Watch this Sbase

In standard notation, there are two ways to represent 2: 10, in base 2, and 2 in every other base. Accordingly, there are three ways to represent 3: 11 in base 2, 10 in base 3, and 3 in every other base. There are four ways to represent 4, five ways to represent 5, and so on. Now, suppose you sum all the digits of all the representations of n in the bases 2 to n, like this:

Σ(2) = 1+02 = 1
Σ(3) = 1+12 + 1+03 = 3 (+2)
Σ(4) = 1+0+02 + 1+13 + 1+04 = 4 (+1)
Σ(5) = 1+0+12 + 1+23 + 1+14 + 1+05 = 8 (+4)
Σ(6) = 1+1+02 + 2+03 + 1+24 + 1+15 + 1+06 = 10 (+2)
Σ(7) = 1+1+12 + 2+13 + 1+34 + 1+25 + 1+16 + 1+07 = 16 (+6)
Σ(8) = 1+0+0+02 + 2+23 + 2+04 + 1+35 + 1+26 + 1+17 + 1+08 = 17 (+1)
Σ(9) = 1+0+0+12 + 1+0+03 + 2+14 + 1+45 + 1+36 + 1+27 + 1+18 + 1+09 = 21 (+4)
Σ(10) = 1+0+1+02 + 1+0+13 + 2+24 + 2+05 + 1+46 + 1+37 + 1+28 + 1+19 + 1+010 = 25 (+4)

It seems reasonable to suppose that as n increases, so the all-digit-sum of n increases. But that isn’t always the case: occasionally it decreases. Here are the sums for n=11..100 (with prime factors when the sum is composite):

