The Apple Mannikin

Back in the 1970s, computer graphics were at a relatively primitive stage. A lot of them were just wireframe, and this very style became iconic of high technology and the futuristic. The 1979 Disney Film ‘The Black Hole’ was notable for having the longest ever CGI sequence in a feature film up until that time, at around a minute and a half. Here it is:

In the cinema, that looked pretty impressive to me at the time, as I’m sure it did others. However, CGI as we’re familiar with it today also existed, as with NASA’s sequence illustrating the Voyager missions, which was however updated with textures from the mission itself. Then there was Sunstone, also from 1979:

A few years later, there was ‘The Works’ in 1984:

However, by then they should’ve known better, because changes were taking place in mathematics which were reaching some kind of climax by that point, namely research into fractals.

I don’t really understand calculus, but I probably inaccurately think of it in two ways: trying to work out where a wiggly line will go next, and finding the slope of a curve at a particular point, with the emphasis on “point”. It’s where my understanding of maths runs out and therefore a bit of a locked gate for me because of what lies beyond in terms of its practical applications, which I can’t access. Nonetheless I am aware that in 1872, Karl Weierstrass announced his discovery of a function expressed by a wiggly line on a graph which was spiky everywhere, no matter how close you zoomed in on it. This is of course the Weierstrass Function, and looks like this:

The zoomed in bit is to show that it’s spiky on every level. Although it’s a line, there’s no curved or straight stretch anywhere along its length where it isn’t changing direction, no matter how small the difference between the values of x is. This is referred to as “nowhere differentiable”. The function can be expressed thus:

where α=the natural logarithm of a divided by the natural logarithm of b. There are plenty of discontinuous functions like this, but this has values at every point. Sometimes it seems like the nineteenth century consists largely of the eighteenth century status quo and simplicity being overturned at every point, just as the seventeenth century feels like a time of rising sophistication after the relative calm of the sixteenth, preceded by the complexity of the Middle Ages, and so on, which of course makes sense from a Marxist and Kuhnian perspective (note the singular).

This was the first of a series of curves, infinite really, which became known as fractals. The standard, and wrong, way of describing a fractal is that it’s self-similar. There are many self-similar fractals, such as the Koch Snowflake:

This starts out as a triangle, to whose sides spikes are added, making a partly concave dodecagon, to whose sides spikes are added, making a four dozen-sided shape and so forth ad infinitum. The above shape, partly blurred by the fact that it isn’t a vector image due to the difficulty of using vector graphics on WordPress, has seven iterations and therefore 12 228 sides, or it would have if it was actually drawn as opposed to being a raster image. And we’re back to computer graphics. However, most fractals are not self-similar in that way. The coastline of this island is fractal. The shorter the ruler used to measure it, the longer it gets, and you could be reduced to measuring between the grains of sand on a beach or the bumps on a cliff face, at which point the tides and whether something counts as wet come into consideration, but it isn’t self-similar. There aren’t lots of “little Britains” just off our coast which themselves have littler Britains off theirs and so on, appealing though the idea might be.

A fractal is actually a shape with a non-integral number of dimensions. Whereas a square has two dimensions and a cube three, and a line one, it’s useful to consider dimensionality as having values in between whole numbers. The Koch Snowflake, for example, has about 1.262 dimensions, and Great Britain 1.21. The reason the number of dimensions a fractal has is not integral is that the “size” of some shapes, such as the measure polytopes of the line segment, square, cube and tesseract, can be thought of as its measure to the power of the number of dimensions it has, and this is in those cases a whole number but in the cases of fractals. The Koch Snowflake is a wiggly line which meets itself, but it comes close to filling the area around the perimeter of a roughly hexagonal shape, so it’s neither one-dimensional – it isn’t a line – nor two-dimensional – it isn’t a hexagon or a star – but somewhere in between. However, although these ideal platonic shapes are self-similar, most fractals are not, but that doesn’t stop them from having a fractional or irrational number of dimensions.

