Digital Disfunction

by in 666, Digit-sums, Mathematics, Narcissistic numbers, Number of the Beast, Powers and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

It’s fun when functions disfunc. The function digit-sum(n^p) takes a number, raises it to the power of p and sums its digits. If p = 1, n is unchanged. So digit-sum(1^1) = 1, digit-sum(11^1) = 2, digit-sum(2013^1) = 6. The following numbers set records for the digit-sum(n^1) from 1 to 1,000,000:

digit-sum(n^1): 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 199, 299, 399, 499, 599, 699, 799, 899, 999, 1999, 2999, 3999, 4999, 5999, 6999, 7999, 8999, 9999, 19999, 29999, 39999, 49999, 59999, 69999, 79999, 89999, 99999, 199999, 299999, 399999, 499999, 599999, 699999, 799999, 899999, 999999.

The pattern is easy to predict. But the function disfuncs when p = 2. Digit-sum(3^2) = 9, which is more than digit-sum(4^2) = 1 + 6 = 7 and digit-sum(5^2) = 2 + 5 = 7. These are the records from 1 to 1,000,000:

digit-sum(n^2): 1, 2, 3, 7, 13, 17, 43, 63, 83, 167, 264, 313, 707, 836, 1667, 2236, 3114, 4472, 6833, 8167, 8937, 16667, 21886, 29614, 32617, 37387, 39417, 42391, 44417, 60663, 63228, 89437, 141063, 221333, 659386, 791833, 976063, 987917.

Higher powers are similarly disfunctional:

digit-sum(n^3): 1, 2, 3, 4, 9, 13, 19, 53, 66, 76, 92, 132, 157, 353, 423, 559, 842, 927, 1192, 1966, 4289, 5826, 8782, 10092, 10192, 10275, 10285, 10593, 11548, 11595, 12383, 15599, 22893, 31679, 31862, 32129, 63927, 306842, 308113.

digit-sum(n^4): 1, 2, 3, 4, 6, 8, 13, 16, 18, 23, 26, 47, 66, 74, 118, 256, 268, 292, 308, 518, 659, 1434, 1558, 1768, 2104, 2868, 5396, 5722, 5759, 6381, 10106, 12406, 14482, 18792, 32536, 32776, 37781, 37842, 47042, 51376, 52536, 84632, 255948, 341156, 362358, 540518, 582477.

digit-sum(n^5): 1, 2, 3, 5, 6, 14, 15, 18, 37, 58, 78, 93, 118, 131, 139, 156, 179, 345, 368, 549, 756, 1355, 1379, 2139, 2759, 2779, 3965, 4119, 4189, 4476, 4956, 7348, 7989, 8769, 9746, 10566, 19199, 19799, 24748, 31696, 33208, 51856, 207198, 235846, 252699, 266989, 549248, 602555, 809097, 814308, 897778.

You can also look for narcissistic numbers with this function, like digit-sum(9^2) = 8 + 1 = 9 and digit-sum(8^3) = 5 + 1 + 2 = 8. 9^2 is the only narcissistic square in base ten, but 8^3 has these companions:

17^3 = 4913 → 4 + 9 + 1 + 3 = 17
18^3 = 5832 → 5 + 8 + 3 + 2 = 18
26^3 = 17576 → 1 + 7 + 5 + 7 + 6 = 26
27^3 = 19683 → 1 + 9 + 6 + 8 + 3 = 27

Twelfth powers are as unproductive as squares:

108^12 = 2518170116818978404827136 → 2 + 5 + 1 + 8 + 1 + 7 + 0 + 1 + 1 + 6 + 8 + 1 + 8 + 9 + 7 + 8 + 4 + 0 + 4 + 8 + 2 + 7 + 1 + 3 + 6 = 108

But thirteenth powers are fertile:

20 = digit-sum(20^13)
40 = digit-sum(40^13)
86 = digit-sum(86^13)
103 = digit-sum(103^13)
104 = digit-sum(104^13)
106 = digit-sum(106^13)
107 = digit-sum(107^13)
126 = digit-sum(126^13)
134 = digit-sum(134^13)
135 = digit-sum(135^13)
146 = digit-sum(146^13)

There are also numbers that are narcissistic with different powers, like 90:

90^19 = 1·350851717672992089 x 10^37 → 1 + 3 + 5 + 0 + 8 + 5 + 1 + 7 + 1 + 7 + 6 + 7 + 2 + 9 + 9 + 2 + 0 + 8 + 9 = 90
90^20 = 1·2157665459056928801 x 10^39 → 1 + 2 + 1 + 5 + 7 + 6 + 6 + 5 + 4 + 5 + 9 + 0 + 5 + 6 + 9 + 2 + 8 + 8 + 0 + 1 = 90
90^21 = 1·09418989131512359209 x 10^41 → 1 + 0 + 9 + 4 + 1 + 8 + 9 + 8 + 9 + 1 + 3 + 1 + 5 + 1 + 2 + 3 + 5 + 9 + 2 + 0 + 9 = 90
90^22 = 9·84770902183611232881 x 10^42 → 9 + 8 + 4 + 7 + 7 + 0 + 9 + 0 + 2 + 1 + 8 + 3 + 6 + 1 + 1 + 2 + 3 + 2 + 8 + 8 + 1 = 90
90^28 = 5·23347633027360537213511521 x 10^54 → 5 + 2 + 3 + 3 + 4 + 7 + 6 + 3 + 3 + 0 + 2 + 7 + 3 + 6 + 0 + 5 + 3 + 7 + 2 + 1 + 3 + 5 + 1 + 1 + 5 + 2 + 1 = 90

One of the world’s most famous numbers is also multi-narcissistic:

666 = digit-sum(666^47)
666 = digit-sum(666^51)

1423 isn’t multi-narcissistic, but I like the way it’s a prime that’s equal to the sum of the digits of its power to 101, which is also a prime:

1423^101 = 2,
976,424,759,070,864,888,448,625,568,610,774,713,351,233,339,
006,775,775,271,720,934,730,013,444,193,709,672,452,482,197,
898,160,621,507,330,824,007,863,598,230,100,270,989,373,401,
979,514,790,363,102,835,678,646,537,123,754,219,728,748,171,
764,802,617,086,504,534,229,621,770,717,299,909,463,416,760,
781,260,028,964,295,036,668,773,707,186,491,056,375,768,526,
306,341,717,666,810,190,220,650,285,746,057,099,312,179,689,
423 →

2 + 9 + 7 + 6 + 4 + 2 + 4 + 7 + 5 + 9 + 0 + 7 + 0 + 8 + 6 + 4 + 8 + 8 + 8 + 4 + 4 + 8 + 6 + 2 + 5 + 5 + 6 + 8 + 6 + 1 + 0 + 7 + 7 + 4 + 7 + 1 + 3 + 3 + 5 + 1 + 2 + 3 + 3 + 3 + 3 + 9 + 0 + 0 + 6 + 7 + 7 + 5 + 7 + 7 + 5 + 2 + 7 + 1 + 7 + 2 + 0 + 9 + 3 + 4 + 7 + 3 + 0 + 0 + 1 + 3 + 4 + 4 + 4 + 1 + 9 + 3 + 7 + 0 + 9 + 6 + 7 + 2 + 4 + 5 + 2 + 4 + 8 + 2 + 1 + 9 + 7 + 8 + 9 + 8 + 1 + 6 + 0 + 6 + 2 + 1 + 5 + 0 + 7 + 3 + 3 + 0 + 8 + 2 + 4 + 0 + 0 + 7 + 8 + 6 + 3 + 5 + 9 + 8 + 2 + 3 + 0 + 1 + 0 + 0 + 2 + 7 + 0 + 9 + 8 + 9 + 3 + 7 + 3 + 4 + 0 + 1 + 9 + 7 + 9 + 5 + 1 + 4 + 7 + 9 + 0 + 3 + 6 + 3 + 1 + 0 + 2 + 8 + 3 + 5 + 6 + 7 + 8 + 6 + 4 + 6 + 5 + 3 + 7 + 1 + 2 + 3 + 7 + 5 + 4 + 2 + 1 + 9 + 7 + 2 + 8 + 7 + 4 + 8 + 1 + 7 + 1 + 7 + 6 + 4 + 8 + 0 + 2 + 6 + 1 + 7 + 0 + 8 + 6 + 5 + 0 + 4 + 5 + 3 + 4 + 2 + 2 + 9 + 6 + 2 + 1 + 7 + 7 + 0 + 7 + 1 + 7 + 2 + 9 + 9 + 9 + 0 + 9 + 4 + 6 + 3 + 4 + 1 + 6 + 7 + 6 + 0 + 7 + 8 + 1 + 2 + 6 + 0 + 0 + 2 + 8 + 9 + 6 + 4 + 2 + 9 + 5 + 0 + 3 + 6 + 6 + 6 + 8 + 7 + 7 + 3 + 7 + 0 + 7 + 1 + 8 + 6 + 4 + 9 + 1 + 0 + 5 + 6 + 3 + 7 + 5 + 7 + 6 + 8 + 5 + 2 + 6 + 3 + 0 + 6 + 3 + 4 + 1 + 7 + 1 + 7 + 6 + 6 + 6 + 8 + 1 + 0 + 1 + 9 + 0 + 2 + 2 + 0 + 6 + 5 + 0 + 2 + 8 + 5 + 7 + 4 + 6 + 0 + 5 + 7 + 0 + 9 + 9 + 3 + 1 + 2 + 1 + 7 + 9 + 6 + 8 + 9 + 4 + 2 + 3 = 1423

