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.
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
Here’s a graph representing the fractional parts of √n for n = 1 to 1832, with frac(√1) at the top left and frac(√1832) at the bottom right:
graph for frac(√n), n = 1..1832
There’s an odd optical illusion making it seem as though each set of triangular waves ends lower on the right than it starts on the left. Otherwise, it’s a dull graph, because the fractional part of √n simply rises towards 0.9999…, then falls to 0 like this:
0 = frac(1) = frac(√1)
0.414213562373… = frac(1.414213562373…) = frac(√2)
0.732050807568… = frac(1.732050807568…) = frac(√3)
0 = frac(2) = frac(√4)
0.236067977499… = frac(2.236067977499…) = frac(√5)
0.449489742783… = frac(2.449489742783…) = frac(√6)
0.645751311064… = frac(2.645751311064…) = frac(√7)
0.828427124746… = frac(2.828427124746…) = frac(√8)
0 = frac(3) = frac(√9)
0.162277660168… = frac(3.162277660168…) = frac(√10)
0.316624790355… = frac(3.316624790355…) = frac(√11)
0.464101615137… = frac(3.464101615137…) = frac(√12)
0.605551275463… = frac(3.605551275463…) = frac(√13)
0.741657386773… = frac(3.741657386773…) = frac(√14)
0.872983346207… = frac(3.872983346207…) = frac(√15)
0 = frac(4) = frac(√16)
0.123105625617… = frac(4.123105625617…) = frac(√17)
0.242640687119… = frac(4.242640687119…) = frac(√18)
0.358898943540… = frac(4.358898943540…) = frac(√19)
0.472135954999… = frac(4.472135954999…) = frac(√20)
But what about the fractional parts of the sum of √n? What does that graph look like? Much more interesting:
frac(sum(√n)), n = 1..1832
Here are the fractional parts for the sum of √n:
0 = frac(1) = frac(sum(√1))
0.414213562373… = frac(02.414213562373…) = frac(sum(√1..√2))
0.146264369941… = frac(04.146264369941…) = frac(sum(√1..√3))
0.146264369941… = frac(06.146264369941…) = frac(sum(√1..√4))
0.382332347441… = frac(08.382332347441…) = frac(sum(√1..√5))
0.831822090224… = frac(10.831822090224…) = frac(sum(√1..√6))
0.477573401289… = frac(13.477573401289…) = frac(sum(√1..√7))
0.306000526035… = frac(16.306000526035…) = frac(sum(√1..√8))
0.306000526035… = frac(19.306000526035…) = frac(sum(√1..√9))
0.468278186204… = frac(22.468278186204…) = frac(sum(√1..√10))
0.784902976559… = frac(25.784902976559…) = frac(sum(√1..√11))
0.249004591697… = frac(29.249004591697…) = frac(sum(√1..√12))
0.854555867161… = frac(32.854555867161…) = frac(sum(√1..√13))
0.596213253935… = frac(36.596213253935…) = frac(sum(√1..√14))
0.469196600142… = frac(40.469196600142…) = frac(sum(√1..√15))
0.469196600142… = frac(44.469196600142…) = frac(sum(√1..√16))
0.592302225760… = frac(48.592302225760…) = frac(sum(√1..√17))
0.834942912879… = frac(52.834942912879…) = frac(sum(√1..√18))
0.193841856420… = frac(57.193841856420…) = frac(sum(√1..√19))
0.665977811419… = frac(61.665977811419…) = frac(sum(√1..√20))
As with square roots, so with cube roots. The graph of frac(∛n) looks like this:
frac(∛n), n = 1..1832
It’s dull again, because the fractional part of ∛n is simply rising towards 0.9999…, then falling to 0. Just more slowly. But the graph of frac(sum(∛n)) looks like this:
frac(sum(∛n))
But roots don’t end with √n and ∛n, of course. Those roots represent n^(1/2) and n^(1/3), respectively, because when x = n^(a/b), n = x^(b/a). What about the graph of sum(n^(4/5)), where n = (n^4/5)^(5/4)? The graph looks like this:
frac(sum(n^(4/5)))
One of the curves in that graph reminds of the cephalopodic tentacles in Jean Delville’s marvellous painting Les Trésors de Sathan (sic) (Treasures of Satan) (1895), which was used on the cover of Morbid Angel’s Blessed Are the Sick (1991). The curve is longer in frac(sum(n^(84/97))):
frac(sum(n^(84/97))) (curve in red)
Les Trésors de Sathan (1895) by Jean Delville as the cover of Blessed Are the Sick (1991)
frac(sum(n^(4/5))) (curve in red)
