Also known as the Carolina parakeet, Conuropsis carolinensis, this bird is now extinct.
Also known as the Carolina parakeet, Conuropsis carolinensis, this bird is now extinct.
• Oil Say — another painting by Tissot
John Atkinson Grimshaw (1836-93), Autumn Morning.
Collins Bird Guide: The Most Complete Guide to the Birds of Britain and Europe (second edition), text and maps by Lars Svensson, illustrations and captions by Killian Mullarney and Dan Zetterström, with a significant contribution by Peter J. Grant, translated by David Christie and Lars Svensson (HarperCollins, 2009)
A literate musician can read a score and hear a symphony in his head. I wonder whether the mega-minds of the future will be able to do something similar with genomes: read a DNA recipe and see the animal or plant cooked from it. The mega-minds will need to know about the oven, that is, the womb, egg or seed, but then musicians need to know about instruments, not just notes. The code can’t exist in isolation: it needs a world to be realized in and a musician’s mind can mimic that world.
But mega-minds aren’t here yet for genetics, so we have to use books like this to see the product of DNA-recipes. Collins Bird Guide is effectively a genetic cook-book or genomic score, but we don’t see the naked genes, just the dish or symphony cooked or played from them. Lars Svensson describes thousands of birds of all shapes, sizes, colours, diets and habitats, from the huge golden eagle, Aquila chrysaetos, which can carry off a lamb, to the tiny goldcrest, Regulus regulus, which isn’t much bigger than a bumblebee. But these two, like all other birds, have a common ancestor: when you see a bird sitting in a tree, it is also, metaphorically speaking, sitting in a genetic tree whose twigs, branches and boughs spring from a single trunk. One DNA-recipe has turned into many under the influence of natural and sexual selection.
Birds, which often come in very distinct male and female forms, offer lots of good examples of sexual selection. One of the most spectacular examples isn’t native to the region covered by the book, but it has been introduced here. And so there are pleasant surprises in store for some European ornithophiles. I once came across a wild-living golden pheasant, Chrysolophus pictus, early one morning in a park in northern England. I thought for a moment that I was hallucinating: the bird has a crest of spun gold, a scarlet breast and belly, and an orange/black “nuchal cape”, or neck-feathers, that “can be raised like a fan when displaying” (“Partridges & Pheasants”, pg. 59). It also has yellow legs, blue wings and a long, attractively patterned tail. “Unmistakable!” notes the book.
That’s true of the ♂, at least. The ♀, whose eyes and brain are responsible for the spectacular appearance of the ♂, is undistinguished and similar to the ♀ of Lady Amherst’s pheasant, Chrysolophus amherstiae, whose ♂ is again “Unmistakable!”, thanks to the sexual selection of its ♀. These closely related species are native to eastern Asia and “occasionally hybridize” in Britain (pg. 59). In other words, their common ancestor was fairly recent and their DNA recipes can still work together. But these hybridizations may also be a function of small populations and restricted habitat in Britain. “Function” is the operative word: birds, like all other forms of life, are mechanisms with inputs, throughputs and outputs. For a pheasant, some of the input is sense-data. The throughput is the processing of sense-data in the brain. The output is behaviour: for example, mating with a less-than-ideal partner under the restricted conditions of Britain.
All this can be modelled mathematically, but in the widest and deepest sense it already is mathematical: the human invention of mathematics, with a small “m”, is a symbolic representation of Mathematics with a big “M”. Mathematical symbols represent entities and operations and are manipulated according to logical rules. This mimics the inter-play of entities in the real world, which are subject to the rules of logic implicit in physics and chemistry. Human mathematics is fallible, albeit self-correcting. The mathematics underlying reality realizes the pipe-dreams of the papacy and is infallible, in the sense that it never disobeys the rules by which it is governed.
