# Monbiot’s Mothbiota

When they opened the trap, I was astonished by the range and beauty of their catch. There were pink and olive elephant hawkmoths; a pine hawkmoth, feathered and ashy; a buff arches, patterned and gilded like the back of a barn owl; flame moths in polished brass; the yellow kites of swallow-tailed moths; common emeralds the colour of a northern sea, with streaks of foam; grey daggers; a pebble prominent; heart and darts; coronets; riband waves; willow beauties; an elder pearl; small magpie; double-striped pug; rosy tabby. The names testify to a rich relationship between these creatures and those who love them. — George Monbiot, “Our selective blindness is lethal to the living world”, *The Guardian*, 20xii2017

# Get Your Prox Off #3

I’ve looked at lot at the fractals created when you randomly (or quasi-randomly) choose a vertex of a square, then jump half of the distance towards it. You can ban jumps towards the same vertex twice in a row, or jumps towards the vertex clockwise or anticlockwise from the vertex you’ve just chosen, and so on.

But you don’t have to choose vertices directly: you can also choose them by distance or proximity (see “Get Your Prox Off” for an earlier look at fractals-by-distance). For example, this fractal appears when you can jump half-way towards the nearest vertex, the second-nearest vertex, and the third-nearest vertex (i.e., you can’t jump towards the fourth-nearest or most distant vertex):

vertices = 4, distance = (1,2,3), jump = 1/2

It’s actually the same fractal as you get when you choose vertices directly and ban jumps towards the vertex diagonally opposite from the one you’ve just chosen. But this fractal-by-distance isn’t easy to match with a fractal-by-vertex:

v = 4, d = (1,2,4), j = 1/2

Nor is this one:

v = 4, d = (1,3,4)

This one, however, is the same as the fractal-by-vertex created by banning a jump towards the same vertex twice in a row:

v = 4, d = (2,3,4)

The point can jump towards second-nearest, third-nearest and fourth-nearest vertices, but not towards the nearest. And the nearest vertex will be the one chosen previously.

Now let’s try squares with an additional point-for-jumping-towards on each side (the points are numbered 1 to 8, with points 1, 3, 5, 7 being the true vertices):

v = 4 + s1 point on each side, d = (1,2,3)

v = 4 + s1, d = (1,2,5)

v = 4 + s1, d = (1,2,7)

v = 4 + s1, d = (1,3,8)

v = 4 + s1, d = (1,4,6)

v = 4 + s1, d = (1,7,8)

v = 4 + s1, d = (2,3,8)

v = 4 + s1, d = (2,4,8)

And here are squares where the jump is 2/3, not 1/2, and you can choose only the nearest or third-nearest jump-point:

v = 4, d = (1,3), j = 2/3

v = 4 + s1, d = (1,3), j = 2/3

Now here are some pentagonal fractals-by-distance:

v = 5, d = (1,2,5), j = 1/2

v = 5 + s1, d = (1,2,7)

v = 5 + s1, d = (1,2,8)

v = 5 + s1, d = (1,2,9)

v = 5 + s1, d = (1,9,10)

v = 5 + s1, d = (1,10), j = 2/3

v = 5 + s1, d = (various), j = 2/3 (animated)

And now some hexagonal fractals-by-distance:

v = 6, d = (1,2,4), j = 1/2

v = 6, d = (1,3,5)

v = 6, d = (1,3,6)

v = 6, d = (1,2,3,4)

v = 6 + central point, d = (1,2,3,4)

v = 6, d = (1,2,3,6)

v = 6, d = (1,2,4,6)

v = 6, d = (1,3,4,5)

v = 6, d = (1,3,4,6)

v = 6, d = (1,4,5,6)

Elsewhere other-accessible:

• Get Your Prox Off — an earlier look at fractals-by-distance

• Get Your Prox Off # 2 — and another

# Loricifera Rising

N.B. The title of this incendiary intervention is a paronomasia on Kenneth Anger’s film

*Lucifer Rising*(1972) (which I ain’t never seen nohow).