Previously pre-posted (please peruse):

• The Hill to Power
• Narcissarithmetic #1
• Narcissarithmetic #2

Voc and Rôle

by in History, Music, Music reviews and tagged , , , , , , , , , , , , , , , | Leave a comment

Medieval music by Vox Vulgaris and Trouvère

At one time, people could never hear their own voices the way others heard them, because our own voices come to us partly through the flesh and bone of our skulls. Then phonographs and tape-recorders were invented and nowadays we all know what we really sound like. But what does the medieval music of groups like Vox Vulgaris and Trouvere really sound like? It comes to us through the flesh and bone of history and we listen to it with uninnocent ears, soaked in a hundred different genres. Medieval music doesn’t stand alone any more, it stands in contrast: acoustic, not amplified; simple, not complex; authentic, not artificial.

Or is it authentic? No, because it’s not the music that comes most readily to hand or to ear any more. Playing it and listening to it are roles you choose, not roles you’re born into, because it’s part of a cultic fringe nowadays. Ferns once ruled the forests; now they’re pushed to the damp or rocky margins by more advanced plants. So this is ferny music: fresh, green and simple, with a glamour of exile and overthrow. You can hear that glamour more strongly in Trouvere, who play slow, sad and sometimes stately music that seems both to celebrate and to lament the Middle Ages. Vox Vulgaris, which literally means “Popular Voice”, celebrate and don’t lament: they’re raucous and almost rocking and sound like a group for inns and peasants’ weddings, not for courts and cathedrals. They’re fun, not bittersweet like Trouvere, who remind me of the Early Music Consort of London. But, like Vox Vulgaris, Trouvere play instrumentals and don’t add lost languages to their lost music.

The music is enough, but they’re surely playing it with a modern accent that would raise smiles or laughter in a real medieval audience. We can’t go back and that is part of why their music is so attractive. It lilts, it longs and it laments, searching for something it will never find. And that is another way Trouvere evoke the Middle Ages:

La royne Blanche comme ung lys,
Qui chantoit à voix de sereine;
Berthe au grand pied, Bietris, Allys;
Harembourges, qui tint le Mayne,
Et Jehanne, la bonne Lorraine,
Qu’Anglois bruslèrent à Rouen;
Où sont-ilz, Vierge souveraine?…
Mais où sont les neiges d’antan!

“Ballade des dames du temps jadis”, François Villon (1431-c.1485)

White Queen Blanche, like a queen of lilies,
With a voice like any mermaiden,—
Bertha Broadfoot, Beatrice, Alice,
And Ermengarde the lady of Maine,—
And that good Joan whom Englishmen
At Rouen doomed and burned her there,—
Mother of God, where are they then?…
But where are the snows of yester−year.

Translation by Rossetti.