The real world is not like the smoothness seen in computer graphics, particularly earlier ones. The three videos at the start of this post are all coolly mathematical and, while difficult to produce, involve simple shapes textures with simple textures. With the aid of fractals, it became easy to generate this kind of picture:

This image dates from around 1982. In ‘The Works’, there is some kind of bumpy terrain and I’m not sure how this was generated. As far as I know, this was first used in a feature film, ‘Star Trek II’, in 1982:

The structure of this clip is quite interesting because it goes from old-style wire frame models through textured rendering of three-dimensional objects and ends with the mapping of a fractally-generated surface. At the end of the Voyager missions to Saturn in late 1980, it was mentioned that the CGI people who had produced the videos of the mission and mapped the textures taken by the Voyagers’ cameras onto models of the planets and moons had left to work on ‘Star Trek II’. I presume this is what they went on to do. Incidentally, this disbanding of the team working on the Voyager projects, which was related to the six-year gap between the Saturn and Uranus encounters, shows the difficulty the kind of societies which send rockets into space have with achieving long-term projects. They couldn’t just keep these people on the payroll for six years while they did nothing, so we get this clip but at what cost? What else didn’t we get and who else was “let go”?

This is a “making of” video of the same:

A further tangential detail: the star field is as seen from ε Indi. Alnitak, Alnilam and Mintaka are seen as lined up near the beginning of the clip, indicating their relatively great distance, and as the commentary mentions, the Sun is visible as part of Ursa Major near the end. The constellation of Indus is opposite that of Ursa Major in the sky – it’s a Southern constellation – and ε Indi is almost twelve light years away. This particular sequence is a milestone in the development of CGI.

Raster scan CGI on flat displays is often quite rationally organised at a fairly low level, in that the screen is seen as a rectangular array of pixels like a graph, with the origin either at a corner or the centre. This means that the famous Mandelbrot Set image – the Apfelmännchen or “apple mannikin” as it’s known in German – is effectively a graph with the X axis running horizontally along the middle of the picture. It’s often difficult to remember that this X axis at the centre is in fact the real number line. These are the actual axes of that graph:

Perhaps surprisingly, zero is near one side of the cardioid (heart shape) whereas intuition would suggest it was at the bottom of Seahorse Valley where the circle and cardioid meet. It can be seen from this graph that the set is based on some kind of calculation involving real numbers, but what about the vertical axis?

The vertical axis represents the so-called imaginary numbers. These are numbers based on a concept which originally arose when it was realised that the square root of minus one seemed to be impossible. Since signs cancel out in multiplication, -1 x -1 is 1, so it clearly isn’t the real number one, and the only option appears to be to invent a second axis and think of numbers as existing on a plane as coördinates. These are known as complex numbers. They have both a real and an imaginary part. The word “imaginary” is used for want of a better term, as in fact these numbers are just as real as “real” numbers. There are also hypercomplex numbers such as quaternions and octonions which are a generalisation of this idea from the plane to space and hyperspace. On the whole, all of these numbers can be added, subtracted and the like, but the operations concerned don’t always have the same properties as those on real numbers. For instance, real number multiplication is commutative: 4 x 5 = 5 x 4. Octonion mutliplication is not, and this is crucial because for reasons I won’t go into here, it leaves the possibility that there is an omniscient observer open – it prevents Bell’s Theorem from being a proof of atheism.

The formula used to generate the Mandelbrot Set is quite simple, but before I get to that I’m briefly going to wallow in complex numbers for a bit. Complex numbers are combinations of real and imaginary numbers, more specifically the sum of a real number and the product of a real and complex number. They are useful, and here are two examples:

AC current is generated by a rotary dynamo and its voltage and current therefore vary as a sine wave – that’s the dynamo spinning. Capacitors and inductors alter this variation. Capacitors delay current, for example. The impedance, which is the opposition of a circuit to a current when voltage is applied. This can be calculated using complex numbers and power consumption can be reduced by doing these calculations to design efficient AC circuits.

An object with a rest mass can never move at the velocity of light because it would have infinite mass. However, if mass can be validly expressed by a complex number, this would not be a problem and therefore if tachyons – particles which only move faster than light – exist, they would have to have complex mass.

