Stand and see, some sun o’clock,
Wet flame devour the headland rock.
Author Archives: Krilling for Company
Breeding Bunnies
The Golden Ratio: The Story of Phi, the Extraordinary Number of Nature, Art and Beauty, Mario Livio (Headline Review 2003)
A good short popular guide to perhaps the most interesting, and certainly the most irrational, of all numbers: the golden ratio or phi (φ), which is approximately equal to 1·6180339887498948482… Prominent in mathematics since at least the ancient Greeks and Euclid, phi is found in many places in nature too, from pineapples and sunflowers to the flight of hawks. Livio catalogues its appearances in both maths and nature, looking closely at the Fibonacci sequence and rabbit-breeding, before going on to debunk mistaken claims that phi also appears a lot in art, music and poetry. Dalí certainly used it, but da Vinci, Debussy and Virgil almost certainly didn’t. Nor, almost certainly, did the builders of the Parthenon and pyramids. Finally, he examines what has famously been called (by the physicist Eugene Wiegner) the unreasonable effectiveness of mathematics: why is this human invention so good at describing the behaviour of the Universe? Livio quotes one of the best short answers I’ve seen:
Human logic was forced on us by the physical world and is therefore consistent with it. Mathematics derives from logic. That is why mathematics is consistent with the physical world. (ch. 9, “Is God a mathematician?”, pg. 252)
It’s not hard to recommend a book that quotes everyone from Johannes Kepler and William Blake to Lewis Carroll, Christopher Marlowe and Jef Raskin, “the creator of the Macintosh computer”, whose answer is given above. Recreational mathematicians should also find lots of ideas for further investigation, from fractal strings to the fascinating number patterns governed by Benford’s law. It isn’t just human beings who look after number one: as a leading figure, 1 turns up much more often in data from the real world, and in mathematical constructs like the Fibonacci sequence, than intuition would lead you to expect. If you’d like to learn more about that and about many other aspects of mathematics, hunt down a copy of this book.
Elsewhere other-posted:
• Roses Are Golden – φ and floral homicide
DeVil to Power
666 is the Number of the Beast described in the Book of Revelation:
13:18 Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.
But 666 is not just diabolic: it’s narcissistic too. That is, it mirrors itself using arithmetic, like this:
666^47 =
5,049,969,684,420,796,753,173,148,798,405,
564,772,941,516,295,265,408,188,117,632,
668,936,540,446,616,033,068,653,028,889,
892,718,859,670,297,563,286,219,594,665,
904,733,945,856 → 5 + 0 + 4 + 9 + 9 + 6 + 9 + 6 + 8 + 4 + 4 + 2 + 0 + 7 + 9 + 6 + 7 + 5 + 3 + 1 + 7 + 3 + 1 + 4 + 8 + 7 + 9 + 8 + 4 + 0 + 5 + 5 + 6 + 4 + 7 + 7 + 2 + 9 + 4 + 1 + 5 + 1 + 6 + 2 + 9 + 5 + 2 + 6 + 5 + 4 + 0 + 8 + 1 + 8 + 8 + 1 + 1 + 7 + 6 + 3 + 2 + 6 + 6 + 8 + 9 + 3 + 6 + 5 + 4 + 0 + 4 + 4 + 6 + 6 + 1 + 6 + 0 + 3 + 3 + 0 + 6 + 8 + 6 + 5 + 3 + 0 + 2 + 8 + 8 + 8 + 9 + 8 + 9 + 2 + 7 + 1 + 8 + 8 + 5 + 9 + 6 + 7 + 0 + 2 + 9 + 7 + 5 + 6 + 3 + 2 + 8 + 6 + 2 + 1 + 9 + 5 + 9 + 4 + 6 + 6 + 5 + 9 + 0 + 4 + 7 + 3 + 3 + 9 + 4 + 5 + 8 + 5 + 6 = 666
666^51 =
993,540,757,591,385,940,334,263,511,341,
295,980,723,858,637,469,431,008,997,120,
691,313,460,713,282,967,582,530,234,558,
214,918,480,960,748,972,838,900,637,634,
215,694,097,683,599,029,436,416 → 9 + 9 + 3 + 5 + 4 + 0 + 7 + 5 + 7 + 5 + 9 + 1 + 3 + 8 + 5 + 9 + 4 + 0 + 3 + 3 + 4 + 2 + 6 + 3 + 5 + 1 + 1 + 3 + 4 + 1 + 2 + 9 + 5 + 9 + 8 + 0 + 7 + 2 + 3 + 8 + 5 + 8 + 6 + 3 + 7 + 4 + 6 + 9 + 4 + 3 + 1 + 0 + 0 + 8 + 9 + 9 + 7 + 1 + 2 + 0 + 6 + 9 + 1 + 3 + 1 + 3 + 4 + 6 + 0 + 7 + 1 + 3 + 2 + 8 + 2 + 9 + 6 + 7 + 5 + 8 + 2 + 5 + 3 + 0 + 2 + 3 + 4 + 5 + 5 + 8 + 2 + 1 + 4 + 9 + 1 + 8 + 4 + 8 + 0 + 9 + 6 + 0 + 7 + 4 + 8 + 9 + 7 + 2 + 8 + 3 + 8 + 9 + 0 + 0 + 6 + 3 + 7 + 6 + 3 + 4 + 2 + 1 + 5 + 6 + 9 + 4 + 0 + 9 + 7 + 6 + 8 + 3 + 5 + 9 + 9 + 0 + 2 + 9 + 4 + 3 + 6 + 4 + 1 + 6 = 666
