Comte Robert de Montesquiou-Fézensac, Henri Lucien Doucet (1879)
Passionately Pre-Posted…
• Portrait of a Peacock — Cornelia Otis Skinner’s biographical sketch of Montesquiou
• Le Paon dans les Pyrénées — review of Julian Barnes’ The Man in the Red Coat (2019)
Elsewhere Other-Accessible…
• The hands of Robert de Montesquiou at Strange Flowers
Currently listening…
• Lupine Katie, Dark Calico (1976)
• Ô Grecq, Mouse Makes Cop (2005)
• Julia Sedburgh, Vacant Reef (1975)
• Fox in Quake, Drifting on Sonic Seas (2002)
• الاصداف, ميوريكس ﮪاوستلم (1995)
• Pidita Kick, I Leap for Hawaii (1998)
• Yuperfen Lagekim, Jeptic (2013)
• Quiggy and the Grin, Soda Family (1983)
• Oxton Bvulpsi, Dealt a Devil’s Hand (2017)
• მარმარილი მანქანა, აბრეშუმის ჭია (1997)
• Dogmatic Cajuns, Little Bit (1996)
• Pamela Kelpstan, Episodic Outtakes (1970)
• Okeäp V, Räucherfisch (1994)
• Bells in the Battery, Hue You (1990)
• Showt Nowt, Our Wild Folkloric Dreams (2012)
• ¡XtrmDrm!, LIVE in ¡Thunderland! (2004)
• Jumjin, Mqoag i Hio (1979)
• Ѫй Ѯяоѱ Ѩѽ, Koxiio (1997)
• Edgar Roslin Quintet, Zeppelin Zoo (2017)
• Uh Bpuerw, Sqiaamn (1983)
• Stellae Vulpis, Umbrian Folk Revival (2006)
• Milirad im Untergang, Ab’xu Mim Myaceleg (1973)
Previously pre-posted:
Toxic Turntable #1 • #2 • #3 • #4 • #5 • #6 • #7 • #8 • #9 • #10 • #11 • #12 • #13 • #14 • #15 • #16 • #17 • #18 • #19 • #20 • #21 • #22 • #23 • #24 •
Greater quaking-grass, Briza maxima (Linnaeus 1753)
In “Wake the Snake”, I looked at the digits of powers of 2 and mentioned a fascinating mathematical phenomenon known as Benford’s law, which governs — in a not-yet-fully-explained way — the leading digits of a wide variety of natural and human statistics, from the lengths of rivers to the votes cast in elections. Benford’s law also governs a lot of mathematical data. It states, for example, that the first digit, d, of a power of 2 in base b (except b = 2, 4, 8, 16…) will occur with the frequency logb(1 + 1/d). In base 10, therefore, Benford’s law states that the digits 1..9 will occur with the following frequencies at the beginning of 2^p:
1: 30.102999%
2: 17.609125%
3: 12.493873%
4: 09.691001%
5: 07.918124%
6: 06.694678%
7: 05.799194%
8: 05.115252%
9: 04.575749%
Here’s a graph of the actual relative frequencies of 1..9 as the leading digit of 2^p (open images in a new window if they appear distorted):
And here’s a graph for the predicted frequencies of 1..9 as the leading digit of 2^p, as calculated by the log(1+1/d) of Benford’s law:
The two graphs agree very well. But Benford’s law applies to more than one leading digit. Here are actual and predicted graphs for the first two leading digits of 2^p, 10..99:
And actual and predicted graphs for the first three leading digits of 2^p, 100..999:
But you can represent the leading digit of 2^p in another way: using an adaptation of the famous Ulam spiral. Suppose powers of 2 are represented as a spiral of squares that begins like this, with 2^0 in the center, 2^1 to the right of center, 2^2 above 2^1, and so on:
←←←⮲
432↑
501↑
6789
If the digits of 2^p start with 1, fill the square in question; if the digits of 2^p don’t start with 1, leave the square empty. When you do this, you get this interesting pattern (the purple square at the very center represents 2^0):
Ulam-like power-spiral for 2^p where 1 is the leading digit
Here’s a higher-resolution power-spiral for 1 as the leading digit:
Power-spiral for 2^p, leading-digit = 1 (higher resolution)