But this infallible mathematics can fail the entities for whom it operates: birds can die young and fail to reproduce or have fewer offspring than their competitors. But this is the fuel of a larger mechanism: evolution, which is a mathematical process. Genes mutate and vary in frequency under the influence of natural and sexual selection, inter alia. Birds offer more good examples of the effects, because they have wings, beaks and feet. These are mathematical mechanisms, shaped by and for the physics of a particular environment: wings have input from the air and provide the output of flight. Or the output of swimming: some wings are adapted for movement underwater, as in the cormorants, or Phalacrocoracidae, whose beaks are adapted for seizing fish and feet for paddling.
You can look through this book and survey the varying geometry of wings, beaks and feet, from gliding gulls to hovering warblers, from seed-cracking finches to flesh-tearing owls, from tiny-toed swifts to wading egrets. The tool-kit of the common ancestor has become many tool-kits and evolution has been morally neutral as it has worked its multiplicative magic. The feet of the odd and endearing wallcreeper, Tichodroma muraria, are adapted to clinging onto vertical rock; the feet of eagles and owls are adapted to puncturing nerve-filled flesh. And presumably each species enjoys using its adaptation. A distinct psychology will accompany each distinct wing, beak and foot, because no organ can change in isolation: it is evolving within the environment of the body, influencing and influenced by other organs, in particular the brain.
But changes in the brain aren’t easily visible. If they were, some parts of evolution would be much less controversial: racial differences in human intelligence, for example. But races differ in other ways: in their attitudes to animals, for example. One generalization is that northern Europeans like listening to songbirds and southern Europeans like shooting them. So it’s not surprising that this book was originally published in Swedish as Fågelguiden, Europas och Medelhavsområdets fåglar i fält (1999). It would also be interesting to see the statistics of ornithological publishing in Europe. Those statistics will reflect genetic differences in the white European race, and so will readers’ reactions to the book.
My interest is partly aesthetic and mathematical, for example, and I quail at the thought of learning the differences between what bird-watchers call “little brown jobs”: the various kinds of warbler are hard enough to tell apart in pictures, let alone in the wild. But things can get even worse at night: Lars Svensson notes of Savi’s warbler, Locustella luscinioides, that “A possible confusion risk at distance and at night in S and C Europe is the mole-cricket” (“Warblers”, pg. 318). Birdsong and bird-cries are another aspect of ornitho-mathematics, but it’s hard to represent them in print: “kru-kih karra-kru-kih chivi trü chivi chih” (clamorous reed warbler, Acrocephalus stentoreus, pg. 322), “glipp-glipp-glipp” (common crossbill, Loxia curvirostra, pg. 386), “trrsh, trre-trre-trre-rrerrerre” (sand martin, Riparia riparia, pg. 258), “pyük…popopo…” (pygmy owl, Glaucidium passerinum, pg. 226), “brrreep, bip bip bip” (red phalarope, Phalaropus fulicarius, pg. 162), and so on.
In an electronic manual of ornithology, you’d be able to hear the songs, rather than imagine them, but electronic manuals, by offering more, in some ways offer less. Because the book has so many species to cover, it can’t describe any species in detail. So there are occasional fleeting comments like this:
Asian Desert Warbler, Sylvia nana V*** [= rare vagrant in northern Europe]… has the peculiar habit of sometimes “tailing” the Desert Wheatear [Oenanthe deserti] (“Warblers”, pg. 310-1)
The accompanying illustration shows a desert warbler standing under a small bush and peering out at a nearby wheatear. It’s anthropomorphic and anthropocentric to be amused by the behaviour, but ornithology is a human invention and humans don’t have to be purely scientific. I get a boy-racer thrill from another “V***” bird, the white-throated needletail, Hirundapus caudacutus:
Big, with heavy compact body, neckless, stub-tailed (shape somewhere between fat cigar and “flying barrel”). Flight impressively fast, the bird seems to draw easily away from other swifts (though these are still fast flyers!). (“Vagrants”, pg. 415)
That I would like to see. In the meantime, I have this book and the multiplex mutational mathematics it captures in pictures and words.