Σ(11) = 35 = 5·7 (+10)
Σ(12) = 34 = 2·17 (-1)
Σ(13) = 46 = 2·23 (+12)
Σ(14) = 52 = 22·13 (+6)
Σ(15) = 60 = 22·3·5 (+8)
Σ(16) = 58 = 2·29 (-2)
Σ(17) = 74 = 2·37 (+16)
Σ(18) = 73 (-1)
Σ(19) = 91 = 7·13 (+18)
Σ(20) = 92 = 22·23 (+1)
Σ(21) = 104 = 23·13 (+12)
Σ(22) = 114 = 2·3·19 (+10)
Σ(23) = 136 = 23·17 (+22)
Σ(24) = 128 = 27 (-8)
Σ(25) = 144 = 24·32 (+16)
Σ(26) = 156 = 22·3·13 (+12)
Σ(27) = 168 = 23·3·7 (+12)
Σ(28) = 171 = 32·19 (+3)
Σ(29) = 199 (+28)
Σ(30) = 193 (-6)
Σ(31) = 223 (+30)
Σ(32) = 221 = 13·17 (-2)
Σ(33) = 241 (+20)
Σ(34) = 257 (+16)
Σ(35) = 281 (+24)
Σ(36) = 261 = 32·29 (-20)
Σ(37) = 297 = 33·11 (+36)
Σ(38) = 315 = 32·5·7 (+18)
Σ(39) = 339 = 3·113 (+24)
Σ(40) = 333 = 32·37 (-6)
Σ(41) = 373 (+40)
Σ(42) = 367 (-6)
Σ(43) = 409 (+42)
Σ(44) = 416 = 25·13 (+7)
Σ(45) = 430 = 2·5·43 (+14)
Σ(46) = 452 = 22·113 (+22)
Σ(47) = 498 = 2·3·83 (+46)
Σ(48) = 472 = 23·59 (-26)
Σ(49) = 508 = 22·127 (+36)
Σ(50) = 515 = 5·103 (+7)
Σ(51) = 547 (+32)
Σ(52) = 556 = 22·139 (+9)
Σ(53) = 608 = 25·19 (+52)
Σ(54) = 598 = 2·13·23 (-10)
Σ(55) = 638 = 2·11·29 (+40)
Σ(56) = 634 = 2·317 (-4)
Σ(57) = 670 = 2·5·67 (+36)
Σ(58) = 698 = 2·349 (+28)
Σ(59) = 756 = 22·33·7 (+58)
Σ(60) = 717 = 3·239 (-39)
Σ(61) = 777 = 3·7·37 (+60)
Σ(62) = 807 = 3·269 (+30)
Σ(63) = 831 = 3·277 (+24)
Σ(64) = 819 = 32·7·13 (-12)
Σ(65) = 867 = 3·172 (+48)
Σ(66) = 861 = 3·7·41 (-6)
Σ(67) = 927 = 32·103 (+66)
Σ(68) = 940 = 22·5·47 (+13)
Σ(69) = 984 = 23·3·41 (+44)
Σ(70) = 986 = 2·17·29 (+2)
Σ(71) = 1056 = 25·3·11 (+70)
Σ(72) = 1006 = 2·503 (-50)
Σ(73) = 1078 = 2·72·11 (+72)
Σ(74) = 1114 = 2·557 (+36)
Σ(75) = 1140 = 22·3·5·19 (+26)
Σ(76) = 1155 = 3·5·7·11 (+15)
Σ(77) = 1215 = 35·5 (+60)
Σ(78) = 1209 = 3·13·31 (-6)
Σ(79) = 1287 = 32·11·13 (+78)
Σ(80) = 1263 = 3·421 (-24)
Σ(81) = 1293 = 3·431 (+30)
Σ(82) = 1333 = 31·43 (+40)
Σ(83) = 1415 = 5·283 (+82)
Σ(84) = 1368 = 23·32·19 (-47)
Σ(85) = 1432 = 23·179 (+64)
Σ(86) = 1474 = 2·11·67 (+42)
Σ(87) = 1530 = 2·32·5·17 (+56)
Σ(88) = 1530 = 2·32·5·17 (=)
Σ(89) = 1618 = 2·809 (+88)
Σ(90) = 1572 = 22·3·131 (-46)
Σ(91) = 1644 = 22·3·137 (+72)
Σ(92) = 1663 (+19)
Σ(93) = 1723 (+60)
Σ(94) = 1769 = 29·61 (+46)
Σ(95) = 1841 = 7·263 (+72)
Σ(96) = 1784 = 23·223 (-57)
Σ(97) = 1880 = 23·5·47 (+96)
Σ(98) = 1903 = 11·173 (+23)
Σ(99) = 1947 = 3·11·59 (+44)
Σ(100) = 1923 = 3·641 (-24)

The sum usually increases, occasionally decreases. In one case, when 87 = n = 88, it stays the same. This also happens when 463 = n = 464, where Σ(463) = Σ(464) = 39,375. Does it happen again? I don’t know. The ratio of sum-ups to sum-downs seems to tend towards 3:1. Is that the exact ratio at infinity? I don’t know. Watch this sbase.

Advertisements

The Kisses of Narcissus

…nec corpus remanet, quondam quod amaverat Echo.*

P. Ovidi Nasonis Metamorphoses, Liber Tertius, 493.

He sits to gaze his selfish pool
Whilst I, his fond, besotted fool,
Grow hateful of the air that sips
The sweetness of his yearning lips.
They yearn for him as mine do yearn,
Or sun or stars above me burn.

The kisses of Narcissus I
Shall never taste: ’tis thus I die.
And tho’ ye’ll hear my voice down time
Recall the burden of this rhyme,
Yet know that I am gone and he
Lies loveless where ye too shall be.


*Nor the form remains, belovèd once of Echo.

