The Thoughts Of A Starfish

Imagine yourself standing in front of a mirror. You know the deal, if you can see. Everything which is to your left is to your right in a mirror and vice versa. But in a flat mirror, your head doesn’t become your feet and if you lie down in front of it, your head and feet are at the same ends as in the reflection. Your back and front have swapped sides of course, and this is related to left and right swapping.

We’re usually roughly bilaterally symmetrical and our eyes are not on flexible stalks. However, our fingertips kind of are. It may just be me, but when I read braille on a box of pills, if the writing is at the back of the box it appears to be mirror writing. I presume this is how everyone experiences braille. From this it appears correct to deduce that if my eyes were on stalks and I were to extend one and look back at myself, I would appear to be seeing a reflection. I’m not sure there’s even a way for such a being to look at itself and see itself the “right” way round. Because we’re nearly bilaterally symmetrical, we consider ourselves to have a left and right, and therefore our perceptual world. Most writing is said to run from left to right or right to left, but some is vertical and some is written in various forms of boustrophedon, where the writing alternates direction and in some cases turns upside down as well. Motorists drive on the right or left side of the road and pedestrians do whatever is the opposite in their part of the world. Clearly, although it can’t be accounted for in geometry or other kinds of mathematics, we are able to consider our world as being the right way round compared to a reflection, but there isn’t anything which says which way round it is. There is no cosmic “THIS SIDE UP” sign or anything else like that.

It’s established then that we tend towards being bilaterally symmetrical animals with two forward-facing eyes which cannot swivel round and are set in sockets. It doesn’t seem inevitable that we would be like this though. Back to the M`ubv, as mentioned in yesterday’s post. These were my imaginary pentaradiately symmetrical aliens with five sexes, although they’ve turned out to have eight, I came up with when I was twelve. At the time, to me the most important difference between them and us is that we’re bilaterally symmetrical and they’re pentaradiate, like starfish. This means they don’t really have left and right sides to their bodies, or a back and front, and to be honest I’m happy to add eyes on stalks to their body plan to see what happens. They do, however, have a top and bottom to their bodies and this raises the question of where the food goes in and if and where it comes out when they’re done with it. Although echinoderms have an oral and aboral surface, with the oral corresponding to the bottom, for a land animal this would be inconvenient as it would mean they’d have to excrete through their heads and it would probably end up all over them. Then again, maybe their etiquette is different and they don’t mind. It still seems unhygienic though. Therefore I’m going to put their mouths and genitals at the tops of their bodies and their main excretory organs at the bottom. This will enable two of them to make love face to face regardless of the genders involved.

The question arises of whether they have strong concepts of left and right. If two of them are facing each other, one might say to the other, “it’s to your left” or “to my right”, but this would be a temporary situation which would change if the conversation was taking place in a different orientation. There’s an Australian aboriginal language which lacks ways of expressing left and right and uses something like compass directions instead. This would, I presume, work fine for the M`ubv and in fact would be less ambiguous, although at first it might be expressed as something more like “towards the mountain” or “towards the coast”. Giving directions would be different. The question arises of whether they’d even have words for the ideas, and also of what would happen with their writing. It isn’t easy to think of a way of writing which wouldn’t in some way be linked to notions of left and right. Vertical writing would still proceed across the page line by line and spiral or circular writing direction would always be clockwise or counterclockwise, at least within the line, or rather circle. It’s possible that concepts could be built up by superimposing or modifying existing characters, but even then at some point a new character would be needed and that would have to be situated elsewhere. When zoologists began to describe the anatomy of radially symmetrical animals, they found themselves introducing the terms “oral” and “aboral”, but these are technical terms and usages which aren’t part of most people’s everyday language. It might turn out that the interior of such an organism is asymmetrical, in which case perhaps “left” and “right” would become largely medical terms. There would also be hazier notions of front, back, forwards and backwards, and these would influence the way position was expressed in their languages. This would presumably go on to influence figurative uses of the same concepts. For instance, we think in terms of progress and setbacks. Would they? They would, though, share ideas of up and down, and of the tops and bottoms of things, upper and lower and so forth. Hence their vertical understanding could be directionally similar to ours, but horizontal understanding would be more like compass directions, refer to landmarks and would be able to express inner and outer and distance from the speaker, but would they even do that? Would their way of expressing other positional concepts influence those too?

This morning I said “I’m going to wash my glasses”, meaning all of the pairs of spectacles I use, and was acutely aware of how English lacks an easy way to express the dual as opposed to the plural. Many languages do have this facility, which they sometimes confine to items which are more often found in pairs but also sometimes more generally. Later stages of a language tend to use these only for the former. Were it common to have five members or organs rather than pairs, there would probably be languages with special inflections for five of something, but the question then arises of whether there’d be a paucal number too. Would there be separate forms for the singular, paucal (two to four items), quintal (five) and plural, or would there be forms for each of the numbers from singular to plural? In this situation, five would be the most frequent number other than singular and plural. English retains traces of the dual in words such as “alternative” and “either”. These might have special forms for five options as well as two, or might have three different forms: one for two to four, one for five and one for many.

