Toxic Turntable #7

Currently listening…

• Slow Exploding Gulls, Salmaris EP (1997)
• Dubioso, Codicil LVI (1968)
• Ubair Yex, Weever (1973)
• Dux Tenebrarum, Quinque Fatuae (2012)
• Arctic Midge, Celsius (1992)
• Ijek Mveodeybda, Terë Conuva (1980)
• Schwarzschrein, Du Bist Dunst (1995)

Previously pre-posted:

Polymorphous Pursuit

Suppose four mice are standing on the corners of a large square. Each mouse begins running at the same speed towards the mouse one place away, reckoning clockwise. The mice will meet at the centre of the square and the path taken by each mouse will be what is known as a pursuit curve:

vertices = 4, mouse-increment = 1

v = 4, mi = 1 (animated)

As I showed in “Persecution Complex”, it’s easy to find variants on the basic pursuit curve. If mi = 2, i.e. each mouse runs towards the mouse two places away, the mice will run in straight lines direct to the centre of the square:

v = 4, mi = 2

v = 4, mi = 2 (animated)

That variant is trivial, but suppose there are eight mice, four starting on the corners of the square and four starting on the midpoints of the sides. Mice starting on the corners will run different pursuit curves to those starting on the midpoints, because the corners are further from the centre than the midpoints are:

v = 4, si = 1, mi = 1

If mi = 3, the pursuit curves look like this:

v = 4, si = 1, mi = 3

v = 4, si = 1, mi = 3 (animated)

Suppose there are twelve mice, four on each corner and two more on each side. If each mouse runs towards the mouse four places away, then the pursuit curves don’t all meet in the centre of the square. Instead, they meet in groups of three at four points equidistant from the centre, like this:

v = 4, si = 2, mi = 4

v = 4, si = 2, mi = 4 (animated)

v = 4, si = 4, mi = 4 (animated)

v = 4, si = 4, mi = 4 (zoom)

Now suppose each mouse become sophisticated and runs toward the combined positions of two other mice, one two places away, the other three places away, like this:

v = 4, si = 1, mi = (2, 3)

v = 4, si = 1, mi = (2, 3) (animated)

These polypursuits, as they could be called, can have complicated central regions:

v = 4, si = 2, mi = (1, 4)

v = 4, si = 2, mi = (1, 4) (animated)

v = 4, si = various, mi = various

And what if you have two teams of mice, running towards one or more mice on the other team? For example, suppose two mice, one from each team, start on each corner of a square. Each mouse on team 1 runs towards the mouse on team 2 that is one place away, while each mouse on team 2 runs towards the mouse on team 1 that is two places away. If the pursuits curves of team 1 are represented in white and the pursuit curves of team 2 in green, the curves look like this:

v = 4 * 2, vmi = 1, vmi = 2

v = 4 * 2, vmi = 1, vmi = 2

v = 4 * 2, vmi = 1, vmi = 2 (animated)

Now suppose the four mice of team 1 start on the corners while the mice of team 2 start at the centre of the square.

v = 4, centre = 4, vmi = 1, cmi = 2 (white team)

v = 4, centre = 4, vmi = 1, cmi = 2 (green team)

v = 4, centre = 4, vmi = 1, cmi = 2 (both teams)

v = 4, centre = 4, vmi = 1, cmi = 2 (animated)

Here are more variants on pursuit curves formed by two teams of mice, one starting on the corners, one at the centre:

v = 4, centre = 4, vmi = (0, 1), cmi = 0

v = 4, centre = 4, vmi = (0, 2), cmi = 0

v = 4, centre = 4, vmi = (0, 3), cmi = 0

White Rites

The blancmange curve is an interesting fractal formed by summing a series of zigzags. It
takes its name from its resemblance to the milk-pudding known as a blancmange
(blanc-manger in French, meaning “white eating”):

Blancmange curve

In successive zigzags, the number of zags doubles as their height halves, i.e. z(i) = z(i-1) * 2, h(i) = h(i-1) / 2. If all the zigzags are represented at once, the construction looks like this:

Zigzags 1 to 10

Zigzags 1 to 10 (animated)

Here is a step-by-step construction, with the total sum of zigzags in white, the present zigzag in red and the previous zigzag in green:

Blancmange curve stage 1

Stage 2

Stage 3

Stage 4

Stage 5

Stage 6

Stage 7

Stage 8

Stage 9

Stage 10

Blancmange curve (animated)

It’s easy to think of variants on the standard blancmange curve. Suppose the number of zags triples as their height is divided by three, i.e. z(i) = z(i-1) * 3, h(i) = h(i-1) / 3:

Blancmange curve for z(i) = z(i-1) * 3, h(i) = h(i-1) / 3

Bash the Pash

An heretical Guardianista keyly critiquizes a core component of Guardianese:

I recently considered nominating for a board position on a professional association to which I belong, so I had a look at the biographical statements of the incumbents. One claimed to be “passionate about helping individuals, businesses, and communities thrive”, another declared “a particular passion for thought leadership and executive profiling”, and another revealed “a passion for social inclusion”. Yet another claimed “a passion about creating valuable career development opportunities for the profession”. The best was the one that stated, without irony, “a passion for working on meaningful projects”.

