1: 111

2: 112

3: 121

4: 122

5: 211

6: 212

7: 221

8: 222

Choice #1 is 111, which means tea every day. Choice #6 is 212, which means coffee on day 1, tea on day 2 and coffee on day 3. Now look at the counting again and the way the numbers change: 111, 112, 121, 122, 211… It’s really base 2 using 1 and 2 rather than 0 and 1. That’s why there are 8 ways to choose two drinks over three days: 8 = 2^3. Next, note that you use the same number of 1s to count the choices as the number of 2s. There are twelve 1s and twelve 2s, because each number has a mirror: 111 has 222, 112 has 221, 121 has 212, and so on.

Now try the number of ways to choose from three kinds of drink (tea, coffee, orange juice) over two days:

11, 12, 13, 21, 22, 23, 31, 32, 33 (c=9)

There are 9 ways to choose, because 9 = 3^2. And each digit, 1, 2, 3, is used exactly six times when you write the choices. Now try the number of ways to choose from three kinds of drink over three days:

111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313, 321, 322, 323, 331, 332, 333 (c=27)

There are 27 ways and (by coincidence) each digit is used 27 times to write the choices. Now try three drinks over four days:

1111, 1112, 1113, 1121, 1122, 1123, 1131, 1132, 1133, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1311, 1312, 1313, 1321, 1322, 1323, 1331, 1332, 1333, 2111, 2112, 2113, 2121, 2122, 2123, 2131, 2132, 2133, 2211, 2212, 2213, 2221, 2222, 2223, 2231, 2232, 2233, 2311, 2312, 2313, 2321, 2322, 2323, 2331, 2332, 2333, 3111, 3112, 3113, 3121, 3122, 3123, 3131, 3132, 3133, 3211, 3212, 3213, 3221, 3222, 3223, 3231, 3232, 3233, 3311, 3312, 3313, 3321, 3322, 3323, 3331, 3332, 3333 (c=81)

There are 81 ways to choose and each digit is used 108 times. But the numbers don’t have represent choices of drink in a café. How many ways can a point inside an equilateral triangle jump four times half-way towards the vertices of the triangle? It’s the same as the way to choose from three drinks over four days. And because the point jumps toward each vertex in a symmetrical way the same number of times, you get a nice even pattern, like this:

vertices = 3, jump = 1/2

Every time the point jumps half-way towards a particular vertex, its position is marked in a unique colour. The fractal, also known as a Sierpiński triangle, actually represents all possible choices for an indefinite number of jumps. Here’s the same rule applied to a square. There are four vertices, so the point is tracing all possible ways to choose four vertices for an indefinite number of jumps:

v = 4, jump = 1/2

As you can see, it’s not an obvious fractal. But what if the point jumps two-thirds of the way to its target vertex and an extra target is added at the centre of the square? This attractive fractal appears:

v = 4 + central target, jump = 2/3

If the central target is removed and an extra target is added on each side, this fractal appears:

v = 4 + 4 midpoints, jump = 2/3

That fractal is known as a Sierpiński carpet. Now up to the pentagon. This fractal of endlessly nested contingent pentagons is created by a point jumping 1/φ = 0·6180339887… of the distance towards the five vertices:

v = 5, jump = 1/φ

With a central target in the pentagon, this fractal appears:

v = 5 + central, jump = 1/φ

The central red pattern fits exactly inside the five that surround it:

v = 5 + central, jump = 1/φ (closeup)

v = 5 + c, jump = 1/φ (animated)

For a fractal of endlessly nested contingent hexagons, the jump is 2/3:

v = 6, jump = 2/3

With a central target, you get a filled variation of the hexagonal fractal:

v = 6 + c, jump = 2/3

And for a fractal of endlessly nested contingent octagons, the jump is 1/√2 = 0·7071067811… = √½:

v = 8, jump = 1/√2

Previously pre-posted: ]]>

Elsewhere other-engageable:

• Place of Glades — a review of *The Oxford Dictionary of British Place Names*, A.D. Mills (1991)

1, 2, 6, 12, 44, 92, 184, 1208, 1256, 4792, 9912, 19832, 39664, 563952, 576464, 4496112, 4499184, 17996528, 17997488, 143972080, 145057520, 145070832, 294967024, 589944560...