The Joy of Six

Ken Libbrecht’s Field Guide to Snowflakes, Kenneth Libbrecht (2006)

If you ask someone to draw up a list of inventions that have shaped the modern world, it’s likely that a very important one will get overlooked: the microscope. But where would medicine, science and technology be without it? It opened a door in the cellar of the universe just as the telescope opened a door in the attic. We stepped through each door into a new world of beauty and strangeness. The difference is that, so far, only one of these new worlds has proved to be inhabited: the microscopic one. But snowflakes remind you that the microcosmos can interest physicists and mathematicians as well as biologists and doctors. It should interest artists and aestheticians too: this book will delight the eye as well as challenge and enrich the mind. Snowflakes are among the most beautiful of all natural objects, reminiscent now of ice-stars, now of crystalline trees, now of stained glass, now of surreal swords. Their symmetry is part of their appeal, but I think it’s also important that their symmetry is never complete: snowflakes partake both of mathematical perfection and of biological imperfection. They’re individual not just because no two are identical but because no two arms of a single snowflake are identical either.

Cover of Ken Lebbrecht's Field Guide to Snowflakes

But any two arms can still be very similar, which is an astonishing fact when you consider that they’re laid down blindly by a purely chemico-physical process. How does one arm know what the others are doing? Well, it doesn’t: it’s subject to the same fluctuations of the same environment. As air-currents move a snow-crystal inside a cloud, the temperature and moisture of the air change constantly and so do the crystal’s patterns of growth: “…since no two snowflakes follow exactly the same path, no two are exactly alike.” But snowflakes can be much more unalike than the uninitiated might guess. Some aren’t actually stars, but bullets, needles, columns and “arrowhead twins”. There are also snowflakes with twelve rather than six arms, and snowflakes large enough to have arms on their arms on their arms on their arms. Kenneth Libbrecht, a professor of physics at Caltech, examines all this and more in a book that should appeal to every age, every level of intelligence, and every level of interest in science. You don’t have to read the text that accompanies the beautiful photographs, but if you do you’ll find your appreciation of the photos deepened and enriched.

Either way, spare a thought for the book’s co-author: the humble but hugely powerful microscope. Unlike bacteria and protozoa, snowflakes were known and admired long before the microscope was invented, but they were never known then in their full beauty and wonder. And you don’t have to confine yourself to books like this to experience that for yourself: Libbrecht has a section on “Observing Snowflakes” that demonstrates once again how some of the best things in life are free, or nearly free. For the price of a simple hand-lens you can find new beauty in winter; with a microscope, you can find even more.

Alda News (Dat’s Fit to Print)

Some more reviews of The Eyes, with commentary by the esteemed Espanish exponent of extremissimity:

I wasn’t half as impressed with this short-story collection as I hoped to be. It’s too well-written to be called bad and too disturbing to be called boring, but of all the stories, only “Ikarus” approaches greatness. The rest begin as vague and confusing messes until they reach that certain moment of horror and atrocity that seem to wake the author up; then they abruptly end. I couldn’t dismiss the impression that Jesús Ignacio Aldapuerta himself took no interest whatsoever in anything he wrote here but for those few paragraphs of shocking perversity. It’s not enough to make The Eyes worth reading. Except for “Ikarus.” This story is like all the others until a nameless man crashes a rocket-powered interceptor called a Bachem Ba-349 Natter into a B-17 bomber. From there the story evokes a surreal atmosphere of cosmic horror and unknowableness as the pilot explores the strange bomber, walking its huge cathedral-like fuselage while the airplane “floated kilometers high over a black, unending sea. Far, very far below, almost directly under him, the reflection of an almost full moon lay flat and corroded on smooth water.” Then he finds an alien device torturing a woman to death. If it had all been like this, I would be calling The Eyes brilliant, but none of the other stories reached this level of fascination for me.

Original review

Jesús say: I… S…. R… U… B… B… I…. S… H…. R… E… V….. I… E… W…. | A… L….. L…. S…. T… O… R…. I… E… S….. G… O… O… D… | Y… A… N…. K… I…. C… A… B… R… O…. N…..