And here, at higher resolution still, are power-spirals for all the possible leading digits of 2^p, 1..9 (some spirals look very similar, so you have to compare those ones carefully):
Power-spiral for 2^p, leading-digit = 1 (very high resolution)
Power-spiral for 2^p, leading-digit = 2
Power-spiral for 2^p, ld = 3
Power-spiral for 2^p, ld = 4
Power-spiral for 2^p, ld = 5
Power-spiral for 2^p, ld = 6
Power-spiral for 2^p, ld = 7
Power-spiral for 2^p, ld = 8
Power-spiral for 2^p, ld = 9
Power-spiral for 2^p, ld = 1..9 (animated)
Now try the power-spiral of 2^p, ld = 1, in some other bases:
Power-spiral for 2^p, leading-digit = 1, base = 9
Power-spiral for 2^p, ld = 1, b = 15
You can also try power-spirals for other n^p. Here’s 3^p:
Power-spiral for 3^p, ld = 1, b = 10
Power-spiral for 3^p, ld = 2, b = 10
Power-spiral for 3^p, ld = 1, b = 4
Power-spiral for 3^p, ld = 1, b = 7
Power-spiral for 3^p, ld = 1, b = 18
Elsewhere Other-Accessible…
• Wake the Snake — an earlier look at the digits of 2^p
In my story “Kopfwurmkundalini”, I imagined the square root of 2 as an infinitely long worm or snake whose endlessly varying digit-segments contained all stories ever (and never) written:
• √2 = 1·414213562373095048801688724209698078569671875376948073…
But there’s another way to get all stories ever written from the number 2. You don’t look at the root(s) of 2, but at the powers of 2:
• 2 = 2^1 = 2
• 4 = 2^2 = 2*2
• 8 = 2^3 = 2*2*2
• 16 = 2^4 = 2*2*2*2
• 32 = 2^5 = 2*2*2*2*2
• 64 = 2^6 = 2*2*2*2*2*2
• 128 = 2^7 = 2*2*2*2*2*2*2
• 256 = 2^8 = 2*2*2*2*2*2*2*2
• 512 = 2^9 = 2*2*2*2*2*2*2*2*2
• 1024 = 2^10
• 2048 = 2^11
• 4096 = 2^12
• 8192 = 2^13
• 16384 = 2^14
• 32768 = 2^15
• 65536 = 2^16
• 131072 = 2^17
• 262144 = 2^18
• 524288 = 2^19
• 1048576 = 2^20
• 2097152 = 2^21
• 4194304 = 2^22
• 8388608 = 2^23
• 16777216 = 2^24
• 33554432 = 2^25
• 67108864 = 2^26
• 134217728 = 2^27
• 268435456 = 2^28
• 536870912 = 2^29
• 1073741824 = 2^30
[...]
The powers of 2 are like an ever-lengthening snake swimming across a pool. The snake has an endlessly mutating head and a rhythmically waving tail with a regular but ever-more complex wake. That is, the leading digits of 2^p don’t repeat but the trailing digits do. Look at the single final digit of 2^p, for example:
• 02 = 2^1
• 04 = 2^2
• 08 = 2^3
• 16 = 2^4
• 32 = 2^5
• 64 = 2^6
• 128 = 2^7
• 256 = 2^8
• 512 = 2^9
• 1024 = 2^10
• 2048 = 2^11
• 4096 = 2^12
• 8192 = 2^13
• 16384 = 2^14
• 32768 = 2^15
• 65536 = 2^16
• 131072 = 2^17
• 262144 = 2^18
• 524288 = 2^19
• 1048576 = 2^20
• 2097152 = 2^21
• 4194304 = 2^22
[...]
The final digit of 2^p falls into a loop: 2 → 4 → 8 → 6 → 2 → 4→ 8…
Now try the final two digits of 2^p:
• 02 = 2^1
• 04 = 2^2
• 08 = 2^3
• 16 = 2^4
• 32 = 2^5
• 64 = 2^6
• 128 = 2^7
• 256 = 2^8
• 512 = 2^9
• 1024 = 2^10
• 2048 = 2^11
• 4096 = 2^12
• 8192 = 2^13
• 16384 = 2^14
• 32768 = 2^15
• 65536 = 2^16
• 131072 = 2^17
• 262144 = 2^18
• 524288 = 2^19
• 1048576 = 2^20
• 2097152 = 2^21
• 4194304 = 2^22
• 8388608 = 2^23
• 16777216 = 2^24
• 33554432 = 2^25
• 67108864 = 2^26
• 134217728 = 2^27
• 268435456 = 2^28
• 536870912 = 2^29
• 1073741824 = 2^30
[...]
Now there’s a longer loop: 02 → 04 → 08 → 16 → 32 → 64 → 28 → 56 → 12 → 24 → 48 → 96 → 92 → 84 → 68 → 36 → 72 → 44 → 88 → 76 → 52 → 04 → 08 → 16 → 32 → 64 → 28… Any number of trailing digits, 1 or 2 or one trillion, falls into a loop. It just takes longer as the number of trailing digits increases.
