• Μεστὰ δὲ πάντα σημείων καὶ σοφός τις ὁ μαθὼν ἐξ ἄλλου ἄλλο. — Πλωτῖνος, Ἐννεάδες Βʹ.γʹ.ζʹ
• • “All teems with symbol; the wise man is he who in any one thing can learn of another.” — Plotinus, Ennead II. 3. 7
Scale Supremacy
Orange-tip butterfly ♂, Anthocharis cardamines
lepidoptera, n. a large order of insects comprising the butterflies and moths. ← Greek λεπιδο-, “scale” + πτερόν, “wing” — Oxford English Dictionary
The Power of Cow
1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925, …
• Narayana’s cows sequence: a(0) = a(1) = a(2) = 1; thereafter a(n) = a(n-1) + a(n-3). […] Number of digits in A061582. — A000930 at the Online Encyclopedia of Integer Sequences
1, 3, 9, 27, 621, 1863, 324189, 961232427, 2718369612621, 6213249182718361863, 1863961227324621324918324189, 32418927183662196121863961227324961232427, 961232427621324918186327183632418927183662196122718369612621, …
• a(1) = 1, a(n) = number obtained by replacing each digit of a(n-1) with three times its value. — A061582 at the OEIS
Vowel-Voided Verse
THE RUSSO-TURKISH WAR
WAR harms all ranks, all arts, all crafts appal;
At Mars’ harsh blast arch, rampart, altar fall!
Ah! hard as adamant a braggart Czar
Arms vassal-swarms, and fans a fatal war!
Rampant at that bad call, a Vandal band
Harass, and harm, and ransack Wallach-land.
A Tartar phalanx Balkan’s scarp hath past,
And Allah’s standard falls, alas! at last.
THE FALL OF EVE
EVE, Eden’s empress, needs defended be;
The Serpent greets her when she seeks the tree.
Serene she sees the speckled tempter creep;
Gentle he seems — perverted schemer deep —
Yet endless pretexts, ever fresh, prefers,
Vervetts her senses, revers when she errs.
Sneers when she weeps, regrets, repents she fell,
Then, deep-revenged, reseeks the nether Hell!
THE APPROACH OF EVENING
IDLING I sit in this mild twilight dim.
Whilst birds, in wild swift vigils, circling skim.
Light wings in sighing sink, till, rising bright.
Night’s Virgin Pilgrim swims in vivid light.
INCONTROVERTIBLE FACTS
NO monk too good to rob, or cog, or plot.
No fool so gross to bolt Scotch collops hot.
From Donjon tops no Oronooko rolls.
Logwood, not lotos, floods Oporto’s bowls.
Troops of old tosspots oft to sot consort.
Box tops our schoolboys, too, do flog for sport.
No cool monsoons blow oft on Oxford dons,
Orthodox, jog-trot, book-worm Solomons!
Bold Ostrogoths of ghosts no horror show.
On London shop-fronts no hop-blossoms grow.
To crocks of gold no Dodo looks for food.
On soft cloth footstools no old fox doth brood.
Long storm-tost sloops forlorn do work to port.
Rooks do not roost on spoons, nor woodcocks snort.
Nor dog on snowdrop or on coltsfoot rolls.
Nor common frog concocts long protocols.
PHILOSOPHY
DULL humdrum murmurs lull, but hubbub stuns.
Lucullus snuffs up musk, mundungus shuns.
Puss purrs, buds burst, bucks butt, luck turns up trumps;
But full cups, hurtful, spur up unjust thumps.