Hence complex numbers do have real world applications. They’re not only pure mathematics.

The Mandelbrot Set can be plotted on a plane without explicitly using complex numbers. There are programming languages, such as FORTRAN, which have complex numbers as a data type just as they do real and integer numbers, and in FORTH and other threaded interpretative languages there are ways of defining words which can perform operations on complex numbers as two values on the stack, but this isn’t necessary to plot the Mandelbrot Set, although it is effectively what’s happening when it’s done. The formula for the set is z_n+1 = z_n² + c, where z and c are a complex numbers and z is an integer. On a computer screen, each pixel is used as an example of c. This is a QBASIC program to generate the Mandelbrot Set in full:

DECLARE SUB InitPalette ()
DECLARE SUB DrawFractal ()
DECLARE SUB SaveScreen (filename AS STRING, w AS INTEGER, h AS INTEGER)

SCREEN 13
CLS
LINE (0, 0)-(320, 200), 15, BF

CONST w = 320
CONST h = 200

CALL InitPalette
CALL DrawFractal
CALL SaveScreen("MANDELBR.RAW", w, h)

REM This is fractal rendering code, the other functions are to make it look nicer
SUB DrawFractal
    DIM sx AS SINGLE
    DIM xy AS SINGLE
    DIM x AS SINGLE
    DIM y AS SINGLE
    DIM x2 AS SINGLE
    DIM y2 AS SINGLE

    DIM p AS INTEGER
    DIM r AS INTEGER
    DIM g AS INTEGER
    DIM B AS INTEGER

    CONST maxi = 100
    CONST colours = 256

    FOR py = 0 TO h - 1
        REM scale Y to -1:+1
        sy = (py / h) * 2! - 1
        FOR px = 0 TO w - 1
            REM scale x to -2.5:1
            sx = (px / w) * 3.5 - 2.5
            vy = 0
            vx = 0
            i = 0
            x = 0
            y = 0
            x2 = 0
            y2 = 0
            WHILE (x2 + y2 < 4) AND (i < maxi)
                xt = x2 - y2 + sx
                y = 2 * x * y + sy
                x = xt
                x2 = x * x
                y2 = y * y
                i = i + 1
            WEND
            c = i / maxi * colours
            PSET (px, py), c
        NEXT
    NEXT
END SUB


REM Changes the palette as the default is not pretty
SUB InitPalette
    DIM red AS LONG
    DIM green AS LONG
    DIM blue AS LONG
    DIM colour AS LONG

    FOR i = 0 TO 63
        blue = i
        green = i / 2
        red = i / 3
        colour = blue * 65536 + green * 256 + red
        PALETTE i, colour
        PALETTE i + 128, colour
    NEXT i
    FOR i = 63 TO 0 STEP -1
        blue = i
        green = i / 2
        red = i / 3
        colour = blue * 65536 + green * 256 + red
        PALETTE i + 64, colour
        PALETTE i + 192, colour
    NEXT i
END SUB

SUB SaveScreen (filename AS STRING, w AS INTEGER, h AS INTEGER)
    OPEN filename FOR OUTPUT AS #1
    FOR y = 0 TO h - 1
        FOR x = 0 TO w - 1
            PRINT #1, CHR$(POINT(x, y));
        NEXT x
    NEXT y
    CLOSE #1
END SUB

This is in QBASIC, Microsoft’s BASIC for DOS. It’s possible to zoom in by changing the ranges of py and sy in the control loops, and also their step value. The above code is both fancy and not optimised.

The first time I plotted the set, it was on an Acorn Electron. This was an eight-bit computer based on BBC Micro architecture, and was notably cut down from the glories of its predecessor. One of the ways in which this was done was by providing only four dynamic RAM chips and using them to store bytes in two halves, which meant that every time the CPU interacted with memory the clock speed was effectively halved – it had to fetch or store one byte in two cycles. It was possible to speed the computer up somewhat by fooling the video hardware into thinking it was using a text mode, because this skipped two scan lines per line of text and there was no need for the CPU to halt when the video hardware accessed video RAM, but this spoilt the display. The video modes I used to display the set were MODE 2 and MODE 0. The first of these is an eight-colour (supposèdly sixteen but half are merely flashing versions of the others) 256 x 160 display, and has the benefit of being colourful while not being very detailed. The other is 640 x 256 in two colours, allowing a lot of detail and to be honest I prefer it. Doing it on a BBC Micro would, for the reasons just mentioned, be faster than on an Electron, but that’s what I had. However, it’s dead easy to speed it up by only calculating the top or bottom half and mirroring it on the other half of the display. I think it took about eight hours to do the whole set.

