Bigger On The Inside

Will be removed on request

“Dimensionally transcendental” was initially a cool-sounding phrase mentioned by, I think, Susan Foreman in the first episode of ‘Doctor Who’. It meant “bigger on the inside”, and definitely sounds like technobabble. TARDIS stands, as we all know, for “Time And Relative Dimensions In Space”, but even in the Whoniverse this is probably a backronym because why would something from Gallifrey have an English initialism? I think most people who think about it would probably say that Susan came up with the abbreviation, which probably explains why it doesn’t make much sense.

The BBC, and also Terry Nation’s estate, are quite protective about their intellectual property with respect to ‘Doctor Who’, which has led to a couple of disputes over the use of the likeness of police boxes and the word “Tardis”. Therefore I’ve posted a picture of a Portaloo up there instead of a Tardis or police box. In 2013, the portable toilet hire company Tardis Environmental came into dispute with the BBC over the use of the word, which was registered as a trademark by the Corporation in 1976. The BBC claimed that the company might end up seeming to be endorsed by them, to which they responded, “we don’t roam the universe in little police boxes from the 1930s, we actually hire out portable toilets and remove waste.”. I think we can all be grateful to them for clearing that up. I suppose it does make sense that the taboo against human excrement is not a positive association for this word. There was also a dispute with the Met. In 2002, after six years, the BBC won a case against the Metropolitan Police who took them to court over their use of the police box in ‘Doctor Who’ merchandise because they claimed that since they were responsible for the original boxes, it rightly belonged to them. I think I’ve seen two or possibly three police boxes, in Glasgow, Bradgate Park and London, this last being the one I’m least confident about, and I don’t think any of them look very like the Tardis. The one in Bradgate Park I’ve seen on a regular basis, and looks like this:

This is a listed building and is apparently still in use. It doesn’t look like a Tardis to me really but it’s a nice shade of blue. It’s 9 646 metres from where I’m sitting right now. The one in Glasgow is rather further away. It was the Met against which the BBC won the case, but the Tardis props are clearly wooden, a different shade of blue and have different windows, at least compared to the one I’m familiar with, so it seems a bit unfair. To be honest I don’t understand why this dispute even happened. It was between two publicly-funded bodies, I think, and seems to be a bit of a waste of money and time. Even if it was BBC Worldwide or BBC Enterprises, the Met was still involved.

Anyway, this is not what I came here to talk about today, but the concept of dimensional transcendentality. I’ve previously mentioned the fact that extremely large spheres are appreciably larger on the inside than their Euclidean volume because space is non-Euclidean – parallel lines always meet, at a distance of many gigaparsecs. This is possible because Euclid’s Fifth Postulate is based on observation rather than axiomatic or deduction, and the observation turned out to be incorrect. A sphere whose radius is equivalent to that of the Universe’s has a volume of five thousand quintillion (long scale) cubic light years, but if it were to be considered a sphere in Euclidean space, its volume would be only four hundred and twenty quintillion cubic light years, a difference of a dozenfold. This is quite counter-intuitive and I’ve ended up checking the calculation about five times to ensure it’s correct, but it starts to indicate how very confounding to the human mind higher dimensions really are.

I want to consider three cases of curved shapes in hyperspace to illustrate what I mean. Well, actually one of them is rotary motion rather than a literal curved shape, and I’ll go into that first. Here’s a circle with a dot in the middle:

(I’m drawing all of these in a ZX Spectrum emulator because Chromebooks rule out the use of more sophisticated graphics programs as far as I know). The circle can be rotated around the dot, so in a sense that dot is the “axis” of rotation of that circle. Now consider this as a cross-section down the middle of a sphere:

This is an axis of symmetry and also of rotation. Spinning the sphere through which this is a cross-section would lead to it turning round this line, which would be the only stationary part of the sphere just as the point is the only stationary part of the circle. Geometrically speaking, these are infinitely thin and infinitely small, so it’s rather abstract, but in the real world the closer you get to the centre of a spinning circle or sphere, the less you’d move.

