Tri Again

All roads lead to Rome, so the old saying goes. But you may get your feet wet, so try the Sierpiński triangle instead. This fractal is named after the Polish mathematician Wacław Sierpiński (1882-1969) and quite a few roads lead there too. You can create it by deleting, jumping or bending, inter alia. Here is method #1:

Sierpinski middle delete

Divide an equilateral triangle into four, remove the central triangle, do the same to the new triangles.

Here is method #2:

Sierpinski random jump

Pick a corner at random, jump half-way towards it, mark the spot, repeat.

And here is method #3:

Sierpinski arrowhead

Bend a straight line into a hump consisting of three straight lines, then repeat with each new line.

Each method can be varied to create new fractals. Method #3, which is also known as the arrowhead fractal, depends on the orientation of the additional humps, as you can see from the animated gif above. There are eight, or 2 x 2 x 2, ways of varying these three orientations, so eight fractals can be produced if the same combination of orientations is kept at each stage, like this (some are mirror images — if an animated gif doesn’t work, please open it in a new window):






If different combinations are allowed at different stages, the number of different fractals gets much bigger:

• Continuing viewing Tri Again.

Vigor Mortis

Front cover of The Best of Black Sabbath
In the Christian religion, the resurrection follows the virgin birth. In the rock-graves at Heysham, the virgin birth follows the resurrection. Or rather: the virgin-births follow the resurrections. There are many of both. The rock-graves at Heysham* are carved in solid rock near the remains of St Patrick’s chapel, an ancient ruin overlooking Morecambe Bay on the coast of Lancashire in England. You may have seen them before, because they appear on the cover of a compilation album by the heavy-metal band Black Sabbath, where they’re filled with ice and look suitably dark and sinister. But the graves are sometimes full of life and activity. In spring, as the rainwater filling them begins to warm, there are resurrections – dozens of them. Tiny crustaceans (a group of animals that includes crabs, shrimps and woodlice) hatch from eggs that have over-wintered in the sediment on the floors of the graves. Some of the crustaceans are called water-fleas, others are called seed-shrimps. Water-fleas, whose scientific name is Daphnia, hop through the water with jerks of their antennae, sieving it for fresh-water plankton. Seed-shrimps, or ostracods, are enclosed in tiny double-sided shells through which their legs protrude. They trundle over the stone sides of the graves, scraping off algae and catching even smaller and simpler animals like rotifers and protozoa.

The rock graves at Heysham (c. 11th century A.D.)

Rock graves at Heysham, Lancs. (c. 1000s)

Water-fleas are famous for parthenogenesis, or their ability to produce offspring without sex. Those that hatch first in spring are female and give birth without mating with any males. A single water-flea in a jar of stagnant water soon becomes a swarm. It’s only later in the year that males are born and the water-fleas mate to produce winter eggs, which sink to the floor of the graves and lie there through the cold weather. The eggs of water-fleas and ostracods can also survive desiccation, or drying-up, and can be blown on the wind to new sites. That is probably how these crustaceans arrived in the rock-graves, which they must have occupied for centuries, through the coldest winters and the hottest summers, dying and being reborn again and again. When a human being or large animal dies, chemical changes in the body make the muscles rigid and wood-like. The scientific term for this is rigor mortis, or the “stiffness of death”. Rigor mortis wears off in time and the body begins to rot. The rock-graves at Heysham are an example of vigor mortis, or the “vigour of death”. Medieval human beings created the graves to bury their dead, but the bodies that were once there have been lost to history. The water-fleas and the seed-shrimps remain, tiny, overlooked and fascinating.

A seed-shrimp or ostracod

A seed-shrimp

A water-flea, Daphnia pulex

A water-flea

*Heysham is pronounced HEE-shum and is an old coastal village near the city of Lancaster, after which Lancashire is named.

A Feast of Fractiles

A rep-tile is a shape that can be divided into copies of itself. One of the simplest rep-tiles is the equilateral triangle, which can be divided into four copies of itself, like this:

Self-dividing equilateral triangle

If, on the other hand, the triangle is subdivided and then one of the copies is discarded, many interesting fractals can be made from this very simple shape:

Fractal triangle creating Sierpinski gasket

Triangle fractal 2

This sequence illustrates how a more complex fractal is created:

Triangle fractal 3 split image 1

Triangle fractal 3 split image 2

Triangle fractal 3 split image 3

Triangle fractal 3 split image 4

Triangle fractal 3 split image 5

Triangle fractal 3 split image 6

Triangle fractal 3 split image 7

Triangle fractal 3 split image 8

And here is the sequence in a single animated gif:

Triangle fractal 3

Triangle fractal 4

Triangle fractal 5

Triangle fractal 6

Triangle fractal 7

Triangle fractal 8

Triangle fractal 9

Triangle fractal 10

Triangle fractal 11

Triangle fractal 12

Triangle fractal 13

Triangle fractal 15

Triangle fractal 16

Triangle fractal 17

Triangle fractal 18

Triangle fractal 19

Triangle fractal 20

Triangle fractal 21

Triangle fractal 22

Triangle fractal 23

Triangle fractal 24

Triangle fractal 25

Triangle fractal 26

Puro Lojo

Los Ojos

He was haunted by eyes. It had begun quite slowly, quite simply: a feeling that he was being watched whenever he went out, that hostile eyes were staring out at him, down on him, from the windows of the town, tracking his progress, feeding the nectar of data to the honeycombs of brains, for savouring later, when he had passed. He started to find quieter, gloomier streets, to stay indoors on sunny days, to keep his curtains drawn. He felt calmer when he knew he wasn’t overlooked, wasn’t being watched, couldn’t be. But then the calm began to evaporate, for he realized that there were eyes even in his house, even here, where he was cocooned in privacy.

The eyes of his books – indeed, the eyes of “books”, the two o’s, the two little eyes staring out between the “b” and the “k”. There were hundreds of eyes, thousands of them, in every book, ready to stare out at him, to watch him, whenever he opened a book and tried to read. Even “eyes” itself, with the twin e’s, seemed to peer at him, if not to stare. It had half-closed lids, ready to open on him, to glare its full. In Spanish, the word opened fully: ojo. The word reminded him of two eyes with a nose between them, with a bindi over the nose, the mystic dot of Hinduism, symbolizing the third eye. In Hindi, though, the word didn’t threaten him: आँख, ānkh. When he’d first started to worry about his books, about the eyes in his books, he’d gone to the public library and spent an hour searching through the dictionaries, making a list of the eye-words he did find threatening, as though he could trap them, confine them, on a single piece of paper.

The “oog” of Dutch. The “öga” of Swedish. The “øje” and “øye” of Danish and Norwegian, which reminded him of the razor-blade slicing the eye in Un Chien Andalou. The “oko” of Czech, Polish, and Russian. The “ojo” of Spanish, the “olho” of Portuguese, the “occhio” of Italian, all from the “oculus” of Latin (“oculo” in the ablative and dative). Then there was the “göz” of Turkish, entirely unrelated, but staring out at him like a cyclops. And what about the “ojú” of Yoruba? It was unrelated to Spanish, but disturbingly similar. Disturbing in a different way was the οφθαλμος, the “ophthalmos”, of ancient Greek. It was such a juicy, gelatinous word, like the juice and jelly of an eye itself, like a mechanism of chemicals and flesh from which the two o’s stared out at him, watching him, judging him, storing data about him for use in his trial, the secret trial that was being prepared for him.

He felt relieved, at first, that modern Greek had a less threatening μάτι, “mati”. But then he discovered that Malay had a word that was very similar, “mata”, though he knew that Malay and Greek were entirely unrelated. He felt his skin prickle as the first hint of a conspiracy trickled into his brain. The lower-case a’s of “matia”, the Greek plural, like the lower-case e’s of “eyes”, were like peering eyes, as though the words were beginning to take on the same form as “ojo” and “oko”, as though they would open fully one day too. He wondered if, one day, all the world’s languages would have eyes in their words for “eye”, would have twin o’s gazing out, glaring out at the reader. The biter bit. The reader read. Or what about a script, a font, a language, that was entirely ocular, entirely based on o’s, that read its readers as they read it? You could do that, could create one. There were enough forms of “o” in the alphabet, or the alphabets, the Latin and French and Czech and Yoruba and Vietnamese.

It was then he realized that he hadn’t looked at the Vietnamese dictionary, having missed it near the end of the shelf. He went to look at it, carrying his eye-list with him. He took the dictionary off the shelf and flicked to the right page. He nearly dropped the book when he saw the word staring up at him: mắt. Like Greek, like Malay. He would discover later, after more research, that the Vietnamese and Malay words probably had a common origin, something he’d half-suspected, after that moment of shock. The language families were spoken in neighbouring regions, after all. But for that moment of shock it had seemed a confirmation of something at work, something beneath the surface, or beneath the lid, peering out, ready to be fully exposed, fully opened, ready to stare its full, to drink his soul.