• from Literary Frivolities, Fancies, Follies and Frolics compiled by by William T. Dobson (1880)
Third Whirled Warp
Here’s a regular hexagon inside a regular triangle, that is, an equilateral triangle:
Regular hexagon inside regular triangle
Imagine that two points are moving around the perimeter of each polygon, with the hex-point moving half as fast as the tri-point (after adjustment for the incommensurate relative lengths of the perimeters). If you trace the midpoint of the twin spinning points, you get this shape:
v3v6, 1 : 1/2, pol
And if you adjust the midpoint path as though the triangle had been stretched into a circle, you get this shape:
v3v6, 1 : 1/2, circ, pol
Here’s the same when the ratio of speeds is 1/2 to 1/3, that is, 1 to 2/3:
v3v6, 1/2 : 1/3, circ, pol
Without the polygons, it looks like this:
v3v6, 1/2 : 1/3, circ
When the ratio of speeds if -1/3 to 2/3, that is, the tri-point is moving counter-clockwise around the triangle, you get this shape:
v3v6, -1/3 : 2/3, pol
When it’s stretched into a circle, you get this:
v3v6, -1/3 : 2/3, circ, pol
It looks like a moustache:
v3v6, -1/3 : 2/3, circ
Here are more midpoint shapes created with a hexagon inside a triangle:
v3v6, 2/2 : 3/3, circ
v3v6, -1/2 : 3/4, circ
v3v6, 1/4 : 1/5, circ
v3v6, -1/4 : 3/4, circ
v3v6, -1/4 : 4/5, circ
v3v6, 2/3 : 3/4, circ
v3v6, 2/3 : 3/5, circ
v3v6, 3/4 : 4/5, circ
v3v6, 3/4 : 4/5, circ
Now try aligning the nested hexagon like this, so that the sides of the hexagon coincide with the middle third of the sides of the triangle:
v3v6, side alignment
With two points moving in a ratio of 1/3 to 1/4, you get this midpoint shape:
v3v6, sided, 1/3 : 1/4, pol
Here it is without the polygons:
v3v6, sided, 1/3 : 1/4
Now try a regular octagon inside a square:
v4v8, 1/2 : 1/3, circ, pol
v4v8, 1/2 : 1/3, circ
v4v8, -1/3 : 3/4, circ
v4v8, 2/3 : 3/5, circ
Now place a triangle inside a hexagon:
v6v3, 1 : 1/4, pol
If you stretch the midpoint path according to perimeter of the triangle, you get this:
v6v3, 1 : 1/4, circ, pol
v6v3, 1 : 1/4, circ
The three stretching shapes remind me of hands in Egyptian art, like this image of King Tutankhamun and Queen Ankhesenamun:
Detail from the Golden Throne of Tutankhamnun
Here are more midpoint paths:
v6v3, 1 : -1/4, circ
v6v3, 1 : 1/2, circ
v6v3, 1 : 1/3, circ
v6v3, -1 : 1/3, circ
v6v3, -1 : 1/4, circ
v6v3, 1 : 1/5, circ
v6v3, 2/3 : 1/4, circ
Now try a square inside an octagon:
v8v4, 2/3 : 1/4, circ, pol
v8v4, 2/3 : 1/4, circ
v8v4, 2/5 : 1/6, circ
v8v4, 2/5 : 3/7, circ
v8v4, 4/5 : 3/7, circ
Elsewhere Other-Accessible…
• First Whirled Warp — an earlier look at this kind of geometry
• Second Whirled Warp — and another earlier look
Sheen Scene
Crossing the bridge,
The old bridge,
I caught my idle eye
On a small sign,
A white sign,
Sheened and set ahigh.
And rainbow flared
As I slantwise stared,
Idle passer-by.