Now consider the hypersphere, i.e. a four-dimensional version of a sphere: that which is to a sphere as a sphere is to a circle. If that rotates, doesn’t that mean its “axis” is a circular portion of a plane bisecting it? Can we even imagine something rotating about a two-dimensional axis? Also, just as two-dimensional objects have lines or points of symmetry and three-dimensional ones lines or planes of symmetry, surely that means that four-dimensional ones can have solids of symmetry? A hypersphere could be divided into two hemihyperspheres along a central sphere touching its surface, and since it’s symmetrical in that way, just as points on or in a sphere describe circles when they spin, doesn’t that mean line segments on or in a hypersphere would describe spheres? I find this entirely unimaginable, but is that a failure of my three-dimensional imagination or a flaw in the idea of hyperspace. It’s probably the former but this brings up a surprising recent finding about the nature of the human brain, which is that small cliques of neurones form which are best modelled topologically in up to eleven dimensions. No, I don’t really understand that either.

This hints the nature of hyperspace is very counter-intuitive, which isn’t that surprising really. Another issue is that of the torus. This is a Clifford Torus:

And this is a flat torus:

By Claudio Rocchini – Own work, CC BY 2.5, https://commons.wikimedia.org/w/index.php?cur
id=1387006

Travelling across the surface of a torus, one would find oneself disappearing off the top or bottom of a map and appearing on the bottom or top of it, or doing the same at the right and left hand sides. This is not like a cylindrical map projection of a planet, where the poles are either at an infinite distance or one would traverse horizontally exactly half way across the map and appear 180° of longitude away vertically but do the same as on a torus horizontally. With a four-dimensional torus, one would be in an apparently three-dimensional warped space forming the analogue of its surface, which you might think of as a cube with linked opposite faces, but the faces could be linked in different ways. One of the dimensions could be like a spherical map, with the concomitant traversal near the faces, or two of them could be, so there seem to be at least two different four-dimensional toroidal analogues. I confess at this point that this may not be what the above two animations represent.

The third problem relates to what ‘Doctor Who’ calls dimensional transcendentality, and it’s this which I’ve only recently heard about, from Numberphile. To illustrate this, I’ll go back to the Spectrum:

These are supposed to be four circles fifty pixels in radius touching each other. Now the question arises of what the biggest circle fitting among those four would be. The answer is quite straightfoward because squares can be drawn around each circle whose diagonals touch at the centre of these four circles. If you think of each circle as having a radius of one, the diagonal of the containing square has a length of the square root of 2² +2², or roughly 2.8. The radius of the circles is one, so subtract that from 1.4, or half the length of that diagonal, and you have 0.6. In other words, the square root of two is involved.

If you then extend this into three dimensions and imagine eight spheres stacked together in a similar manner, there’s a bit more room. The hypotenuse of a right angled triangle from the centre of an outer sphere to the inner one’s is then the square root of the sum of the squares of the three sides, which is root three, so the radius of the inner sphere is just over 73% of the outers’. This makes sense intuitively, for the last time, because it’s easy to understand that the diagram above shows a cross-section of the equators of all the spheres and therefore the minimum space between them, so a larger sphere is possible than one with the same circumference as the central circle in two dimensions.

The radius of the hyperspheres at the centres of analogous arrangements in higher dimensions is always going to be one less than the square root of the number of dimensions involved. At four dimensions, the central hypersphere’s radius is one less than root four, also known as 2-1, which is one, so rather surprisingly perhaps, it’s possible to fit seventeen equally sized spheres into a hypercubic arrangement. At five dimensions, the central “sphere” is actually 23% larger in radius, as root 5 minus 1. This is actually nearly three times the size in terms of a five-dimensional “bulk”, if that’s the right word. At nine dimensions, even the radius is double that of the surrounding hyperspheres, which makes it five hundred and twelve times larger altogether. There’s no limit to the increase in radius at all. I find this highly counterintuitive.

Moreover, these sphere analogues don’t even occupy the whole space. What does is a peculiar pointed shape which starts off like a square with concave sides in two dimensions (whose bottom point I’ve accidentally cut off) and a kind of inwardly-curved octahedron in three. In three, it has to be greater than the area of the largest circle in six different directions. In four, it resembles a concave version of a cross polytope, which is the higher-dimensional counterpart to the octahedron. Cross polytopes always have twice the number of vertices as they have dimensions, whereas measure polytopes, also known as hypercubes, always have twice the number of faces as dimensions.