After his visit to the library, he found it more and more difficult to read English and Spanish, both with eye-like eye-words, both full of o’s. He was haunted by a line from Lovecraft, from “The Shadow over Innsmouth”: “I could not escape the sensation of being watched from ambush on every hand by sly, staring eyes that never shut.” But in English, in Spanish, they were not in ambush. They were there openly, eagerly, greedily. He sought refuge in French and German, whose eye-words were asymmetric, un-eye-like, and where “o” was blessedly far down the list of letter-frequencies. English ran e-t-a-o. All those to’s and of’s and not’s. Spanish was even worse: e-a-o. All those masculine endings, those past tenses, those no’s and lo’s. But French ran e-a-s-i-t-n-r-u-l-o. German was even better: e-n-i-s-t-r-a-d-h-u-g-m-c-l-b-o. He began to sellotape his English and Spanish books shut, as though each were an eye that he was closing by force, blinding so that its myriad inner eyes could not watch him. He felt much calmer reading French and German, much better able to cope when he came across a reference to eyes, and he even laughed aloud when, returning to À Rebours, he read of how the ancestral portraits of des Esseintes alarmaient avec leurs yeux fixes, “startled one with their fixed gaze”.

The disaster, when it came, came without warning. He picked up a French guide to butterflies one afternoon, meaning to browse through it before lunch, and almost at once came across a slip of paper handwritten on both sides. The writing was neat but small and he had to concentrate to read it. It was all part of the trap, he realized too late, to make him focus, to bite more deeply on the poison bait that had been dangled before him. Ice began to form around his viscera as he read, but he could not stop himself until he had finished:

Die einzelnen Worte schwammen um mich; sie gerannen zu Augen die mich anstarrten und in die ich wieder hineinstarren muß: Wirbel sind sie, in die hinabzusehen mich schwindelt, die sich unaufhaltsam drehen und durch die hindurch man ins Leere kommt.

With sick horror, he noted that the passage contained exactly two o’s, near the beginning and near the end, like the grotesque eyes of a distorted, teratomorphic face. Then he turned the slip over and found that the other side contained a translation in English, full of o’s, full of eyes, staring at him, eager to drink the emotion in his face, the realization that he had been trapped. Again, he could not prevent himself from reading to the end:

Single words floated round me; they congealed into eyes that stared at me and that I was forced to stare back into – whirlpools that gave me vertigo and, reeling ceaselessly, led into the void.

He let out an involuntary cry. Where had the slip come from? Who was the author of the German? Who had translated it? Who had written the two languages down, hidden the slip in the book? He looked again and realized that it was his own writing, slow, careful, half-disguised, but unmistakable, now that he looked. There was a conspiracy, yes, there was, and he was the author of it, spinning a web for himself, plotting his own destruction, with hidden motives, hidden hatred. He moaned. Things were moving in his head. The slip was a linguistic key, turning a lock in his subconscious, releasing a phrase that he himself had hidden there. Esse Est Percipi. Berkeley’s great dictum. “To Be Is To Be Perceived”. He knew the truth now. He could not escape, could find no refuge, draw no curtain, close no door, find no darkness to hide in. The universe itself was an eye all around him and he was its eternal focus, naked always, visible always, pierced through and through by a torturer’s gaze that created its own object of torture.

Central Government

A magic square is a square of numbers in which all rows and columns and both diagonals add to the same number, or the magic total. The 3×3 magic square, also known as the Lo Shu square (“scroll of the River Lo” square), uses the numbers 1 to 9 and has a magic total of 15. I haven’t seen it explicitly stated anywhere on the net, perhaps because it’s trivially obvious to proper mathematicians, but in this and other 3×3 magic squares, the magic total must be three times the central number. Here is the proof:

4 9 2
3 5 7
8 1 6
a b c
d e f
g h i

1. a + b + c = a + e + i = b + e + h = c + e + g

2. 3(a + b + c) = (a + e + i) + (b + e + h) + (c + e + g)

3. 3a + 3b + 3c = 3e + a + i + b + h + c + g

4. 2a + 2b + 2c = 3e + g + h + i

5. 2a + 2b + 2c – (g + h + i) = 3e

6. 3e = a + b + c = magic total

Update: In fact, this fact about 3×3 squares is mentioned a lot on the web. See, for example, Negative Magic Squares, which describes a proof discovered by Māori mathematicians in 736 B.C.E.