But those are tiny numbers compared to 6^(6^6). That means 6^46,656 and equals roughly 2·6591… x 10^36,305. It’s 36,306 digits long and its full digit-sum is 162,828. However, 666 lies concealed in those digits too. To see how, consider the function Σ(x1,xn), which returns the sum of digits 1 to n of x. For example, π = 3·14159265…, so Σ(π1,π4) = 3 + 1 + 4 + 1 = 9. The first 150 digits of 6^(6^6) are these:
26591197721532267796824894043879185949053422002699
24300660432789497073559873882909121342292906175583
03244068282650672342560163577559027938964261261109
… (150 digits)
If x = 6^(6^6), then Σ(x1,x146) = 666, Σ(x2,x148) = 666, and Σ(x2,x149) = 666.
There’s nothing special about these patterns: infinitely many numbers are narcissistic in similar ways. However, 666 has a special cultural significance, so people pay it more attention and look for patterns related to it more carefully. Who cares, for example, that 667 = digit-sum(667^48) = digit-sum(667^54) = digit-sum(667^58)? Fans of recreational maths will, but not very much. The Number of the Beast is much more fun, narcissistically and otherwise:
666 = digit-sum(6^194)
666 = digit-sum(6^197)
666 = digit-sum(111^73)
666 = digit-sum(111^80)
666 = digit-sum(222^63)
666 = digit-sum(222^66)
666 = digit-sum(333^58)
666 = digit-sum(444^53)
666 = digit-sum(777^49)
666 = digit-sum(999^49)
Previously pre-posted (please peruse):
• More Narcissisum
• Digital Disfunction
• The Hill to Power
• Narcissarithmetic #1
• Narcissarithmetic #2
The Ess in Lancashire
More Musings on Music
• Black Mother Nirvana — from the potency of Purushmedh to the blast-beat barrage of Bodhisattva
• Groaning and Grieving — the ashen passion of Slough of Despond
• Pummelling Putridity — probing the purulence and putrefaction of Paraphistomiasis
Elsewhere other-posted:
• More Musings on Music
Brit Bot Book
Reader’s Digest Field Guide to the Wild Flowers of Britain, J.R. Press et al, illustrated Leonora Box et al (1981)
This is probably the best introduction to British wild flowers that I’ve seen: drawings, photographs and text complement each other perfectly over more than four hundred pages. Despite being compact, it’s a little heavy to be a good field guide, but it would be useful in every British field, wasteland and marsh. From Indian balsam (Impatiens glandulifera) to flowering-rush (Butomus umbellatus) by way of green alkanet (Pentaglottis sempervirens), it’s got a lot, if not the lot (no Mycelis muralis, or wall lettuce, for example). The drawings are skilful, detailed, and often show the plant growing with different species in its habitat, which prepares the eye for identifying it in situ. The drawings also often have the adventitious additions that make David N. Pegler’s Pocket Guide to Mushrooms and Toadstools more enjoyable too, like the half-brick with Canadian fleabane (Conyza canadensis), the chewing-gum wrapper with sea mayweed (Matricaria maritima) and the frog with water violet (Hottonia palustris).
The drawings dominate the page devoted to each plant, but there’s a small photograph of a living specimen too, though “small” doesn’t always mean undramatic. Sea thrift (Armeria maritima) is shown growing quietly on a cliff-top with swirling sea and towering rocks beyond and below it. The photo sums up the book: wild flowers are often delicate and unobtrusive, but they illustrate some grand themes of evolution and biology, from ecological webs to mimicry, parasitism and toxicology: dead-nettles (Lamium spp.) mimick nettles, broomrape (Orobanche spp.) parasitizes broom, clover and more, and lots of British plants can kill you, sicken you or drive you insane, from hemlock (Conium maculatum) to henbane (Hyoscyamus niger). The book explores some grand themes of culture too: the text mixes serious botany with folklore, cuisine, herbalism, and literature. Pignuts (Conopodium majus) appear in The Tempest, for example, and in Ireland “were thought to be the food of leprachauns”. The etymologies aren’t always trustworthy — the “-ard” of “mustard” doesn’t mean ardente, “burning” — but that makes the book itself part of folklore and adds to the plants’ appeal. Highly recommended in this first edition.