To take the first step towards the answer, you need to put the numbers into binary:

1, 10, 110, 1100, 101100, 1011100, 10111000, 10010111000, 10011101000, 1001010111000, 10011010111000, 100110101111000, 1001101011110000, 10001001101011110000, 10001100101111010000, 10001001001101011110000, 10001001010011011110000, 1000100101001101011110000, 1000100101001111010110000, 1000100101001101011011110000, 1000101001010110011011110000, 1000101001011001101011110000, 10001100101001101011011110000, 100011001010011101011011110000...

The second step is compare those binary numbers with these binary numbers, which represent 1 to 30:

1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110...

To see what’s going on, take the first five numbers from each set:

• 1, 10, 110, 1100, 101100...

• 1, 10, 11, 100, 101...

What’s going on? If you look, you can see the *n*-th binary number of set 1 contains the digits of all binary numbers <= *n* in set 2. For example, 101100 is the 5th binary number in set 1, so it contains the digits of the binary numbers 1 to 5:

101100 ← 1

101100 ← 10

101100 ← 11

101100← 100

101100 ← 101

Now try 1256 = 10,011,101,000, the ninth number in set 1. It contains all the binary numbers from 1 to 1001:

10011101000 ← 1 (n=1)

10011101000 ← 10 (n=2)

10011101000 ← 11 (n=3)

10011101000 ← 100 (n=4)

10011101000 ← 101 (n=5)

10011101000 ← 110 (n=6)

10011101000 ← 111 (n=7)

10011101000← 1000 (n=8)

10011101000 ← 1001 (n=9)

But where do grandmothers come in? They come in via this famous toy:

It’s called a Russian doll and the way all the smaller dolls pack inside the largest doll reminds me of the way all the smaller numbers 1 to 1010 pack into 1001010111000. But in the Russian language, as you might expect, Russian dolls aren’t called Russian dolls. Instead, they’re called matryoshki (матрёшки, singular матрёшка), meaning “little matrons”. However, there’s a mistaken idea in English that in Russian they’re called babushka dolls, from Russian бабушка, *babuška*, meaning “grandmother”. And that’s what I thought, until I did a little research.

But the mistake is there, so I’ll call these babushka numbers or grandmother numbers:

1, 2, 6, 12, 44, 92, 184, 1208, 1256, 4792, 9912, 19832, 39664, 563952, 576464, 4496112, 4499184, 17996528, 17997488, 143972080, 145057520, 145070832, 294967024, 589944560...

They’re sequence A261467 at the *Online Encyclopedia of Integer Sequences*. They go on for ever, but the biggest known so far is 589,944,560 = 100,011,001,010,011,101,011,011,110,000 in binary. And here is that binary babushka with its binary babies:

100011001010011101011011110000 ← 1 (n=1)

100011001010011101011011110000 ← 10 (n=2)

100011001010011101011011110000 ← 11 (n=3)

100011001010011101011011110000 ← 100 (n=4)

100011001010011101011011110000 ← 101 (n=5)

100011001010011101011011110000 ← 110 (n=6)

100011001010011101011011110000 ← 111 (n=7)

100011001010011101011011110000 ← 1000 (n=8)

100011001010011101011011110000 ← 1001 (n=9)

100011001010011101011011110000 ← 1010 (n=10)

100011001010011101011011110000 ← 1011 (n=11)

100011001010011101011011110000 ← 1100 (n=12)

100011001010011101011011110000 ← 1101 (n=13)

100011001010011101011011110000 ← 1110 (n=14)

100011001010011101011011110000 ← 1111 (n=15)

100011001010011101011011110000← 10000 (n=16)

100011001010011101011011110000 ← 10001 (n=17)

100011001010011101011011110000 ← 10010 (n=18)

100011001010011101011011110000 ← 10011 (n=19)

100011001010011101011011110000 ← 10100 (n=20)

100011001010011101011011110000 ← 10101 (n=21)

100011001010011101011011110000 ← 10110 (n=22)

100011001010011101011011110000 ← 10111 (n=23)

100011001010011101011011110000 ← 11000 (n=24)

100011001010011101011011110000 ← 11001 (n=25)

100011001010011101011011110000 ← 11010 (n=26)

100011001010011101011011110000 ← 11011 (n=27)

100011001010011101011011110000 ← 11100 (n=28)

100011001010011101011011110000 ← 11101 (n=29)