# B a Pal

As a keyly committed core component of the counter-cultural community (I wish!), I like to post especially edgy and esoteric material to Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral *on the 23rd of each month*. And today I may be posting the especially edgiest and esoterickest material *ever* dot dot dot

After all, this entry at the *Online Encyclopedia of Integer Sequences* is about numbers that are palindromes in two particularly pertinent bases:

A060792Numbers that are palindromic in bases 2 and 3.0, 1, 6643, 1422773, 5415589, 90396755477, 381920985378904469, 1922624336133018996235, 2004595370006815987563563, 8022581057533823761829436662099, 392629621582222667733213907054116073, 32456836304775204439912231201966254787, 428027336071597254024922793107218595973 (A060792 at OEIS, with more entries)

And here are the underlying palindromes:

0: 0 ↔ 0

1: 1 ↔ 1

6643: 1100111110011 ↔ 100010001

1422773: 101011011010110110101 ↔ 2200021200022

5415589: 10100101010001010100101 ↔ 101012010210101

90396755477: 1010100001100000100010000011000010101 ↔ 22122022220102222022122

381920985378904469: 10101001100110110110001110011011001110001101101100110010101 ↔ 2112200222001222121212221002220022112

1922624336133018996235: 11010000011100111000101110001110011011001110001110100011100111000001011 ↔

122120102102011212112010211212110201201021221

2004595370006815987563563: 110101000011111010101010100101111011110111011110111101001010101010111110000101011 ↔ 221010112100202002120002212200021200202001211010122

8022581057533823761829436662099: 1100101010000100101101110000011011011111111011000011100001101111111101101100000111011010010000101010011 ↔ 21000020210011222122220212010000100001021202222122211001202000012

392629621582222667733213907054116073: 10010111001111000100010100010100000011011011000101011011100000111011010100011011011000000101000101000100011110011101001 ↔ 122102120011102000101101000002010021111120010200000101101000201110021201221

32456836304775204439912231201966254787: 11000011010101111010110010100010010011011010101001101000001000100010000010110010101011011001001000101001101011110101011000011 ↔ 1222100201002211120110022121002012121101011212102001212200110211122001020012221

428027336071597254024922793107218595973: 101000010000000110001000011111100101011110011100001110100011100010001110001011100001110011110101001111110000100011000000010000101 ↔ 222001200110022102121001000200200202022111220202002002000100121201220011002100222

0: 0 ↔ 0

1: 1 ↔ 1

6643: 1100111110011 ↔ 100010001

1422773: 101011011010110110101 ↔ 2200021200022

5415589: 10100101010001010100101 ↔ 101012010210101

90396755477: 1010100001100000100010000011000010101 ↔ 22122022220102222022122

381920985378904469: 10101001100110110110001110011011001110001101101100110010101 ↔ 2112200222001222121212221002220022112

1922624336133018996235: 11010000011100111000101110001110011011001110001110100011100111000001011 ↔

122120102102011212112010211212110201201021221

2004595370006815987563563: 110101000011111010101010100101111011110111011110111101001010101010111110000101011 ↔ 221010112100202002120002212200021200202001211010122

8022581057533823761829436662099: 1100101010000100101101110000011011011111111011000011100001101111111101101100000111011010010000101010011 ↔ 21000020210011222122220212010000100001021202222122211001202000012

392629621582222667733213907054116073: 10010111001111000100010100010100000011011011000101011011100000111011010100011011011000000101000101000100011110011101001 ↔ 122102120011102000101101000002010021111120010200000101101000201110021201221

32456836304775204439912231201966254787: 11000011010101111010110010100010010011011010101001101000001000100010000010110010101011011001001000101001101011110101011000011 ↔ 1222100201002211120110022121002012121101011212102001212200110211122001020012221

428027336071597254024922793107218595973: 101000010000000110001000011111100101011110011100001110100011100010001110001011100001110011110101001111110000100011000000010000101 ↔ 222001200110022102121001000200200202022111220202002002000100121201220011002100222

# Foam Comb

# Nail Supremacy

Ὁ γαρ ἡδονής και ἀλγηδόνος ἧλος, ὃς πρὸς το σώμα τήν ψυχην προσηλοῖ, μέγιστον κακὸν ἔχειν ἔοικε, τὸ τα αἰσθητά ποιεῖν ἐναργέστερα τῶν νοητῶν, καὶ καταβιάζεσθαι καὶ πάθει μᾶλλον ἢ λόγῳ κρίνειν τήν διάνοιαν.

• ΠΡΟΒΛΗΜΑ Β’. Πώς Πλάτων ἔλεγε τον θεὸν άεὶ γεωμετρεῖν.

Nam voluptatis et doloris ille clavus, quo animus corpori affigitur, id videtur maximum habere malum, quod sensilia facit intelligibilibus evidentiora, vimque facit intellectui, ut affectionem magis quam rationem in judicando sequatur.

• QUÆSTIO II: Qua ratione Plato dixerit, Deum semper geometriam tractare.