Some 3×3 magic squares using entirely prime numbers (except for 1 in the first square):

00043 00001 00067
00061 00037 00013
00007 00073 00031 mt = 111 = 37 x 3

00071 00005 00101
00089 00059 00029
00017 00113 00047 mt = 177 = 59 x 3

00083 00029 00101
00089 00071 00053
00041 00113 00059 mt = 213 = 71 x 3

00103 00007 00109
00079 00073 00067
00037 00139 00043 mt = 219 = 73 x 3

00107 00011 00149
00131 00089 00047
00029 00167 00071 mt = 267 = 89 x 3

00139 00007 00163
00127 00103 00079
00043 00199 00067 mt = 309 = 103 x 3

12841 09769 15013
14713 12541 10369
10069 15313 12241 mt = 37623 = 12541 x 3

12721 07753 17167
16993 12547 08101
07927 17341 12373 mt = 37641 = 12547 x 3

13183 08059 16417
15787 12553 09319
08689 17047 11923 mt = 37659 = 12553 x 3

Lulu Lunatic Luz

It’s disturbing what you can find online…

Tales of Silence & Sortilege, Simon Whitechapel, Paperback, 111 Pages

May 28, 2012

If you love weird fantasy, if you love the English language, even if you don’t love Clark Ashton Smith, you should read this book. The back cover describes it as “the darkest and most disturbing fantasy” of this millennium, but that’s either sarcastic or tragically optimistic, because what these stories really are is beautiful. The breath of snow-wolves is described as “harsh-spiced.” A mysterious gargoyle leaning from the heights of a great cathedral has “wings still glistening with the rime of interplanetary flight.” Hummingbirds are “gem-feathered… their glittering breasts dusted with the gold of a hundred pollens.” If you cannot appreciate such imagery, then perhaps you are dead to beauty, or simply dead. These tales are very short, but some of them have stayed with me for years, such as “The Treasure of the Temple,” in which a thief seems to lose the greatest fortune he could ever have found by stealing a king’s ransom in actual treasure. Most of the stories are brilliant, one or two is only good, but the masterpieces are “Master of the Pyramid” and “The Return of the Cryomancer.” The sense of loss and mystery evoked by these two companion stories is almost physically painful, it is so haunting. There is nothing like these stories being published today. Reading them, I feel the excitement and wonder that fans of Weird Tales magazine must have known long ago when new stories would appear by H.P. Lovecraft, Clark Ashton Smith, and Robert E. Howard. Simon Whitechapel doesn’t imitate these authors so much as apply their greatest lessons to new forms of fantasy. This is one of the cheapest books I own, but I accord it one of my most valuable. It is easily the best work of art you will find in any form on Lulu. I cannot recommend it highly enough.

The Roses of Hsūlag-Beiolă, Simon Whitechapel, Paperback, 154 Pages

Jun 8, 2012

This collection of weird fantasy is filled with mystery, wonder and a sense of the ineffable. Not every story is a mind-blowing masterpiece, but the best of them are absolutely spectacular. Even the worst are good and all are haunting in one way or another. My two favorites were: 1. “The Mercy of the Osmomancer,” wherein a knight on a mission to investigate the tower of a scent-wizard encounters demons made of smells and even learns the language of odors… 2. “The Swans,” in which a pawnbroker tracks down all the known paintings of a seemingly insane artist who paints his canvases entirely black, nothing but black, for reasons best and most poetically left to Simon Whitechapel to explain… Any fan of H.P. Lovecraft, Robert E. Howard, Clark Ashton Smith, Edgar Allan Poe, Comte de Lautréamont, Charles Baudelaire or William S. Burroughs will find something wonderful to love in here. I sure did.

Even more disturbing is the thought that this individual may be able to pass themself off as normal in real life: there are no spelling mistakes or solecisms. (Then again, perhaps I’m reviewing my own books in my sleep. (But I wouldn’t compare myself to B*rr**ghs, surely? (Unless it’s a bluff or double-bluff. (Disturbing, as I said. (I agree.)))))

Bat’s the Way to Do It!

I think Britain would be much better off without three things that start with “c”: cars, canines, and coos (sic (i.e., pigeons)). But perhaps I should add another c-word to the list: cats. I like cats, but there’s no doubt that, in terms of issues around negative components/aspects of conservation/bio-diversity issues vis-à-vis the feline community/demographic, they’re buggers for killing wildlife:

A recent survey by the Mammal Society was based on a sample of 1,000 cats, countrywide, over the summer of 1997. The results included only “what the cat brought in” and ignored what it ate or left outside. Leaving aside this substantial hidden kill, it still concluded that cats killed about 230,000 bats a year. This is equivalent to more than the entire population of any species other than the two most common pipistrelles. If these 1,000 cats are typical, and there is no reason to believe that they are not, cats kill many more bats than all natural predators combined. They are one of the biggest causes of bat mortality in Britain, perhaps the biggest. (Op. cit., chapter 6, “Conservation”, pg. 139)

