Restoring Pluto And Elevating Cynthia

I was going to blog about the larger asteroids at this point, but in recent days it’s been borne in upon me that there’s a current issue in astronomy, perhaps over-emphasised but definitely there, over whether Pluto was unfairly demoted. The reason I mention this now is Steve’s comment about what the difference between Phobos and Deimos and asteroids might be. It’s a very good question and I’ll address this first.

Phobos and Deimos, the moons of Mars, are a little puzzling. There are two hypotheses about where they come from. One is that they’re main-belt asteroids which were captured by Mars. At first glance this sounds very sensible and logical. After all, Mars is next to the asteroid belt, it could be expected to gather up a few stones from it from time to time and the pair seem to be only the latest representatives of a whole series which have scarred Mars with chains of craters as they broke up and impacted. However, there are problems with it. Firstly, the common type of asteroid found near the edge of the belt closest to Mars is different from the type of asteroid Phobos and Deimos would be if they are asteroids. That type is found near Jupiter. This is due to the inner belt being warmer than the outer belt, so the composition differs because temperature makes a difference to them. Secondly, both moons have almost perfectly circular orbits over the Martian equator, and if they were captured, they would usually have come in at a high angle to the equator and have markèdly elliptical orbits. This can be seen with Nereid, Neptune’s third largest moon, and Saturn’s moon Phoebe orbits backwards compared to most other bodies in the system. Therefore, if Mars’s moons are asteroidal in origin, something needs to be evoked to explain that. A simpler explanation would be that they emerged from the cloud which was forming Mars. This would be spinning in the same plane as any moons which formed from it, and if they were formed in situ they would be more likely to have almost circular orbits. However, as Steve astutely pointed out, the actual nature of the bodies themselves is very close to being asteroidal, and in fact is asteroidal, so maybe it doesn’t matter in most ways. In the sense of the physical nature of the two moons, they basically are asteroids. The way in which they aren’t is to do with their history and orbits, which may not be a sensible thing to focus on. The only thing which goes against this is that both are directly affected by orbiting Mars. Phobos has streaks because of the tidal forces of its planet, and Deimos accumulates fragments and dust from itself as it moves through its rather short orbit. If they were orbiting in the asteroid belt itself, neither of these things would be happening. All that said, I can totally see the argument that they are in fact just asteroids in an unusual place which are also moons rather than minor planets. So I agree with you Steve.

This connects to a wider issue which affects Pluto, and it also affects a number of other worlds in the system which if addressed could solve the problem of knowing what to call the big round things in our Solar System. It could also address the peculiarity of our own “moon”. The 2006 CE definition of a planet by the International Astronomical Union is:

The IAU members gathered at the 2006 General Assembly agreed that a “planet” is defined as a celestial body that

(a) is in orbit around the Sun,

(b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and

(c) has cleared the neighbourhood around its orbit.

This definition was motivated by the discovery of a number of relatively large trans-Neptunian objects. Eris, discovered at the start of the previous year, has now been established to have a diameter of 2326 kilometres and a mass of 1.6466 x 10²² kilogrammes. Sedna, discovered in 2003, has a diameter of around a thousand kilometres and an unknown mass because unlike Eris it seems to have no moons. Sedna is less of a threat to the status quo but Eris was initially thought to be larger than it has now turned out to be. For comparison, Pluto is 2376.6 kilometres and it has a mass of 1.303 x 10²² kilogrammes, so it’s actually slightly larger than Eris but also less massive, so the question arose of whether it would be acceptable to admit a potential host of newly discovered planets, thereby reducing the “specialness” of planets, or to invent a new category. This last idea, of “dwarf planets”, seems very odd to me because the category of “minor planet” had existed for a very long time up until that point and instead of inventing an entirely new class of object, it would’ve made more sense, if they were going to do this. Whether or not I agree with the decision, there seems to be no merit in creating a whole new category of “planet” when “minor planet” already existed. I honestly don’t know why they did this.

Many people have disagreed with the decision to demote Pluto. It did elevate Ceres, previously considered a mere asteroid, at the same time. Before that point, for most of its history since discovery Ceres was considered an asteroid, but it’s the only body in the asteroid belt which has managed to make itself round due to its own gravity (there might be other bodies which just happen to be round-ish through chance because asteroids are irregular and could hypothetically be many shapes, including spheroidal), so it probably does deserve special recognition.