This isn’t a bad book, just an exceedingly oversold one. It’s the first and thus far only English-language collection of stories by the late Jesús Ignacio Aldapuerta, the so-called “Andalusian de Sade” who specialized in scatological excess. In truth this book’s gross-out quotient is about equal to that found in the writing of better-known practitioners of Sadean literature like James Havoc and Simon Whitechapel, even if the back-cover description proclaims that “to read all the stories of Aldapuerta’s infamous THE EYES is, perhaps in fact, to become mad, or worse” and that “Once read, they will be with you always.”

If the introduction by Lucía Teodora is to be believed, Jesús Ignacio Aldapuerta was a petty thief obsessed with pornography who immolated himself (or was murdered) in 1987. THE EYES, originally translated into English by Aldapuerta himself in 1986, is representative of his many unsavory obsessions, and possibly of his actual crimes. The prose, alas, is only intermittently effective, which may have something to do with the translation, or simply the fact that Aldapuerta, who died at age thirty-seven, still had a ways to go before fully hitting his stride as a writer.

The eleven stories collected here all pivot on death and perversion, more often than not in the form of sociopathic individuals who happen upon the aftermath of horrific accidents that inflame those individuals’ psychoses. Particularly representative examples include “Ikarus,” about a Nazi pilot who discovers a woman enclosed in some kind of bizarre torture-machine, “Yin & Yang,” in which a man makes weird patterns with the flesh and organs of some frozen corpses he discovers in a crashed plane, and “Orphea,” featuring a nut who fellates himself with a woman’s severed head.

The most effective of THE EYES’ stories, and the only one that really lives up to the grandiose back-cover claims, is the startling and repellent “Armful.” It’s about an incarcerated pervert who literally devours a little girl he (rather improbably) finds locked up with him. The poetic grotesquerie of the tale is very much in the vein of the aforementioned James Havoc, yet with a verve unique to Mr. Aldapuerta, who was a sick fuck without question but also a (somewhat) talented one.

Original review

Jesús say: I… S….. A… L… S… O….. R…. U… B…. B… I….. S… H…. | H… E…. N… O… T….. G… E… T…. J… O… K… E….. | W… A… N…. K… I…. Y… A….. N…. K… I…..


Take De Sade’s “120 Days of Sodom”, add a dash of George Bataille’s “Madame Edwarda”, “Blue of Noon”, “The Dead Man” and garnish with the ‘grand finale’ of André Pieyre de Mandiargues’ “Portrait of an Englishman in his Château” and you have a rough idea of the bloodsport and delights to be found herein. (Don’t forget your “Lobster Bib” and a Big Grin before you dig in!). Positively ‘lip-smacking’!

Original review

Jesús say: I… S…. G… O…. O… D…. R… E… V…. I… W….. | N… I… C…. E…. O… N… E…..

The Call of Cthuneus

Cuneiform, adj. and n. Having the form of a wedge, wedge-shaped. (← Latin cuneus wedge + -form) (Oxford English Dictionary)

This fractal is created by taking an equilateral triangle and finding the centre and the midpoint of each side. Using all these points, plus the three vertices, six new triangles can be created from the original. The process is then repeated with each new triangle (if the images don’t animate, please try opening them in a new window):

triangle_div2

If the centre-point of each triangle is shown, rather than the sides, this is the pattern created:

triangle_div2_dots

Triangles in which the sides are divided into thirds and quarters look like this:

triangle_div3

triangle_div3_dots

triangle_div4

triangle_div4_dots

And if sub-triangles are discarded, more obvious fractals appear, some of which look like Lovecraftian deities and owl- or hawk-gods:


cthuneus1

cthuneus2

cthuneus3

For the Love of Mycology

Mushrooms, Roger Phillips, assisted by Derek Reid, Ronald Rayner, Geoffrey Kibby, and Alick Henrici, designed by Jill Bryan (MacMillan 2006)