Go with the Floe

by in Fractals, Geometry, Mathematics and tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Fractals are shapes that contain copies of themselves on smaller and smaller scales. There are many of them in nature: ferns, trees, frost-flowers, ice-floes, clouds and lungs, for example. Fractals are also easy to create on a computer, because you all need do is take a single rule and repeat it at smaller and smaller scales. One of the simplest fractals follows this rule:

1. Take a line of length l and find the midpoint.
2. Erect a new line of length l x lm on the midpoint at right angles.
3. Repeat with each of the four new lines (i.e., the two halves of the original line and the two sides of the line erected at right angles).

When lm = 1/3, the fractal looks like this:

stick1

(Please open image in a new window if it fails to animate)

When lm = 1/2, the fractal is less interesting:

stick2

But you can adjust rule 2 like this:

2. Erect a new line of length l x lm x lm1 on the midpoint at right angles.

When lm1 = 1, 0.99, 0.98, 0.97…, this is what happens:

stick3

The fractals resemble frost-flowers on a windowpane or ice-floes on a bay or lake. You can randomize the adjustments and angles to make the resemblance even stronger:

frostfloe

Ice floes (see Owen Kanzler)

Ice floes (see Owen Kanzler)

Frost on window (see Kenneth G. Libbrecht, )

Frost on window (see Kenneth G. Libbrecht)

Magna Mater Marina

by in Aesthetics, Biology, Book Reviews, Marine Biology, Science, The Sea and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Front cover of The Illustrated World Encyclopedia of Marine Fish and Sea CreaturesThe Illustrated World Encyclopedia of Marine Fish and Sea Creatures, Amy-Jane Beer and Derek Hall (Lorenz Books, 2007)

Books about marine life need to be big, like this one, because the sea is a big place and has been occupied for far longer than the land. You’ll learn here that some land creatures have even returned to it, like the ancestors of cetaceans (whales et al), sirenians (dugongs and manatees), and sea-snakes. The saltiness of human blood means that we each carry around a miniature ocean of our own, symbol of our own marine ancestry. The Illustrated World Encyclopedia of Marine Fish and Sea Creatures is an excellent guide to the remainers and the returners of our ancient home. It isn’t a proper scientific encyclopedia, but you can get a good sense of the richness and variety of marine life here, from jellyfish to electric rays by way of the deep-water sea-cucumber, Irpa abyssicola, and the very strange tripod fish, Bathypterois grallator.

Bathypterois grallator

The tripod fish, Bathypterois grallator

That scientific name literally means “the deep-wing stilt-walker”, because the tripod fish lives very deep, up to 3·5km down, and props itself up on extended fin-rays to save energy on swimming. Its tiny prey float towards to it on the current: it isn’t an active hunter. It’s also hermaphroditic, so that each fish can fertilize its own eggs if, thanks to depth and darkness, it doesn’t find a mate. Unlike many other deep-sea fish, however, it isn’t particularly ugly or grotesque and wouldn’t easily find place in an H.P. Lovecraft story. Vampyroteuthis infernalis, or “the vampire squid from hell”, definitely would. It looks rather like an animated umbrella, with dark webs between its tentacles and huge, light-thirsty eyes.

Sea anemones by Ernst Haeckel

Sea anemones by Ernst Haeckel

Elsewhere there’s proof that the sea contains not just abysmal ugliness but sublime beauty too, from cone shells (Conus spp.) and jewel “anemones” (Corynactis viridis) (really a form of coral, the book notes) to gorgeous fish like the copperband butterflyfish (Chelmon rostratus) and the Moorish idol (Zanclus cornutus). And the greater blue-ringed octopus (Hapalochlaena lunulata) is beautiful too, despite the “toxin in its saliva estimated to be 10,000 times more deadly than cyanide”. There isn’t enough here about plankton, which can be strange, ugly, and beautiful, but plankton could fill several encyclopedias, and this one does incorporate some more recent scientific discoveries, including the marine life that doesn’t depend ultimately on sunlight, however deep down dark it lives. The giant beardworm, Riftia pachyptila, lives in symbiosis with sulphide-digesting bacteria at hydrothermal vents on the ocean floor. It’s not part of the sun-chain and might have homologues beneath the ice-cap of Jupiter’s moon Europa. Life needs liquid, so far as we can see, and certainly on earth it had to get its start there. This book is an excellent introduction to the great biological cradle that is the sea and would be an ideal gift for a budding marine biologist or scientifically inclined sailer or fisherman.

