This Means RaWaR

by in Notices, Randomness | 2 Comments

The Overlord of the Über-Feral says: Welcome to my bijou bloguette. You can scroll down to sample more or simply:

• Read a Writerization at Random: RaWaR

• ¿And What Doth It Mean To Be Flesh?

• მათემატიკა მსოფლიოს მეფე

Gweel & Other Alterities – Incunabula’s new edition

Tales of Silence & Sortilege – Incunabula’s new edition

If you’d like to donate to O.o.t.Ü.-F., please click here.

Versūs Vīlēs Veneficī

by in Language, Latin, Literature, Poetry, Transgression... and tagged , , , , , , , | Leave a comment

Satanas

To Jorge Santayana

ECCE! Princeps infernorum,
Rex veneficus amorum
Vilium et mortiferorum,
   Ecce! regnat Lucifer:
Animis qui dominatur,
Quibus coelum spoliatur;
Qui malignus bona fatur,
   Cor corrumpens suaviter.

Fructus profert; inest cinis:
Profert flores plenos spinis:
Vitae eius mors est finis:
   Crux est eius requies.
Qualis illic apparebit
Cruciatus, et manebit!
Quantas ista quot habebit
   Mors amaritudines!

Iuventutis quam formosa
Floret inter rosas rosa!
Venit autem vitiosa
   Species infamiae:
Veniunt crudeles visus,
Voces simulati risus;
Et inutilis fit nisus
   Flebilis laetitae.

Quanto vitium splendescit,
Tanto anima nigrescit;
Tanto tandem cor marcescit,
   Per peccata dulcia.
Gaudens mundi Princeps mali
Utitur veneno tali,
Voluptate Avernali;
   0 mellita vitia!

Gaudet Princeps huius mundi
Videns animam confundi;
Cordis amat moribundi
   Aspectare proelium.
Vana tentat, vana quaerens,
Cor anhelum, frustra moerens;
Angit animae inhaerens
   Flamma cor miserrimum.

Gaudet Rector tenebrarum
Immolare cor amarum;
Satiare furiarum
   Rex sorores avidas.
Vae! non stabit in aeternum
Regnum, ait Rex, infernum:
Sed, dum veniat Supernum,
   Dabo vobis victimas.

• Lionel Johnson (1867-1902), text courtesy Laudator Temporis Acti

Fibonacci Friday Factors

by in Arithmetic, Base Behavior, Fibonacci Sequence, Integer Sequences, Mathematics, Ulam Spirals and tagged , , , , , , | Leave a comment

Today’s a Phiday Friday or Φiday Friday or Φriday, so let’s have some more Fibonacci Fun. Here is the famous Fibonacci sequence, where each number (after seeding with “0, 1”) is formed by adding the previous two numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, … — A000045 at the Online Encyclopedia of Integer Sequences (OEIS)

It’s obvious that the numbers get bigger for ever and that no number repeats except 1. But what happens to the final digit of the Fibonacci numbers, as underlined here?:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, …

If you think about it, you’ll realize that the final digit has to repeat. Look for the “0, 1, 1” re-appearing:

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, … — A003893 at the OEIS, “a(n) = Fibonacci(n) mod 10”

As you’ll see, all the numbers 0 to 9 appear in that sequence. But what about the final two digits of the Fibonacci sequence? Do all the numbers 0 to 99 appear before the sequence repeats?

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 44, 33, 77, 10, 87, 97, 84, 81, 65, 46, 11, 57, 68, 25, 93, 18, 11, 29, 40, 69, 9, 78, 87, 65, 52, 17, 69, 86, 55, 41, 96, 37, 33, 70, 3, 73, 76, 49, 25, 74, 99, 73, 72, 45, 17, 62, 79, 41, 20, 61, 81, 42, 23, 65, 88, 53, 41, 94, 35, 29, 64, … — A105471 at the OEIS, “a(n) = Fibonacci(n) mod 100”

And what about the the final three digits and the numbers 0 to 999?