In 1981, Roger Phillips began his career in natural history publishing with a book on mushrooms. In 2006, he was back for another bite at the chanterelle. And it would have been a fitting way to end his career, because this is one of the most important books ever published on fungi. It puts its best photo forward for hundreds of pages and hundreds of species, all the way from the massive, like the Giant Puffball, Calvatia gigantea, which can be bigger than a man’s head, to the minute, like the Conifer Disco, Lachnellula subtilissima, which is smaller than a baby’s fingernail. En route, it takes in the gorgeous, the gaudy, and the grotesque, like the Angel’s Wings, Pleurocybella porrigens, the Vermilion Waxcap, Hygrocybe miniata, and the Goliath Webcap, Cortarius praestans. With the g-crew come the delicious, the deadly, and the delicate: the Oyster Mushroom, Pleurotus ostreatus, the Destroying Angel, Amanita virosa, and the Milky Bonnet, Hemimycena lactea. And let’s not forget the phantasmagoric, the phosphorescent, and the phallic: the Devil’s Fingers, Clathrus archeri, the Jack O’ Lantern, Omphalotus illudens, and the Stinkhorn, Phallus impudicus. Which is Latin for “shameless dick”. Fungi can also look like ears, brains, and birds’-nests: the Jelly Ear, Auricularia auricula-judaei, the Morel, Morchella esculenta, and the Common Bird’s Nest, Crucibulum laeve. Oh, and they can look like cages, clubs, and coral too: the Red Cage, Clathrus ruber, the Giant Club, Clavariadelphus pistillarius, and the Violet Coral, Clavaria zollingeri.

And that covers only their appeal, or offence, to the eye and the taste-buds: they can also appeal to, or offend, the nose and fingertips. On olfactory side there are the Coconut Milkcap, Lactarius glyciosmus, the Pear Fibrecap, Inocybe fraudans, the Geranium Brittlegill, Russula fellea, the Mousepee Pinkgill, Entoloma incanum, the Iodine Bolete and Bonnet, Bolitus impolitus and Mycena filopes, and the “Stinking” set: the Brittlegill, Russula foetens, the Dapperling pair Lepiota cristata and L. felina, and the Earthfan, Thelephora palmata. On the tactile side, there are the various Velvets: the Bolete, Suillus variegatus, the Brittlegill, Russula violeipes, the Shank, Flammulina velvutipes, the Shield, Pluteus umbrosus, the Tooth, Hydnellum spongiosipes, and the Toughshank, Kuehneromyces mutabilis. There are too many shaggies, slimies, and slipperies to list, like the Shaggy Parasol, Macrolepiota rhadoces, the Slimy Waxcap, Hygrocybe irrigata, and the Slippery Jack, Suillus luteus. All in all, mushrooms make me muse on Middle-earth. Tolkien’s world is full of richness and variety. So is the world of fungi. The folk and things of Middle-earth can be beautiful or ugly, delicate or sturdy, colourful or drab, tasty or deadly, lovers of light or dwellers in dark. Mushrooms, toadstools, and their smaller relatives are the same. You could find one or more species in this book to match all of Tolkien’s creations: men, wizards, hobbits, elves, dwarves, orcs, trolls, ents, and more. The Cortinarius genus is hobbit-like, for example: stocky, sturdy, and coloured mostly in earthy ochres, yellows, and reds. More elf- and wizard-like are the genera Lepiota and Macrolepiota: these mushrooms are taller and more attractively proportioned. For pre-Tolkienean elves, look to the small and slender Micromphale, Omphalina and Mycena genera, shaped like little umbrellas, bonnets, and parachutes.