Mentem Mutantes Moleculæ Matutinæ
Performativizing Papyrocentricity #80
Papyrocentric Performativity Presents…
• Pulsating Portal to Punk Paradise… – Puke, Pills & Pussy: On the Road with America’s Wildest Punk-Rock Performers, Olga Trebor (2025)
• Stains for Brains – The Secret Lives of Stones, Hettie Judah (2022)
• Mysterious Chess – Selected Poems, Jorge Luis Borges, edited Alexander Coleman (1999)
• Bees Please Me – The Bee Bible: 50 Ways to Keep Bees Buzzing, Sally Coulthard (2019)
• N.N-K.P-T.L.Q.P.w.P.M.a.A. – Noxious N*gg*r-Killer Pre-Teen Lactation Queens Party with Pope Muhammad at Auschwitz, Simon Whitechapel (2023)
• Absorbing Absinthesis – The Dedalus Book of Absinthe, Phil Baker (2001; 2006)
• Benny Bowden – Shaman of the Radical Right: The Life and Mind of Jonathan Bowden, Edward Dutton (2025)
• Kyle Away the Powers – A Year in Numbers: 365 Astonishing Maths Facts, Kyle D. Evans (2023)
• Warriors, Come Out to SLAY… – Encyclopedia Psychopathica: Top Tips, Tactics, and Targetting Techniques for Successful Serial Slayers, Dr Samuel P. Salatta (2025)
• Sinister Slooooow Slayer… – Slo-Mo Psycho: The Sinister Story of the Stockport Slayer, Dr Zachariah Zialli (2021)
• Maximal Munch Meisterwerk – Crunch: An Ode to Crisps, Natalie Whittle (2024)
• Northanger Abyss… – Jane in Blood: Castration, Clitoridolatry and Communal Cannibalism in the Novels of Jane Austen, Dr Miriam B. Stimbers (2025)
Or simply…
Moniliform Maths
2 = 1/2 + 2/4 + 3/8 + 4/16 + 5/32…
sum(np / 2n)
2 = prime = sum(n / 2n)
6 = 2·3 = sum(n2 / 2n)
26 = 2·13 = sum(n3 / 2n)
150 = 2·3·52 = sum(n4 / 2n)
1082 = 2·541 = sum(n5 / 2n)
9366 = 2·3·7·223 = sum(n6 / 2n)
94586 = 2·47293 = sum(n7 / 2n)
1091670 = 2·3·5·36389 = sum(n8 / 2n)
14174522 = 2·7087261 = sum(n9 / 2n)
204495126 = 2·3·11·41·75571 = sum(n10 / 2n)
• A000629 Number of necklaces of partitions of n+1 labeled beads.
1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646, 185603174638656822266, 5355375592488768406230
• moniliform ← French moniliforme (1800 or earlier) ← classical Latin monīle necklace
sum(np / 3n)
3/4 = prime / (22) = sum(n / 3n)
3/2 = prime / prime = sum(n2 / 3n)
33/8 = (3·11) / (23) = sum(n3 / 3n)
15 = 3·5 = sum(n4 / 3n)
273/4 = (3·7·13) / (22) = sum(n5 / 3n)
1491/4 = (3·7·71) / (22) = sum(n6 / 3n)
38001/16 = (3·53·239) / (24) = sum(n7 / 3n)
17295 = 3·5·1153 = sum(n8 / 3n)
566733/4 = (3·188911) / (22) = sum(n9 / 3n)
2579313/2 = (3·11·47·1663) / prime = sum(n10 / 3n)
sum(np / 4n)
4/9 = (22) / (32) = sum(n / 4n)
20/27 = (22·5) / (33) = sum(n2 / 4n)
44/27 = (22·11) / (33) = sum(n3 / 4n)
380/81 = (22·5·19) / (34) = sum(n4 / 4n)
4108/243 = (22·13·79) / (35) = sum(n5 / 4n)
17780/243 = (22·5·7·127) / (35) = sum(n6 / 4n)
269348/729 = (22·172·233) / (36) = sum(n7 / 4n)
4663060/2187 = (22·5·107·2179) / (37) = sum(n8 / 4n)
10091044/729 = (22·2522761) / (36) = sum(n9 / 4n)
218374420/2187 = (22·5·11·23·103·419) / (37) = sum(n10 / 4n)
sum(np / 5n)
5/16 = prime / (24) = sum(n / 5n)
15/32 = (3·5) / (25) = sum(n2 / 5n)
115/128 = (5·23) / (27) = sum(n3 / 5n)
285/128 = (3·5·19) / (27) = sum(n4 / 5n)
3535/512 = (5·7·101) / (29) = sum(n5 / 5n)
26355/1024 = (3·5·7·251) / (210) = sum(n6 / 5n)
458555/4096 = (5·91711) / (212) = sum(n7 / 5n)
1139685/2048 = (3·5·75979) / (211) = sum(n8 / 5n)
25492435/8192 = (5·17·443·677) / (213) = sum(n9 / 5n)
316786305/16384 = (3·5·11·1919917) / (214) = sum(n10 / 5n)
sum(np / 6n)
