Currently listening…
• UnderHill, EnCrypt (2008)
• The Whelms, Malt Does More (1981)
• Nhyul, 883·307·173 (1993)
• Exqvies, Deinococcus radiodurans (2010)
• Bunu Zakr, Mbigak II (1968)
• The Glamours, Lottery in Babylon (2008)
• Incgo, DeMTR (1993)
• Aokka, Ophicléide (1985)
• Diamond Head, Canterbury (1983)
• Z Dirqenav, Z (1972)
Previously pre-posted:
Toxic Turntable #1 • #2 • #3 • #4 • #5 • #6 • #7 • #8 • #9 •
One of the pleasures of exploring an ancient city like York or Chester is that of learning new routes to the same destination. There are byways and alleys, short-cuts and diversions. You set off intending to go to one place and end up in another.
Maths is like that, even at its simplest. There are many routes to the same destination. I first found the fractal below by playing with the L-triomino, or the shape created by putting three squares in the shape of an L. You can divide it into four copies of the same shape and discard one copy, then do the same to each of the sub-copies, then repeat. I’ve decided to call it the hourglass fractal:
Hourglass fractal (animated)
Hourglass fractal (static)
Then I unexpectedly came across the fractal again when playing with what I call a proximity fractal:
Hourglass animated (proximity fractal)
(Static image)
Now I’ve unexpectedly come across it for a third time, playing with a very simple fractal based on a 2×2 square. At first glance, the 2×2 square yields only one interesting fractal. If you divide the square into four smaller squares and discard one square, then do the same to each of the three sub-copies, then repeat, you get a form of the Sierpiński triangle, like this:
Sierpiński triangle stage 1
Sierpiński triangle #2
Sierpiński triangle #3
Sierpiński triangle #4
Sierpiński triangle animated
(Static image)
The 2×2 square seems too simple for anything more, but there’s a simple way to enrich it: label the corners of the sub-squares so that you can, as it were, individually rotate them 0°, 90°, 180°, or 270°. One set of rotations produces the hourglass fractal, like this:
Hourglass stage 1
Hourglass #2
Fractal #3
Hourglass #4
Hourglass #5
Hourglass #6
Hourglass animated
(Static image)
Here are some more fractals from the 2×2 square created using this technique (I’ve found some of them previously by other routes):
(Static image)
(Static image)
(Static image)
(Static image)
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κατ’ ἐπακολούθημα δὲ καὶ περὶ τῆς ἐγκυκλίου καλουμένης παιδείας, εἰς ὅσα εὔχρηστος, περί τε ἀστρολογικῆς καὶ μαθηματικῆς καὶ μαγικῆς γοητείας τε ἐπιδραμητέον. αὐχοῦσι γὰρ δὴ καὶ ἐπὶ ταῖσδε οἱ Πανέλληνες ὡς μεγίσταις ἐπιστήμαις. — Κλήμης ὁ Ἀλεξανδρεύς, Στρώματα.
“By consequence, also we must treat of what is called the curriculum of study — how far it is serviceable; and of astrology, and mathematics, and magic, and sorcery. For all the Greeks boast of these as the highest sciences.” — Clement of Alexandria, The Stromata, Book II.
It’s like watching a seed grow. You take a number and count how many 0s it contains, then how many 1s, how many 2s, 3s, 4s and so on. Then you create a new number by writing the count of each digit followed by the digit itself. Then you repeat the process with the new number.
Here’s how it works if you start with the number 1:
1
The count of digits is one 1, so the new number is this:
→ 11
The count of digits for 11 is two 1s, so the next number is:
→ 21
The count of digits for 21 is one 1, one 2, so the next number is:
→ 1112
The count of digits for 1112 is three 1s, one 2, so the next number is:
→ 3112
The count of digits for 3112 is two 1s, one 2, one 3, so the next number is:
→ 211213
What happens after that? Here are the numbers as a sequence:
1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 114213 → 31121314 → 41122314 → 31221324 → 21322314…
That’s all you need, because something interesting happens with 21322314. The digit count is two 1s, three 2s, two 3s, one 4, so the next number is:
→ 21322314
In other words, 21322314 is what might be called a self-descriptive number: it describes the count of its own digits. That’s why I think this procedure is like watching a seed grow. You start with the tiny seed of 1 and end in the giant oak of 21322314, whose factorization is 2 * 3^2 * 13 * 91121. But there are many more self-descriptive numbers in base ten and some of them are much bigger than 21322314. A047841 at the Online Encyclopedia of Integer Sequences lists all 109 of them (and calls them “autobiographical numbers”). Here are a few, starting with the simplest possible:
22 → two 2s → 22
10213223 → one 0, two 1s, three 2s, two 3s → 10213223
10311233 → one 0, three 1s, one 2, three 3s → 10311233
21322314 → two 1s, three 2s, two 3s, one 4 → 21322314
21322315 → two 1s, three 2s, two 3s, one 5 → 21322315
21322316 → two 1s, three 2s, two 3s, one 6 → 21322316*
1031223314 → one 0, three 1s, two 2s, three 3s, one 4 → 10
31223314
3122331415 → three 1s, two 2s, three 3s, one 4, one 5
→ 3122331415
3122331416 → three 1s, two 2s, three 3s, one 4, one 6
→ 3122331416*
*And for 21322317, 21322318, 21322319; 3122331417, 3122331418, 3122331419.