For the nail of pain and pleasure, which fastens the soul to the body, seems to do us the greatest mischief, by making sensible things more powerful over us than intelligible, and by forcing the understanding to determine them rather by passion than by reason.

• Plutarch’s *Symposiacs*, QUESTION II: What is Plato’s Meaning, When He Says that God Always Plays the Geometer?

# Fractal + Star = Fractar

Here’s a three-armed star made with three lines radiating at intervals of 120°:

Triangular fractal stage #1

At the end of each of the three lines, add three more lines at half the length:

Triangular fractal #2

And continue like this:

Triangular fractal #3

Triangular fractal #4

Triangular fractal #5

Triangular fractal #6

Triangular fractal #7

Triangular fractal #8

Triangular fractal #9

Triangular fractal #10

Triangular fractal (animated)

Because this fractal is created from a series of star, you could call it a fractar. Here’s a black-and-white version:

Triangular fractar (black-and-white)

Triangular fractar (black-and-white) (animated)

(Open in a new window for larger version if the image seems distorted)

A four-armed star doesn’t yield an easily recognizable fractal in a similar way, so let’s try a five-armed star:

Pentagonal fractar stage #1

Pentagonal fractar #2

Pentagonal fractar #3

Pentagonal fractar #4

Pentagonal fractar #5

Pentagonal fractar #6

Pentagonal fractar #7

Pentagonal fractar (animated)

Pentagonal fractar (black-and-white)

Pentagonal fractar (bw) (animated)

And here’s a six-armed star:

Hexagonal fractar stage #1

Hexagonal fractar #2

Hexagonal fractar #3

Hexagonal fractar #4

Hexagonal fractar #5

Hexagonal fractar #6

Hexagonal fractar (animated)

Hexagonal fractar (black-and-white)

Hexagonal fractar (bw) (animated)

And here’s what happens to the triangular fractar when the new lines are rotated by 60°:

Triangular fractar (60° rotation) #1

Triangular fractar (60°) #2

Triangular fractar (60°) #3

Triangular fractar (60°) #4

Triangular fractar (60°) #5

Triangular fractar (60°) #6

Triangular fractar (60°) #7

Triangular fractar (60°) #8

Triangular fractar (60°) #9

Triangular fractar (60°) (animated)

Triangular fractar (60°) (black-and-white)

Triangular fractar (60°) (bw) (animated)

Triangular fractar (60°) (no lines) (black-and-white)

A four-armed star yields a recognizable fractal when the rotation is 45°:

Square fractar (45°) #1

Square fractar (45°) #2

Square fractar (45°) #3

Square fractar (45°) #4

Square fractar (45°) #5

Square fractar (45°) #6

Square fractar (45°) #7

Square fractar (45°) #8

Square fractar (45°) (animated)

Square fractar (45°) (black-and-white)

Square fractar (45°) (bw) (animated)

Without the lines, the final fractar looks like the plan of a castle:

Square fractar (45°) (bw) (no lines)

And here’s a five-armed star with new lines rotated at 36°:

Pentagonal fractar (36°) #1

Pentagonal fractar (36°) #2

Pentagonal fractar (36°) #3

Pentagonal fractar (36°) #4

Pentagonal fractar (36°) #5

Pentagonal fractar (36°) #6

Pentagonal fractar (36°) #7

Pentagonal fractar (36°) (animated)

Again, the final fractar without lines looks like the plan of a castle:

Pentagonal fractar (36°) (no lines) (black-and-white)

Finally, here’s a six-armed star with new lines rotated at 30°:

Hexagonal fractar (30°) #1

Hexagonal fractar (30°) #2

Hexagonal fractar (30°) #3

Hexagonal fractar (30°) #4

Hexagonal fractar (30°) #5

Hexagonal fractar (30°) #6

Hexagonal fractar (30°) (animated)

And the hexagonal castle plan:

Hexagonal fractar (30°) (black-and-white) (no lines)

# Chough’s Enough

# Oh My Guardian #8

“When it comes to Harry Potter, JK Rowling just can’t leave it alone. This is not necessarily a bad thing – fans have got to see Harry and friends all grown-up in the Cursed Child plays – but she’s also managed to muddy the waters by her constant rejigging of the original narrative furniture.” — Fantastic Beasts isn’t racist, but JK Rowling should stop tweaking the source material, Hannah Flint, *The Guardian*, 28ix2018.

• Oh My Guardian #7 — the previous entry in this award-winning series

• Reds under the Thread

*more*on mixed metaphors… in terms of

*The Guardian*…

• All posts interrogating issues around the

*Guardian*-reading community and its affiliates