Boldly Breaking the Boundaries

In “M.I.P. Trip”, I looked at fractals like this, in which a square is divided repeatedly into a pattern of smaller squares:
2x2inner

2x2inner_static


3x3innera

3x3innera_static


3x3innerb

3x3innerb_static


As you can see, the sub-squares appear within the boundaries of the original square. But what if some of the sub-squares appear beyond the bounds of the original square? Then a new family of fractals is born, the over-fractals:

fractal2x2a

fractal2x2a_static


fractal2x2b

fractal2x2b_static


fractal2x2c

fractal2x2c_static


fractal2x2d

fractal2x2d_static


fractal2x2e

fractal2x2e_static


fractal3x3a

fractal3x3a_static


fractal3x3b

fractal3x3b_static


fractal3x3c

fractal3x3c_static


fractal3x3d


fractal3x3e


fractal3x3f


fractal3x3g


fractal3x3h


fractal3x3i


fractal3x3j


fractal3x3k


fractal3x3l


fractal3x3m


fractal3x3n


fractal4x4a


fractal4x4c


fractal4x4b

He Say, He Sigh, He Sow #27

“When you run and jump on rocks, your entire brain and body are at work; you stretch your back better than with yoga; every muscle in your body is involved; no two movements will be identical (unlike running in gyms); you become yourself.” — Nassim Taleb, Opacity: A Philosophical Notebook.

Cry’ Me A Shiver

It’s not true that Cryogénie are best experienced live. That would imply their music can be experienced some other way. It can’t. The live experience is the only experience. And it’s guaranteed unique. These French avant-gardists aren’t the only band to hand out earplugs on the door, but they don’t do it for the conventional reason: that they play so loud.

In fact, they don’t play loud. They don’t play soft either. In the conventional sense, they don’t play at all. Here’s an interview from 2008 with Tïurbeau magazine:


Tïurbeau: I’ve got your latest album in front of me now. Words fail me.

Alexandre: And us too.

François: As usual.

Tïurbeau: Then one has to ask: why do you bother to release albums?

Alexandre: We see it, you could say, as a little ritual, something solid, something material––

François: Something permanent.

Alexandre: Yes, something permanent, to mark the occasion, that will remain with our audience. Often, we hear, they will buy an album after they have attended a concert, as a souvenir, almost. And they will truly play it!

Tïurbeau: They will play thirty-seven minutes of silence?

Cryogénie, Nix Sonica (2008)

Cryogénie, Nix Sonica (2008)*

François: Yes. The silence creates a space, a kind of opening in the present, for memories of the concert.

Alexandre: Yes, for memories, exactly so. Although, of course, in one sense we have pride in the irreproducibility of our music, in another sense we are recording every moment we are on stage. On the brain.

François: On the brains of the audience.

Alexandre: We are recording memories.

Tïurbeau: And the albums are designed to trigger the memories?

Alexandre: Trigger?

Tïurbeau: Bring the memories back.

Alexandre: Ah, yes, exactly so. The albums are a focus for memories of a concert.

François: Almost talismans.

Tïurbeau: In a magical sense?

Alexandre: Yes, why not? For us, experience is the ultimate magic. In the moment, but also in memory.

Tïurbeau: And does this relate to the sensory restrictions of your concerts, the way you try to turn down some senses in order to heighten the sense you are seeking to stimulate?

François: Yes, exactly so. Earplugs.

Alexandre: No aftershave, no perfume.

François: And please shower carefully before you attend.

Alexandre: Yes, shower carefully. And we ourselves, we will take care of the light. Remove it, make the scene very dark. You are not at a Cryogénie concert for pleasing your ears, your nose, eyes, mouth. Non, vous êtes là pour la chair!

François: Oui, pour la chair.

Tïurbeau: For the flesh.

Alexandre: Yes, the flesh. And how do we stimulate the flesh when we may not use another mode, not exploit another sense? No vibration, no infra-bass even. Then what?

François: Yes, this was the question we faced in our formative days.

Tïurbeau: And the answer…

Cryogénie, Rois du Froid (1996)

Cryogénie, Rois du Froid (1996)**

Alexandre: The cold!

François: Cold.

Alexandre: Please remember a question in the Gay Science of Nietzsche: Ist es nicht kälter geworden?