Cover of British Bats by John D. Altringham

That is the unhappy conclusion in John D. Altringham’s very interesting and educative book British Bats (HarperCollins, 2003). Accordingly, I’d rather have fewer cats and more bats. Anyone but a cat-fanatic – and cat-fanatics are found in one or two places – should feel the same, and even the fanatics might reconsider if they read this book. The cat family contains some of the most beautiful and athletic animals on earth; the bat family contains some of the strangest and most interesting. In fact, all bats are strange: they’re mammals capable of sustained powered flight. Little else unites them: in chapter two, Altringham describes the huge variety of bats around the world. They live in many places and live off many things. Some drink nectar, some drink blood; some eat fruit, some eat fish. Some roost in caves, some in trees. Some hibernate, some migrate. Some use echolocation and some don’t. Bats are much more varied than cats and scientifically speaking are much more interesting.

Although echolocation isn’t universal, it is the most interesting aspect of bats’ behaviour and it’s used by all the species found in Britain, from the big ones, like the noctule and greater horseshoe bat, Nyctalus noctula and Rhinolophus ferrumequinum, to the small ones, like the whiskered bat and the pipistrelles, Myotis mystacinum and Pipistrellus spp.[1] Two of the pipistrelles are in fact most easily distinguished by the frequencies they call at: the 45 kHz pipistrelle, Pipistrellus pipistrellus, and the 55 kHz pipistrelle, Pipistrellus pygmaeus. As their English names suggest, one calls at an average of 45 kilo-Herz, or 45,000 cycles a second, and the other at an average of 55. The two species weren’t recognized as separate until recently: they look almost identical, although the 55 kHz is “on average… very slightly smaller”, and they forage for food in the same places, although the 55 kHz is “more closely associated with riparian habitat” (that is, it feeds more over rivers and other bodies of water). But examine their calls on a spectrograph, an electronic instrument for visually representing sounds, and there’s a much more obvious difference. This is a good example of how much the scientific study of bats depends on technology. Human beings didn’t need science to know about and understand the ways a cat uses its senses, because they’re refinements of what we use ourselves. We might marvel at the acuity of a cat’s eyes or ears, as we might marvel at the acuity of a dog’s nose, but we know for ourselves what seeing, hearing, and smelling are like.

Echolocation is something different. Bats don’t just see with their ears, as it were: they illuminate with their mouths, pouring out sound to detect objects around them. And the sound has to be very loud: “The intensity of a pipistrelle’s call, measured 10 centimetres in front of it, is as much as 120 decibels: that is the equivalent of holding a domestic smoke alarm to your ear.”[2] The “inverse square law”, whereby the intensity of sound (or light) falls in ratio to the square of the distance it travels, means that the returning echoes are far, far fainter than the original call. It’s as though Motörhead, playing at full volume, could hear someone at the back of the crowd unwrapping a toffee. How do bats call very loudly and hear very acutely? How do they avoid deafening themselves and drowning their own echoes? These are some of the questions bat-researchers have investigated and Altringham gives a fascinating summary of the answers. For example, they avoid deafening themselves by switching off their ears as they call. They’ve had to solve many other tricky acoustic problems to perfect their powers of echolocation.

Or rather evolution has had to solve the problems. The DNA of bats has changed in many ways as they evolved from the common mammalian ancestor (which also gave rise to you, me, and the author of this book) and those changes in DNA represent changes in their neurology, anatomy, and appearance. It’s easy to see that hearing is important for bats, because their eyes are relatively small and their ears are often large and rigid and come in a great variety of shapes. What isn’t easy to see is what those ears are supplying: the bat-brain and its astonishing ability to process and classify sound-data as though it were light-data. Bats can create sound-pictures of their surroundings in complete darkness. Of course, the feline or human ability to create light-pictures is astonishing too, but we’re too familiar with it to remember that easily. Bats aren’t just marvels in themselves: they should encourage us to marvel at ourselves and what our own brains can do. The digestive system of a bat, cat, or human needs food; the nervous system of a bat, cat, or human needs data. That’s what our sense-organs are there for and in principle it doesn’t matter whether we create a picture of the world with our eyes or with our ears.

