things to contfrac(a1/b1) beside just reversing it. What about the permutations of contfrac(a1/b1), for example? If length(contfrac(a1/b1)) = n, the permutations can generate up to n! (factorial n) new a2/b2 for the y co-ordinate (if all the numbers of contfrac(a1/b1) are different, you’ll get n! permutations). The resultant fractal is the first of the FracTeasels above (note that a2/b2 isn’t multipled by two):

FracTeasel #1 from fract-L for y = perm(contfrac(a1/b1))


If you think about it, you’ll see that the fractal from permed contfrac(a1/b1) contains the fractal from reversed contfrac(a1/b1). It also contains the second FracTeasel:

FracTeasel #2


How so? Because the second FracTeasel — let’s call it the stemmed FracTeasel — is created by shifting some numbers in contfrac(a1/b1) and leaving others alone. For example:

contfrac(940/1089) = 1, 6, 3, 4, 5, 2 → 1, 4, 3, 2, 5, 6 = contfrac(1008/1243)

So the function is finding one particular permutation of contfrac(a1/b1) to generate a2/b2, not all permutations. And so the function creates the stemmed FracTeasel, which carries an infinite number of seedheads on the same stem. To show that, here’s an animated gif zooming in on the bend of the fract-L for the stemmed FracTeasel:

Zooming the FracTeasel (animated at ezGif)


Elsewhere Other-Accessible…

I Like Gryke — a first look at the limestone fractal
Lime Time — more on the limestone fractal

Matchin’ Fraction

I wondered whether contfrac(a/b), the continued fraction of a/b, ever matched the digits of a and b in some base. The answer was yes. But I haven’t found any examples in base 10:

3,1,2 = contfrac(3/12) in base 9 = contfrac(3/11) in base 10
4,1,3 = contfrac(4/13) in b16 = contfrac(4/19) in b10
5,1,45/14 in b25 = 5/29
6,1,56/15 in b36 = 6/41
25,22/52 in b9 = 2/47 → 23,2
7,1,67/16 in b49 = 7/55
8,1,78/17 in b64 = 8/71
9,1,89/18 in b81 = 9/89
A,1,9A/19 in b100 = 10/109 → 10,1,9
42,1,34/213 in b8 = 4/139 → 34,1,3
4,1,2,3,341/233 in b8 = 33/155
1,17,1,2,3117/123 in b14 = 217/227 → 1,21,1,2,3
3A,33/A3 in b28 = 3/283 → 94,3
3,5,A,235/A2 in b34 = 107/342 → 3,5,10,2
3,1,4,1,4,1,5314/1415 in b8 = 204/781
2,1,36,3,2213/632 in b12 = 303/902 → 2,1,42,3,2
3,2,11,2,2,2321/1222 in b9 = 262/911 → 3,2,10,2,2,2
4H,44/H4 in b65 = 4/1109 → 277,4
6,2,1,3,J62/13J in b35 = 212/1349 → 6,2,1,3,19
8,3,3,1,D83/31D in b22 = 179/1487 → 8,3,3,1,13
93,1,89/318 in b27 = 9/2222 → 246,1,8
1,3A,1,1,4,2,213A1/1422 in b12 = 2281/2330 → 1,46,1,1,4,2,2
C7,1,BC/71B in b21 = 12/3119 → 259,1,11
1,2,2,1,O,F122/1OF in b50 = 2602/3715 → 1,2,2,1,24,15
2,1,1,5,55211/555 in b28 = 1597/4065 → 2,1,1,5,145
3,1,1,A,K,6311/AK6 in b29 = 2553/8996 → 3,1,1,10,20,6
1,2[70],1,3,912[70]/139 in b98 = 9870/9907 → 1,266,1,3,9
1,E,4,1,M,71E4/1M7 in b100 = 11404/12207 → 1,14,4,1,22,7
LG,5,4L/G54 in b28 = 21/12688 → 604,5,4
G4,1,FG/41F in b64 = 16/16463 → 1028,1,15

A Pox on Tox

This is 2026. Headlines like this are NOT acceptable:


As soon as I saw that, I thought: “Harmful”? Harmful schmarmful! It should be toxic

Funnily enough, when I looked at the story itself, it was toxic. That is, it was “toxic”:

That’s the way it should be. Or rather, the way it should be when English isn’t used the way it should be. As I’ve said pre-previously:

There’s only one word for it: toxic. The proliferation of this word is an incendiarily irritating abjectional aspect of contemporary culture. My visit to Google Ngram has confirmed my worst suspicions:

Use of “toxic” in English over the decades

• Continue reading Get Your Tox Off

Back to Frac’

Here’s a second serendipitous fractal:

A serendipitous fractal on a fract-L


It looks like (and is related to) the limestone fractal and I found it similarly serendipitously. This time I was looking at continued fractions, a simple yet subtle and seductive way of representing non-integer numbers like 2/3 and 7/9 (or √2 and π). To generate a continued fraction from a/b < 1, you divide a/b into 1 and take away the integer part. Then you repeat with the remainder until nothing is left (or, as with irrationals like 1/√2 and 1/π, you've calculated long enough for your needs). The integers at each stage are the numbers of the continued fraction. Here is the working for contfrac(2/3), the continued fraction of 2/3:

int(1/(2/3)) = int(3/2) = int(1.5) = 1
3/2 – 1 = 1/2
int(1/(1/2)) = int(2) = 2
2 – 2 = 0

contfrac(2/3) = 1, 2

By working backwards with (1, 2), you can use the continued fraction to reconstruct the original number a/b. Start with a/b = 0/1:

1 / (0/1 + 2) = 1 / ((0+2*1)/2) = 1 / (2/1) = 1/2
1 / (1/2 + 1) = 1 / ((1+2*1)/2) = 1 / (3/2) = 2/3

And here’s the working for contfrac(7/9), the continued fraction of 7/9:

int(1/(7/9)) = int(9/7) = int(1.285714…) = 1
9/7 – 1 = 2/7
int(1/(2/7)) = int(7/2) = int(3.5) = 3
7/2 – 3 = 1/2
int(1/(1/2)) = int(2) = 2
2 – 2 = 0

contfrac(7/9) = 1, 3, 2

And here’s the reconstruction of 7/9 from its continued fraction, starting again with a/b = 0/1:

1 / (0/1 + 2) = 1 / ((0+2*1)/2) = 1 / (2/1) = 1/2
1 / (1/2 + 3) = 1 / ((1+2*3)/2) = 1 / (7/2) = 2/7
1 / (2/7 + 1) = 1 / ((2+7*1)/7) = 1 / (9/7) = 7/9

From that simple algorithm arise subtle and seductive things. Look at some continued fractions, cf(a/b), for a/b in simplest form (giving only the first few reciprocals, 1/b, because cf(1/b) = b). Interesting patterns appear, e.g. when a/b uses adjacent or nearly adjacent Fibonacci numbers:

cf(1/3) = 3 = cf(0.333333333…)
cf(2/3) = 1,2 = cf(0.666666666…)
cf(1/4) = 4 = cf(0.25)
cf(3/4) = 1,3 = cf(0.75)
cf(1/5) = 5 = cf(0.2)
cf(2/5) = 2,2 = cf(0.4)
cf(3/5) = 1,1,2 = cf(0.6)
cf(4/5) = 1,4 = cf(0.8)
cf(5/6) = 1,5 = cf(0.833333333…)
cf(2/7) = 3,2 = cf(0.285714285…)
cf(3/7) = 2,3 = cf(0.428571428…)
cf(4/7) = 1,1,3 = cf(0.571428571…)
cf(5/7) = 1,2,2 = cf(0.714285714…)
cf(6/7) = 1,6 = cf(0.857142857…)
cf(3/8) = 2,1,2 = cf(0.375)
cf(5/8) = 1,1,1,2 = cf(0.625)
cf(7/8) = 1,7 = cf(0.875)
cf(2/9) = 4,2 = cf(0.222222222…)
cf(4/9) = 2,4 = cf(0.444444444…)
cf(5/9) = 1,1,4 = cf(0.555555555…)
cf(7/9) = 1,3,2 = cf(0.777777777…)
cf(8/9) = 1,8 = cf(0.888888888…)
cf(3/10) = 3,3 = cf(0.3)
cf(7/10) = 1,2,3 = cf(0.7)
cf(9/10) = 1,9 = cf(0.9)
cf(2/11) = 5,2 = cf(0.181818181…)
cf(3/11) = 3,1,2 = cf(0.272727272…)
cf(4/11) = 2,1,3 = cf(0.363636363…)
cf(5/11) = 2,5 = cf(0.454545454…)
cf(6/11) = 1,1,5 = cf(0.545454545…)
cf(7/11) = 1,1,1,3 = cf(0.636363636…)
cf(8/11) = 1,2,1,2 = cf(0.727272727…)
cf(9/11) = 1,4,2 = cf(0.818181818…)
cf(10/11) = 1,10 = cf(0.909090909…)
cf(5/12) = 2,2,2 = cf(0.416666666…)
cf(7/12) = 1,1,2,2 = cf(0.583333333…)
cf(11/12) = 1,11 = cf(0.916666666…)
cf(2/13) = 6,2 = cf(0.153846153…)
cf(3/13) = 4,3 = cf(0.230769230…)
cf(4/13) = 3,4 = cf(0.307692307…)
cf(5/13) = 2,1,1,2 = cf(0.384615384…)
cf(6/13) = 2,6 = cf(0.461538461…)
cf(7/13) = 1,1,6 = cf(0.538461538…)
cf(8/13) = 1,1,1,1,2 = cf(0.615384615…)
cf(9/13) = 1,2,4 = cf(0.692307692…)
cf(10/13) = 1,3,3 = cf(0.769230769…)
cf(11/13) = 1,5,2 = cf(0.846153846…)
cf(12/13) = 1,12 = cf(0.923076923…)
cf(3/14) = 4,1,2 = cf(0.214285714…)
cf(5/14) = 2,1,4 = cf(0.357142857…)
cf(9/14) = 1,1,1,4 = cf(0.642857142…)
cf(11/14) = 1,3,1,2 = cf(0.785714285…)
cf(13/14) = 1,13 = cf(0.928571428…)
cf(2/15) = 7,2 = cf(0.133333333…)
cf(4/15) = 3,1,3 = cf(0.266666666…)
cf(7/15) = 2,7 = cf(0.466666666…)
cf(8/15) = 1,1,7 = cf(0.533333333…)
cf(11/15) = 1,2,1,3 = cf(0.733333333…)
cf(13/15) = 1,6,2 = cf(0.866666666…)
cf(14/15) = 1,14 = cf(0.933333333…)
cf(3/16) = 5,3 = cf(0.1875)
cf(5/16) = 3,5 = cf(0.3125)
cf(7/16) = 2,3,2 = cf(0.4375)

After investigating some of those patterns, I wondered what happened when you reversed the continued fraction cf(a/b) and used those reversed numbers backward (that is, used the numbers of cf(a/b) forward) to generate another and different a/b. And a/b will always be different unless cf(a/b) is a palindrome, like cf(5/12) = 2,2,2 or cf(5/13) = 2,1,1,2 or cf(4/15) = 3,1,3. Note that a continued fraction never ends in 1, so that when reversing, say, cf(5/8) = (1, 1, 1, 2), you need an adjustment from (2, 1, 1, 1) to (2, 1, 1+1) = (2, 1, 2). Here’s a little of what happens when you reverse cf(a1/b1) to generate a2/b2:

cf(1/2) = 2 → 2 = cf(1/2)
1/2 = 0.5 : 0.5 = 1/2
cf(1/3) = 3 → 3 = cf(1/3)
1/3 = 0.333333333 : 0.333333333 = 1/3
cf(2/3) = 1, 2 → 2, 1 → 3 = cf(1/3)
2/3 = 0.666666666 : 0.333333333 = 1/3
cf(3/4) = 1, 3 → 3, 1 → 4 = cf(1/4)
3/4 = 0.75 : 0.25 = 1/4
cf(2/5) = 2, 2 → 2, 2 = cf(2/5)
2/5 = 0.4 : 0.4 = 2/5
cf(3/5) = 1, 1, 2 → 2, 1, 1 → 2, 2 = cf(2/5)
3/5 = 0.6 : 0.4 = 2/5
cf(4/5) = 1, 4 → 4, 1 → 5 = cf(1/5)
4/5 = 0.8 : 0.2 = 1/5
cf(5/6) = 1, 5 → 5, 1 → 6 = cf(1/6)
5/6 = 0.833333333 : 0.166666666 = 1/6
cf(2/7) = 3, 2 → 2, 3 = cf(3/7)
2/7 = 0.285714286 : 0.428571428 = 3/7
cf(3/7) = 2, 3 → 3, 2 = cf(2/7)
3/7 = 0.428571429 : 0.285714286 = 2/7
cf(4/7) = 1, 1, 3 → 3, 1, 1 → 3, 2 = cf(2/7)
4/7 = 0.571428571 : 0.285714286 = 2/7
cf(5/7) = 1, 2, 2 → 2, 2, 1 → 2, 3 = cf(3/7)
5/7 = 0.714285714 : 0.428571429 = 3/7
cf(6/7) = 1, 6 → 6, 1 → 7 = cf(1/7)
6/7 = 0.857142857 : 0.142857143 = 1/7
cf(3/8) = 2, 1, 2 → 2, 1, 2 = cf(3/8)
0.375 : 0.375
cf(5/8) = 1, 1, 1, 2 → 2, 1, 1, 1 → 2, 1, 2 = cf(3/8)
0.625 : 0.375
cf(7/8) = 1, 7 → 7, 1 → 8 = cf(1/8)
0.875 : 0.125
cf(2/9) = 4, 2 → 2, 4 = cf(4/9)
0.222222222 : 0.444444444
cf(4/9) = 2, 4 → 4, 2 = cf(2/9)
0.444444444 : 0.222222222