François: “Has it not become colder?”

Alexandre: And we want, if you attend a Cryogénie concert, for you to say: Ja! Oui! Yes! Kälter, kälter! Plus froid, plus froid! Colder, colder!

Tïurbeau: The triumph of the chill?

François: Yes. Triumph of the chill!

Alexandre: I don’t understand.

François: [Explains briefly in French]

Alexandre: Ah, yes, a triumph.

Tïurbeau: And with the concept came the name?

Alexandre: Yes, and so we had our name also. Cryogénie. With several meanings. Cryogénie is “creation of cold”, but also, for us, “genius of cold”, “spirit of cold”. Remember the concept of ritual. Our concerts, you might say, are rituals of cold, invocations of cold.

François: And: “If it’s too cold, you’re too old!”

Alexandre: Yes, so it’s said. Of course, in truth we welcome all ages, but if you are in poor health, perhaps better not to attend.

François: Nevertheless, visits to the pharmacy surely increase after we have passed through a city.

Tïurbeau: How cold do you go?

Alexandre: Ah, we prefer not to speak of that. No numbers, no statistics. You are there for the music, not to watch le thermomètre.

François: We get cold enough for our purposes.

Tïurbeau: That sounds rather sinister!

François: Yes, perhaps so. But would that not be the ultimate experience, to die pour une grêlodie, for a grêlodie?

Tïurbeau: Grêlodie?

Alexandre: It’s a joke, un calembour, a mixing of words.

François: A pun. In French, grêle is “hail”, you know, the little balls of ice, and mélodie is “melody”, of course, and so you have grêlodie, for a tune as performed by Cryogénie, a tune of ice, a tune of cold.

Tïurbeau: But not literal hail?

Alexandre: No, not literal. Though sometimes the breath of our audience will freeze and fall as a kind of snow. It makes a sound, that, a very delicate sound, le chuchotement des étoiles, comme on dit en Sibérie.

François: Yes, the whisper of the stars, as they say in Siberia. But of course, no-one will hear it, if they have followed their instructions.

Alexandre: Earplugs in!

Cryogénie, Blanchette (2003)

Cryogénie, Blanchette (2003)***

François: But the snow, the breath-snow, can be felt on the skin as it falls. This is acceptable, though it is an indirect effect of our music, not something we have planned for.

Tïurbeau: I have felt it. In the middle section of “Frissonique”, particularly.

Alexandre: Yes, and in “Bruitmal”.

François: When the framplifiers are cooking, as you might say.

Tïurbeau: Framplifiers? Can you explain for the benefit of our readers?

François: It is from froid and amplificateur. Framplicateur, framplifier. Amplifiers of cold, or generators of cold.

Tïurbeau: That is one of the most widely discussed aspects of your music, isn’t it? Your equipment.

Alexandre: Yes.

François: Yes, certainly.

Tïurbeau: But you’re rather secretive about it, aren’t you?

Alexandre: Yes!

François: You discuss, we are sphinxes.

Tïurbeau: Silent?

François: Yes. We have our – what is the term? – our trade-secrets. It’s not in our interests to expose our techniques. Nor in yours, we think.

Tïurbeau: You want to preserve that air of mystery?

Alexandre: Yes, precisely so. The experience is more strong when you don’t understand.

François: Like magic.

Alexandre: Yes, magic. We perform a ritual. The invocation of the cold. We invoke the cold and we throw the cold, we throw it on the audience.

François: Waves of cold. Cryorrhythms. Chords of cold, congelations, grêlodies, chills, thrills, rivers of shivers. That is the Cryogénie experience.

Tïurbeau: But there’s some serious technology behind the experience, isn’t there?

Alexandre: Yes.

François: Yes.

Tïurbeau: And you’re saying no more?

Alexandre: Yes, no more.

François: It’s not in our interests to explain. Or yours.

Tïurbeau: Nothing?

Alexandre: Nothing.

Tïurbeau: Not even a little?

François: Well, maybe a little. We had problems, in the early days, with unwanted noise, from the equipment.

Alexandre: Just a little.

François: I mean, if you think of a refrigerator, there is noise, of course. And we didn’t want noise, we wanted silence, pure silence.

Tïurbeau: A blank canvas, sensorily speaking?