100011001010011101011011110000 ← 11110 (n=30)

Babushka numbers exist in higher bases, of course. Here are the first thirteen in base 3 or ternary:

1 contains 1 (c=1) (n=1)

12 contains 1, 2 (c=2) (n=5)

102 contains 1, 2, 10 (c=3) (n=11)

1102 contains 1, 2, 10, 11 (c=4) (n=38)

10112 contains 1, 2, 10, 11, 12 (c=5) (n=95)

101120 contains 1, 2, 10, 11, 12, 20 (c=6) (n=285)

1021120 contains 1, 2, 10, 11, 12, 20, 21 (c=7) (n=933)

10211220 contains 1, 2, 10, 11, 12, 20, 21, 22 (c=8) (n=2805)

100211220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100 (c=9) (n=7179)

10021011220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101 (c=10) (n=64284)

1001010211220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102 (c=11) (n=553929)

1001011021220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110 (c=12) (n=554253)

10010111021220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111 (c=13) (n=1663062)

Look at 1,001,010,211,220 (n=553929) and 1,001,011,021,220 (n=554253). They have the same number of digits, but the babushka 1,001,011,021,220 manages to pack in one more baby:

1001010211220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102 (c=11) (n=553929)

1001011021220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110 (c=12) (n=554253)

That happens in binary too:

10010111000 contains 1, 10, 11, 100, 101, 110, 111, 1000, 1001 (c=9) (n=1208)

10011101000 contains 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010 (c=10) (n=1256)

What happens in higher bases? Watch this space.

]]>• Early Riser — *Decline and Fall*, Evelyn Waugh (1928)

• The Future is Fascist — *Futurism*, Richard Humphreys (1999 Tate Publishing)

• Mystery and Meaning — *Dictionary of Plant Names*, Allen J. Coombes (1985)

• Noshing on Noxiousness — *Nekro-Noxious: Toxic Tales of True Transgression in Miami Municipal Mortuary*, Norberto Fetidescu (TransVisceral Books 2018)

Or Read a Review at Random: RaRaR

]]>Illustrations from

**Goop to pay out over unproven health benefits of vaginal eggs**

Goop, the new age lifestyle and publishing company founded by the [actress] Gwyneth Paltrow, has agreed to pay a substantial settlement over unproven claims about the health benefits of its infamous vaginal eggs. Goop’s website still claims that inserting the eggs into the vagina helps “cultivate sexual energy, clear chi pathways in the body, intensify femininity, and invigorate our life force”.

Its $66 Jade Egg and $55 Rose Quartz egg are still offered for sale on the site, but the company has agreed to pay $145,000 to settle allegations that it previously made unscientific claims about the eggs, and a herbal essence that it had said helped tackle depression.

It also agreed to refund customers who purchased the products from January to August last year. During that period it claimed the eggs could balance hormones, regulate menstrual cycles, prevent uterine prolapse, and increase bladder control, according to officials in Santa Clara part of a group of California district attorneys who filed the lawsuit. — Goop to pay out over unproven health benefits of vaginal eggs, *The Guardian*, 5ix2018.

N.B. The title of this incendiary intervention is a paronomasia on the old British advertising slogan “Go to work on an egg.” ]]>

I’ve never been able to get into the band Sleep and, not being a keyly committed core component of the hive-mind, I’m not a fan of dopesmoking either. But this is a good cover by the artist Arik Roper, with a nice

To engage issues around the title of this incendiary intervention, see here:

]]>capno-, capn-, capnod- (Greek: smoke; vapor; sooty) — Wordquests

“There was a certain edgy excitement to turning on the computer every morning and immediately checking to see what Mark had thrown down in terms of an ideas-gauntlet.” – Simon Reynolds in the foreword to

K-Punk: The Collected and Unpublished Writings of Mark Fisher (2004–2016), edited by Darren Ambrose, Repeater Books 2018.

Elsewhere other-engageable:

• Ex-term-in-ate!