And if you plot x = a1/b1 and y = (a2/b2 * 2) on a fract-L, that is, a graph whose horizontal and vertical arms represent 0 to 1, you get the fractal right at the beginning:

Fract-L for x = a1/b1 and y = (a2/b2 * 2), where a2/b2 is generated from reversed(cf(a1/b1))


You need to use (a2/b2 * 2) because a2/b2 from reversed(cf(a1/b1)) is always <= 0.5, so using raw a2/b2 generates this graph:

Fract-L for x = a1/b1 and y = a2/b2 (i.e. a2/b2 is unadjusted)


Why is it always true that a2/b2 <= 0.5? For two reasons. First, a/b > 0.5 always generate continued fractions that start with 1, like cf(2/3) = 1, 2 or cf(3/4) = 1, 3 or cf(3/5) = 1, 1, 2. Second, as previously mentioned, no continued fraction ends with 1. Therefore a reversed cf(a1/b1), where the final number, n > 1, moves to the beginning, will never begin with 1 and the a2/b2 generated from reversed(cf(a1/b1)) will always be less than 0.5 (or equal to it in the solitary case of cf(1/2) = 2).

Now let's look at the development of the fractal as a1/b1 uses larger and larger denominators:

Fract-L for x = a1/b1 and y = (a2/b2 * 2) for a1/b1 <= 6/7


Fract-L for for a1/b1 <= 14/15


Fract-L for a1/b1 <= 30/31


Fract-L for a1/b1 <= 62/63


Fract-L for a1/b1 <= 126/127


Fract-L for a1/b1 <= 254/255


Fract-L for a1/b1 <= 357/358


Fract-L for a1/b1 <= 467/468


Animated fract-L for x = a1/b1 and y = (a2/b2 * 2) (animated at ezGif)


The fractal changes subtly when you restrict the b1 of a1/b1 in some way, say using multiples of 2, 3, 4, 5…:

Fract-L for x = a1/b1 and y = (a2/b2 * 2) for b1 = n = 2, 3, 4, 5, 6, 7, 8…


Fract-L for b1 = 2n = 2, 4, 6, 8, 10…


Fract-L for b1 = 3n = 3, 6, 9, 12, 15…


Fract-L for b1 = 4n


Fract-L for b1 = 5n


Fract-L for b1 = 6n


Animated fract-L for b1 = 1n..12n (animated at ezGif)


Finally, here are fract-Ls when b1 is a triangular, square, hexagonal or octagonal number:

Fract-L for x = a1/b1 and y = (a2/b2 * 2) for triangular(b1) = 3, 6, 10, 15, 21, 28,…


Fract-L for square(b1) = 4, 9, 16, 25, 36, 49,…


Fract-L for hexagonal(b1) = 6, 15, 28, 45, 66, 91,…


Fract-L for octagonal(b1) = 8, 21, 40, 65, 96, 133,…


Elsewhere Other-Accessible…

Back to Drac’ — a parallel pun for a pre-previous fractal
I Like Gryke — a first look at the limestone fractal
Lime Time — more on the limestone fractal

Dog-Gristle Politics

As Reform seeks a statement victory in Gorton and Denton, with a candidate who has made dog-whistle nativism his calling card, Manchester has again become a political crucible. […] As Nigel Farage’s intellectual outriders speak of a “politics of home”, which casts doubt over certain citizens’ right to be considered British, a billionaire tax exile in Monaco has added grist to their mill. — “Guardian view on Sir Jim Ratcliffe: Britain does not need political lectures from a billionaire tax exile”, The Guardian, 12ii26