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?
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.
I identify as √-37 (the square root of -37) in bases 5 (on Mondays and Wednesdays), 17 (Tuesdays, Thursdays and alternate Saturdays) and 89 (Fridays, Sundays and alternate Saturdays).
My preferred pronouns are 6.08276i, 雲 and მსოფლიოს მეფე.
Cover of X-ÆON by the Italian psychedelic-rock band GIÖBIA
I don’t like the music, but I do like the cover.
• GIÖBIA at Bandcamp
Fascinating. Surprising. Inspiring. Baffling. Frustrating. Disappointing. Those are the things recreational maths has been to me down the decades. But sinister? Disturbing? Frightening?
No, rec-math had never been that to me. Not until a few days ago, when I was looking at the patterns created by the function digmul(n), which multiplies the digits of an integer. For example, if you ignore zeros, digmul(237) = 2*3*7 = digsum(7203) = digsum(611170) = 42. Here’s an un-sinister graph for digmul(n) of the integers 1 to 1353 (in each line, the digmul(n) is adjusted to fit into the same height, ht_mx, by finding the biggest digmul(n), mul_mx, on that line, then using ht = ht_mx * digmul(n)/mul_mx):
Graph of digmul(n) for n = 1..1353 (click for larger)
That’s not sinister, although the patterns are vaguely reminiscent of shark-fins. But then I looked at digmul(n) for the multiples of 2,3,4,5… Here are the graphs for multiples of 3, 5 and 8:
digmul(3n) for n = 3..4056
digmul(5n) for n = 5..6760
digmul(8n) for n = 8..10816
Again, nothing sinister. Then I pressed a key and the graph for digmul(9n), the digmul of the multiples of 9, appeared on the screen:
digmul(9n) for n = 9..12168
Gravestones! That’s what the patterns now reminded me of. Gravestones rising from or sinking into the soft earth of an ancient graveyard. And I must admit I felt unease and even fear for a second or two. Even now I don’t like the patterns when I think of them as gravestones, even though the curves are attractive. It hasn’t helped that I came across a piece of sinister recmath for the first time shortly before Halloween.
Old Mortlake Burial Ground, Richmond-upon-Thames, London (Wikipedia)
But what’s going on to produce those mathematical stelomorphs? (As you could call shapes that look like gravestones, using Greek στήλη, stēlē, a funerary monument.) Well, I don’t understand things enough, but it’s not a coincidence that 9 = 10-1 and the graph of digmul(9n) was for base 10. This are graphs for digmul((base-1)n) in bases 8, 20 and 29:
digmul(7n) in base 8 for n = 7..9464
digmul(19n) in base 20 for n = 19..25688
digmul(28n) in base 29 for n = 28..37856
The gravestones get wider as the base gets higher (the maximum width in base b is b-1). Let’s return to base 10 and have a look at the gravestone marked with a red dot in a detail from the graph for digmul(9n):
digmul(9n) in base 10
The marked gravestone represents these multiples of 9 (with the 9n for the much lower patches of earth directly to its left and right):
0128 = digmul(8820)
1152 = digmul(8829)
1536 = digmul(8838)
1792 = digmul(8847)
1920 = digmul(8856)
1920 = digmul(8865)
1792 = digmul(8874)
1536 = digmul(8883)
1152 = digmul(8892)
0072 = digmul(8901)
As you can see, there’s symmetry in the digmul(9n), because in base 10 some runs of consecutive 9n contain numbers whose digits are permutations of higher or lower 9n in that run. For example, digmul(8829) = digmul(8892) = 1152 because 8829 and 8892 have the same digits slightly rearranged. Similar runs must occur in other bases, b, for digmul((b-1)n).
What happens when you sum the Farey sequence, that is the “sequence of completely reduced fractions that have denominators less than or equal to n“?
1/2 + 1/3 + 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… → ?