Graphische Sammlung Albertina, Vienna.
The Belgian symbolist Fernand Khnopff (1858-1921) is one of my favourite artists; Caresses (1896) is one of his most famous paintings. I like it a lot, though I find it more interesting than attractive. It’s a good example of Khnopff’s art in that the symbols are detached from clear meaning and float mysteriously in a world of their own. As Khnopff used to say: On n’a que soi – “One has only oneself.” But he was clearly inspired by the story of Oedipus and the Sphinx, which is thousands of years old. Indeed, an alternate title for the painting is The Sphinx.
Even older than the Oedipus story is another link to the incestuous themes constantly explored by Khnopff, who was obsessed with his sister Marguerite and portrayed her again and again in his art. That’s her heavy-jawed face rubbing against the heavy-jawed face of the oddly nippled man, but Khnopff has given her the body of a large spotted felid. Many people misidentify it as a leopard, Panthera pardus. It’s actually a stranger and rarer felid: a cheetah, Acinonyx jubatus, which occupies a genus of its own among the great cats. And A. jubatus, unlike P. pardus, is an incestuous animal par excellence:
Cheetahs are very inbred. They are so inbred that genetically they are almost identical. The current theory is that they became inbred when a “natural” disaster dropped their total world population down to less than seven individual cheetahs – probably about 10,000 years ago. They went through a “Genetic Bottleneck”, and their genetic diversity plummeted. They survived only through brother-to-sister or parent-to-child mating. (Cheetah Extinction)
It must have been a large disaster. Perhaps cheetahs barely survived the inferno of a strike by a giant meteor, which would make them a cat out of hell. In 1896, they became a cat out of Bel too when Khnopff unveiled Caresses. Back then, biologists could not analyse DNA and discover the ancient history of a species like that. So how did Khnopff know the cheetah would add extra symbolism to his painting? Presumably he didn’t, though he must have recognized the cheetah as unique in other ways. All the same, I like to think that perhaps he had extra-rational access to scientific knowledge from the future. As he dove into the subconscious, Khnopff used symbols like weights to drag himself and his art deeper and darker. So perhaps far down, in the mysterious black, where time and space lose their meaning, he encountered a current of telepathy bearing the news of the cheetah’s incestuous nature. And that’s why he chose to give his sphinx-sister a cheetah’s body.
The Pocket Guide to The Trees of Britain and Northern Europe, Alan Mitchell, illustrated by David More (1990)
Leafing through this book after I first bought it, I suddenly grabbed at it, because I thought one of the illustrations was real and that a leaf was about to slide off the page and drop to the floor. It was an easy mistake to make, because David More is a good artist. That isn’t surprising: good artists are often attracted to trees. I think it’s a mathemattraction. Trees are one of the clearest and commonest examples of natural fractals, or shapes that mirror themselves on smaller and smaller scales. In trees, trunks divide into branches into branchlets into twigs into twiglets, where the leaves, well distributed in space, wait to eat the sun.
When deciduous, or leaf-dropping, trees go hungry during the winter, this fractal structure is laid bare. And when you look at a bare tree, you’re looking at yourself, because humans are fractals too. Our torsos sprout arms sprout hands sprout fingers. Our veins become veinlets become capillaries. Ditto our lungs and nervous systems. We start big and get small, mirroring ourselves on smaller and smaller scales. Fractals make maximum and most efficient use of space and what’s found in me or thee is also found in a tree, both above and below ground. The roots of a tree are also fractals. But one big difference between trees and people is that trees are much freer to vary their general shape. Trees aren’t mirror-symmetrical like animals and that’s another thing that attracts human eyes and human artists. Each tree is unique, shaped by the chance of its seeding and setting, though each species has its characteristic silhouette. David More occasionally shows that bare winter silhouette, but usually draws the trees in full leaf, disposed to eat the sun. Trees can also be identified by their leaves alone and leaves too are fractals. The veins of a leaf divide and sub-divide, carrying the raw materials and the finished products of photosynthesis to and from the trunk and roots. Trees are giants that work on a microscopic scale, manufacturing themselves from photons and molecules of water and carbon dioxide.