François: Yes, a blank canvas, for us to paint with cold. So there was that problem to solve. The noise, unwanted noise.

Tïurbeau: And you solved it?

François: Yes, I think we did.

Alexandre: I think so.

Tïurbeau: But the earplugs are still necessary?

Alexandre: Yes, necessary, we think. Because, of course, with silent equipment there is still the movement of people, our movement on the stage, movement of the audience.

François: And the whisper of the stars, with some other effects. There are many things to create noise at a concert. We cannot eliminate them all, or we choose not to, because the earplugs are in themselves symbolic. To use them, you say: “See? I choose to close this door, this sensory mode.”

Alexandre: And you give yourself to us, to Cryogénie, to exploit another sense.

François: To submit you to our chill.

Tïurbeau: Esclaves du froid?

François: Yes, very good. Slaves of cold! But equally we are the slaves.

Alexandre: Yes, esclaves du froid. I like it. Perhaps we will write a song of that title one day.


Elsewhere Other-Engageable:

Rois du Froid — Cryogénie’s official site


*Sonic Snow.
**Kings of Cold.
***Little White One.

Lette’s Roll

A roulette is a little wheel or little roller, but it’s much more than a game in a casino. It can also be one of a family of curves created by tracing the path of a point on a rotating circle. Suppose a circle rolls around another circle of the same size. This is the resultant roulette:
roulette1

roulette1static
The shape is called a cardioid, because it looks like a heart (kardia in Greek). Now here’s a circle with radius r rolling around a circle with radius 2r:
roulette2

roulette2static

That shape is a nephroid, because it looks like a kidney (nephros in Greek).

This is a circle with radius r rolling around a circle with radius 3r:
roulette3

roulette3static
And this is r and 4r:
roulette4

roulette4static
The shapes above might be called outer roulettes. But what if a circle rolls inside another circle? Here’s an inner roulette whose radius is three-fifths (0.6) x the radius of its rollee:
roulette5

roulette5static
The same roulette appears inverted when the inner circle has a radius two-fifths (0.4) x the radius of the rollee:
roulette5a
But what happens when the circle rolling “inside” is larger than the rollee? That is, when the rolling circle is effectively swinging around the rollee, like a bunch of keys being twirled on an index finger? If the rolling radius is 1.5 times larger, the roulette looks like this:
roulette6
If the rolling radius is 2 times larger, the roulette looks like this:
roulette2over

Here are more outer, inner and over-sized roulettes:

roulette_outer

roulette_inner

roulette_over

And you can have circles rolling inside circles inside circles:

roulette7

roulette0616

roulette0616all

And here’s another circle-in-a-circle in a circle:

roulette07c015c

He Say, He Sigh, He Sow #26

“What he said often had double and even triple meanings so that, while the rest of us speak and think in single notes, he thought in chords.” — Robert Trivers on W.D. Hamilton, Vignettes of Famous Evolutionary Biologists, Large and Small, Unz Review, 27/iv/2015.

Playing the Double Base

Here’s some mathematical nonsense:

10 > 12
100 > 122
1000 > 1222

How can 1000 > 1222? Well, it makes perfect sense in what you might call a double base. In this base, every number is identified by a unique string of digits, but the strings don’t behave as they do in a standard base.

To see how this double base works, first look at 9 in standard base 2. To generate the binary digits from right to left, you follow the procedure x mod 2 and x = x div 2, where (x mod 2) returns the remainder when x is divided by 2 and (x div 2) divides x by 2 and discards the remainder:

9 mod 2 = 1 → ...1
9 div 2 = 4
4 mod 2 = 0 → ..01
4 div 2 = 2
2 mod 2 = 0 → .001
2 div 2 = 1
1 mod 2 = 1 → 1001

So 9[b=10] = 1001[b=2]. To adapt the procedure to base 3, simply use x mod 3 and x = x div 3:

32 mod 3 = 2 → ...2
32 div 3 = 10
10 mod 3 = 1 → ..12
10 div 3 = 3
3 mod 3 = 0 → .012
3 div 3 = 1
1 mod 3 = 1 → 1012

So 32[b=10] = 1012[b=3].

