Autonomata

“Describe yourself.” You can say it to people. And you can say it to numbers too. For example, here’s the number 3412 describing the positions of its own digits, starting at 1 and working upward:


3412 – the 1 is in the 3rd position, the 2 is in the 4th position, the 3 is in the 1st position, and the 4 is in the 2nd position.

In other words, the positions of the digits 1 to 4 of 3412 recreate its own digits:


3412 → (3,4,1,2) → 3412

The number 3412 describes itself – it’s autonomatic (from Greek auto, “self” + onoma, “name”). So are these numbers:


1
21
132
2143
52341
215634
7243651
68573142
321654798

More precisely, they’re panautonomatic numbers, because they describe the positions of all their own digits (Greek pan or panto, “all”). But what if you use the positions of only, say, the 1s or the 3s in a number? In base ten, only one number describes itself like that: 1. But we’re not confined to base 10. In base 2, the positions of the 1s in 110 (= 6) are 1 and 10 (= 2). So 110 is monautonomatic in binary (Greek mono, “single”). 10 is also monautonomatic in binary, if the digit being described is 0: it’s in 2nd position or position 10 in binary. These numbers are monoautonomatic in binary too:


110100 = 52 (digit = 1)
10100101111 = 1327 (d=0)

In 110100, the 1s are in 1st, 2nd and 4th position, or positions 1, 10, 100 in binary. In 10100101111, the 0s are in 2nd, 4th, 5th and 7th position, or positions 10, 100, 101, 111 in binary. Here are more monautonomatic numbers in other bases:


21011 in base 4 = 581 (digit = 1)
11122122 in base 3 = 3392 (d=2)
131011 in base 5 = 5131 (d=1)
2101112 in base 4 = 9302 (d=1)
11122122102 in base 3 = 91595 (d=2)
13101112 in base 5 = 128282 (d=1)
210111221 in base 4 = 148841 (d=1)

For example, in 131011 the 1s are in 1st, 3rd, 5th and 6th position, or positions 1, 3, 10 and 11 in quinary. But these numbers run out quickly and the only monautonomatic number in bases 6 and higher is 1. However, there are infinitely long monoautonomatic integer sequences in all bases. For example, in binary this sequence at the Online Encyclopedia of Integer Sequences describes itself using the positions of its 1s:


A167502: 1, 10, 100, 111, 1000, 1001, 1010, 1110, 10001, 10010, 10100, 10110, 10111, 11000, 11010, 11110, 11111, 100010, 100100, 100110, 101001, 101011, 101100, 101110, 110000, 110001, 110010, 110011, 110100, 111000, 111001, 111011, 111101, 11111, …

In base 10, it looks like this:


A167500: 1, 2, 4, 7, 8, 9, 10, 14, 17, 18, 20, 22, 23, 24, 26, 30, 31, 34, 36, 38, 41, 43, 44, 46, 48, 49, 50, 51, 52, 56, 57, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 75, 77, 80, 83, 86, 87, 89, 91, 94, 95, 97, 99, 100, 101, 103, 104, 107, 109, 110, 111, 113, 114, 119, 120, 124, … (see A287515 for a similar sequence using 0s)

In any base, you can find some sequence of integers describing the positions of any of the digits in that base – for example, the 1s or the 7s. But the numbers in the sequence get very large very quickly in higher bases. For example, here are some opening sequences for the digits 0 to 9 in base 10:


3, 10, 1111110, … (d=0)
1, 3, 10, 200001, … (d=1)
3, 12, 100000002, … (d=2)
2, 3, 30, 10000000000000000000000003, … (d=3)
2, 4, 14, 1000000004, … (d=4)
2, 5, 105, … (d=5)
2, 6, 1006, … (d=6)
2, 7, 10007, … (d=7)
2, 8, 100008, … (d=8)
2, 9, 1000009, … (d=9)

In the sequence for d=0, the first 0 is in the 3rd position, the second 0 is in the 10th position, and the third 0 is in the 1111110th position. That’s why I’ve haven’t written the next number – it’s 1,111,100 digits long (= 1111110 – 10). But it’s theoretically possible to write the number. In the sequence for d=3, the next number is utterly impossible to write, because it’s 9,999,999,999,999,999,999,999,973 digits long (= 10000000000000000000000003 – 30). In the sequence for d=5, the next number is this:


1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005 (100 digits long = 105 – 5).

And in fact there are an infinite number of such sequences for any digit in any base – except for d=1 in binary. Why is binary different? Because 1 is the only digit that can start a number in that base. With 0, you can invent a sequence starting like this:


111, 1110, 1111110, …

Or like this:


1111, 11111111110, …

Or like this:


11111, 1111111111111111111111111111110, …

And so on. But with 1, there’s no room for manoeuvre.

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Russell in Your Head-Roe

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world. — Bertrand Russell, “The Study Of Mathematics” (1902)


The title of this incendiary intervention is of course a paronomasia on these lines from Led Zeppelin’s magisterial “Stairway to Heaven”:

“If there’s a bustle in your hedgerow, don’t be alarmed now:
It’s just a spring-clean for the May Queen…”

And “head-roe” is a kenning for “brain”.