Performativizing Papyrocentricity #15
Papyrocentric Performativity Presents:
Brought to Book – A Book of English Essays, selected by W.E. Williams (Pelican 1942)
Glamourdämmerung – Treasures of Nirvana, Gillian G. Gaar (Carlton 2011)
Highway to Hell – The Road, Cormac McCarthy (2006)
Solids and Shadows – An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes, Koji Miyazaki (Wiley-Interscience 1987) (posted @ Overlord of the Über-Feral)
Magna Mater Marina – The Illustrated World Encyclopedia of Marine Fish and Sea Creatures, Amy-Jane Beer and Derek Hall (Lorenz Books 2007) (@ O.o.t.Ü.-F.)
Or Read a Review at Random: RaRaR
Solids and Shadows
An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes, Koji Miyazaki (Wiley-Interscience 1987)
Two, three, four – or rather, two, three, ∞. Polygons are closed shapes in two dimensions (e.g., the square), polyhedra closed shapes in three dimensions (the cube), and polytopes closed shapes in four or more (the hypercube). You could spend a lifetime exploring any one of these geometries, but unless you take psychedelic drugs or brain-modification becomes much more advanced, you’ll be able to see only two of them: the geometries of polygons and polyhedra. Polytopes are beyond imagining but you can glimpse their shadows here – literally, because we can represent polytopes by the shadows they cast in 3-space or by the shadows of their shadows in 2-space.
A four-dimensional shape in two dimensions (see Tesseract)
Elsewhere Miyazaki doesn’t have to convey wonder and beauty by shadows: not only is this book full of beautiful shapes, it’s beautifully designed too and the way it alternates black-and-white pages with colour actually increases the power of both. It isn’t restricted to pure mathematics either: Miyazaki also looks at the modern and ancient art and architecture inspired by geometry, and at geometry in nature: the dodecahedral pollen of Gypsophilum elegans (Showy Baby’s-Breath), for example, and the tetrahedral seeds of the Water Chestnut (Trapa spp.), which the Japanese spies and assassins called the ninja used as natural caltrops. A regular tetrahedron always lies on a flat surface with a vertex facing directly upward, and when a pursued ninja scattered the sharply pointed tetrahedral seeds of the Water Chestnut, they were regular enough to injure “the soles of feet of his pursuers”.
Polyhedral plankton by Ernst Haeckel
The slightly odd English there is another example of what I like about this book, because it proves the parochialism of language and the universality of mathematics. Miyazaki’s mathematics, as far as I can tell, is flawless, like that of many other Japanese mathematicians, but his self-translated English occasionally isn’t. Japanese mathematics was highly developed before Japan fell under strong Western influence. It would continue to develop if the West disappeared tomorrow. Language is something we have to absorb intuitively from the particular culture we’re born into, but mathematics is learnt and isn’t tied to any particular culture. That’s why it’s accessible in the same way to minds everywhere in the world. Miyazaki’s pictures and prose are an extended proof of all that, and the book is actually more valuable because it was written by a Japanese speaker. I think it’s probably more attractively designed for the same reason: the skill with which the pictures have been selected and laid out reflects something characteristically Japanese. Elegance and simplicity perhaps sum it up, and elegance and simplicity are central to mathematics and some of the greatest art.