In spite of this definition, which is quite unpopular, a paper has recently been published on the subject arguing that Pluto, among other worlds, does in fact merit planethood. The paper can be found here. It’s sixty-eight pages long and I haven’t read the whole thing but the general gist of it seems to be that there used to be a scientifically arrived-at understanding of what a planet was, but over a period in the early twentieth century when astronomers focussed more on what was happening outside the Solar System, the popular uneducated public understanding of what a planet was took over. I have to say this doesn’t reflect my perception of what happened based on my knowledge of astronomy. I’m aware of the controversy about the canals, the discovery of Pluto, the idea that Mercury always faced the Sun and so on, all ideas which resulted from astronomical research at around that time. I’m aware of the research that was being done at the time about stellar evolution and the realisation that there were other galaxies, but it really doesn’t seem like they were concentrating that much on that more than this Solar System, but anyway, that’s what this paper claims.

Further, it claims that because they adopted a kind of folk understanding of what a planet was, it had led to them adopting earlier, non-scientific ideas about it. So for example, the public was really into astrology and had only recently got used to the idea that the Sun was at the centre of the Solar System rather than Earth. The authors of the paper give examples of how scientific classifications differ from public ones. For instance, most people think of fruit and vegetables as two different things but when it comes to botany, vegetables include fruits, which are the reproductive organs of plants, so from a culinary viewpoint fruit and veg are separate but scientifically they aren’t. To this I would add a couple of things which are I hope relevant to astronomy. One is that I think of a lot of things as fruit, such as tomatoes, aubergines, courgettes, peppers and tomatoes, which other people seem to think of as vegetables because it makes sense to me to think of them nutritionally and in terms of flavour in that way. The other is that the culinary arts are also sciences, and it seems a bit hierarchical to see them as inferior to botany for some reason. After all, we all need to eat. Applying that to astronomy and planets, that would mean that although some things are planets and some things aren’t according to astronomers of a particular vintage, that doesn’t mean there isn’t another branch of science which would view them differently. For instance, everything is subject to the laws of physics, and geology would seem to apply pretty much equally to planets, moons and asteroids in their own way. They’re just bodies in space like everything else. Therefore, I’m not convinced about this. Also, the general public were specifically irritated at the idea of Pluto not being a planet any more, so I don’t see how exactly they were using the public view of what planets were if they managed to annoy so many non-astronomers with their assertion that Pluto wasn’t one.

What seems to have happened is that the problem crept up on astronomers and they kind of panicked and made a fairly slapdash and hasty decision. As various large bodies were discovered on the edge of the Solar System, they became uncomfortable with the idea that they were probably going to end up with a very long list of planets, which seemed unwieldy and not very “neat”, and they also perceived it as an imposition on education that people were going to have to learn about so many worlds. They seemed to feel like this would be regarded as off-putting. The paper compares the situation with how mammals are defined. The official definition of a mammal is now rather abstruse, because it actually hinges on how many bones are in the jaws and the ears, but this is partly because of the need to identify fossil mammals. The widely-used definition is “animals who suckle from their mothers as infants, maintain a different body temperature from their environment, are often covered in fur or hair and mostly give birth to live young”, and the first criterion is the most important. There are exceptions to most of these. For instance, some hibernating mammals don’t keep their body temperatures above their surroundings and humans, whales and elephants are largely hairless, but this is a fairly good definition. However, claim the authors, astronomers have taken a weird approach to planets, having concentrated on whether they dominate their local region, which is in any case vague because what’s local? They’ve also looked at how they move. If mammals had been defined in this way, as warm-blooded vertebrates who walk in herds on land for example, a lot of mammals would’ve been excluded. Bats and whales would then not be mammals and any mammal who has a largely solitary life, such as leopards or sloths, would not then count as mammals either.

Looking at the history of the idea of planets, for a long time any round object in the sky which didn’t appear to stay in the same place was a planet. This used to include Cynthia and the Sun, when people thought Earth was at the centre of the Universe, and it didn’t include Earth. Later on, the four largest moons of Jupiter were discovered and also referred to as planets, and even the thick parts of the rings on either side of Saturn due to the poor quality of telescopes at the time. Later still, Ceres was called a planet because it seemed to fit into Bode’s Law, and turned up where it was expected. By that time, however, the known satellites had been relegated to moons, and soon after Ceres was also demoted because it was realised that there were thousands of other bodies between Mars and Jupiter, some even quite large.