Lots of things happen. But one simple thing is that the sum exceeds one integer after another, like this:
fr = 0
0 + 1/2 = 1/2 (count=1)
1/2 + 1/3 = 5/6 (count=2)
5/6 + 2/3 = 1_1/2 > 1 (c=3)
1_1/2 + 1/4 = 1_3/4 (c=4)
1_3/4 + 3/4 = 2_1/2 > 2 (c=5)
2_1/2 + 1/5 = 2_7/10 (c=6)
2_7/10 + 2/5 = 3_1/10 > 3 (c=7)
3_1/10 + 3/5 = 3_7/10 (c=8)
3_7/10 + 4/5 = 4_1/2 > 4 (c=9)
4_1/2 + 1/6 = 4_2/3 (c=10)
4_2/3 + 5/6 = 5_1/2 > 5 (c=11)
5_1/2 + 1/7 = 5_9/14 (c=12)
5_9/14 + 2/7 = 5_13/14 (c=13)
5_13/14 + 3/7 = 6_5/14 > 6 (c=14)
If you mark the steps at which the sum exceeds n on a square spiral, you get this pattern:
Square spiral for sum(1/2, 1/3, 2/3, 1/4, 3/4, 1/5, …) > 1, 2, 3, 4, 5,…
At a higher resolution, it looks like this:
Square spiral for sum(reduced(a/b)) > 1, 2, 3, 4, 5,… x 4
But it’s not obviously interesting. So try summing all fractions a/b, reduced and unreduced:
1/2 + 1/3 + 1/4 + 2/4 + 3/4 + 1/5 + 2/5 + 3/5 + 4/5 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6, …
fr = 0
0 + 1/2 = 1/2 (count=1)
1/2 + 1/3 = 5/6 (count=2)
5/6 + 2/3 = 1_1/2 > 1 (c=3)
1_1/2 + 1/4 = 1_3/4 (c=4)
1_3/4 + 2/4 = 2_1/4 > 2 (c=5)
2_1/4 + 3/4 = 3 (c=6)
3 + 1/5 = 3_1/5 > 3 (c=7)
3_1/5 + 2/5 = 3_3/5 (c=8)
3_3/5 + 3/5 = 4_1/5 > 4 (c=9)
4_1/5 + 4/5 = 5 (c=10)
5 + 1/6 = 5_1/6 > 5 (c=11)
5_1/6 + 2/6 = 5_1/2 (c=12)
5_1/2 + 3/6 = 6 (c=13)
6 + 4/6 = 6_2/3 > 6 (c=14)
6_2/3 + 5/6 = 7_1/2 > 7 (c=15)
7_1/2 + 1/7 = 7_9/14 (c=16)
7_9/14 + 2/7 = 7_13/14 (c=17)
7_13/14 + 3/7 = 8_5/14 > 8 (c=18)
8_5/14 + 4/7 = 8_13/14 (c=19)
8_13/14 + 5/7 = 9_9/14 > 9 (c=20)
9_9/14 + 6/7 = 10_1/2 > 10 (c=21)
10_1/2 + 1/8 = 10_5/8 (c=22)
Now the square spiral looks like this:
Square spiral for sum(unreduced(a/b)) > 1, 2, 3, 4, 5,…
After that, I tried experimenting with different denominators (the b of a/b). If the denominators are the multiples of 2, the reduced fractions a/2b create this spiral:
Square spiral for sum(reduced(a/2b))
But the unreduced fractions a/2b create something very different:
Square spiral for sum(unreduced(a/2b))
If the denominators are the odd numbers > 1 for reduced(a/b) and unreduced(a/b), you get two very different spirals that look like the two produced by a/2b:
Square spiral for sum(reduced(a/(2b+1)))
Square spiral for sum(unreduced(a/(2b+1)))
If the denominators are the Fibonacci numbers, you get this:
Square spiral for sum(reduced(a/b=fib_num))
And this:
Square spiral for sum(unreduced(a/fib_num=b))
The ratio of the areas of successive nested squares is tending towards φ = 1.618033988749894848…
But denominators as Fibonacci numbers aren’t obviously interesting, let’s return to multiples. When I tried denominators of 7b and 8b, the spirals turned chiral. That is, they were either right-handed or left-handed (chiral/ity roughly means “handed/ness”). There was a right-handed spiral:
Square spiral for sum(unreduced(a/9b))
And a left-handed spiral:
Square spiral for sum(unreduced(a/7b))
After that, I tried applying tests to a/b < 1. If you sum only those unreduced fractions a/b < 6/7, you get a three-armed left-handed spiral:
Square spiral for sum(unreduced(a/b) < 6/7)
And so on:
Square spiral for sum(unreduced(a/b) < 1/8)
Square spiral for sum(unreduced(a/b) < 8/9)
If sum(unreduced(a/b) < 6/7) produced a three-armed left-handed spiral, sum(unreduced(a/b) < 10/11) produced a three-armed right-handed spiral:
Square spiral for sum(unreduced(a/b) < 10/11)
And tests for a/b < 11/13 and a/b < 12/13 produced spirals of opposite chirality:
Square spiral for sum(unreduced(a/b) < 11/13)