Another four-dimensional shape in two dimensions (see 120-cell)
More Narcissisum
The number 23 is special, inter alia, because it’s prime, divisible by only itself and 1. It’s also special because its reciprocal has maximum period. That is, the digits of 1/23 come in repeated blocks of 22, like this:
1/23 = 0·0434782608695652173913 0434782608695652173913 0434782608695652173913…
But 1/23 fails to be special in another way: you can’t sum its digits and get 23:
0 + 4 + 3 + 4 + 7 = 18
0 + 4 + 3 + 4 + 7 + 8 = 26
0 + 4 + 3 + 4 + 7 + 8 + 2 + 6 + 0 + 8 + 6 + 9 + 5 + 6 + 5 + 2 + 1 + 7 + 3 + 9 + 1 + 3 = 99
1/7 is different:
1/7 = 0·142857… → 1 + 4 + 2 = 7
This means that 7 is narcissistic: it reflects itself by manipulation of the digits of 1/7. But that’s in base ten. If you try base eight, 23 becomes narcissistic too (note that 23 = 2 x 8 + 7, so 23 in base eight is 27):
1/27 = 0·02620544131… → 0 + 2 + 6 + 2 + 0 + 5 + 4 + 4 = 27 (base=8)
Here are more narcissistic reciprocals in base ten:
1/3 = 0·3… → 3 = 3
1/7 = 0·142857… → 1 + 4 + 2 = 7
1/8 = 0·125 → 1 + 2 + 5 = 8
1/13 = 0·076923… → 0 + 7 + 6 = 13
1/14 = 0·0714285… → 0 + 7 + 1 + 4 + 2 = 14
1/34 = 0·02941176470588235… → 0 + 2 + 9 + 4 + 1 + 1 + 7 + 6 + 4 = 34
1/43 = 0·023255813953488372093… → 0 + 2 + 3 + 2 + 5 + 5 + 8 + 1 + 3 + 9 + 5 = 43
1/49 = 0·020408163265306122448979591836734693877551… → 0 + 2 + 0 + 4 + 0 + 8 + 1 + 6 + 3 + 2 + 6 + 5 + 3 + 0 + 6 + 1 + 2 = 49
1/51 = 0·0196078431372549… → 0 + 1 + 9 + 6 + 0 + 7 + 8 + 4 + 3 + 1 + 3 + 7 + 2 = 51
1/76 = 0·01315789473684210526… → 0 + 1 + 3 + 1 + 5 + 7 + 8 + 9 + 4 + 7 + 3 + 6 + 8 + 4 + 2 + 1 + 0 + 5 + 2 = 76
1/83 = 0·01204819277108433734939759036144578313253… → 0 + 1 + 2 + 0 + 4 + 8 + 1 + 9 + 2 + 7 + 7 + 1 + 0 + 8 + 4 + 3 + 3 + 7 + 3 + 4 + 9 = 83
1/92 = 0·010869565217391304347826… → 0 + 1 + 0 + 8 + 6 + 9 + 5 + 6 + 5 + 2 + 1 + 7 + 3 + 9 + 1 + 3 + 0 + 4 + 3 + 4 + 7 + 8 = 92
1/94 = 0·01063829787234042553191489361702127659574468085… → 0 + 1 + 0 + 6 + 3 + 8 + 2 + 9 + 7 + 8 + 7 + 2 + 3 + 4 + 0 + 4 + 2 + 5 + 5 + 3 + 1 + 9 + 1 + 4 = 94
1/98 = 0·0102040816326530612244897959183673469387755… → 0 + 1 + 0 + 2 + 0 + 4 + 0 + 8 + 1 + 6 + 3 + 2 + 6 + 5 + 3 + 0 + 6 + 1 + 2 + 2 + 4 + 4 + 8 + 9 + 7 + 9 + 5 = 98
Previously pre-posted (please peruse):
• Digital Disfunction
• The Hill to Power
• Narcissarithmetic #1
• Narcissarithmetic #2
Court in the Act
Bombshell: The Hits and More, The Primitives (1994)
In all walks of life, from pop music to drug-dealing, some people achieve far more success than their talents deserve and some people achieve far less. Paul Court, the song-writer for the late-’eighties-and-a-bit-of-the-’nineties indie group The Primitives, is one of the second group. And perhaps drug-dealing describes his largely unrewarded talents too. Like a drug, music is designed to alter your consciousness and some of the songs on this compilation album are perfect little pills of pop, filling your brain with a two- or three-minute rush of jingly-jangly melodic pleasure. And maybe jungly pleasure too: The Primitives were a primitive band in the garage-and-bubblegum-pop tradition, particularly when they played live. Female vox, occasional male backing vocals, guitar, bass and drums, and that was it. There was no pretension about them, but they achieved the kind of a-lot-in-a-little simplicity that only an intelligent and skilful songwriter can give a band.
“Crash”, their most famous song, both opens and closes the album (apart from the doubly unexpected hidden track). It appears first as the album track, then as a demo, and some of the other songs come in a second version, whether demo or acoustic. I enjoy the chance to hear the different interpretations, but this padding does reflect the brevity of their career, which stretched from about 1987 to about 1992. Unfortunately, a twice-misspelt “Way Behing Me” and the appearance of “Secrets (Demo)” as the already-heard album track rather than the demo also reflect the sloppiness of the German company that put the compilation out. Court deserved better. There’s further proof of that in the single cover version, “As Tears Go By” by the Rolling Stones. It’s given the light treatment of the early Primitives and isn’t anywhere near as good as Court’s own compositions, I’d say.
Perhaps that’s why he chose it, and perhaps the darker songs on their final album, “Glamour”, reflect his frustration at not achieving the success that seemed to await him in the beginning. But there was a big obstacle ahead of him: although bands with attractive female singers can get attention more easily, they find it harder to get taken seriously. The Primitives never did drop any bombshells in the end and I suspect that the title of this compilation is a self-ironizing acknowledgment of that, as well as a reference to Tracy’s gleaming blonde locks.
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 