The 2006 definition also has a rather silly consequence which a few people have noticed: it means Earth isn’t a planet! As I’ve mentioned before, from the Sun’s perspective Cynthia doesn’t orbit Earth, but the two weave in and out of each other’s orbits. I’m not completely clear what the astrological influence is supposed to be, but I think it’s the emphasis on orbits, i.e. the kind of definition which would’ve excluded bats, whales and leopards from being mammals. Whatever the definition of a mammal is, it seems to make more sense to use their anatomy and physiology than other more dubious criteria. Both of the definitions I mentioned above do this. The first is rather abstract and strange to most people, although there are good reasons for it – mammal jaws and teeth survive better than the rest of their bodies so it’s like identifying a body by dental records – but both of them focus on what their bodies are like, which seems entirely sensible compared to that fictional other definition.

What, then, is proposed as a more sensible definition of a planet? Well, it’s closer in spirit to that way of defining a mammal. A planet is a geologically active body. I have to admit I’m not sure about this because of various things, such as “eggshell planets”, and I’d also want planets to be round and I can’t tell if they also stipulated that. What it means (I’ll get back to eggshell planets in a moment) is that Pluto’s Sputnik Planitia which is created by frozen nitrogen and is active even though the Sun isn’t strong enough at that distance to have that effect. In talking about asteroids, I’ve mentioned the fact that the larger ones tend to be layered like Earth is, but the smaller ones are either rubble piles or mixtures of different minerals and other substances which aren’t separated out in the same way. A geological process has done this sorting in the larger ones, and consequently Ceres, for example, could count as a planet: it has been geologically active.

This applies also to some moons. Io, the innermost large moon of Jupiter, is intensely active with continual volcanic eruptions, to the extent that it’s thought to “turn itself inside out” every few years – some much of its interior is spewed onto the surface that the former surface becomes the interior and proceeds to get thrown out itself a few years later. This is because of the tidal forces effectively “wringing” the moon all the time, with the other large moons in the Jovian system along with Jupiter itself wreaking havoc on the place. By this standard, Io is definitely a planet, albeit a planet which is also a moon.

I’ll now permit myself a digression into eggshell planets. An eggshell planet is a surprising kind of “planet” which kind of “does nothing”. It isn’t necessarily possible to tell from a distance which planets are like this. Earth’s crust is divided into plates, and other planets have a thick, solid layer all the way round, but there is another possibility or which at least three examples may have been found already. This is where the crust is thin and fragile, and so cannot have plates or thick layers, and also can’t even support mountains or hills, so the surface is solid and also smooth, and nothing happens there – no volcanic eruptions, continental drift or erosion, because there’s nothing to erode. The question arises of whether this even counts as a planet under this new definition, since it isn’t geologically active. However, there are no such planets in our Solar System as far as anyone knows, and they’re probably quite rare.

There are three categories of planets suggested in this new definition: terrestrial planets; giant planets; satellite and dwarf planets. The last category is the largest. It includes the large moons of Jupiter, Ceres, Titan, Pluto, Charon, Eris and Sedna, and in fact there are more than a gross of these. Far from the expected response, apparently people tend to be quite excited at the idea that there are so many planets around the Sun. The giant planets include Jupiter, Saturn, Uranus and Neptune, so no surprises there, although this clear-cut division may be an artifact of how our own Solar System is, with its complete absence of the very commonest type of planet, the mini-Neptune, intermediate between Earth and Neptune in size.

There are five planets in the terrestrial category rather than four, because once the criterion for dominating its orbit has been removed, Cynthia becomes eligible, which makes me very happy! Cynthia is not even in the same group as the satellite and dwarf planets, but a planet just like Mars and Mercury. This also means that the Apollo astronauts landed on another planet, not just our moon. As well as that, Earth now has no moon!

It seems that the process leading to the decision to redefine planets was not very scientifically grounded and was in fact rather acrimonious. The orbital dynamics people took umbrage at the geophysical definition and there were only a few days available for debate, forcing people to take sides quickly without due consideration. Planetary scientists were underrepresented because they’re apparently not officially astronomers, which is a bit astonishing. Another motivation was to keep the number of official planets low because the IAU didn’t expect the alternative to go down well with the public because previously, i.e. in Victorian times, they’d felt more comfortable with a small number of planets. They were used to seven at that point, including the Sun and Cynthia. This is probably no longer the case, so in 2006 they made a decision based on misjudging the mood of the general public.

To finish, I’m going to make a commitment. Henceforth I will be referring to every spheroidal body in the Solar System as a planet, although I will also acknowledge what kind of planet it is, such as a moon or dwarf planet. And Pluto is a planet!

I’d be delighted to hear your views on this.

The Ring Earth

Unknown property rights – removed on request

Intellectual property rights unknown. Please contact for removal.