6/25 = (2·3) / (52) = sum(n / 6n)
42/125 = (2·3·7) / (53) = sum(n2 / 6n)
366/625 = (2·3·61) / (54) = sum(n3 / 6n)
4074/3125 = (2·3·7·97) / (55) = sum(n4 / 6n)
11334/3125 = (2·3·1889) / (55) = sum(n5 / 6n)
189714/15625 = (2·3·7·4517) / (56) = sum(n6 / 6n)
3706518/78125 = (2·3·181·3413) / (57) = sum(n7 / 6n)
82749954/390625 = (2·3·7·1970237) / (58) = sum(n8 / 6n)
2078250726/1953125 = (2·3·31·1061·10531) / (59) = sum(n9 / 6n)
11598884682/1953125 = (2·3·7·11·232·47459) / (59) = sum(n10 / 6n)
sum(np / 7n)
7/36 = prime / (22·32) = sum(n / 7n)
7/27 = prime / (33) = sum(n2 / 7n)
91/216 = (7·13) / (23·33) = sum(n3 / 7n)
70/81 = (2·5·7) / (34) = sum(n4 / 7n)
2149/972 = (7·307) / (22·35) = sum(n5 / 7n)
3311/486 = (7·11·43) / (2·35) = sum(n6 / 7n)
285929/11664 = (7·40847) / (24·36) = sum(n7 / 7n)
220430/2187 = (2·5·7·47·67) / (37) = sum(n8 / 7n)
1359337/2916 = (7·17·11423) / (22·36) = sum(n9 / 7n)
5239157/2187 = (7·11·68041) / (37) = sum(n10 / 7n)
sum(np / 8n)
8/49 = (23) / (72) = sum(n / 8n)
72/343 = (23·32) / (73) = sum(n2 / 8n)
776/2401 = (23·97) / (74) = sum(n3 / 8n)
10440/16807 = (23·32·5·29) / (75) = sum(n4 / 8n)
174728/117649 = (23·21841) / (76) = sum(n5 / 8n)
3525192/823543 = (23·32·11·4451) / (77) = sum(n6 / 8n)
11870648/823543 = (23·41·36191) / (77) = sum(n7 / 8n)
319735800/5764801 = (23·32·52·19·9349) / (78) = sum(n8 / 8n)
9686934584/40353607 = (23·1210866823) / (79) = sum(n9 / 8n)
326084753016/282475249 = (23·32·11·16273·25301) / (710) = sum(n10 / 8n)
sum(np / 9n)
9/64 = (32) / (26) = sum(n / 9n)
45/256 = (32·5) / (28) = sum(n2 / 9n)
531/2048 = (32·59) / (211) = sum(n3 / 9n)
1935/4096 = (32·5·43) / (212) = sum(n4 / 9n)
34983/32768 = (32·132·23) / (215) = sum(n5 / 9n)
381465/131072 = (32·5·72·173) / (217) = sum(n6 / 9n)
9725787/1048576 = (32·67·1272) / (220) = sum(n7 / 9n)
35420535/1048576 = (32·5·787123) / (220) = sum(n8 / 9n)
1160703963/8388608 = (32·47·409·6709) / (223) = sum(n9 / 9n)
21129845715/33554432 = (32·5·11·2213·19289) / (225) = sum(n10 / 9n)
sum(np / 10n)
10/81 = (2·5) / (34) = sum(n / 10n)
110/729 = (2·5·11) / (36) = sum(n2 / 10n)
470/2187 = (2·5·47) / (37) = sum(n3 / 10n)
7370/19683 = (2·5·11·67) / (39) = sum(n4 / 10n)
142870/177147 = (2·5·7·13·157) / (311) = sum(n5 / 10n)
1114190/531441 = (2·5·7·11·1447) / (312) = sum(n6 / 10n)
30495890/4782969 = (2·5·3049589) / (314) = sum(n7 / 10n)
953934190/43046721 = (2·5·11·569·15241) / (316) = sum(n8 / 10n)
3728765410/43046721 = (2·5·372876541) / (316) = sum(n9 / 10n)
145739620510/387420489 = (2·5·11·1324905641) / (318) = sum(n10 / 10n)
The Hex Crystals
To coin a phrase: Never Mind the Bollocks — Here’s the Hex Crystals! And what is a hex crystal? It’s what I call a shape that’s created algorithmo inside a hexagon and looks like a crystal:
A hex crystal
Here are some more hex-crystals:
I came across hex-crystals when I was looking at an interesting little geometrical question. How does sum(vd), the sum of distances to the vertices of a square, vary from different points, (x,y), inside the square? Say the square is created inside a circle of radius = 500 units and centered on (x,y) = (0,0). When the point is at (0,0), the center of the square, sum(vd) is obviously 2000, because the four vertices all fall on the perimeter of the circle at 500 units from the center and 4 * 500 = 2000:
0
sum(vd) = 2000 = sum of distances to vertices from (0,0)