I also did Seahorse Valley, which doesn’t benefit from any kind of reflection hack, and it took twenty hours. Not having a printer, I recorded it on a VHS cassette, which I still have somewhere. There are ways of speeding it up a lot, such as writing it in machine code and using fixed point arithmetic rather than floating point, and not bothering to calculate large blank bits of the picture, but I didn’t do those. Around the same time as I was doing this, a DOS program called FRACTINT was developed which managed to do it using only integer arithmetic and was therefore much faster.

The Mandelbrot Set is among the most complex objects in the Universe, or rather the Multiverse. It’s often claimed that the human brain is this, but in fact this object is far more complex because no matter how far you zoom in, there’s always more. It can also be extended into four dimensions because each point on it can be used to generate a Julia Set, such as this:

This can be thought of as a two-dimensional cross-section of an analogue of the Mandelbrot Set, as it varies continuously according to the location of the point used. One perhaps surprising fact about the set is that it took about fifteen years to prove that it was actually a fractal, over the period when it was all the rage and everyone was calling it one, when in fact it wasn’t known to be one. Also around this time it was proven to be as complex as it possibly could be.

The Mandelbrot Set itself does not reflect anything in the physical world, or rather the formula used to generate it seems not to have any practical application, but there is another set which is very similar and does represent something real, in the area of magnetism. This formula:

is quite reminiscent of the Mandelbrot Set’s, and describes what happens inside a magnetic material when it’s heated to the point where it’s completely demagnetised. A cold magnet is magnetic all the way through. All of its constituent parts which are individually magnetic are lined up. As it’s heated, the random movement of the atoms begins to dislodge the alignments in apparently random places, but they can in fact be predicted using this formula. If you plot this in the same way as the Mandelbrot Set, you get something like this:

From here. Will be removed on request.

There’s a blog mentioning this here.

Because of the visual effectiveness of fractal and this kind of imaging, it was suggested at some point that the nature of mathematical discovery had changed, because it was now possible to visualise much of what woul previously have seemed highly obscure. This has been seen as both bad and good. It’s good in that it makes maths more accessible and appealing, but it may also lead towards a bias towards the kind of maths that can do this kind of thing.

Finally, it occurs to me that my metaphor for consciousness being a property of matter like magnetism could be extended meaningfully to model a dying brain. What if the way consciousness works involves a whole, fully awake and living brain as one of the stable states, the “dark bits” as it were on the Mandelbrot Set, but that on falling asleep, having a seizure, being starved of oxygen or under the influence of drugs it fragments consciousness physically into little areas throughout the brain which are individually conscious but not unified? And what if this could be modelled mathematically? I don’t know where I’m going with this but it sounds promising.

Sex, Pentamory And The Single Fibonacci Number

Sarada recently experimented with writing a novel where the word count for each chapter followed the Fibonacci sequence. It was called ‘Tapestry’. Although it didn’t work as a novel format, it reminded me somewhat of ‘The Curious Incident Of The Dog In The Nighttime’, whose chapters use prime numbers rather than the usual sequence, and also the probably accidental diminishing length of Jeanette Winterson’s novel titles where each one was two words shorter than the last, although on examination this seems to be a myth. I have also attempted to use the Fibonacci series in my writing, when I was twelve: I tried to imagine aliens called the “M`ubv” who had fivefold symmetry and five sexes.