We eat or sculpt what they manufacture, as Alan Mitchell describes in the text of this book:
The name “Walnut” comes from the Anglo-Saxon for “foreign nut” and was in use before the Norman Conquest, probably dating from Roman times. It may refer to the fruit rather than the tree but the Common Walnut, Juglans regia, has been grown in Britain for a very long time. The Romans associated their god Jupiter (Jove) with this tree, hence the Latin name juglans, “Jove’s acorn (glans) or nut”… The wood [of Black Walnut, Juglans nigra] is like that of Common Walnut and both are unsurpassed for use as gunstocks because, once seasoned and worked, neither moves at all and they withstand shock particularly well. They are also valued in furniture for their good colour and their ability to take a high polish. (entry for “Walnuts”, pg. 18)
That’s from the first and longer section, devoted to “Broadleaved Trees and Palms”; in the second section, “Conifers”, devoted to pines and their relatives, maths appears in a new form. Pine-cones embody the Fibonacci sequence, one of the most famous of all number sequences or series. Start with 1 and 1, then add the pair and go on adding pairs: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… That’s the Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci (c.1170-c.1245). And if you examine the two spirals created by the scales of a pine-cone, clockwise and counter-clockwise, you’ll find that there are, say, five spirals in one direction and eight in another, or eight and thirteen. The scales of a pineapple and petals of many flowers behave in a similar way. These patterns aren’t fractals like branches and leaves, but they’re also about distributing living matter efficiently through space. Mitchell doesn’t discuss any of this mathematics, but it’s there implicitly in the illustrations and underlies his text. Even the toxicity of the yew is ultimately mathematical, because the effect of toxins is determined by their chemical shape and its interaction with the chemicals in our bodies. Micro-geometry can be noxious. Or nourishing:
The Yews are a group of conifers, much more primitive than those which bear cones. Each berry-like fruit has a single large seed, partially enclosed in a succulent red aril which grows up around it. The seed is, like the foliage, very poisonous to people and many animals, but deer and rabbits eat the leaves without harm. Yew has extremely strong and durable wood [and the] Common Yew, Taxus baccata, is nearly immortal, resistant to almost every pest and disease of importance, and immune to stress from exposure, drought and cold. It is by a long way the longest-living tree we have and many in country churchyards are certainly much older than the churches, often thousands of years old. Since the yews pre-date the churches, the sites may have been holy sites and the yews sacred trees, possibly symbols of immortality, under which the Elders met. (entry for “Yews”, pg. 92)
This isn’t a big book, but there’s a lot to look at and read. I’d like a doubtful etymology to be true: some say “book” is related to “beech”, because beech-bark or beech-leaves were used for writing on. Bark is another way of identifying a tree and another aspect of their dendro-mathematics, in its texture, colours and patterns. But trees can please the ear as well as the eye: the dendrophile A.E. Housman (1859-1936) recorded how “…overhead the aspen heaves / Its rainy-sounding silver leaves” (A Shropshire Lad, XXVI). There’s maths there too. An Aspen sounds like rain in part because its many leaves, which tremble even in the lightest breeze, are acting like many rain-drops. That trembling is reflected in the tree’s scientific name: Populus tremula, “trembling poplar”. Housman, a Latin professor as well as an English poet, could have explained how tree-nouns in Latin are masculine in form: Alnus, Pinus, Ulmus; but feminine in gender: A. glutinosa, P. contorta, U. glabra (Common Alder, Lodgepole Pine, Wych-Elm). He also sums up why trees please in these simple and ancient words of English:
Give me a land of boughs in leaf,
A land of trees that stand;
Where trees are fallen, there is grief;
I love no leafless land.