When is sum(vd) at a maximum? When the point is on one or another of the vertices, which are at (+/-354,+/-354) units in relation to the center at (0,0):
sum(vd) = 2414 = sum of distances to vertices from (354,-354)
More precisely, the sum is 2414.213562373… = 1000 * (√2 + 1) units and the vertices are at (+/-353.55339…, +/-353.55339…) units, as simple geometry dictates for a square inside a circle of radius 500. Accordingly, sum(vd) varies between exactly 2000 and 2414.213562373… as the point moves inside the square:
sum(vd) = 2165 from (132,256)
sum(vd) = 2182 from (-135,271)
sum(vd) = 2069 from (177,51)
I wondered what shapes appeared as one traced the route of a point jumping, say, 1/2 towards the vertices according to tests on sum(vd). For example, if the point starts at (0,0) at time t0) and sum(vd) at time ti has to be alternately greater and less than sum(vd) at ti-1 for successive jumps, you get this shape:
jump = 1/2, test = sum(vd,ti) >,< sum(vd,ti-1)
You can use the binary number 10bin to represent the test on sum(vd) at ti-1 and ti-1, i.e. the test at jump 1 is sum(vd,ti) > sum(vd,ti-1), at step 2 is sum(vd,ti) < sum(vd,ti-1), and so on. Using the same test and a jump of 1/3, you get this shape:
jump = 1/3, test = sum(vd,ti,10bin)
Now the shape is clearly a fractal. So are some of the other shapes I found by applying the same kind of tests to a point jumping inside a pentagon:
vertex = 5, jump = 55/144 = fib(10) / fib(12), test on sum(vd) = 10bin
v = 5, j = 55/144, test = 10010bin
v = 5, j = 55/144, test = 11000bin
When test = 10010bin, you read the binary number left-to-right and check for s1><s0,s2<s1,s3<s2,s4>s3,s5<s4. Then you apply the same tests to subsequent jumps, i.e., you return to the beginning of the binary number and read it left-to-right again. Now let’s apply similar tests to hexagons and create some hex-crystals:
v = 6, j = 1/2, test = 10bin
Various hex-crystals (animated gif courtesy EZgif)
I searched an array to calculate the possible routes, so the same test yielded different results depending on dp, the depth of the search. This is because tl, the length of the test, fits more or less well into dp by dp modulo tl, that is, by whether tl is a factor of dp. For example, when the test is 110 and tl = 3, you get this with dp = 9:
v = 6, j = 1/2, test = 110, dp = 9
And you get this when dp = 10 (i.e., dp = 9+1):
v = 6, j = 1/2, test = 110bin, dp = 10dec
Here are some more hex-crystals:
test = 1100bin
test = 1110bin
test = 10010bin
test = 11010bin
test = 11100bin
test = 101000, dp = 12
test = 101100bin
test = 111100bin
test = 111100, dp = 11
test = 1110010bin
test = 1111100bin
test = 10010110bin
test = 10011110bin
test = 11000110bin
test = 11001110bin
test = 11010110bin
test = 11100110bin
test = 11101000bin
test = 11110010bin
test = 100101000bin
test = 100111110bin
test = 110011110bin
test = 110111000bin
test = 1001101010bin
test = 1001111000bin
test = 1001111010bin
test = 1010011110bin
test = 1011101110bin
test = 1101010000bin
test = 1110001110bin
test = 1110101000bin
test = 1110101010bin
test = 1111100010bin
j = 1/3, test = 1 (i.e., for all jumps sum(vd) at ti > sum(vd) at ti-1, center point
j = 2/3, test = 11100bin
j = 2/5, test = 10010bin
Finally, here are some hex-crystals based on a test of sorted distances from (x,y), i.e. how the vertices rank by distance from (x,y):
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 