Male noctule (Nyctalus noctula) calling from tree-roost to attract mates

Male noctule calling from tree-roost

In practice, there are some very important differences between sound and light. Light works instantly and powerfully on a terrestrial scale; sound takes its time and is much more easily diluted or blocked. A hunting cat can scan an illuminated or unilluminated environment for free, because it doesn’t have to generate the light it sees by or the sound it hears by. Hunting bats have to pay when they scan their environment, because they’re using energy to create sound and induce echoes. Once they’ve got their data, both cats and bats have to pay to process it: it takes energy to run a brain. But bat-brains are solving more complicated problems than cat-brains: Altringham describes the questions a flying bat has to answer when it detects the echo of an insect:

How far away is the insect?… How big is it?… In which direction does it lie?… How fast is it flying and in what direction?… What is it?… (ch. 3, “The Biology of Temperate Bats”, pp. 42-3)

Like insect-eating birds, bats can answer all these questions in mid-flight, but what is relatively easy for birds, using their eyes, is a much greater computational problem for bats, using their ears. “Computational” is the key word: brains are mathematical mechanisms and process sense-data using algorithms that run on chemicals and electricity. Bats were intuitively using mathematical concepts like doppler shift and frequency modulation (as in FM radio) millions of years before man invented mathematics, but man-made mathematics is an essential tool in the study of echolocation. For example, the concept of wavelength, or the distance between one crest of a sound-wave and the next, is very important in understanding how bats perceive objects. Light has very short wavelengths, so humans and other visual animals can easily resolve small objects. Sound has much longer wavelengths, so bats find it hard to resolve small objects. But some find it harder than others: Daubenton’s bat, Myotis daubentonii, and other Myotis spp. “can resolve distances down to about 5 millimetres when given tasks to perform in the laboratory”. But horseshoe bats, Rhinolophus spp., “can do little better than 12 millimetres.”

Why this difference? You have to look at the nature of the sound being produced by the different species: the Myotis spp. use “high frequency FM calls”; the Rhinolophus use “predominantly CF [constant frequency] calls”. The mathematical nature of the call determines the bats’ powers of perception. Calls can also determine how easily a bat can identify an insect: “relatively long calls can have a ‘flutter detector’… If a call is 50 milliseconds long, then within one echo a bat can detect the full wingbeat of insects beating their wings at more than 50 Hz.”[3] So bats can tell one kind of insect from another, something like the way a blindfolded human can tell a bumblebee from a mosquito. But insects aren’t passive as prey and one of the most interesting sections of the book describes how they try to avoid being eaten. Some moths have “ultrasound detectors” and if a moth hears a calling bat, it “will either stop flying and drop toward the ground, or begin a series of rapid and unpredictable manoeuvres involving dives, loops and spirals”.[4] This kind of ecological interaction creates an “evolutionary arms race”: each side evolves to become better at capture or evasion.

The moth/bat air-battle is reminiscent of the air-battles of the Second World War, which involved radar trying to detect bombers and bombers trying to evade radar. One defensive technique was jamming, or attempts to interfere with radar signals or drown them in noise. Some moths may use this technique too. The tiger moths, the Arctiidae, don’t try to escape detection. Instead, they “emit their own, loud clicks”[5], perhaps to interfere with echolocation or startle a predatory bat. Alternatively, Altringham suggests, the clicks may be the aural equivalent of “bright warning colours and patterns”: the moths may be warning bats of their unpalatability. If so, it would be another example of the difference between the costs of sight and the costs of sound. An unpalatable insect in daylight doesn’t have to pay for its warning colours, after the initial investment of creating them, and doesn’t have to know when a predator is watching. An unpalatable insect in the dark, on the other, can’t send out a constant audible warning: it has to select its moment and know when a predator is nearby. Unless, that is, some insects use passive signals of unpalatability, like body modifications that create a distinctive echo.

Bat-researchers don’t know the full story: there is still a lot to learn about bats’ hunting techniques and the ways insects try to defeat them. But “cost” is a word that comes up again and again in this book, which is partly a study in bio-economics. Bats have to pay a lot for echolocation and flight, but flight is a more general phenomenon in the animal kingdom, so the economics of bat flight also illuminates (insonates?) bird and insect flight. Altringham points out a very important but not very obvious fact: that flight is expensive by the unit of time and cheap by the unit of distance. Movement on foot is the opposite: it’s expensive by the unit of distance and cheap by the unit of time. Bats, birds, and insects expend more energy per second in flight, but can travel further and faster in search of food or new habitats. However, bats don’t all fly in the same way: a bat expert can identify different species by their wings alone. The wings vary in “wing loading”, which is “simply the weight of the bat divided by the total area of its wings. Bats with a high wing loading are large and heavy in relation to their wing area, bats with small bodies and large wings have a low wing loading”.[6] Then there’s “aspect ratio”, the “ratio of wingspan to average wing width”, or, because “bats have such an irregular wing shape”, “wingspan squared divided by wing area.”