Square spiral for sum(unreduced(a/b) < 12/13)
And what about irrational numbers? I tried summing the fractional part of various roots and got spirals like these:
Square spiral for sum(frac(n^2/3)))
Square spiral for sum(frac(n^3/5)))
And finally (for now), I tried adding roots of fr to fr itself. If fr1 = 1 and fri+1 = fri + sqrt(fri), you get this:
Square spiral for fr = fr + fr^1/2
When it’s the fifth root of fr added to fr, you get this:
Square spiral for fr += fr^1/5
Ay, but I meant not thee; I meant not her,
Whom all the pines of Ida shook to see
Slide from that quiet heaven of hers […] — from Tennyson’s “Lucretius” (1868)
Is Tennyson describing the descent of the goddess Cybele in the form of a meteorite on Mount Ida in Phrygia? I need to do more research.
Los Poetas
Somos los dueños
De todos los sueños
De la noche o del día.
Reiteramos
Y siempre bordamos
Esta melodía:
El mundo es el suyo,
El sol es el tuyo,
La luna es la mía.
• Clark Ashton Smith at The Eldritch Dark
Poets
We are masters
Of all dreams
Night ones and day ones.
We repeat
And always embroider
this melody:
World is theirs,
Sun is yours,
Moon is mine.
• Translation by: Ramón Cabrales at The Eldritch Dark
What happens when you divide the space between 0 and 1 into finer and finer slices using the fractions 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5…? Some interesting things. In the images below, the white line on the left with a white dot below it represents 0 and the white line on the right with a white dot below it represents 1. The single line in the middle of the first image represents 1/2 between 0 and 1, then the lines in the second image represent 1/4, 1/3, 1/2, 2/3, 3/4 between 0 and 1. And so on:
1/2 between 0 and 1 (click for larger)
1/4, 1/3, 1/2, 2/3, 3/4 between 0 and 1
a/b <= 7/8 between 0 and 1
a/b <= 15/16 between 0 and 1
a/b <= 31/32 between 0 and 1
a/b <= 63/64 between 0 and 1
a/b <= 127/128 between 0 and 1
a/b <= 255/256 between 0 and 1
Animation of a/b <= 255/256 between 0 and 1 (created at ezGif)
The lines are dividing space into finer and finer slices, but the lines aren’t distributed evenly. Some gaps close quickly, some slowly. Here’s some color for the slicing (when the colored circles below the lines are solid, they mark a/b <= 7/8; when they're hollow, they represent 1/9 <= a/b <= 11/12). The lines are now adjusted in height to the fraction a/b they represent, so they create a wedge rising from left to right:
1/2 between 0 and 1
a/b <= 1/3 between 0 and 1
a/b <= 2/3 between 0 and 1
a/b <= 1/4 between 0 and 1
a/b <= 3/4 between 0 and 1
a/b <= 4/5 between 0 and 1
a/b <= 5/6 between 0 and 1
a/b <= 6/7 between 0 and 1
[…]
↓
[…]
a/b <= 11/12 between 0 and 1
a/b <= 19/20 between 0 and 1
a/b <= 29/30 between 0 and 1
a/b <= 39/40 between 0 and 1
a/b <= 49/50 between 0 and 1
a/b <= 99/100 between 0 and 1
Animation of a/b <= 99/100 between 0 and 1 (created at ezGif)
Here’s how the colored lines representing a/b <= 11/12 get swamped as b gets larger and larger:
a/b <= 11/12 between 0 and 1
a/b <= 15/16 between 0 and 1
a/b <= 31/32 between 0 and 1
a/b <= 63/64 between 0 and 1
a/b <= 127/128 between 0 and 1
a/b <= 199/200 between 0 and 1
a/b <= 255/256 between 0 and 1
Animation of a/b <= 255/256 between 0 and 1 (created at ezGif)
Post-Performative Post-Scriptum
In fact, thrice isn’t slice as twice, because the line marking the slice of 1/2 gets swamped after the lines marking 1/3 and 2/3, which get swamped after 1/4 and 3/4, and so on.