Elsewhere other-posted:

• Guise and Molls — review of Octopus: The Ocean’s Intelligent Invertebrate (2010)
• Mental Marine Music — the band who supplied the title of this review

A Diet of Quiet

by in Music and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

The astonishment one feels on learning of the success of the movement takes second place only to the astonishment that this success has been achieved almost without notice from the mainstream and the underground press alike. Nation-wide weeklies like the NME [New Musical Express] haven’t condescended as yet to feature a single band even vaguely affiliated to the movement, and the fanzines that spring up like mushrooms around any spasmodic “new genre” emptying of popular music’s overclogged bowels have found little nourishment in the only truly fresh development in the field since the earliest days of rap. So far only one or two of the bands on the very fringes of the movement have received any serious attention. Yet Dwaal, who by their own admission are one of the “baby” acts in Quiet, have sold nearly 20,000 of their debut album Slow Hearts in the eight months since its release; the really big boys like Morpheus and Dark could expect to shift this amount in the first week of release of their next albums. Tickets for tours by either of the last mentioned have been known to sell out within an hour of going on sale, and rumours of an imminent ocean-hop by Murmur, the Chicago band seen by many as the kings of Quiet, had the telephones at the offices of London’s GigWise booking agency tied up solid for nearly two days.

So, how has Quiet not merely got off the ground but got well on the way to the outer reaches of the Solar System without collecting a write-up or a review worthy of the name? Why do the names of the most important bands in the movement still mean nothing to the majority of music fans? Well, for a start, Quiet — in every sense — lives up to the name bestowed on by Hank O’Dowell, Murmur’s lead guitarist. To see how this is so, it’s necessary to take a look at its history.

• Continue reading A Diet of Quiet…

Lute to Kill

by in A.E. Housman, Poetry and tagged , , , , , , , , , , , , , , | Leave a comment

A little-known Housman poem that should be better-known:

Breathe, my lute, beneath my fingers
    One regretful breath,
One lament for life that lingers
    Round the doors of death.
For the frost has killed the rose,
And our summer dies in snows,
    And our morning once for all
    Gathers to the evenfall.

Hush, my lute, return to sleeping,
    Sing no songs again.
For the reaper stays his reaping
    On the darkened plain;
And the day has drained its cup,
And the twilight cometh up;
    Song and sorrow all that are
    Slumber at the even-star.

A.E. Housman (1859-1936) — see also Breathe, my lute at Wikilivres.

Performativizing Papyrocentricity #14

by in Art, Über-Transgression..., Book Reviews, Philosophy, Politics, Post-Transgressive Fiction and tagged , , , , , , , , , , , , , , , , | Leave a comment

Papyrocentric Performativity Presents:

Scheming Demon – The Screwtape Letters, C.S. Lewis (1942)

Ai Wei to Hell – How to Read Contemporary Art, Michael Wilson (Thames & Hudson, 2013)

Toxic Twosome – Doll, Peter Sotos and James Havoc (TransVisceral Books, 2013)

Know Your Limaçons – The Penguin Dictionary of Curious and Interesting Geometry, David Wells (1991) (posted @ Overlord of the Über-Feral)

Pestilent, Pustulent and Pox-Pocked – various books by Dr Miriam B. Stimbers (@ O.o.t.Ü.-F.)

Or Read a Review at Random: RaRaR

The Brain in Pain

by in Free Will, Mathematics, Philosophy, Probability, Psychology, Randomness, Theology and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

You can stop reading now, if you want. Or can you? Are your decisions really your own, or are you and all other human beings merely spectators in the mind-arena, observing but neither influencing nor initiating what goes on there? Are all your apparent choices in your brain, but out of your hands, made by mechanisms beyond, or below, your conscious control?

In short, do you have free will? This is a big topic – one of the biggest. For me, the three most interesting things in the world are the Problem of Consciousness, the Problem of Existence and the Question of Free Will. I call consciousness and existence problems because I think they’re real. They’re actually there to be investigated and explained. I call free will a question because I don’t think it’s real. I don’t believe that human beings can choose freely or that any possible being, natural or supernatural, can do so. And I don’t believe we truly want free will: it’s an excuse for other things and something we gladly reject in certain circumstances.

Continue reading The Brain in Pain…