It’s mathematics again: there are no explicit numbers in a bat’s life, but everything it does, from echolocating to flying, from eating to mating, is subject to mathematical laws of physics, ecology, and economics. Bats have to invest time and energy and make a profit to survive and have offspring. As warm-blooded, fast-moving animals with high energy needs, they’re usually nearer famine than feast, which is one reason they migrate or hibernate to avoid or survive through cold weather and scarcity. They also vary their diet during the year, to take advantage of changes in the abundance of one insect species or another, and seek out specialized feeding niches. Daubenton’s bat, for example, “habitually feeds very low over water”, using echolocation to catch not just flying insects but floating ones too. That is why it needs smooth water to feed over: ponds, lakes, canals and placid streams and rivers. The floating insects are easier to echolocate on a smooth surface, rather like, for humans, a black spider on a white wall. Once spotted, they “are gaffed with the large feet or the tail mechanism and quickly transferred to the mouth as the bat continues its flight”.[7]

Long-eared bat (Plecotus auritus) gleaning harvestman

Long-eared bat gleaning harvestman

One of the photos in the colour section in the middle of the book shows a Daubenton’s bat mirrored in smooth water, having just scooped up prey from the surface. Other photos show other species roosting, perching, or in flight, but the book also has excellent black-and-white illustrations mixed with the text, hand-drawn using a speckled or pointillist technique that suits bats very well. I particularly like the drawings on pages 48, 67 and 101. The first shows a long-eared bat, Plecotus auritus, “gleaning”, or snapping up, a “harvestman” (a long-legged relation of the spiders) from a leaf (ch. 3); the second shows a “male noctule calling from his tree roost to attract mates” (ch. 3); and the third shows a tawny owl trying to catch another long-eared bat (ch. 4).

Owls could be called the avian equivalents of bats: they’re specialized nocturnal hunters with very sharp hearing, but I think they’re both less interesting and more attractive. Bats, with their leathery wings, sometimes huge ears, and oddly shaped noses, are strange rather than attractive and some people find them repulsive. But some people, or peoples, find them divine or lucky: the introduction describes the Mayan bat-god Zotz, with his leaf-shaped nose modelled on that of the phyllostomids, or vampire bats.[8] The Chinese use a ring of five bats to symbolize the “five great happinesses: health, wealth, good luck, long life and tranquillity.”[9] Altringham blames the less positive image of bats in European cultures partly on Bram Stoker’s Dracula, which was first published in 1897. Before then, he says, “bats were not linked with witches, vampires and the evil side of the supernatural in any significant way.”[10] Dracula may have done for bats what the novel Jaws (1974) and its cinematic offspring did for sharks: encouraged human beings to harm the animal fictionally and falsely depicted as villainous.

Daubenton's bats (Myotis daubentonii) in a summer roost

Roosting Daubenton’s bats

If so, British Bats is partly redressing the balance. You can learn a lot from this book about both biology in general and bat-biology in particular. It stimulates the mind, pleases the eye, describes the appearance, ecology, and range of all British species, and points the way to further reading and research. So let’s not hear it for John D. Altringham! Without specialized equipment, that is, but that equipment is getting cheaper and more widely available all the time: you don’t have to be a professional zoologist to record and analyse bat-calls any more. There is still a lot for zoologists, both amateur and professional, to learn about bats. Okay, some of the research – like fitting miniature radio-transmitters to wild bats – seems intrusive and smacks of Weber’s Entzauberung, or “disenchantment”, but the more we know about bats, the more we will be able to help conserve them and their habitats. Bats aren’t villains: cats are. I like both kinds of mammal, but I hope we can find some way in future to help stop the latter preying so heavily on the former. If this book helps publicize the problem, it will be valuable for bat-conservation just as it is already valuable for bat-science. In short, no more brick-bats for Brit-bats: we should control our cats better.

Reviewer’s note: Any scientific mistakes, misinterpretations or misunderstandings in this review are entirely your responsibility.


1. sp = species, singular; spp = species, plural.

2. ch. 3, “The Biology of Temperate Bats”, pg. 40

3. Ibid., pg. 45

4. ch. 4, “An Ecological Synthesis”, pg. 98

5. Ibid., pg. 99

6. Ibid., pg. 71

7. ch. 5, “British Bats, Past and Present”, pg. 117

8. ch. 1, “Introduction”, pg. 10. “Phyllostomid” is scientific Greek for “leaf-mouthed clan”.

9. Ibid., pg. 11

10. Ibid., pg. 9

Damsels and Dragons

If I were asked to nominate a great work of 21st-century art, I would not choose anything by the likes of Damien Hirst or the architect Frank Gehry (responsible for the giant metal midden in Bilbao known as the Guggenheim Museum). Instead, I’d put forward something by Klaas-Douwe B. Dijkstra, Richard Lewington, and British Wildlife Publishing of Gillingham in Dorset. They’re not big names like Hirst and Gehry and they’re not earning big money or exercising big influence. And they’re unlike Hirst and Gehry in another way: they’ve created a genuinely beautiful and intellectually stimulating piece of art.