For the dark side of Tolkien’s world, look everywhere: almost every group of fungi can supply poisons and sicken or slay the incautious or ignorant. But the deadliest of all are the Amanitas. There’s something suitably and sardonically Sauronic about the modus operandi of the Deathcap, Amanita phalloides:

Poisoning by the Deathcap is characterized by a delay of 6 to 24 hours between ingestion and the onset of symptoms, during which time the cells of the liver and kidney are attacked… The next stage is one of prolonged and violent vomiting and diarrhoea accompanied by severe abdominal pains, lasting for a day or more. Typically this is followed by an apparent recovery, when the victim may be released from hospital or think their ordeal is over, but death results from kidney and liver failure in a few days. (pg. 144-45)

No antidote has yet been discovered to the amatoxins, as the most dangerous compounds are called, and the mortality rate from Amanita poisoning is “still up to 90%”. The Fly Agaric, Amanita muscaria, with its red, white-spotted cap, is the most famous in the genus, but not responsible for the most fatalities. It’s trippily toxic: “a strong hallucinogen and intoxicant, and used as such by the Sami of northern Scandinavia” (pg. 140). Phillips suggests that the Sami began to use A. muscaria by “observing its effects on reindeer”, which “like it so much that all one has to do to round up a wandering herd is to scatter pieces of Fly Agaric on the ground.” Elsewhere in Europe, it was used against flies: the common English name “comes from the practice of breaking the cap into platefuls of milk… to stupefy flies.” Fungi are not plants and form a separate kingdom in biological classification, but they are like plants in the way they can be either delicious, deadly, or dementing.

But if some weren’t so delicious, some others wouldn’t have dealt death so often: the Amanitas are similar in appearance to the Wood mushrooms in the genus Agaricus and can be found in similar places. Agaricus contains some of the most widely eaten of all mushrooms, including the Cultivated Mushroom, A. bisporus, “believed to be the wild form of the many cultivated crop varieties” (pg. 242). But literally cultivated mushrooms don’t compare to wild-grown: I can still remember the richness and flavour of some Field Mushrooms, A. campestris, I picked near the witches’ haunt of Pendle Hill in Lancashire. My other gastro-mycological excursions have included wild-grown puffball and a large Oyster Mushroom that had sprouted from the wood of a sea-side ice-cream stand. It fell off under its own weight, or I wouldn’t have carried it off: Oyster Mushrooms aren’t just good to eat, they’re also good to look at and I would have left it undisturbed otherwise. But picking a mushroom is rather like picking an apple or pear: the visible part is a fruiting body that sprouts from the thread-like hyphae growing in soil, wood, compost, or dung. So you don’t necessarily kill a fungus by picking the part you can see, though you do obviously interfere with its reproduction. The part you can see is what this book is about: unlike David N. Pegler’s Pocket Guide to Mushrooms and Toadstools, there are no drawings of the microscopic spores, merely descriptions: for example, “9-12×5-7μ, elliptical to almond-shaped. Spore-print dark purplish-brown. Chrysocystidia absent. Cheilocystidia lageniform, thin-walled” is in the entry for the Blueleg Brownie, Psilocybe cyanescens.

The fungus itself is described as “hallucinogenic” and “said to be extremely strong” (pg. 253). This book isn’t just for those seeking succulence: it can guide the searcher for synaesthesia too. The Liberty Cap or Magic Mushroom, Psilocybe semilanceata, doesn’t just open the doors of perception: it can throw down the walls of the senses too and make you hear sights or taste colours. The psycho-active psilocybes are all covered and described, but I’ve preferred to leave psycho-mycology alone and get my mental thrills from the look of, and language about, fungi. The scientific names, as always, are interesting, informative, and occasionally uninspired: with a common name like Angel’s Wings, Pleurocybella porrigens has a disappointing scientific name. But there’s a surprisingly complex descriptive vocabulary to learn if you’re interested in acquiring an expertise in these apparently primitive plant-alikes. You’ll even have to dabble in chemistry: the simplest way to distinguish some species is to dip them. The “chrysocystidia” mentioned above are cells “that turn yellowish” – Greek chrysos, “golden”, is hyperbolic – in “alkali solutions”. That’s from the glossary on page 13, but the weird and wonderful words – chlamydospore, dendrophyses, gloeocystidia, lageniform, merulioid, sphaeropedunculate – aren’t illustrated, only defined. This isn’t a textbook of mycology, but an identification guide. And I wouldn’t say it was a work of art like Pegler’s Pocket Guide. It’s well-designed and aesthetically pleasing, but photographs have a superficiality, even a triviality, that Pegler’s drawings don’t. Yes, you can see exactly how the fungi look from a photograph, but there’s no room for the wit and quirkiness I described in my review of the Pocket Guide: the closest you get to the extra-mycological touches I described there is an occasional pine-cone, as in the photos for the Pine Milkcap, Lactarius musteus, the Pinecone Cap, Strobilurus tenacellus, and the Rosy Spike, Gomphidius roseus.

