Ancient Symmetry

This post has an odd inspiration. A few days ago one of my FB friends changed her profile pic to the face of a cartoon character of some kind, whom I don’t recognise. Whenever I looked at her photo, it reminded me of the head of an animal who is probably very obscure, although I have no idea if this is true or not. My usual problem. It’s not really worth saying that a drawing of such an animal is shown above, and as you can see the head in this illustration does look rather cartoony in nature. There are better illustrations than that which show what I mean, such as this:

MEB shot of the chaetognath Sagitta
Date July 2006
Source Own work
Author Zatelmar

This works slightly better, as it shows the jaws more clearly, which to me look like long hair, although this requires them to be a bit blurry. The jaws give them their scientific name, but the common name is arrow worms.

Arrow worms, also known as chætognaths, are marine swimming animals who are members of a minor animal phylum. I want to explain this first. There are eight major multicellular animal phyla: arthropods, nematodes, flatworms, cnidaria, segmented worms, molluscs, chordates (us) and echinoderms. Then there are something like three or four dozen phyla which only contain a few species. Of these, one, the brachiopods, actually used to be pretty dominant, and another, the priapulids, is almost the smallest of all nowadays with something like eight species but used to be more common than chordates. Animal phyla each have a basic body plan of a particular kind. For instance, arthropods tend to have hard external skeletons and jointed legs, and for some reason lack cilia, which are the little hairs cells often have in other species. It’s yet another example of the 80:20 rule, as most phyla only contain a few species but a few phyla contain most species. Leaving aside sponges and certain other unusual animals such as placozoa and mesozoa, the metazoa, multicellular animals with organs, fall into a small number of superphyla, a couple of which I’ve mentioned before. These are the two-layered largely gelatinous coelenterates, including the cnidaria (corals, jellyfish, sea anemones and some others) and a smaller phylum called the ctenophores or comb jellies such as the sea gooseberry, a kind of basal group of primitive or simple-seeming animals such as flatworms and roundworms (which form another superphylum including animals such as rotifers), a group called the lophophorates, who are distinctive in having a “hand” of ciliated hollow tentacles which they feed themselves with (this includes the brachiopods and bryozoa), protostomes and deuterostomes. I’ve mentioned the last two before on here, but I may as well again. Protostomes, who include arthropods, molluscs and segmented worms such as earthworms, are called that because their embryos develop their mouths before their ani, and their zygotes divide in a spiral pattern with a kind of leading edge producing more specialised cells behind them. Deuterostomes have their mouths develop second, after their ani, hence the name, and divide by simple radial cleavage, and in chordates at least, which includes us vertebrates, at least up until the 32-cell stage can be divided into individual cells each of which becomes a clone. This principle was used in Brave New World. The genes that govern the development of one end of the body in protostomes govern the other end in deuterostomes, so the body plan of a human is an upside down version of that of a housefly and vice versa (which spoils a certain Cronenberg movie a bit but could also have been the making of it).

Because they’re closer to home, deuterostomes are interesting. They include two major phyla and a larger number of minor ones. The major ones are the chordates, including the vertebrates and sea squirts, salps, lancelets and larvaceans, and the echinoderms, including starfish, sea urchins, sea cucumbers, sea potatoes, sea lilies, brittle stars and a few others. Echinoderms are the smallest major phylum and are bloody weird compared to the others, because for example they have pentaradiate symmetry and something called a water vascular system which controls suction organs and means they can only survive in sea water. You’ll have noticed the title for this post so I’ll go into their symmetry.

Although most zoologists believe chordates evolved from echinoderms, a few think it was the other way round. Sea lilies share a particularly striking feature with chordates. They have a stalk which supports the rest of their body, which has tentacles and of course a mouth and anus. Chordates, particularly vertebrates, are notable for having ani or cloacæ which open before the end of their body, and frequently have tails. This may be due to the two being related, but it’s not clear whether the sea lilies, or rather their ancestors, are the ancestors of vertebrates or if chordates gave rise to sea lilies. Whichever way it was, there seems to be an intermediate form between echinoderms and chordates which is asymmetrical and has a tail-like stalk, thus:

Description
English: Cothurnocystis
Source
Own work
Author
Haplochromis

Whether or not they’re our ancestors or our descendants, so to speak, is quite significant. If we are descended from these, our bilateral symmetry re-appeared after a phase of pentaradiate symmetry. If, on the other hand, they are descended from us (as in chordates), they represent a branch which lost its bilateral symmetry and then evolved pentaradiate symmetry. The second version is clearly more parsimonious, but wouldn’t it be interesting if the first version is what really happened? It could then mean either that the genes leading to bilateral symmetry are latent in early echinoderms but not expressed in the adults or that it’s a good idea to have a bilaterally-symmetrical body or that pressure makes that more probable. The homalozoa are nowadays not considered a clade because some of them seem to be sea lilies.