The art-work is called the Field Guide to the Dragonflies of Britain and Europe (2006). Dijkstra oversees the detailed, expert, and fascinating text, Lewington supplies the detailed, accurate, and beautiful drawings, complemented by photographs of dragonflies and damselflies in the wild. Lots of people don’t know the difference between these two suborders of the Odonata, but their common names reflect their appearance: the Zygoptera, or damselflies, are delicate and fold their wings at rest; the Anisoptera, or dragonflies, are robust and always hold their wings at right angles to their bodies. Both come in a huge variety of colours, pure and mixed, as their common names prove: damselflies include the Azure, the Goblet-Marked, the Orange White-legged, the Scarce Blue-Tailed and the Scarce Emerald; dragonflies include the Green and Mosaic Darners, the Banded, Red-Veined, Scarlet, Violet-marked and Yellow-winged Darters, the Orange-spotted Emerald, and the Four-spotted Skimmer. There’s also Somatochlora metallica, the Brilliant Emerald dragonfly, which looks as though it’s made of bright green metal or enamel.

These rich colours, with the complex venation of their wings, have made the Odonata a popular subject for artists and jewellers: for example, the art nouveau master René Lalique (1860-1945) made dragonfly mascots for cars. Unfortunately, the book doesn’t cover the Odonata in art: it’s a scientific text, a microcosm of the macrocosm of biology. Biology depends on accurate description and classification, so odonatology has a rich vocabulary: antehumeral stripes, arculus, carina, clypeus, diapause, discoidal cell, gynomorph, medial supplemental vein, pronotum, pseudopterostigma, siccation, and so on. Even the segments of the abdomen are numbered, from S1, just below the wings, to S9 and S10 at the tip of the tail, where the females have their almost clockwork genitalia. Males have theirs beneath S2, so mating in the Odonata is a complicated, almost tantric, business, as some of the photographs prove. Nomenclature in the Odonata is a complicated, almost incantatory business: Calopteryx splendens, virgo, xanthostoma; Enallagma cyathigerum; Pyrrhosoma nymphula; Anax parthenope, imperator; Ophiogomphus cecilia; Onychogomphus forcipatus; Libellula quadrimaculata; Sympetrum depressiusculum; Zygonyx torridus.

That nomenclature, and that sex-life, are two of the ways that the Odonata are CASean creatures; that is, their complexity, strangeness, and beauty remind me of the work of “the Emperor of Dreams”, the Californian writer Clark Ashton Smith (1893-1961). The obsessive, minutely detailed nature of the book is CASean too, and some of its subjects might literally be emperors in dreams: the common name of the Anax genus is the Emperors. One of these Emperors answered a CASean question I had as I leafed through the book: distribution tides washed back and forth across the little map of Europe that accompanied each specific description, submerging here Britain and Ireland, there France and Spain, here Germany and Scandinavia, there Greece and Turkey, and sometimes all of them at once.

But the strange, isolated island of Iceland, though included on every map, always seemed redundant, like a wall-flower at the Odonatan dance. “Was it a dragon- and damselfly desert?” I wondered. Then I came across Anax epihippiger, or the Vagrant Emperor: “A. epihippiger is the only dragonfly ever recorded on Iceland.” From the magnificent to the minute, from damselflies in the burning deserts of Morocco to dragonflies amid the frosty volcanoes of Iceland, it’s all here in a book that truly does deserve to represent European civilization in the twenty-first century. But doesn’t, alas.

Ink For Your Elf

The Majikalph Script

Majikalph was created by Simon Whitechapel in 2012 to combine his interests in artificial alphabets and recreational mathematics. It is based on the patterns created when lines are drawn between numbers of various 4×4 magic squares. In a magic square, every row, column, and diagonal of numbers adds to the same total. In the 4×4 magic square below, the most interesting patterns are created when each number is connected to the number 2 or 4 places higher than it (e.g. 2 goes to 4 or 6; 13 goes to 15 or 1).

Majikalph is used for writing English and is written from right to left. There is no distinction between upper and lower case. No character of the script is invented: each is based on one or another of the 880 possible 4×4 magic squares (for further information, please see

The sample text is an extract from Tennyson’s The Princess (1847):

Oh, hark, oh, hear! how thin and clear,
And thinner, clearer, farther going!
Oh, sweet and far from cliff and scar
The horns of Elfland faintly blowing!
Blow, let us hear the purple glens replying:
Blow, bugle; answer, echoes, dying, dying, dying.

Alfred, Lord Tennyson (1809-1892).

Sample Text