But David Pegler covered far fewer species in a smaller and more subjective book. His science was stronger because he included images of spores, but Roger Phillips has contributed more to mycology, let alone to other fields of natural history. If I had to choose between the two books, I would choose the Pocket Guide, because it’s richer and earthier, and also more minor, in a way that suits its topic better. Fortunately, I don’t have to choose: both books are available for mycophiles and both help explain what is fascinating about fungi. But there are universal aspects to their appeal, beside the particularity of their fungality: maths, the Magistra Mundi, or Mistress of the World, reigns among mushrooms as She reigns everywhere else. Like beetles, though rather more so, fungi are topological variants on a theme: evolution has shaped, squeezed, slendered, squattened, and swollen them over millions of years to produce the huge variety on display in this book. I think architecture can illuminate how they grow: fungi face some of the same problems as architects in erecting and securing their fruiting bodies, but they’re working with less sturdy material. Fungal flesh doesn’t have the toughness and flexibility of wood or the solidity and sturdiness of stone, but it can do surprising things: the Pavement Mushroom, Agaricus bitorquis, is “sometimes found growing through asphalt” (pg. 241).

“Pavement Mushroom”, like “Orange Peel Fungus”, “Purple Stocking Webcap”, “Rooting Poisonpie”, and “Snaketongue Truffleclub”, is one of the odd common names that may catch your eye in the detailed index, which offers specific and generic names, including the outmoded ones that Phillips wanted to update from his early book. But he’s expanded as well as revised, adding some oversea species that “travellers might find on their visits abroad” (introduction, pg. 6). Or might find unexpectedly at home: the Plantpot Dapperling, Leucocoprinus birnbaumii, is a “tropical species that can be found in heated greenhouses” and is shown growing with a potted cactus on page 135. Not illustrated, but mentioned in the entry for the Deathcap, is the “tropical fungus Galerina sulcipes”, which “has a higher α-amanitin content” and is “occasionally found in hothouses” (pg. 145). That would be a sinister note to end on, so instead I’ll end on the Scarlet Elfcup, Sacroscypha austriaca. This is one of my favourite fungi in the book. It is indeed scarlet, it does indeed look like a cup, in the early stages of its growth at least, and its common name is a reminder of why mushrooms are associated with magic and fungi with the fantastic. They can appear very suddenly in unexpected places and have a special association with the melancholy and mystery of autumn. The more elaborate and evolved plant and animal kingdoms are more obvious and found in more places, but they couldn’t exist without fungi, which “break down leaf litter and dead wood and thus ensure that the surface of the world has a fertile layer of soil rather than being a heap of detritus” (pg. 6). In other words: no fungi, no flowers, firs, or figs. In short: no mushrooms, no man. The fungal kingdom isn’t, and can’t be, conscious of the debt owed to it by the other two kingdoms, but this book can be seen as part payment. To see the inhabitants of that mycological Middle-earth in all their variety and strangeness, look no further, because you’ll find no fungaller.