Those Delvillean curves are examples of how, as the a/b of n^(a/b) climbs from 1/b to (b-1)/b, the graph of frac(sum(n^(a/b))) changes in interesting ways. Here are the graphs for n^(1..10/11):
frac(sum(n^(1/11)))
frac(sum(n^(2/11)))
frac(sum(n^(3/11)))
frac(sum(n^(4/11)))
frac(sum(n^(5/11)))
frac(sum(n^(6/11)))
frac(sum(n^(7/11)))
frac(sum(n^(8/11)))
frac(sum(n^(9/11)))
frac(sum(n^(10/11)))
Here’s an animation of those graphs:
animation of frac(sum(n^(1/11..10/11))) (created at ezGif)
And here are graphs for n^(72/83) and n^(82/83), with a Delvillean curve in the graph of 72/83 and domes in the graph of 82/83:
frac(sum(n^(72/83)))
frac(sum(n^(82/83)))
As the denominators get bigger, so do the domes:
frac(sum(n^(1000/1009)))
frac(sum(n^(1001/1009)))
frac(sum(n^(1002/1009)))
frac(sum(n^(1003/1009)))
frac(sum(n^(1004/1009)))
frac(sum(n^(1005/1009)))
frac(sum(n^(1006/1009)))
frac(sum(n^(1007/1009)))
frac(sum(n^(1008/1009)))
Here’s an animation of those graphs:
animation of frac(sum(n^(1000/1009..1008/1009))) (created at ezGif)
Previously Pre-Posted…
• Think Frinc — an earlier look at fraction-patterns
• Altars of Mathness — and similar patterns from integer-digits
Elsewhere Other-Accessible…
• Jean Delville at Wikipedia
• Blessed Are the Sick at BandCamp
Thinking it over, I’ve decided that trircle is a much better name than ciangle:
↓
A Sierpiński trircle, or triangle converted into a circle
A trircle is a triangle converted into a circle; a ciangle is a circle converted into a triangle. Or another regular polygon. The trircle reminded me that circularized fractals are fungible, because the Sierpiński triangle can be converted into another regular polygon like a square or pentagon or hexagon. You can go viâ the trircle, but you don’t have to. The point is that the Sierpiński triangle has a center and points lying at some distance and some angle 0° through 360°, so you can easily adjust the points to fit inside any other regular polygon:
A Sierpiński triangle again
↓
A Sierpiński trare, or triangle converted into square
↓
A Sierpiński trentagon
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A Sierpiński trexagon
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A Sierpiński treptagon
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A Sierpiński troctogon
Here’s an animation of the conversions:
Sierpiński triangle → square, pentagon, hexagon, heptagon, octagon (animated at ezGif)
A Sierpiński carpet, with points lying at 0° through 360° inside a square, is similarly fungible:
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A Sierpiński carpet converted into a circle
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A Sierpiński carpet converted into a triangle
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A Sierpiński carpet again
↓
A Sierpiński carpet converted into a pentagon
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↓
↓
And an animation of the carpet conversions:
Sierpiński carpet → triangle, pentagon, hexagon, heptagon, octagon (animated at ezGif)
Finally, and fungibly, the fractal that I call the centered Sierpiński triangle:
A centered Sierpiński triangle
↓
A centered Sierpiński trircle, or triangle converted into a circle
↓
↓
↓
↓
↓
And the final animation:
Centered Sierpiński triangle → square, pentagon, hexagon, heptagon, octagon (animated at ezGif)
Cover of The Sea Nymphs’ On The Dry Land (2016), with radiolarian skeletons
Good band-name, beautiful cover, but I don’t like the music. It’s interesting to compare a trimmed version of the cover:
Trimmed version of The Sea Nymphs’ On the Dry Land
Elsewhere Other-Accessible…
• On the Dry Land at Bandcamp
• Radiolaria at Wikipedia
Five feathers found,
Then pearls of rain
Gliding the face
Of a puddle.
Yet the hornèd snails
Are wiser than we
In the wet-wefted wonder
Whereby the worlds be.