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 597, 584, 181, 765, 946, 711, 657, 368, 25, 393, 418, 811, 229, 40, 269, 309, 578, 887, 465, 352, 817, 169, 986, 155, 141, 296, 437, 733, 170, 903, 73, 976, 49, 25, 74, 99, 173, 272 … — A248740 at the OEIS, “a(n) = Fibonacci(n) mod 1000”

In fact, some numbers do go missing as the final block of digits gets longer. But all that is based on the representation of the Fibonacci numbers in base 10. What about other bases? I had a look at that question and came up with some interesting patterns when I represented the final-block numbers on an Ulam-like spiral, where numbers are represented as squares on a spiral rotating counter-clockwise. This is the spiral for powers of 2 (the red square marks the center of the spiral and the number 1):

Spiral of final Fib-digits modulo 2^p

Fib-spiral mod 2^p (smaller scale)

Fib-spiral mod 2^p (smaller scale still)

What fraction of numbers are missing from the spiral? Watch this space. In the meantime, here’s the Fib-spiral for powers of 3:

Fib-spiral mod 3^p

It’s completely filled, because no numbers are missing (the red square marks “1” at the center of the spiral). What about powers of 4? Well, we’ve already seen that Fib-spiral, because all powers of 4 are also powers of 2:

Fib-spiral mod 4^p

The Fib-spiral for powers of 5 is the same as the Fib-spiral for powers of 3: it’s completely filled again. But the Fib-spiral for powers of 6 is interesting:

Fib-spiral mod 6^p

Fib-spiral mod 6^p (smaller scale)

Fib-spiral mod 6^p (smaller scale still)

And here are more Fib-spirals and more interesting patterns:

Fib-spiral mod 7^p

Fib-spiral mod 10^p — identical to the Fib-spiral for 2^p

Fib-spiral mod 11^p

Fib-spiral mod 11^p (smaller scale)

Fib-spiral mod 13^p

Fib-spiral mod 13^p (smaller scale)

Fib-spiral mod 17^p

Fib-spiral mod 17^p (smaller scale)

Fib-spiral mod 19^p

Fib-spiral mod 19^p (smaller scale)

Fib-spiral mod 41^p

Fib-spiral mod 41^p (smaller scale)

Fib-spiral mod 47^p

Elsewhere Other-Accessible…

• Friday is Φday — a first look at Phiday on Friday

Run Fun

by in Arithmetic, Base Behavior, Mathematics and tagged , | Leave a comment

I was looking at numbers where the digit-runs of n in one base, b1, represented the digits of n in another base, b2. To understand digit-runs, take the number 775555111 (in whatever base). The function digitrun(775555111) = 2,4,3, because reading 775555111 left-to-right you see two 7s, four 5s, three 1s. Here’s one pattern I found:

10000011111 in base 2 = 1055 in base 10. The digit runs of 10000011111 are 1,5,5 and 15530 = 105510 = 100000111112.
10000022222 in base 3 = 59291 in base 10. The digit runs of 10000022222 are again 1,5,5 and 155241 = 5929110 = 100000222223.
1000009999910 → 1,5,5 and 15599998 = 1000009999910

The pattern isn’t difficult to understand — what are 30+2 and 99998+2? — and it’s easy to prove that it applies to all standard bases. But I wondered how AI would go about the proof. Here’s the result:

For base b ≥ 2, let the largest digit be d = b − 1. Then

100000dddddb

has digit runs

1, 5, 5

so the run-length encoding is always

155.

The value of 155 is interpreted in base

b5 − 2,

because

155b5−2 = (b5−2)2 + 5(b5−2) + 5.

Thus

100000dddddb = 155b5−2.