It’s more usual in science fiction to imagine three sexes. This is done, for example, in ‘Delta’, a short story by Christine Renard and Claude Chenisse, and in Iain M Banks’s ‘The Player Of Games’. Five sexes probably wouldn’t work and even three might be difficult, for a couple of reasons. Two sexes increases the genetic diversity of a species by allowing genomes to mix, so there’s a good reason for that to happen. One sex is also viable because it allows an otherwise unoccupied environment to be populated by a single individual. This doesn’t work with the “lesbian lizards” of course, also known as New Mexican whiptails, who are a species of entirely female American lizards who, however, don’t ovulate unless they have sex with each other. Three sexes would mean that an individual would need to encounter two other individuals, each of a different sex, which seems to present a further barrier to reproduction which has nothing to do with fitness but is just to do with luck.

The idea behind the M`ubv was that the fact that they had five sexes was linked to them having pentamerous symmetry, like starfish or sea urchins, so that just as bilateral animals often have two sexes per species, pentamerous animals would be likewise pentamorous, as it were. I chose five because it was in the Fibonacci sequence, as is three. Another way to go with this would be to imagine a triplanar species with three sexes or an eight-fold one with eight sexes. However, this assumes a correlation between symmetry of body plan and number of sexes which may not exist. As well as being a Fibonacci number, two is simply the first integer after one and there are no echinoderms (starfish etc) who have five sexes, because if there were they would probably have died out almost immediately. This brings up the question of why the Fibonacci sequence turns up so much in the Universe, and it is the Universe and not just among living things, and also whether there could by any means be a connection between it and the number of sexes. And at this point I have to go off on a tangent and explain what I mean by “number of sexes”.

There is a sense in which the apparen number of sexes is not an integer. In fact it could even be considered not to be a real number. As with gender, sex could be seen not so much as a spectrum as a landscape with two peaks, female and male. There are other conditions which don’t fit neatly into those categories and they have varying degrees of intensity, but they don’t fit into a scale between female and male either because considered as merely a third possible condition they work fine as intermediates, but when one tries to relate them to each other the variation is more multidimensional. To illustrate, males with complete androgen insensitivity are “superfemale” because their androgens are converted to an oestrogenic form and their bodies don’t respond to androgens at all, but there is a range of sensitivity to androgen between that and typically male bodies, so that is on a scale, but guevedoces (I know that’s a slur but the other term is hard to remember) start off female and become male at puberty. These are different ways of being intersex, and it means a mere one-dimensional line is not enough. Moreover, all sexual variations are effectively from female rather than from male. It’s biologically impossible for boy babies to become cis women adults. This means that mathematically, women are similar to zero and Turner Syndrome people (a single X chromosome with no other sex chromosome) are even closer, and everything else is an addition, or rather, a modification from that basic body plan. The variations might make sense as regions on a two-dimensional graph, or even one with a larger number of dimensions.

Interestingly, there is a way of generalising Fibonacci numbers onto the complex number plane, and by this point I’m building up quite a number of further things people might not know about, so I’ll talk about those too. Unfortunately I have very little idea what other people know.

Firstly, there’s the Fibonacci sequence. This is a series where each member is the sum of the previous two integers, so 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3+5=8 and so on. It can be extended to negative numbers, and similar sequences exist, such as Lucas numbers, which start with 2+1=3, 1+3=4, 3+4=7, 4+7=11 etc. They turn up in all sorts of places, notably on the number of spirals either way on pine cones, composite flower inflorescences and leaf numbers on stalks. The Lucas numbers tend to do the same. An important feature of this sequence is that the proportions between the numbers and their immediate predecessors in the series approaches a limit known as φ, phi, which is approximately 1.618 but is an irrational number like π. One of the most notable features of this proportion, also known as the Golden Ratio and used in such areas as architecture to create the impression of beauty, is that the reciprocal is equivalent to the number minus one.