That’s the tail of the snake. At the other end, the head of the snake, the digits don’t fall into a loop (because of the carries from the lower digits). So, while you can get only 2, 4, 8 and 6 as the final digits of 2^p, you can get any digit but 0 as the first digit of 2^p. Indeed, I conjecture (but can’t prove) that not only will all integers eventually appear as the leading digits of 2^p, but they will do so infinitely often. Think of a number and it will appear as the leading digits of 2^p. Let’s try the numbers 1, 12, 123, 1234, 12345…:
• 16 = 2^4
• 128 = 2^7
• 12379400392853802748... = 2^90
• 12340799625835686853... = 2^1545
• 12345257952011458590... = 2^34555
• 12345695478410965346... = 2^63293
• 12345673811591269861... = 2^4869721
• 12345678260232358911... = 2^5194868
• 12345678999199154389... = 2^62759188
But what about the numbers 9, 98, 987, 986, 98765… as leading digits of 2^p? They don’t appear as quickly:
• 9007199254740992 = 2^53
• 98079714615416886934... = 2^186
• 98726397006685494828... = 2^1548
• 98768356967522174395... = 2^21257
• 98765563827287722773... = 2^63296
• 98765426081858871289... = 2^5194871
• 98765430693066680199... = 2^11627034
• 98765432584491513519... = 2^260855656
• 98765432109571471006... = 2^1641098748
Why do fragments of 123456789 appear much sooner than fragments of 987654321? Well, even though all integers occur infinitely often as leading digits of 2^p, some integers occur more often than others, as it were. The leading digits of 2^p are actually governed by a fascinating mathematical phenomenon known as Benford’s law, which states, for example, that the single first digit, d, will occur with the frequency log10(1 + 1/d). Here are the actual frequencies of 1..9 for all powers of 2 up to 2^101000, compared with the estimate by Benford’s law:
1: 30% of leading digits ↔ 30.1% estimated
2: 17.55% ↔ 17.6%
3: 12.45% ↔ 12.49%
4: 09.65% ↔ 9.69%
5: 07.89% ↔ 7.92%
6: 06.67% ↔ 6.69%
7: 05.77% ↔ 5.79%
8: 05.09% ↔ 5.11%
9: 04.56% ↔ 4.57%
Because (inter alia) 1 appears as the first digit of 2^p far more often than 9 does, the fragments of 123456789 appear faster than the fragments of 987654321. Mutatis mutandis, the same applies in all other bases (apart from bases that are powers of 2, where there’s a single leading digit, 1, 2, 4, 8…, followed by 0s). But although a number like 123456789 occurs much frequently than 987654321 in 2^p expressed in base 10 (and higher), both integers occur infinitely often.
As do all other integers. And because stories can be expressed as numbers, all stories ever (and never) written appear in the powers of 2. Infinitely often. You’ll just have to trim the tail of the story-snake.
“In a very real sense, the Holocaust, as the ultimate moral and aesthetic obscenity, was also the ultimate drum-solo.” — Simon Whitechapel, 31i18
ვარდები
მე, ზამთრისაგან ჯაჭვაწყვეტილი,
ნაცნობ ბაღისკენ მივემართები,
სად ფერად უცხო, ყნოსვად კეთილი,
ზამთარ და ზაფხულ ჰყვავის ვარდები.
Roses
Unchained from winter,
I walk to a long-known garden,
Where, sweet-scented and bright,
Roses grow winter and summer through.
• ვარდები, გალაკტიონ ტაბიძე
• “Roses”, Galaktion Tabidze — a translation into English
The Rain It Raineth Every Day (1889) by Norman Garstin (1847–1926)
“U2 are a band who believe in meetings, so everything has a meeting. With Depeche, it’s an exception to have a meeting at all. When I did visuals for The Joshua Tree 30th anniversary tour, I also filmed the show in Mexico City. After the concert, U2 all came to look at the footage. At midnight! For two hours! I mean, Depeche – you couldn’t get them to watch a minute. It’s an incredible difference in attitude. But that’s also the charm of Depeche. They don’t do many interviews, there’s no big plans, they just make a record and tour.”
• ‘They had soul’: Anton Corbijn on 40 years shooting Depeche Mode, The Intermzinator, 1vi21
Post-Performative Post-Scriptum
The peri-parapractic paronomasia in the title of this post corely references key Kulturkampf “Mods vs Rockers” in terms of 1960s Britain.
Here’s a pentagon:
Stage #1
And here’s the pentagon with smaller pentagons on its vertices:
Stage #2
And here’s more of the same:
Stage #3
Stage #4
Stage #5
Stage #6
Stage #7
Stage #8
Animated fractal
At infinity, the smaller pentagons have reached out like arms to exactly fill the gaps between themselves without overlapping. But how much smaller is each set of smaller pentagons than its mother-pentagon when the gaps are exactly filled? Well, if the radius of the mother-pentagon is r, then the radius of each daughter-pentagon is r * 1/(φ^2) = r * 0·38196601125…
But what happens if the radius relationship of mother to daughter is r * 1/φ = r * 0·61803398874 = r * (φ-1)? Then you get this fractal:
Stage #1
Stage #2
Stage #3
Stage #4
Stage #5
Stage #6
Stage #7
Stage #8
Stage #9
Animated fractal
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 