Then there’s the question of grammatical gender and noun classes. Although Indo-European and Afro-Asiatic languages are characterised by a form of gender which distinguishes between females and males as well as other items in the same class, many languages, such as the Niger-Congo family, instead have a similar arrangement to grammatical gender which, however, doesn’t always make a distinction between the sexes. The situation here is that there are eight biological sexes. A possible origin for Indo-European grammatical gender is that women have historically tended to be referred to by what they are seen to be while men have tended to be referred to by what they are seen to do. Hence the old system of having separate classes for adjectival and agent nouns has turned into something referred to as grammatical gender but not necessarily having much to do with gender in the social sense. For the M`ubv, this would depend on whether a similar cultural tendency existed for all eight “genders”, plus perhaps the inanimate. On the other hand, variation of that type might lead to a situation where there wasn’t much distinction between them. If the association did occur, though, languages with at least nine genders would be common, and these could extend beyond the species. This would have the advantage of reducing ambiguity because there could be separate pronouns and grammatical forms, if this is what they had, for up to nine separate referents in the same phrase. Considering pronouns, this means that for each person there could easily be thirty-six distinct pronouns, and in the case of the second and first the question of clusivity and differently-gendered groups could also arise. This might also extend to verbs.

All of this would seem intuitive. So far I’ve been assuming that these are kind of five-sided human beings with eyes on stalks because it makes things simpler. In fact, these concessions to humanoid appearance would probably be unrealistic. The question also arises of what different insights this species might have compared to us. Although it can be conjectured that they might lack quotidian concepts of left and right, the chances are they would have additional concepts which we lack, and this brings me to the embodiment thesis, which is significant for us too, and presumably any embodied being.

So far so science fictiony, but there is a more mundane point to be made here. A popular philosophical slogan which I even use myself sometimes is, “I am my brain”, but in fact I’m not. I am more likely to be either less than or more than my brain, both for social reasons and because I am a body. Experiments have shown that if a subject smiles or frowns artificially, they tend to get positive or negative sentences more quickly respectively. Logical behaviourism attempted to claim that verbal thinking was nothing other than sotto voce vocalisation. If you open your mouth and think the word “bubble”, you will tend to get a sensation in your throat, which is said to be caused by the slight movement of the speech muscles. These are both aspects of what’s known as “embodied cognition”. If getting there is half the fun, the distance probably appears to be shorter, but if you’re on your way home from a hard day’s work it’s probably longer even if it’s the same route. We tend to outsource mathematical thought when we count on our fingers, and when we write notes we’re outsourcing our memories. There’s a sense, nowadays, in which our memory has become part of the internet, but this is relatively innocuous in principle considering that probably the first outsourcing of memory was the development of spoken or signed language. Chimpanzees are noted for having much better short term memories than humans, possibly because they don’t usually use our kind of language, although they do have their own signing to some extent. In linguistic terms, we talk about warming to people or being cold, or by contrast, cool. Our emotions partly depend on the physical sensations associated with them such as heartrate, blushing, shaking or breathlessness. This is one reason for suspecting that artificial intelligence would need to be embodied if it were at all humanoid, as emotions inform reasoning. We talk about being down, and oppressed, or oppressed, all of these being spatial metaphors. Looking (and there’s another sensory metaphor) at cognition in this way contrasts with the previous “computing”-type metaphors popular in cognitive science. If you’ve ever been subject to the comment “if we cut your hands off, you wouldn’t be able to talk”, that particular element of language will not escape you, and similar gesticulations, perhaps toned down in public, can occur with internal monologue, which is therefore not really all that internal. Pacing up and down is another aid to thought. Then there are the mirror neurons which activate when we do physical things ourselves and also when we experience others doing them. It also means that we may not in fact have a genuine impression of psychophysical dualism (a soul and a body) unless we already tend to live in our heads a lot.

Beyond embodied cognition lies enclothed cognition. A cis friend of mine once observed that she felt more feminine when wearing a dress, a phrase which is meaningless to me but I take her word for it. Clearly something like clothing sensitivity and the topic I addressed in this post makes the influence of such things very evident. Physically wearing a lab coat or a uniform can help someone adopt a genuine role – looking the part is important to the person who looks it.

We don’t know everything about how our bodies influence how we think because we are not often subjectively disembodied, although we can become depersonalised, as I often did. When it comes to contrasting a human body with a possible or actual non-human one, such as a dog with a much better sense of smell, or a bat or dolphin moving in three dimensions and using echolocation virtually as an extra sense, or for that matter a M`ubv or other hypothetical different body plan, there would still be aspects of such entities’ being which are inaccessible to us which make fundamental differences, such as the idea of left, right, front and back, and perhaps in the case of bats and dolphins even top and bottom to some extent, and all sorts of other things. And we have deficits ourselves which they do not have, such as the superior chimpanzee short term memory. Put all these together and the very concept of intelligence seems to have holes in it, and from a vegan perspective this is quite positive. But it still makes me wonder what obvious, and for others’ intuitive, aspects of reality we’re missing out on.

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.