This is an ammonite fossil showing, as in so many other places in nature, the logarithmic spiral. Thisses diameter increases by φ every quarter turn. This is also true of the arms of many spiral galaxies, presumably including our own, meaning that to a limited extent we already have a map of the Milky Way, something I covered in The Galactic Mandela. It can be concluded, for example, that a coördinate system centred on the supermassive black hole Sagittarius A at the Galactic centre with us at a θ (angular) location of 0° and 27 000 light years from the Galactic centre will be in an arm which will spiral out to approximately 43 700 light years 90° on in the direction of the spiral, and that 180° on the other side lies a region similar to our own within an arm which will expand to beyond the 50 000 light year radius before it wraps round far enough to be on our side, although the edge of the Galaxy is not sharp.

Complex numbers are fairly easy to explain, starting with the real number line. Numbers from -∞ to +∞ can be considered as arrayed along a horizontal line, with the conceivable ones close to zero, which could also be seen as the origin. Much of arithmetic can be considered as forming groups of various kinds involving these numbers and the various operations, which are referred to as real, but square roots are different. Two minuses make a plus, so the square of -2 is either four or ±4, depending on how precise you want to be. √4 is plainly 2, and √2 plainly an irrational number starting 1.414…, but √-1 is not 1 or -1 because “two minuses make a plus”. The solution to this is to treat numbers as if they’re a two-dimensional graph, and incidentally there’s a more technical use of the word “graph” which I’m not using here. This is a plain boring old line graph like what you’d see with blood pressure or stock market prices. That is, the real number line is the X axis and the imaginary number line, which is the line on which i, or √-1 is located, is the Y axis. Complex numbers are located on this plane. Incidentally, I think it’s rather unfortunate that imaginary and real numbers are called that because they make it sound like real numbers are real and imaginary one’s aren’t, whereas in fact both are equally real or unreal. It’s also possible to take it further and add dimensions to this graph and create quaternions and octonions, and these are also important, and it so happens personally important to me because I think they have a bearing on the existence of God, but that’s not for here. Imaginary and complex numbers are still useful, for instance in calculations involving AC circuits, and more significantly, if anything can travel faster than light it will have to have a mass only expressible as such a number.

How does this relate to Fibonacci numbers, you may ask? Well, if you treat the number plane as a bit of graph paper whose origin is at zero, you can draw a Fibonacci spiral on it and get the complex correspondents to the real Fibonacci numbers, and if you get the proportions correct it will intersect the real number line at the values of thos numbers, both positive and negative. This presumably means in turn that there’s a link between φ & π in some way.

Back to sexes. If we consider each intersex condition to be a way of being sexed differently, it’s feasible to think of the number of sexes as usefully complex, in the sense that they have coördinates on a graph, or perhaps in a multidimensional space. However, collapsing this to the number of sexes being two, it means that that number is a real number rather than an integer: there are not 2 sexes but 2.0 of them. It’s difficult to talk about this while being sensitive to people’s feelings, but also important because of the emotional dimension of meaning. This is never going to be about cold numbers to some people because of their own identity and the way the world has treated them. Nonetheless, I am going to talk about the number of sexes as if it were two.

The pentaradiately-symmetrical M`ubv had five sexes, which I did in fact name: female, carrier (the one who gets pregnant or lays eggs), male, hermaphrodite and gynandromorph. The last is particularly significant as regards symmetry because a gynandromorph is often a bilaterally-symmetrical animal, such as an insect, who is female on one side and male on the other. For an animal with five-fold symmetry there are a large number of possibilities here. Assuming two sexes, there seem to be thirty-two possibilities, and assuming three (including carrier) there would apparently be 243. These would include hermaphrodites, but the number is still rather large. Given this arrangement, it isn’t so much that there need to be five sexes for successful sexual reproduction as that different sectors of the body would have different genitals of the three kinds involved: that is, they wouldn’t be symmetrical in that aspect. This also means that the genitals couldn’t be in the midline of the body, or in this case the axis of symmetry. Also, it isn’t as simple as there actually being 243 or thirty-two sexes because some of them would be effectively identical to each other. Looking at them as binary integers, the sexes 11000, 10001, 01100 would all be the same, only differing in the sense that one might be born upside down compared to the other, although since internal organs are often far from symmetrical it could correspond to the locations of the genitals relative to the organism’s innards. Assuming they have a culture, it’s likely that they’d consider these things to be significant, or maybe that number of variations would simply make the distinctions seem irrelevant. The advantage of considering the sexes in this way rather than in terms of five different types of reproductive system or gametes is that provided there is a female and a male, or a female, carrier and a male, reproduction would still be possible and it doesn’t create enormous sexual overheads for a species likely to lead to their extinction. It’s also possible that whereas all these combinations exist theoretically, in practice they don’t, or that some are much more common than others. By this point it has ceased to be trivial to consider how many sexes there could conceivably (pun intended) be in this situation.