• Don’t Do Dot…

• Prior Analytics

• Spike-U-Like?

draconic, adj. /drəˈkɒnɪk/ pertaining to, or of the nature of, a dragon. [Latin *draco*, *-ōnem*, < Greek δράκων dragon] — *The Oxford English Dictionary*

In Curvous Energy, I looked at the strange, beautiful and complex fractal known as the dragon curve and showed how it can be created from a staid and sedentary square:

A dragon curve

Here are the stages whereby the dragon curve is created from a square. Note how each square at one stage generates a pair of further squares at the next stage:

Dragon curve from squares #1

Dragon curve from squares #2

Dragon curve from squares #3

Dragon curve from squares #4

Dragon curve from squares #5

Dragon curve from squares #6

Dragon curve from squares #7

Dragon curve from squares #8

Dragon curve from squares #9

Dragon curve from squares #10

Dragon curve from squares #11

Dragon curve from squares #12

Dragon curve from squares #13

Dragon curve from squares #14

Dragon curve from squares (animated)

The construction is very easy and there’s no tricky trigonometry, because you can use the vertices and sides of each old square to generate the vertices of the two new squares. But what happens if you use lines rather than squares to generate the dragon curve? You’ll discover that less is more:

Dragon curve from lines #1

Dragon curve from lines #2

Dragon curve from lines #3

Dragon curve from lines #4

Dragon curve from lines #5

Each line at one stage generates a pair of further lines at the next stage, but there’s no simple way to use the original line to generate the new ones. You have to use trigonometry and set the new lines at 45° to the old one. You also have to shrink the new lines by a fixed amount, 1/√2 = 0·70710678118654752… Here are further stages:

Dragon curve from lines #6

Dragon curve from lines #7

Dragon curve from lines #8

Dragon curve from lines #9

Dragon curve from lines #10

Dragon curve from lines #11

Dragon curve from lines #12

Dragon curve from lines #13

Dragon curve from lines #14

Dragon curve from lines (animated)

But once you have a program that can adjust the new lines, you can experiment with new angles. Here’s a dragon curve in which one new line is at an angle of 10°, while the other remains at 45° (after which the full shape is rotated by 180° because it looks better that way):

Dragon curve 10° and 45°

Dragon curve 10° and 45° (animated)

Dragon curve 10° and 45° (coloured)

Here are more examples of dragon curves generated with one line at 45° and the other line at a different angle:

Dragon curve 65°

Dragon curve 65° (anim)

Dragon curve 65° (col)

Dragon curve 80°

Dragon curve 80° (anim)

Dragon curve 80° (col)

Dragon curve 135°

Dragon curve 135° (anim)

Dragon curve 250°

Dragon curve 250° (anim)

Dragon curve 250° (col)

Dragon curve 260°

Dragon curve 260° (anim)

Dragon curve 260° (col)

Dragon curve 340°

Dragon curve 340° (anim)

Dragon curve 340° (col)

Dragon curve 240° and 20°

Dragon curve 240° and 20° (anim)

Dragon curve 240° and 20° (col)

Dragon curve various angles (anim)

Previously pre-posted:

• Curvous Energy — a first look at dragon curves

]]>• Kev Neuys, *Eloquence Foss* (1964)

• Zetic Load, *Loadstar* (1991)

• Vi Jubilatus, *Quinconce* EP (1979)

• Ylikipojga, *Vazhwevac Xviwv* (1991)

• Voïde Thyroïde, *I by the Yellow Door* (1996)

• Weepster, *Glows the Ghost* (1983)

• Nous les Revenants, *L’Iodisme* (1988)

• Jiji è Vgelu, *Live in Rheims* (2016)

• Hex Dwi, *Cats is Simplicity* (1994)

• Quixotic Plovers, *Imagine Us* (2006)

• Rantique + Mizao, *Oklahama + Ecclesia Sanctorum* (split EP) (2008)

• Elizabeth Dobie, *Uppers/Downers* (1981)

• Zed Follows Wye, *Goats in Etruria* (2000)

• Hyssop-773, *Abeoma* (1987)

• Francesco Manfredini, *12 Concerti* (1997)

• Eark, *Xeno-Zoo* (1993)

Previously pre-posted:

• Toxic Turntable #1

• Toxic Turntable #2

• Toxic Turntable #3

• Toxic Turntable #4

• Toxic Turntable #5

• Toxic Turntable #6

• Toxic Turntable #7

• Toxic Turntable #8

• Toxic Turntable #9

• Toxic Turntable #10

• Toxic Turntable #11

• Toxic Turntable #12

• Toxic Turntable #13

• Toxic Turntable #14

• Toxic Turntable #15