Papyrocentric Performativity Presents…
• Slopwith Pilot – Biggles Breaks the Silence, W.E. Johns (1949)
• Di Another Day – Diana: The People’s Princess, Nicholas Owen, foreword by Trevor MacDonald (Welbeck 2020)
• Short-Changed at Sea – The Lonely Sea: Collected Short Stories, Alistair MacLean (1985)
• The Hyper-Heretical Hum of Horror… – The Definitive Guide to Horror Movies: 383 Films to Scare You to Death, edited by Kim Newman and James Marriott (Welbeck 2025)
• Vinum Verborum — Valorizing the Verbiviniculture of Clark Ashton Smith
• InS(p)ect(er)s and Others — The Snow Ghost and Other Tales: Classic Japanese Ghost Stories, translated by Lafcadio Hearn et al (Viking Classics 2023)
• Slayers on the Slate, Huh? – 21st Century Psychos: The Most Horrifying Killers of the Modern Era, Al Cimino (2025)
• Rock’n’Road – Lipstick and Leather: On the Road with the World’s Most Notorious Rock Stars, Kim Hawes (Sandstone Press 2023)
• Writefade – Nightshade, Michael Connelly (Orion 2025)
• Veer’d Tales… – S.H.A.D.O.W. Over Innsmouth, H.P. Liebeskraft
• Parabolic Papyrocinematicity – Needful Things, Stephen King (1991)
• Princess Di-Vid… – Shadows Of A Princess: An Intimate Account by Her Private Secretary, Patrick Jephson (2000)
Or simply…
Here are the simplified fractions in order of increasing denominator and numerator:
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, 1/9, 2/9, 4/9, 5/9, 7/9, 8/9, 1/10, 3/10, 7/10, 9/10, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11, 1/12, 5/12, 7/12, 11/12, 1/13, 2/13, 3/13, 4/13, 5/13, 6/13, 7/13, 8/13, 9/13, 10/13, 11/13, 12/13, …
You get 1/4 and 5/12, but no 2/4 or 3/12 and so on, because the fractions are simplified and 2/4 = 1/2 and 3/12 = 1/4. Now suppose you map those simplified fractions on a graph of x = y = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, … The graph is obviously just a straight line at 45°:
Graph of x = y = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, …
But suppose you adjust some of the simplified fractions for y, so that the list looks like this (adjusted fractions appear in red):
1/2, 1/3, 2/3 → 1/3, 1/4, 3/4 → 1/4, 1/5, 2/5, 3/5 → 2/5, 4/5 → 1/5, 1/6, 5/6 → 1/6, 1/7, 2/7 → 3/7, 3/7 → 2/7, 4/7 → 2/7, 5/7 → 3/7, 6/7 → 1/7, 1/8, 3/8, 5/8 → 3/8, 7/8 → 1/8, 1/9, 2/9 → 4/9, 4/9 → 2/9, 5/9 → 2/9, 7/9 → 4/9, 8/9 → 1/9, 1/10, 3/10, 7/10 → 3/10, 9/10 → 1/10, 1/11, 2/11 → 5/11, 3/11 → 4/11, 4/11 → 3/11, 5/11 → 2/11, 6/11 → 2/11, 7/11 → 3/11, 8/11 → 4/11, 9/11 → 5/11, 10/11 → 1/11, 1/12, 5/12, 7/12 → 5/12, 11/12 → 1/12, 1/13, 2/13 → 6/13, 3/13 → 4/13, 4/13 → 3/13, …
What does the graph for straight x and adjusted y look like? Much more interesting:
Graph of x = 1/2, 1/3, 2/3, …, y = contfrac_rev(1/2), contfrac_rev(1/3), contfrac_rev(2/3), …
That graph is what you could call a fractional fractal, because it uses fraction to create a fractal where parts resemble the whole and parts of the parts resemble the whole, down and down for ever (as I’ll show). I call it a limestone fractal or gryke fractal, because it looks like the fissured limestone pavements in the Yorkshire Dales, where the fissures are called grykes:
Fissured limestone pavement, Yorkshire Dales (Wikipedia)
A closer look at a limestone pavement (Dales Rocks)
As you plot the fractions for x and adjust the fractions for y, the gryke graph grows like this:
The gryke graph grows (animated at ezGif)
But how were the fractions for y adjusted to create the gryke fractal? With a very simple function called contfrac_rev(a/b). The function takes what’s called the continued fraction of a/b, then reverses it to derive a new fraction (sometimes it’s new, at least). For example, here’s the continued fraction of 13/17 and how you use the continued fraction to reconstruct 13/17:
continued_fraction(13/17) = [1,3,4] = cf(13/17)
fr = 0
1/4 = 1/(fr + 4) = 1/(0 + 4)
4/13 = 1/(fr + 3) = 1/(1/4 + 3)
13/17 = 1/(fr + 1) = 1/(4/13 + 1)13/17 = 0.764705882352941176470588…
The continued fraction, [1,3,4], is read from right to left when the original fraction is being reconstructed. So reversing the continued fraction means reading it from left to right, like this:
fr = 0
1 = 1/(fr + 1) = 1/(0 + 1)
1/4 = 1/(fr + 3) = 1/(1 + 3)
4/17 = 1/(fr + 4) = 1/(1/4 + 4)4/17 = 0.235294117647058823529411…
(Technical note: The reversed fraction of cf_rev(a/b) is always < 1/2, except for cf_rev(1/2) = 1/2. That's why the gryke graph above actually represents 0 to 1 on the x-axis and 0 to 0.5 on the y-axis.)
Now let's look at how the fractal in the graph can be magnified for ever and always yield new versions of the original graph. You can call it hiking the gryke, because you hike or increase the resolution of the graph to reveal the finer details. The first step is to mark the gryke site to be hiked. Here's the full graph, with a green square showing the section to be hiked or magnified or zoomed:
Gryke to hike
↓
Gryke hiked
You can combine the zoom-to-be and the zoomed section in a single image, where the full fractal is on the left and the zoomed section on the right:
Gryke-to-hike and Gryke-hiked (click for larger)
Here are more hikes of the gryke, or zooms of the full fractal:
Zoom #2
Zoom #3
Zoom #4
Zoom #5
Zoom #5+1
Zoom #5+2
Zoom #6
And here’s an animated gryke-hike or zoom into a single section of the fractal:
Hiking the Gryke (animated at ezGif)
Previously Pre-Posted…
• I Like Gryke — a first look at the gryke fractal
• Worms in Terms of Perms — permutating the gryke fractal
From The Ultimate Christmas Cracker (2019), compiled by John Julius Norwich:
Useful Palindromes
What other words would spring to the lips when breaking the news of the death of a prize herd of cattle, smitten with infective epilepsy?
Stiff, O dairyman! In a myriad of fits!
Dr Barnardo making out the menus:
Desserts I desire not, so long no lost one rise distressed.
The rebuke of the high-principled lady to her rich Norman lover:
Diamond light, Odo, doth gild no maid.