Base b n = 100000(b−1)(b−1)(b−1)(b−1)(b−1)b Run Lengths Compressed Form Base-10 Value of n
2 100000111112 1,5,5 15530 1,055
3 100000222223 1,5,5 155241 59,291
4 100000333334 1,5,5 1551022 1,049,599
5 100000444445 1,5,5 1553123 9,765,624
6 100000555556 1,5,5 1557774 60,466,175
7 100000666667 1,5,5 15516805 282,475,248
8 100000777778 1,5,5 15532766 1,074,266,111
9 100000888889 1,5,5 15559047 3,486,270,320
10 1000009999910 1,5,5 15599998 10,000,099,999
11 100000AAAAA11 1,5,5 155161049 26,094,908,050
12 100000BBBBB12 1,5,5 155248830 62,191,972,031
13 100000CCCCC13 1,5,5 155371291 136,858,491,292
14 100000DDDDD14 1,5,5 155537822 281,793,968,543
15 100000EEEEE15 1,5,5 155759373 547,918,611,874
16 100000FFFFF16 1,5,5 1551048574 1,099,512,627,199
17 100000GGGGG17 1,5,5 1551419855 2,141,423,647,200
18 100000HHHHH18 1,5,5 1551889566 4,058,363,974,751
19 100000IIIII19 1,5,5 1552476097 7,502,570,010,432
20 100000JJJJJ20 1,5,5 1553199998 13,120,000,000,000

(Here A = 10, B = 11, …, J = 19.)

Proof for All b ≥ 2

The number

100000(b−1)(b−1)(b−1)(b−1)(b−1)b

has value

b10 + (b−1)(b4 + b3 + b2 + b + 1).

Since

b4 + b3 + b2 + b + 1 = (b5 − 1)/(b − 1),

this becomes

b10 + (b5 − 1) = b10 + b5 − 1.

Now let

B = b5 − 2.

Then

155B = B2 + 5B + 5.

Substituting B = b5 − 2:

(B2 + 5B + 5) = (b5−2)2 + 5(b5−2) + 5

= b10−4b5+4+5b5−10+5

= b10 + b5−1.

Hence

b10 + b5 − 1 = (b5 − 2)2+5(b5 − 2)+5

for every b ≥ 2.

Therefore,

100000(b−1)5b = 155b5−2.

This is exactly the pattern observed in bases 2 to 20, and it holds for all bases.

Partition Fission

by in Arithmetic, Mathematics, Partitions and tagged , , , , | Leave a comment

Simple but seductive. That’s how I’d describe partitions. Except that they’re not so simple. There are hidden depths in the task of finding how many ways an integer can be expressed as the sum of smaller integers (and as the sum of itself). Here are the partitions of 4, for example:

4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1
4 (1) = 1+3 (2) = 2+2 (3) = 1+1+2 (4) = 1+1+1+1 (5), ∴ partcount(4) = 5

There are five partitions of 4, because an integer is counted as its own partition. Accordingly, partcount(4) = 5. But partcount(n) doesn’t return values in a predictable way:

partcount(1) = 1 ← 1
partcount(2) = 2 ← 2 = 1+1
partcount(3) = 3 ← 3 = 1+2 = 1+1+1
partcount(4) = 5 ← 4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1
partcount(5) = 7 ← 5 = 1+4 = 2+3 = 1+1+3 = 1+2+2 = 1+1+1+2 = 1+1+1+1+1
partcount(6) = 11 ← 6 = 1+5 = 2+4 = 3+3 = 1+1+4 = 1+2+3 = 2+2+2 = 1+1+1+3 = 1+1+2+2 = 1+1+1+1+2 = 1+1+1+1+1+1
partcount(7) = 15 ← 7 = 1+6 = 2+5 = 3+4 = 1+1+5 = 1+2+4 = 1+3+3 = 2+2+3 = 1+1+1+4 = 1+1+2+3 = 1+2+2+2 = 1+1+1+1+3 = 1+1+1+2+2 = 1+1+1+1+1+2 = 1+1+1+1+1+1+1
partcount(8) = 22 ← 8 = 1+7 = 2+6 = 3+5 = 4+4 = 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4 = 2+3+3 = 1+1+1+5 = 1+1+2+4 = 1+1+3+3 = 1+2+2+3 = 2+2+2+2 = 1+1+1+1+4 = 1+1+1+2+3 = 1+1+2+2+2 = 1+1+1+1+1+3 = 1+1+1+1+2+2 = 1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525 — the number of partitions of n, A000041 at the Online Encyclopedia of Integer Sequences