But what happens if you mix bases and use (x mod 3) and (x div 2), like this?:

2 mod 3 = 2 → .2
2 div 2 = 1
1 mod 3 = 1 → 12

3 mod 3 = 0 → .0
3 div 2 = 1
1 mod 3 = 1 → 10

So 10 > 12, i.e. 10[b=3,2] > 12[b=3,2].

5 mod 3 = 2 → ..2
5 div 2 = 2
2 mod 3 = 2 → .22
2 div 2 = 1
1 mod 3 = 1 → 122

6 mod 3 = 0 → ..0
6 div 2 = 3
3 mod 3 = 0 → .00
3 div 2 = 1
1 mod 3 = 1 → 100

So 100 > 122.

11 mod 3 = 2 → ...2
11 div 2 = 5
5 mod 3 = 2 → ..22
5 div 2 = 2
2 mod 3 = 2 → .222
2 div 2 = 1
1 mod 3 = 1 → 1222

12 mod 3 = 0 → …0
12 div 2 = 6
6 mod 3 = 0 → ..00
6 div 2 = 3
3 mod 3 = 0 → .000
3 div 2 = 1
1 mod 3 = 1 → 1000

And 1000 > 1222. Here are numbers 1 to 32 in this double base:

1 = 1
12 = 2
10 = 3
121 = 4
122 = 5
100 = 6
101 = 7
1212 = 8
1210 = 9
1221 = 10
1222 = 11
1000 = 12
1001 = 13
1012 = 14
1010 = 15
12121 = 16
12122 = 17
12100 = 18
12101 = 19
12212 = 20
12210 = 21
12221 = 22
12222 = 23
10000 = 24
10001 = 25
10012 = 26
10010 = 27
10121 = 28
10122 = 29
10100 = 30
10101 = 31
121212 = 32

Given a number represented in this mixed base, how do you extract the underlying n? Suppose the number takes the form n = (digit[1]..digit[di]), where digit[1] is the first and leftmost digit and digit[di] the final and rightmost digit. Then this algorithm will extract n:

n = 1
for i = 2 to di
..n = n * 2
..while n mod 3 ≠ digit[i]
....n = n + 1
..endwhile
next i
print n

For example, suppose n = 12212[b=3,2]. Then di = 5 and the algorithm will work like this:

n = 1
n = n * 2 = 2.
2 mod 3 = 2 = digit[2]
2 * 2 = 4
4 mod 3 = 1 ≠ digit[3]
5 mod 3 = 2 = digit[3]
5 * 2 = 10
10 mod 3 = 1 = digit[4]
10 * 2 = 20
20 mod 3 = 2 = digit[5]

Therefore 12212[b=3,2] = 20[b=10].

Now try some more mathematical nonsense:

21 > 100
111 > 1,000
1,001 > 10,000
10,001 > 100,000

How can numbers with d digits be greater than numbers with d+1 digits? Easily. In this incremental base, the base adjusts itself as the digits are generated, like this:

5 mod 2 = 1 → .1
5 div 2 = 2
2 mod (2 + 1) = 2 mod 3 = 2 → 21

The first digit generated is 1, so the base increases to (2 + 1) = 3 for the second digit. Compare the procedure when n = 4:

4 mod 2 = 0 → ..0
4 div 2 = 2
2 mod 2 = 0 → .00
2 div 2 = 1
1 mod 2 = 1 → 100

So 21 > 100, because 4 is a power of 2 and all the digits generated by (x mod 2) are 0 except the final and leftmost. 2 + 0 = 2. Now try n = 33:

33 mod 2 = 1 → ...1
33 div 2 = 16
16 mod (2+1) = 16 mod 3 = 1 → ..11
16 div 3 = 5
5 mod (3+1) = 5 mod 4 = 1 → .111
5 div 4 = 1
1 mod (4+1) = 1 mod 5 = 1.

33[b=10] = 1111[b=2,3,4,5].