As I mentioned before, there are several minor phyla among the deuterostomes, notably the acorn worms. If I remind you that glans is the Latin word for acorn, you might get an impression of what they look like. Priapulids are a little similar, though unrelated beyond the fact that they’re also animals.

Then there are the arrow worms. As you can see from the diagram, arrow worms are somewhat like lancelets, so it might be thought that they’re deuterostomes. They’re kind of like little fish, and for a long time now they have indeed been considered deuterostomes, which of course fish are too. They’re almost completely invisible, making it easy for them to swim up to and pounce on plankton and small crustaceans with their “claw jaws”, hence the name chætognatha. They’re ecologically important as they constitute food for very many species, including fish and whales. In the oldest zoology textbook I have, which is Victorian, they’re considered a kind of roundworm, which they obviously aren’t but science marches on. As soon as I found out they were thought of as deuterostomes, it made sense to me. They do seem like our kind of animal more than crustaceans or leeches. However, it turned out that the truth is considerably more interesting.

Arrow worms, although traditionally classed as deuterostomes, are in terms of their biochemistry closer to protostomes. Also, their claw-like jaws bear some resemblance to lophophores. Therefore, it’s possible that they are in fact close to the ancestors of most bilaterally symmetrical animals, that is, ancestral to protostomes, deuterostomes and lophophorates, and they even have things in common with roundworms, so they might even be related to those. Roundworms, incidentally, are bilaterally symmetrical at one end and radially symmetrical at the other.

There is a concept, in zoology, of the Urbilaterian, that is, the last common ancestor of all bilaterally symmetrical animals. Arrow worms don’t fossilise well so it’s difficult to find out much information about how old they are or how they evolved from that source. However, it’s long been known that there are mysterious isolated fossils of tiny jaws called conodonts, and it’s now thought that they are probably arrow worm mouthparts. These date back to the end of the Precambrian, so if they are arrow worm mouthparts it indicates that the phylum is indeed extremely old. The earliest known bilateral fossil is from the Ediacaran, the 94-million year period after the snowball earth period known as the Cryogenian when glaciers reached almost to the Equator, an animal called Kimberella who lived 550-odd million years ago. It used to be thought that the flatworms, who also don’t produce many fossils but do leave fossilised eggs in their parasitic forms sometimes, were ancestral to the other bilaterians, but in fact they don’t seem to have appeared until shortly before the evolution of the dinosaurs, although they may have existed since about the time the first vertebrates evolved. This leads to the controversy of whether the Urbilaterian was a simple or complex animal. It is known that it had eyes, because genes controlling eye development are similar in all bilaterally symmetrical animals, although they develop at opposite ends of the body in the protostomes and deuterostomes. However, it isn’t clear that all the compounds they contain were connected to vision originally, because they could be more to do with controlling physiology or behaviour at different times of day or month. It may or may not have had separate anus and mouth – flatworms only have one digestive opening and the fact that protostomes and deuterostomes use opposite orifices suggests otherwise. Likewise, the nervous system may have been organised into chains of ganglia or diffusely distributed throughout the body. None of this is clear.

I personally think it may be significant that it lived fairly soon after the global ice age. It isn’t clear either whether the glaciers actually met in the middle or if there was liquid ocean at the equator when this happened. The earliest known sponges date from this period, and there were also the type of amœbæ who had shells, and since these now protect them against predators it suggests there were also organisms who ate them, which might mean there was a food chain with at least three stages. It’s possible, I think, that the pressures caused to life by the Cryogenian may have stimulated evolution, possibly via a mass extinction freeing up niches, so maybe it was the ultimate cause of the evolution of bilateral animals.

It would be remiss of me not to mention another form of symmetry, shown by these Ediacaran animals:

Unknown attribution

Yes, there were triplanar animals at this time and no other: animals with triangular symmetry. They didn’t prevail over the bilateral animals, but the question arises of whether they could’ve done. Is it just a stroke of luck that this form of symmetry didn’t come to dominate the way bilateral symmetry did, or is there a disadvantage which isn’t clear from this body plan? It happens to some extent among flowering plants – monocotyledonous flowers sometimes have threefold elements in their parts although not actually triplanar symmetry. Is there some disadvantage to these which isn’t obvious?

Our daughter has my brown curly hair, chin cleft and blue eyes. Other apes have opposable thumbs and complex brains. Other chordates have muscle blocks and a notochord at some time in their life cycle. Other deuterostomes have stem cells which commit themselves late. More broadly, with the exception of echinoderms who are descended from them, the majority of animals are bilaterally symmetrical. This shows how we are all related and that all life on this planet is family, with the same kind of obligations as we have to other family members. This is a particularly wide family resemblance.

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.