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.
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.
What’s the point of writing 2/10? It’s equal to 1/5, just as 4/10 = 2/5, 6/10 = 3/5 and 8/10 = 4/5. But there are times when the full-fat form can be mathematically — and æsthetically — appropriate. For example, you get a nice series of triangular waves — and an optical illusion — when you use all the rational fractions to create a graph representing the size of the fraction, a/b, as 0 < a/b < 1. If the even denominators are in white and the odd denominators in red, the graph looks like this:
Graph of 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, …
(click for larger)
The optical illusion is that the lines of waves appear to sink from left to right. Otherwise the graph has no complexity. But suppose you take the continued fraction of a/b, reverse it, and calculate a new fraction from the reversal. If you use the fractions in simplest form, you get this graph:
Graph of a2/b2 = reversed(contfrac(a1/b1)) for 1/2, 1/3, 2/3, 1/4, 3/4, …
But if you use all the rational fractions, simplified and unsimplified, you get a graph with pyramids and ships, like this:
Graph of a2/b2 = reversed(contfrac(a1/b1)) for 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, …
The full pyramids are few, alas. But the other ones decay in interesting ways.
Fall-Crumbs’ Fulcrums
Lichen starred
The chevrons
Of a sign
And she drank
The ivy’s
Nectar’d whine.
Said the physick’d world
Of a leaf that twirled
In the web of an autumn spider:
“Dandelion spheres
Are worth all tears
To a girl with the sun inside her.”
Volucella zonaria, the Hornet Mimic Hoverfly (info and image from Wikipedia)
Possly the cleverest or funniest knock-knock joke ever created. But you might not get it.
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Knock-knock.
Who’s there?
Philip Glass.
• Joke by the photo-realist artist Chuck Close