Now consider a nine-dimensional stack of hyperspheres intersecting with our three dimensional space at one of its equators, with the centres of the hyperspheres aligned at the vertices of a nine-dimensional measure polytope. This would appear to be a stack of eight spheres, so this can be simplified by cutting off the outer spheres and converting them to hemidemisemispheres, if that’s the word, stacked together. Similar slicing could occur in hyperspace. So, it’s converted to a cube, then you put a door in the middle of one of the faces of the cube and find that it opens into a space which is quite a bit larger than the volume of the cube. The trimmed cube is only an eighth of the volume of the original, but it contains a “sphere” which is four thousand and ninety six times larger. With a mere four dimensions this becomes a mere eight times the size. This is starting to sound very like dimensional transcendentality.

The term has two words in it. “Dimensional” is fairly straightforward if one sticks to a simple definition instead of the non-integral dimensions used with fractal geometry. “Transcendental” brings to mind transcendental meditation, which is probably one reason for using it along with the fact that it was also used to refer to a particular set of numbers. What, then, are transcendental numbers?

A transcendental number is defined as a number which is not the root of a non-zero polynomial of finite degree with rational coefficients. The numbers e and π are both transcendental. All such numbers are irrational, that is, they cannot be expressed as the ratio of two whole numbers, since all rational numbers can be expressed in the way transcendental numbers can’t. Π is sometimes approximated by such values as 22/7, but these are not accurate values of the constant in question. Since the value is in fact involved in calculations of these volumes and hypervolumes, there might be a way of including the word “transcendental” in the description of this property of being “larger on the inside”. The square root of two is involved in two dimensions, but that’s merely irrational and not transcendental because it can be expressed using algebra – it’s a square root. This also means that the method of calculating the volume of a central sphere within a stack of hyperspheres is not transcendental either, so a good bet for including the concept would be to use π instead.

Although I can see that π is useful in calculating the surface area of the shape between the spheres, I don’t know what this thing is called. There’s a gallery of similar shapes here but they don’t include this one. I find it hard to believe this thing neither has a name nor has been extensively studied. I can assert various things about it. Its volume is greater than the largest sphere it can contain. It’s also greater than six times the spheres which can be placed touching the equators of the spheres it can occur within. I don’t know if the central sphere overlaps with its neighbours in the points. Each of its eight curved surfaces has an area equal to ½(πr²), meaning that its total surface area is equal to a sphere whose diameter is equal to the length of its largest diameters. Similar criteria apply to its higher dimensional friends. Hence I could perhaps be allowed to say that it’s dimensionally transcendental because its volume or hypervolume, or the volume of its hypersurface in higher-dimensional space can be calculated using the transcendental number π. And it can be, as I will now show.

Up until now I’ve been describing the central spheres and hyperspheres as if they’re three dimensional, and it is possible to lodge three dimensional spheres in there if you want, although it would be rather a waste of space. However, the actual volume of a four-dimensional space is not its bulk but its surface. I’m going to consider this nameless shape as having a length of two units, which is the same as the cube it’s found inside. The surface area of a sphere is 4πr² and the circumference of a circle is 2πr. If it just carries on like this, it makes the volume of the hypersurface of this shape in four dimensions 8πr³ (spot today’s deliberate mistake with the volume of a sphere half the size of the Universe, incidentally). This means the volume of this shape is a bit more than twenty-five cubic metres, which is equivalent to that of a cube 2.9 metres on a side. For a nine-dimensional version, this would be over eight hundred cubic metres, which is a nine-metre cube. That’s about the size of a three-story house.

The TARDIS is of course bigger than that, although as far as I can remember Nu-Who has never shown its real internal size. If the door was located at a point where it was at the end of one of the projections and located in three-dimensional space, it would be accessible to a three-dimensional being. In fact it could have up to six such doors, though if it had there’d be one in the roof and another underneath it, and there could also be two other doors opening into four-dimensional space. If, however, it had nine dimensions, it could have a total of eighteen doors, only a third of which would be accessible from normal space and the majority of which wouldn’t even open into four-dimensional hyperspace.

I think it makes more sense for the police box to be closer to a cube than just a cuboid, for the sake of neatness, so maybe the chamæleon circuit should’ve got stuck on the Bradgate Park police box after all, with two secret trap doors and two hypersecret doors for which there is no name because they’re ana and kata 3-space.

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.