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.
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
The Koch Snowflake is a fractal named after the Swedish mathematician Helge von Koch (1870-1924). It’s simple to make, attractive to see:
And here’s how you making it, starting with an equilateral triangle:
Make the Flake #0
Make the Flake #1: Erect a 1/9-sized equilateral triangle on middle of each side
Make the Flake #2: Then a 1/81-sized equilateral triangle on the middle of each straight line created by #1
Make the Flake #3: And so on.
Make the Flake #4
Make the Flake #5
Make the Flake #6
Make the Flake #0-6 (animated at ezGif)
In the end, the Koch snowflake has an infinitely long perimeter around a finite area (see discussion at Wikipedia). The Koch anti-snowflake or un-flake also combines an infinitely long perimeter and finite area:
Koch un-Flake
You make the un-flake like this:
Make the Un-Flake #0
Make the Un-Flake #1
Make the Un-Flake #2
Make the Un-Flake #3
Make the Un-Flake #4
Make the Un-Flake #5
Make the Un-Flake #6
Make the Un-Flake #0-6 (animated at ezGif)
And you can combine the Koch snowflake and the Koch anti-snowflake like this:
Make the Flake+Un-Flake #0
Make the Flake+Un-Flake #1
Make the Flake+Un-Flake #2
Make the Flake+Un-Flake #3
Make the Flake+Un-Flake #4
Make the Flake+Un-Flake #5
Make the Flake+Un-Flake #6
Make the Flake+Un-Flake #0-6 (animated at ezGif)
Here’s another variation, what you might call the Koch-Cross flake:
Koch-Cross Flake #0
Koch-Cross Flake #1
Koch-Cross Flake #2
Koch-Cross Flake #3
Koch-Cross Flake #4
Koch-Cross Flake #5
Koch-Cross Flake #6
Koch-Cross Flake #0-6 (animated at ezGif)
And the Koch-Cross un-flake:
Koch-Cross un-Flake
And combined Koch-Cross Flake+un-Flake:
Combined Koch-Cross Flake+un-Flake
Here’s an animation of the combined Koch-Cross Flake+un-Flake:
Combined Koch-Cross Flake+un-Flake (animated at ezGif)
And the combined Koch-Cross Flake+un-Flake created on the sides of a square:
Combined Koch-Cross Flake+un-Flake on square
And a final variant of the infinitely many on offer:
Variant Koch snowflake stage #1
Variant Koch snowflake stage #6
Un-Flake of variant Koch snowflake
Combined variant Koch Flake+un-Flake
“The chancellor’s unsuccessful strategy for spiking the AfD’s guns has been to ape its draconian positions on immigration.” — The Guardian view on Friedrich Merz: Germany’s chancellor must change course to halt the rise of the AfD, The Guardian, 21ix26