For a bilaterally symmetrical animal with the alternatives of a vulva or penis to one side of the plane of symmetry, there are four possibilities. This is because bilateral animals have a front and back to their bodies and a left and right side. If a triplanar animal with two possible sexual outcomes per sector existed, it (there is a pronoun problem here!) would not have a much higher number of possibilities due to its rotational symmetry. It would also have four possible sexes: two female sectors and one male, two male sectors and one female, entirely female and entirely male. Any other possibilities may be phantoms, as they would effectively be descriptions of the horizontal orientations of the animal rather than sexes or genders, although if there was a custom that certain triplanar individuals always moved with their single male sector at the back or their single female sector at the front, they would then be gender and the number would increase to a potential eight. Once the sectoral possibilities correspond to two sexes per sector in a pentaradiate organism, it gets quite a bit more difficult to work out. But of the apparent thirty-two possible sexes, there are a simpler number of types, such as purely female, purely male, a sexual segment separated by two of the other sex, a sexual segment separated by one, and so on. There are in fact eight sexes considered this way, some of which are complementary to each other which might make consummate mating between them easier. Unlike four, eight is in the Fibonacci series. There’s an interesting pattern here which amounts to how many different possible bit patterns there are per type of symmetry, and beyond that how many there are of higher number bases such as three.

The question remains of whether there could be any kind of link between the Fibonacci sequence and the number of sexes, or between that and probable external symmetries in living organisms. Most organisms on this planet have either 1.0 or 2.0 sexes, although such cases as eusocial insects arguably have more because they include ostensibly female individuals who are the worker caste or soldier versions who defend the colony. This could be imagined in a microcosm, where some kind of cosy “nuclear family” consists of a queen, a drone and a worker, and this could also be where the carrier comes in. If you introduce a separate carrier to the M`ubv the situation becomes quite confusing, although I would expect there’s a way of simplifying it.

In order to work out if there is a link, it might be productive to investigate why the Fibonacci sequence turns up so often in the first place. One cause, among plants, is that it leads to an optimum spacing of leaves to photosynthesise. A 1/φ of a circle is, rather pleasingly about 137.5°, though this is probably coincidence (where have I heard that before). This means that leaves growing out of the side of a stalk will be able to optimise their light-gathering power if situated at this angle relative to each other, which in turn means that a rosette of leaves or leaflets, that is, leaves situated in a flattened arrangement like a plantain, will also be optimised if they have a Fibonacci number of leaves. This explains, for example, why four-leaved clovers are rare compared to three-leaved ones. Even so, this is not directly encoded in the DNA by some gene which forces clover to have three leaves as opposed to two or four, but is actually caused by the point at which levels of plant growth hormone are lowest in a circular arrangement. It could be caused in other ways. For instance, if a plant stalk twisted 360° in a day and grew a leaf every fourteen hours and forty-nine minutes, it would end up with this kind of arrangement.

It isn’t clear to me whether this applies to animals, although logarithmic spirals do turn up all over the animal kingdom. I should probably explain about protostomes and deuterostomes at this point. The more complex multicellular animals can be divided into two superphyla: deuterostomes and protostomes. Deuterostomes develop their anus before their mouth and protostomes develop the mouth first. This is governed by the same genes working back to front in one taxon compared to the other. Incidentally, this means that the Jeff Goldblum/David Cronenburg movie ‘The Fly’ should’ve depicted Seth growing compound eyes on his buttocks, which seems even more Cronenburgian than the actual version. We’re deuterostomes and flies are protostomes. Other protostomes include molluscs and segmented worms, whereas other deuterostomes include arrow worms, acorn worms and sea urchins. There are other differences between the two groups, notably radial and spiral cleavage. A human zygote has radial cleavage. It splits in half down the middle, then the daughter cells split at right angles to the original cleavage, then those cells split in another plane and the intermediate result is a ball of cells where imaginary sections pass through the nuclei of the cells. Early deuterostome embryos can be separated into separate organisms up until the thirty-two cell stage, and they will develop into identical clones. This is alluded to in Brave New World, except that for some reason that goes up to ninety-six in the finished product, a process known as “Bokanovskification” in the novel, and I’ve never been able to discover whether that refers to a real person or not.