And here’s what happens when you seed a sequence with a number containing all possible digits in base ten:
1234567890 → 10111213141516171819 → 101111213141516171819 → 101211213141516171819 → 101112213141516171819…
That final number is self-descriptive:
101112213141516171819 → one 0, eleven 1s, two 2s, one 3, one 4, one 5, one 6, one 7, one 8, one 9 → 101112213141516171819
So some numbers are self-descriptive and some start a sequence that ends in a self-descriptive number. But that doesn’t exhaust the possibilities. Some numbers are part of a loop:
103142132415 → 104122232415 → 103142132415…
104122232415 → 103142132415 → 104122232415…
1051421314152619 → 1061221324251619 → 1051421314152619…
5142131415261819 → 6122132425161819 → 5142131415261819…
106142131416271819 → 107122132426171819 → 106142131416271819…
10512223142518 → 10414213142518 → 10512213341518 → 10512223142518…
51222314251718 → 41421314251718 → 51221334151718 →
51222314251718
But all that is base ten. What about other bases? In fact, nearly all self-descriptive numbers in base ten are also self-descriptive in other bases. An infinite number of other bases, in fact. 22 is a self-descriptive number for all b > 2. The sequence seeded with 1 is identical in all b > 4:
1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 114213 → 31121314 → 41122314 → 31221324 → 21322314 → 21322314…
In bases 2, 3 and 4, the sequence seeded with 1 looks like this:
1 → 11 → 101 → 10101 → 100111 → 1001001 → 1000111 → 1101001 → 1101001… (b=2) (1101001[2] = 105 in base 10)
1 → 11 → 21 → 1112 → 10112 → 1010112 → 2011112 → 1011122 → 1011122… (b=3) (1011122[3] = 854 in base 10)
1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 1110213 → 101011213 → 201111213 → 101112213 → 101112213… (b=4) (101112213[4] = 71079 in base 10)
In base 2 there are only two self-descriptive numbers (and no loops):
111 → three 1s → 111… (b=2) (111 = 7 in base 10)
1101001 → three 0s, four 1s → 1101001… (b=2) (1101001 = 105 in base 10)
So if you apply the “count digits” procedure in base 2, all numbers, except 111, begin a sequence that ends in 1101001. Base 3 has a few more self-descriptive numbers and also some loops:
22 → 22… (b >= 3)
10111 → one 0, four 1s → 10111… (b=3)
11112 → four 1s, one 2 → 11112…
100101 → three 0s, three 1s → 100101… (b=3)
1011122 → one 0, four 1s, two 2s → 1011122… (b=3)
2021102 → two 0s, two 1s, three 2s → 2021102… (b=3)
10010122 → three 0s, three 1s, two 2s → 10010122…
2012112 → 10101102 → 10011112 → 2012112…
10011112 → 2012112 → 10101102 → 10011112…
10101102 → 10011112 → 2012112 → 10101102…
A question I haven’t been able to answer: Is there a base in which loops can be longer than these?
103142132415 → 104122232415 → 103142132415…
10512223142518 → 10414213142518 → 10512213341518 → 10512223142518…
A question I have been able to answer: What is the sequence when it’s seeded with the title of this blog-post? T4K1NGS3LF13S is a number in all bases >= 30 and its base-30 form equals 15,494,492,743,722,316,018 in base 10 (with the factorization 2 * 72704927 * 106557377767). If T4K1NGS3LF13S seeds a sequence in any b >= 30, the result looks like this:
T4K1NGS3LF13S → 2123141F1G1K1L1N2S1T → 813213141F1G1K1L1N1S1T → A1122314181F1G1K1L1N1S1T → B1221314181A1F1G1K1L1N1S1T → C1221314181A1B1F1G1K1L1N1S1T → D1221314181A1B1C1F1G1K1L1N1S1T → E1221314181A1B1C1D1F1G1K1L1N1S1T → F1221314181A1B1C1D1E1F1G1K1L1N1S1T → G1221314181A1B1C1D1E2F1G1K1L1N1S1T → F1321314181A1B1C1D1E1F2G1K1L1N1S1T → F1222314181A1B1C1D1E2F1G1K1L1N1S1T → E1421314181A1B1C1D1E2F1G1K1L1N1S1T → F1221324181A1B1C1D2E1F1G1K1L1N1S1T → E1421314181A1B1C1D1E2F1G1K1L1N1S1T
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 