And there are fractals — self-similarities at smaller and smaller scales — hidden in that simple arithmetic. Take the partitions of 8:

8 = 1+7 = 2+6 = 3+5 = 4+4 = 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4 = 2+3+3 = 1+1+1+5 = 1+1+2+4 = 1+1+3+3 = 1+2+2+3 = 2+2+2+2 = 1+1+1+1+4 = 1+1+1+2+3 = 1+1+2+2+2 = 1+1+1+1+1+3 = 1+1+1+1+2+2 = 1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1 (c=22)

By definition, the sum of each partition of 8 is the same: 8. But the products — the result of multiplying the numbers of a partition — rise and fall wildly, hitting a maximum of 18 and a minimum of 1:

1.7 = 7
2.6 = 12
3.5 = 15
4.4 = 16
1.1.6 = 6
1.2.5 = 10
1.3.4 = 12
2.2.4 = 16
2.3.3 = 18
1.1.1.5 = 5
1.1.2.4 = 8
1.1.3.3 = 9
1.2.2.3 = 12
2.2.2.2 = 16
1.1.1.1.4 = 4
1.1.1.2.3 = 6
1.1.2.2.2 = 8
1.1.1.1.1.3 = 3
1.1.1.1.2.2 = 4
1.1.1.1.1.1.2 = 2
1.1.1.1.1.1.1.1 = 1

It’s interesting to ask when partitions(n) yield the biggest product (the answer is here). It’s also interesting to create graphs of prod(part(n)), the products of the partitions of n. You’ll see something I call partition fission. The graphs start to fissure into what look like fins or sails, and then each fin or sail starts to fissure too:

Graph for multiples of partitions(8) (partcount(8) = 22)

Graph for prod(part(9)) (partcount = 30)

prod(part(10)) (partcount = 42)

prod(part(11)) (partcount = 56)

prod(part(12)) (partcount = 77)

prod(part(13)) (partcount = 101)

prod(part(14)) (partcount = 135)

prod(part(15)) (partcount = 176)

prod(part(16)) (partcount = 231)

Those graphs are all on the same scale. The two graphs below have been adjusted to capture many more partitions and show the fractality coming into full flower:

prod(part(20)) (partcount = 627)

prod(part(28)) (partcount = 3718)

Finally, here’s an animated gif of the graphs for the partition-products of 8 to 16:

Animated gif for prod(part(8..16)) (animation at EZgif) (click for larger image)

Κοινόκοσμος καὶ Κωματόκοσμοι

by in Ancient Greek, Consciousness, Language, Literature, Philosophy, Quotations and tagged , , , , , , , | Leave a comment

• ὁ Ἡράκλειτός φησι τοῖς ἐγρηγορόσιν ἕνα καὶ κοινὸν κόσμον εἶναι τῶν δὲ κοιμωμένων ἕκαστον εἰς ἴδιον ἀποστρέφεσθαι. — Σέξτος Ἐμπειρικός, Πρὸς μαθηματικούς

• • Heraclitus said that for the waking is one common world, but the sleeping turn aside each into a world of his own. — Sextus Empiricus, fl. 150 A.D., Adversus Mathematicos / Against the Mathematicians (or: Against the Professors), VII. 129

Post-Performative Post-Scriptum

Κοινόκοσμος, Koinokosmos, “common-world”, “shared cosmos” ← κοινός, koinós, “common”, “shared” + κόσμος, kosmos, “world”, “order”, “universe”; καὶ, kai, “and”; Κωματόκοσμοι, Kōmatokosmoi, “sleep-worlds” ← Greek κῶμα, kôma, “deep sleep” + κόσμος, kosmos

The Bird Dimension

by in Art, Geometry, Mathematics, Optical Illusions, Post-Weird, Surrealism and tagged , , , , , , , | 2 Comments

M.C. Escher, Another World / Andere Wereld (1947)

This is almost my favorite image by Escher. But I’d like a frEscher perspective in it: I think the bird should be looking in the other direction, out into the impossibly overlapping universes, not into the cupola and at the viewer.