Curiouser and Cuneuser

This fractal is created by taking an equilateral triangle, then finding the three points halfway, i.e. d = 0.5, between the centre of the triangle and the midpoint of each side. Using all these points, plus the three vertices, seven new triangles can be created from the original. The process is then repeated with each new triangle:

7triangle

When sub-triangles are discarded, more obvious fractals appear, including this tristar, again using d = 0.5:

tristar

However, a simpler fractal is actually more fertile. This cat’s-cradle is created when d = 0.5:

catscradle

But as d takes values from 0.5 to 0, a very familiar fractal begins to appear: the Sierpiński triangle:

catscradle_expanding

When the values of d become negative, from -0.1 to -1, this is what happens:

catscradle_expanding_to_cuneus

Pre-previously posted (please peruse):

Curious Cuneus

Beyond Gold: A Weevil

Cover of Living Jewels by Poul Beckmann

Living Jewels: The Natural Design of Beetles, Poul Beckmann (2001)

Richard Dawkins wrote about the Blind Watchmaker, but the Blind Watchmaker often works in collaboration. This book is about his brother, the Blind Jeweller, who creates the cases for the watchwork of beetles. Sometimes those cases are gorgeous, sometimes they’re grotesque, and sometimes they’re both at once. Beetle #77 in this survey, Phanaeus igneus floridanus, is a squat giant with a huge curving horn on its head, but its thorax and abdomen shimmer with metallic purple, green, red, and gold. If that beetle’s a glam-rock sumo-wrestler, then beetle #49, Julodis hiritiventris sanguinipilig (sic – should be hirtiventris sanguinipilis), is pure punk: green legs and a long dark-blue body scattered with tufts of yellow-orange bristles. Elsewhere you’ve got New Romantics with elaborately patterned bodies and sweeping, dandyish antennae (Rosenbergia straussi and Batus barbicornis), death-metal-heads with gleaming black bodies and fearsome-looking but completely harmless horns (Xylotrupes gideon and Allomyr(r?)hina dichotomus taiwana), and even Status-Quo-ites wearing what looks like worn, work-stained denim (various Eupholus species).

It’s entertaining to look through this book and imagine whose backing band or album cover a particular beetle should play in or sit on, but sometimes you won’t be able to match a beetle to a band, because there are more kinds of beetle than musical genres. Beetles, or rather evolution, has invented more than human beings have, but the same forces have been at work. Topologically speaking, a doughnut is identical to a tea-cup, because one is a distorted variant of the other. Similarly, all the beetles in this book are distorted topological variants of each other: like genres of popular music, they’re variants on a theme. Evolution hasn’t altered the ingredients of beetles, just the quantities used to cook each species: changing the width and shape of the thorax, the length and design of the antennae and legs, and so on. But topology isn’t psychology, and just as glam-rock sounds quite different to punk, though the common ancestor is clearly there if you listen, so a doughnut looks quite different to a teacup and Phanaeus igneus floridanus looks quite different to Julodis hirtiventris sanguipilis.

There’s much more to beetles than their appearance, of course, but one of this book’s shortcomings, because it’s a coffee-table conversation-piece rather than a scientific survey, is that it tells you almost nothing about the ecology and behaviour behind the photographs. And the book’s title is misleading, in fact, because the jewels aren’t living: all the photos are of dead beetles on white backgrounds. The book also tells you very little about the meaning and history of the (sometimes misspelt) scientific names, even though these are fascinating, beautiful, and grotesque in their own right. Instead, there’s a brief but interesting – and occasionally wrong: Chrysophora isn’t Latin – introduction, then page after page of the gorgeous and grotesque photographs people will be buying this book for. Finally, there are some brief “Beetle Profiles”, describing where individual species were caught and how their family lives and feeds. I would have liked to know much more, though the beetles’ beauty is in some ways increased by its mystery and by what might be called the futility of its appearance. Countless millions of these beetles have lived and died without any human brain ever appreciating their beauty and strangeness, and if human beings disappeared from the planet they would continue to live and die unappreciated. They’re not here for us, but without us they could never be recognized as the living jewels they are. Some might draw metaphysical conclusions from that and conclude that they are here for us after all, but I draw a mathematical conclusion: mathematics governs the evolution of both beetles and brains, which is why beetles can appeal to us so strongly.

Living Jewels – Website accompanying the book and its sequel.

Cover of Living Jewels 2 by Poul Beckmann