In days gone by, job applicants listed hobbies. These days, it appears candidates are expected to declare, not merely interests or things they like doing, but things they are allegedly passionate about. — Being enthusiastic is no longer enough. Now we must all be passionate, Paul Begley, The Guardian, 13/vii/2016.

Elsewhere other-engageable:

Ex-term-in-ate!
Titus Graun — Heresy, Homotextuality, Hive-Mind
Oh My Guardian

Walking Winkle

Flat periwinkle, Littorina obtusa (Linnaeus, 1758)*

*Possibly.

Toxic Turntable #6

Currently listening…

• Anaïd, Magna Est (1969)
• Linnet, Venussong (1977)
• Šrúsma, Ðrioso (1994)
• Milchstraße, Der Schwarze Zwerg (2001)
• Ser, Acanthocalycium (1989)
• Zélote, Lac d’Angoisse (1992)

Previously pre-posted:

Boustrophedon (pronounced “bough-stra-FEE-dun” or “boo-stra-FEE-dun”) is an ancient Greek word literally meaning “as the ox turns (in ploughing)”, that is, moving left-right, right-left, and so on. The word is used of writing that runs down the page in the same way. To see what that means, examine two versions of the first paragraph of Clark Ashton Smith’s story “The Demon of the Flower” (1933). The first is written in the usual way, the second is written boustrophedon:

Not as the plants and flowers of Earth, growing peacefully beneath a simple sun, were the blossoms of the planet Lophai. Coiling and uncoiling in double dawns; tossing tumultuously under vast suns of jade green and balas-ruby orange; swaying and weltering in rich twilights, in aurora-curtained nights, they resembled fields of rooted servants that dance eternally to an other-worldly music.

Not as the plants and flowers of Earth, growing peacefully
.iahpoL tenalp eht fo smossolb eht erew ,nus elpmis a htaeneb
Coiling and uncoiling in double dawns; tossing tumultuously
;egnaro ybur-salab dna neerg edaj fo snus tsav rednu
swaying and weltering in rich twilights, in aurora-curtained
ecnad taht stnavres detoor fo sdleif delbmeser yeht ,sthgin
eternally to an other-worldly music.

Boustrophedon writing was once common and sometimes the left-right lines would also be mirror-reversed, like this:

You could also use the term “boustrophedon” to describe the way this table of numbers is filled:

The table begins with “1” in the top left-hand corner, then moves right for “2”, then down for “3”, then right-and-up for “4”, “5” and “6”, then right for “7”, then left-and-down for “8”, “9” and “10”, and so on. You could also say that the numbers snake through the table. I’ve marked the primes among them, because I was interested in the patterns made by the primes when the numbers were represented as blocks on a grid, like this:

Primes are in solid white (compare the Ulam spiral). Here’s the boustrophedon prime-grid on a finer scale:

(click for full image)

And what about other number-tests? Here are the even numbers marked on the grid (i.e. n mod 2 = 0):

n mod 2 = 0

And here are some more examples of a modulus test:

n mod 3 = 0

n mod 5 = 0

n mod 9 = 0

n mod 15 = 0

n mod various = 0 (animated gif)

Next I looked at reciprocals (numbers divided into 1) marked on the grid, with the digits of a reciprocal marking the number of blank squares before a square is filled in (if the digit is “0”, the square is filled immediately). For example, in base ten 1/7 = 0.142857142857142857…, where the block “142857” repeats for ever. When represented on the grid, 1/7 has 1 blank square, then a filled square, then 4 blank squares, then a filled square, then 2 blank squares, then a filled square, and so on:

1/7 in base 10

And here are some more reciprocals (click for full images):

1/9 in base 2

1/13 in base 10

1/27 in base 10

1/41 in base 10

1/63 in base 10

1/82 in base 10

1/101 in base 10

1/104 in base 10

1/124 in base 10

1/143 in base 10

1/175 in base 10

1/604 in base 8

1/n in various bases (animated gif)