Protostomes are different. After the second division, the second plane of cells is rotated with respect to the first, and this continues in an arrangement where there’s a kind of crown of cells at one end of the embryo giving rise to daughter cells which seem to have somewhat different functions to one another as the generations proceed. This is called “spiral cleavage” because of the spiral arrangement of the cells in the nascent embryo, and there is no such plane as there would be in deuterostomes. Instead, there is an axis of symmetry. Due to this situation, clones cannot be produced in the same way from a protostome ball of cells, partly because the fate of each stem cell is fixed early on. If part of a snail embryo were to survive and develop on its own, it might become a heart, a piece of shell or an eyestalk, but it would never become a complete snail.

At this point I’m going to take an ignorant leap of faith and speculate that the spirals found in many protostomes, such as the way octopus tentacles roll up and snail shells curl round, are related to this spiral cleavage process, although since there are also such structures as rolled up fern leaves and ram’s horns in non-protostomes I may well be wrong. That said, my ultimate aim is to justify the idea of pentaradiate organisms with many sexes, and that’s science fiction rather than science. In any event, if the spiral cleavage process were to lead to some kind of flower-like animal, and these do exist though not among protostomes – crinoids, sea anemones and entoprocts are examples – it could well end up developing from an embryo growing in a logarithmic spiral. The signals involved in animal development could resemble those of plant growth. This could then quite easily lead to bilateral, triplanar, pentaradiate and octoradiate animals whose planes of symmetry are in the Fibonacci series in a direct mathematical link, in the same way as a daisy has a Fibonacci number of rays (“petals”) or a three-leaved clover has that number of leaves.

The oddity here, if this is the case, is that the only pentaradiate phylum is deuterostomal – the echinoderms. Nor is it at all clear why they have this symmetry, although it’s been noted that an odd number of sides means that weak edges are counteracted by solid plates on the opposite side, in for example sea urchins. The problem with this is that triplanar symmetry would probably make their structure even stronger, and although there have been triplanar animals they all died out more than five hundred million years ago.

But what if there is another way in which an animal could develop that did involve spiral cleavage and ultimately led to a pentaradiate body? Kind of like a molluscan version of an echinoderm. Here, five-fold symmetry develops where in each sector the fixed fate of stem cells includes those which will eventually become sectors of the reproductive system, leading to an adult with two different possibilities in each plane of symmetry. If development were anything like it is in humans, and it may well not be, that would mean different hormones being present to modulate the development of the organs in different directions. It needn’t be like that though, because different organs end up at the same level in different parts of the human body, such as the liver on the right and the stomach on the left.

Just one more thing about Fibonacci numbers in the living world. Certain things probably are related to it, such as the fivefold symmetry of dicotyledonous flowering plants, so the inside of an apple with the seeds in a pentagram-shaped arrangement, the fivefold transverse symmetry of a banana, which is a monocotyledon and could be expected to have different symmetry, and possibly also that of echinoderms does seem to be connected. But another major example, of the five digits on the limbs of many vertebrates including ourselves, is more questionably relevant. The trouble is that we tend to see patterns where there are none. Insects have six legs, but that’s two times three. Is that a significant Fibonacci number? Likewise with the number of sexes: there just are two, and that may be all there is to it. On the other hand, that may be a kind of “stump” created in accordance with some relevant mathematical principle. Neither that sequence nor Lucas numbers are an explanation for everything.

Next time I plan to talk about how the way someone is embodied might influence their thought and language, using this as an example.