Here are numbers 1 to 60 in this incremental base (note how 21 > 100, 111 > 1000, 1001 > 10000 and 10001 > 100000):

1 = 1
10 = 2
11 = 3
100 = 4*
21 = 5*
110 = 6
101 = 7
1000 = 8*
111 = 9*
210 = 10
121 = 11
1100 = 12
201 = 13
1010 = 14
211 = 15
10000 = 16*
221 = 17
1110 = 18
1001 = 19*
2100 = 20
311 = 21
1210 = 22
321 = 23
11000 = 24
1101 = 25
2010 = 26
1011 = 27
10100 = 28
421 = 29
2110 = 30
1201 = 31
100000 = 32*
1111 = 33
2210 = 34
1021 = 35
11100 = 36
2001 = 37
10010 = 38
1211 = 39
21000 = 40
1121 = 41
3110 = 42
2101 = 43
12100 = 44
1311 = 45
3210 = 46
1221 = 47
110000 = 48
2201 = 49
11010 = 50
2011 = 51
20100 = 52
1321 = 53
10110 = 54
10001 = 55*
101000 = 56
2111 = 57
4210 = 58
1421 = 59
21100 = 60

And here are numbers 256 to 270 (Note how 8,421 > 202,100 > 100,000,000):

100000000 = 256*
11221 = 257
101110 = 258
32101 = 259
202100 = 260*
13311 = 261
41210 = 262
10321 = 263
1111000 = 264
24201 = 265
131010 = 266
23011 = 267
320100 = 268
8421 = 269*
52110 = 270

Extracting n from a number represented in this incremental base is trickier than for the double base using (x mod 3) and (x div 2). To see how to do it, examine 11221[b=incremental]. The fifth and rightmost digit is 1, so the base increases to (2 + 1) = 3 for the fourth digit, which is 2. The base increases to (3 + 2) = 5 for the third digit, which is 2 again. The base increases to (5 + 2) = 7 for the second digit, 1. But the first and rightmost digit, 1, represents (x div 7) mod (7 + 1 = 8). So n can be extracted like this:

digit[1] * 7 = 1 * 7 = 7
7 mod 7 = 0 ≠ digit[2]
8 mod 7 = 1 = digit[2]
8 * 5 = 40
40 mod 5 = 0 ≠ digit[3]
41 mod 5 = 1 ≠ digit[3]
42 mod 5 = 2 = digit[3]
42 * 3 = 126
126 mod 3 = 0 ≠ digit[4]
127 mod 3 = 1 ≠ digit[4]
128 mod 3 = 2 = digit[4]
128 * 2 = 256
256 mod 2 = 0 ≠ digit[5]
257 mod 2 = 1 = digit[5]

So 11221[b=8,7,5,3,2] = 257[b=10].

Now try 8421[b=incremental]. The fourth and rightmost digit is 1, so the base increases to (2 + 1) = 3 for the third digit, which is 2. The base increases to (3 + 2) = 5 for the second digit, 4. But the first and rightmost digit, 8, represents (x div 5) mod (5 + 4 = 9). So n can be extracted like this:

digit[1] * 5 = 8 * 5 = 40
40 mod 5 = 0 ≠ digit[2]
41 mod 5 = 1 ≠ digit[2]
42 mod 5 = 2 ≠ digit[2]
43 mod 5 = 3 ≠ digit[2]
44 mod 5 = 4 = digit[2]
44 * 3 = 132
132 mod 3 = 0 ≠ digit[3]
133 mod 3 = 1 ≠ digit[3]
134 mod 3 = 2 = digit[3]
134 * 2 = 268
268 mod 2 = 0 ≠ digit[4]
269 mod 2 = 1 = digit[4]

So 8421[b=9,5,3,2] = 269[b=10].

Performativizing Papyrocentricity #37

Papyrocentric Performativity Presents:

Maths and Marmosets – The Great Mathematical Problems: Marvels and Mysteries of Mathematics, Ian Stewart (Profile Books 2013)

Be Ear Now – Sonic Wonderland: A Scientific Odyssey of Sound, Trevor Cox (Vintage 2015)

Exquisite Bulgarity – The Future of Architecture in 100 Buildings, Mark Kushner (Simon & Schuster 2015)

Stellar StoryDiscovering the Universe: The Story of Astronomy, Paul Murdin (Andre Deutsch 2014)

Terms of EndrearmentShe Literally Exploded: The Daily Telegraph Infuriating Phrasebook, Christopher Howse and Richard Preston (Constable 2007)


Or Read a Review at Random: RaRaR

Oh My Guardian

“The Naz and Matt Foundation was announced at a special service held in London for Nazim, two weeks after his funeral. The service featured contributions from a gay Muslim, gay Hindu, a gay vicar, a trainee Rabbi and a lesbian interfaith minister.” — My boyfriend killed himself because his family couldn’t accept that he was gay, The Guardian, 21/iii/2015.