Bat out of L

Pre-previously on Overlord-in-terms-of-the-Über-Feral, I’ve looked at intensively interrogated issues around the L-triomino, a shape created from three squares that can be divided into four copies of itself:

An L-triomino divided into four copies of itself


I’ve also interrogated issues around a shape that yields a bat-like fractal:

A fractal full of bats


Bat-fractal (animated)


Now, to end the year in spectacular fashion, I want to combine the two concepts pre-previously interrogated on Overlord-in-terms-of-the-Über-Feral (i.e., L-triominoes and bats). The L-triomino can also be divided into nine copies of itself:

An L-triomino divided into nine copies of itself


If three of these copies are discarded and each of the remaining six sub-copies is sub-sub-divided again and again, this is what happens:

Fractal stage 1


Fractal stage 2


Fractal #3


Fractal #4


Fractal #5


Fractal #6


Et voilà, another bat-like fractal:

L-triomino bat-fractal (static)


L-triomino bat-fractal (animated)


Elsewhere other-posted:

Tri-Way to L
Bats and Butterflies
Square Routes
Square Routes Revisited
Square Routes Re-Revisited
Square Routes Re-Re-Revisited

Performativizing Papyrocentricity #59

Papyrocentric Performativity Presents:

Mobile MetalBattleground: The Greatest Tank Duels in History, ed. Steven J. Zaloga (Osprey Publishing 2011)

Allum’s Album – The Collector’s Cabinet: Tales, Facts and Fictions from the World of Antiques, Marc Allum (Icon Books 2013)

Aschen PassionDeath in Venice and Other Stories, Thomas Mann, translated by David Luke (1988)


Or Read a Review at Random: RaRaR

Tridentine Math

The Tridentine Mass, the 1962 version of which has been officially declared the (authorized) extraordinary form of the Roman Rite of Mass (Extraordinary Form for short; Latin: forma extraordinaria), is the Roman Rite Mass which appears in typical editions of the Roman Missal published from 1570 to 1962. — Tridentine Mass, Wikipedia

A 30°-60°-90° right triangle, with sides 1 : √3 : 2, can be divided into three identical copies of itself:

30°-60°-90° Right Triangle — a rep-3 rep-tile…


And if it can be divided into three, it can be divided into nine:

…that is also a rep-9 rep-tile


Five of the sub-copies serve as the seed for an interesting fractal:

Fractal stage #1


Fractal stage #2


Fractal stage #3


Fractal #4


Fractal #5


Fractal #6


Fractal #6


Tridentine cross (animated)


Tridentine cross (static)


This is a different kind of rep-tile:

Noniamond trapezoid


But it yields the same fractal cross:

Fractal #1


Fractal #2


Fractal #3


Fractal #4


Fractal #5


Fractal #6


Tridentine cross (animated)


Tridentine cross (static)


Elsewhere other-available:

Holey Trimmetry — another fractal cross

Alpha Beat a Gamer

Abstract: The game of chess is the most widely-studied domain in the history of artificial intelligence. The strongest programs are based on a combination of sophisticated search techniques, domain-specific adaptations, and handcrafted evaluation functions that have been refined by human experts over several decades. In contrast, the AlphaGo Zero program recently achieved superhuman performance in the game of Go, by tabula rasa reinforcement learning from games of self-play. In this paper, we generalise this approach into a single AlphaZero algorithm that can achieve, tabula rasa, superhuman performance in many challenging domains. Starting from random play, and given no domain knowledge except the game rules, AlphaZero achieved within 24 hours a superhuman level of play in the games of chess and shogi (Japanese chess) as well as Go, and convincingly defeated a world-champion program in each case. — “Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm”, 5/XII/2017.

Tott ist Rot

• Lautlos und fein rann der rostrot gefärbte Sand durch die gläserne Enge, und da er in der oberen Höhlung zur Neige ging, hatte sich dort ein kleiner, reißender Strudel gebildet. — Thomas Mann, Der Tod in Venedig (1912)

• Silently, subtly, the rust-red sand trickled through the narrow glass aperture, dwindling away out of the upper vessel, in which a little whirling vortex had formed. — “Death in Venice” (translated by David Luke)

Oh My Guardian #4

• The past 16 years have involved a lot of questioning and reflecting, both in terms of what it means to be “good”, but also on the various racist myths about Muslims. — Let’s be clear: Muslims are neither good nor bad. We’re just human, Farah in terms of Elahi, The Guardian, 14/xii/2017.


Elsewhere other-available:

Oh My Guardian #1
Oh My Guardian #2
Oh My Guardian #3
Reds under the Thread

Square Routes Re-Re-Revisited

This is an L-triomino, or shape created from three squares laid edge-to-edge:

When you divide each square like this…

You can create a fractal like this…

Stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Stage #9


Stage #10


Animated fractal


Here are more fractals created from the triomino:

Animated


Static


Animated


Static


Animated


Static


And here is a different shape created from three squares:

And some fractals created from it:

Animated


Static


Animated


Static


Animated


Static


And a third shape created from three squares:

And some fractals created from it:

Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Previously pre-posted (please peruse):

Tri-Way to L
Square Routes
Square Routes Revisited
Square Routes Re-Revisited