Narcischism

What have bits to do with splits? A lot. Suppose you take the digits 12345, split them in all possible ways, then sum the results, like this:

12345 → (1234 + 5) + (123 + 45) + (123 + 4 + 5) + (12 + 345) + (12 + 34 + 5) + (12 + 3 + 45) + (12 + 3 + 4 + 5) + (1 + 2345) + (1 + 234 + 5) + (1 + 23 + 45) + (1 + 23 + 4 + 5) + (1 + 2 + 345) + (1 + 2 + 34 + 5) + (1 + 2 + 3 + 45) + (1 + 2 + 3 + 4 + 5) = 5175.

That’s a sum in base 10, but base 2 is at work below the surface, because each set of numbers is the answer to a series of binary questions: split or not? There are four possible places to split the digits 12345: after the 1, after the 2, after the 3 and after the 4. In (1 + 2 + 3 + 4 + 5), the binary question “Split or not?” is answered SPLIT every time. In (1234 + 5) and (1 + 2345) it’s answered SPLIT only once.

So the splits are governed by a four-digit binary number ranging from 0001 to 1111. When the binary digit is 1, split; when the binary digit is 0, don’t split. In binary, 0001 to 1111 = 01 to 15 in base 10 = 2^4-1. That’s for a five-digit number, so the four-digit 1234 will have 2^3-1 = 7 sets of sums:

1234 → (123 + 4) + (12 + 34) + (12 + 3 + 4) + (1 + 234) + (1 + 23 + 4) + 110 (1 + 2 + 34) + (1 + 2 + 3 + 4) = 502.

And the six-digit number 123456 will have 2^5-1 = 31 sets of sums. By now, an exciting question may have occurred to some readers. Does any number in base 10 equal the sum of all possible numbers formed by splitting its digits?

The exciting answer is: 0. In other words: No. To see why not, examine a quick way of summing the split-bits of 123,456,789, with nine digits. The long way is to find all possible sets of split-bits. There are 2^8-1 = 255 of them. The quick way is to sum these equations:

1 * 128 + 10 * 64 + 100 * 32 + 1000 * 16 + 10000 * 8 + 100000 * 4 + 1000000 * 2 + 10000000 * 1
2 * 128 + 20 * 64 + 200 * 32 + 2000 * 16 + 20000 * 8 + 200000 * 4 + 2000000 * 2 + 20000000 * 1
3 * 128 + 30 * 64 + 300 * 32 + 3000 * 16 + 30000 * 8 + 300000 * 4 + 3000000 * 3
4 * 128 + 40 * 64 + 400 * 32 + 4000 * 16 + 40000 * 8 + 400000 * 7
5 * 128 + 50 * 64 + 500 * 32 + 5000 * 16 + 50000 * 15
6 * 128 + 60 * 64 + 600 * 32 + 6000 * 31
7 * 128 + 70 * 64 + 700 * 63
8 * 128 + 80 * 127
9 * 255

Sum = 52,322,283.

52,322,283 has eight digits. If you use the same formula for the nine-digit number 999,999,999, the sum is 265,621,761, which has nine digits but is far smaller than 999,999,999. If you adapt the formula for the twenty-digit 19,999,999,999,999,999,999 (starting with 1), the split-bit sum is 16,562,499,999,987,400,705. In base 10, as far as I can see, numbers increase too fast and digit-lengths too slowly for the binary governing the split-sums to keep up. That’s also true in base 9 and base 8:

Num = 18,888,888,888,888,888,888 (b=9)
Sum = 16,714,201,578,038,328,760

Num = 17,777,777,777,777,777,777 (b=8)
Sum = 17,070,707,070,625,000,001

So what about base 7? Do the numbers increase slowly enough and the digit-lengths fast enough for the binary to keep up? The answer is: 1. In base 7, this twenty-digit number is actually smaller than its split-bit sum:

Num = 16,666,666,666,666,666,666 (b=7)
Sum = 20,363,036,303,404,141,363

And if you search below that, you can find a number that is equal to its split-bit sum:

166512 → (1 + 6 + 6 + 5 + 1 + 2) + (16 + 6 + 5 + 1 + 2) + (1 + 66 + 5 + 1 + 2) + (166 + 5 + 1 + 2) + (1 + 6 + 65 + 1 + 2) + (16 + 65 + 1 + 2) + (1 + 665 + 1 + 2) + (1665 + 1 + 2) + (1 + 6 + 6 + 51 + 2) + (16 + 6 + 51 + 2) + (1 + 66 + 51 + 2) + (166 + 51 + 2) + (1 + 6 + 651 + 2) + (16 + 651 + 2) + (1 + 6651 + 2) + (16651 + 2) + (1 + 6 + 6 + 5 + 12) + (16 + 6 + 5 + 12) + (1 + 66 + 5 + 12) + (166 + 5 + 12) + (1 + 6 + 65 + 12) + (16 + 65 + 12) + (1 + 665 + 12) + (1665 + 12) + (1 + 6 + 6 + 512) + (16 + 6 + 512) + (1 + 66 + 512) + (166 + 512) + (1 + 6 + 6512) + (16 + 6512) + (1 + 66512) = 166512[b=7] = 33525[b=10].

So 33525 in base 7 is what might be called a narcischist: it can gaze into the split-bits of its own digits and see itself gazing back. In base 6, 1940 is a narcischist:

12552 → (1 + 2 + 5 + 5 + 2) + (12 + 5 + 5 + 2) + (1 + 25 + 5 + 2) + (125 + 5 + 2) + (1 + 2 + 55 + 2) + (12 + 55 + 2) + (1 + 255 + 2) + (1255 + 2) + (1 + 2 + 5+ 52) + (12 + 5 + 52) + (1 + 25 + 52) + (125 + 52) + (1 + 2 + 552) + (12 + 552) + (1 + 2552) = 12552[b=6] = 1940[b=10].

In base 5, 4074 is a narcischist:

112244 → (1 + 1 + 2 + 2 + 4 + 4) + (11 + 2 + 2 + 4 + 4) + (1 + 12 + 2 + 4 + 4) + (112 + 2 + 4 + 4) + (1 + 1 + 22 + 4 + 4) + (11 + 22 + 4 + 4) + (1 + 122 + 4 + 4) + (1122 + 4 + 4) + (1 + 1 + 2 + 24 + 4) + (11 + 2 + 24 + 4) + (1 + 12 + 24 + 4) + (112 + 24 + 4) + (1 + 1 + 224 + 4) + (11 + 224 + 4) + (1 + 1224 + 4) + (11224 + 4) + (1 + 1 + 2 + 2 + 44) + (11 + 2 + 2 + 44) + (1 + 12 + 2 + 44) + (112 + 2 + 44) + (1 + 1 + 22 + 44) + (11 + 22 + 44) + (1 + 122 + 44) + (1122 + 44) + (1 + 1 + 2 + 244) + (11 + 2 + 244) + (1 + 12 + 244) + (112 + 244) + (1 + 1 + 2244) + (11 + 2244) + (1 + 12244) = 112244[b=5] = 4074.

And in base 4, 27 is:

123 → (1 + 2 + 3) + (12 + 3) + (1 + 23) 123[b=4] = 27.

And in base 3, 13 and 26 are:

111 → (1 + 1 + 1) + (11 + 1) + (1 + 11) = 111[b=3] = 13.

222 → (2 + 2 + 2) + (22 + 2) + (2 + 22) = 222[b=3] = 26.

There are many more narcischists in all these bases, even if you exclude numbers with zeroes in them, like these in base 4:

1022 → (1 + 0 + 2 + 2) + (10 + 2 + 2) + (1 + 02 + 2) + (102 + 2) + (1 + 0 + 22) + (10 + 22) + (1 + 022) = 1022[b=4] = 74.

1030 → (1 + 0 + 3 + 0) + (10 + 3 + 0) + (1 + 03 + 0) + (103 + 0) + (1 + 0 + 30) + (10 + 30) + (1 + 030) = 1030[b=4] = 76.

1120 → (1 + 1 + 2 + 0) + (11 + 2 + 0) + (1 + 12 + 0) + (112 + 0) + (1 + 1 + 20) + (11 + 20) + (1 + 120) = 1120[b=4] = 88.