Not to be confused with Ringworld or yesterday’s post, this is about whether a doughnut-shaped planet could exist, but just to clear that up, Ringworld was a concept thought up by Larry Niven for his ‘Known Space’ series of a megastructure consisting of a ring-shaped terrain orbiting a star and given day and night by rectangular shades orbiting further in. It would require as yet undiscovered materials, in other words unobtainium, not to be confused with unobtainium, to be built, although a more diffuse ring of habitats or indeed planets in a single orbit is entirely feasible. This is not that.

Relatively small bodies in this Solar System are representative of the possible shapes which can be achieved by given amounts or quantities of solids, liquids, gases and composite matter made thereof. The fourth state of matter is entirely different but planets are not made of plasma, practically by definition. The smallest approximately object gravitationally obliged to be round is of course the Death Star moon Mimas:

Even this isn’t that close to being perfectly round because of the relatively huge crater Herschel, but it can also be seen that it has a noticeably rough outline in this picture. It’s allowed to have fairly large mountains and deep craters. Mimas is around four hundred kilometres in diameter, although it deviates by about twenty from this, but that’s still pretty round when considering that the deepest trench in our ocean is almost twenty kilometres lower than the highest of our mountains, compared to sea level which is not perfectly spherical itself. Mimas is in absolute terms as close to being round as Earth is, given that our mountains, valleys, trenches, continents and abyssal plains were not scaled down on a Mimas-sized model of our planet.

The next Saturnian moon down from that size appears to be Hyperion:

So far as I can tell, just as Mimas is the smallest roughly round world in our Solar System, so is Hyperion the largest object which is a long way from being round at 360 x 266 x 205 kilometres. It’s within about thirty kilometres of Mimas’s smallest diameter, yet it manages to be rather irregular. It looks more like a pebble of pumice than a planet, and of course it’s neither. Its mean density is only just over half that of water, which is actually lower than pumice, and also lower than that of Mimas, which is about 1.15 times water’s. There must be a complicated relationship between strength, rigidity and density which decides the shape of objects of about Hyperion’s and Mimas’s size.

This house has many Escher prints owing to my family’s joint enthusiasm for the artist. One of my favourites, which I’m sad to say is not on any of our walls, is ‘Double Planetoid’. This shows two intersecting tetrahedra, one completely unaltered by technology, the other completely covered in building. Of this, Escher says:

‘Two regular tetrahedrons that penetrate one another, float through space like a planetoid. The light-coloured one is inhabited by human beings who have completely transformed their region into a complex of houses, trees and roads. The darker tetrahedron has, of course, remained in its natural state, with rocks on which plants and prehistoric animals are living. The two bodies fit together to make a whole but they have no knowledge of each other.’

M C Escher

It would be possible to make a real copy of ‘Double Planetoid’ somewhere in space, at its approximate scale. In fact it would also be possible to scale it up to at least twenty-seven kilometres on a side if it were carved from granite, even if it had Earth’s gravity. However, it probably wouldn’t arise without intelligent manipulation being involved somewhere, except maybe in an infinite Universe or a parallel world somewhere. It would also not generally be possible for ordinary matter to exert sufficient gravity to make a real version of this rather than a model.

I bring this up because clearly an object like Hyperion could be sculpted into a particular shape, although in its case this would probably already be constrained by gravity so it might end up quite rounded off, but one like Mimas couldn’t. Again, there are likely forms something could take other than round, usually just irregular and lumpy, when they’re fairly small, and many of these are seen in asteroids and small moons. 433 Eros, for example, is often described as “sausage-shaped”:

(I’m not sure that’s how I’d describe that). The asteroid Cleopatra is one of several described as dumbbell-shaped:

Once an object is the size of a planet, though, the options for possible shapes closes down a lot. Considering this as an Earthling, the tallest possible cylindrical column of granite is said to be about the height of the Matterhorn, around half that of Everest at 4 478 metres, and the tallest possible pyramid of granite is 13 400 metres, on this planet. However, this can’t be strictly true because if the height of Everest and the depth of the Marianas Trench are added together the total comes to 19 882 metres, so given a wide enough base this can be exceeded. The diameter of the geoid – the shape of this planet defined by the level water would reach given only the influence of gravity and rotation on this planet, which approximately means sea level – varies by over thirty kilometres between the poles and the equator, which is again somewhat more than twice 13.4 kilometres. Twice is fine because we’re talking diameter rather than radius, but more than twice suggests there are other influences, such as rotation. Steel can, if I recall correctly, form a cylindrical column up to thirty kilometres high and there are a few specialised substances which could be used to build a tower which officially reaches into space, but they’re exotic and would have to be specially synthesised.

This full-disc image of Jupiter was taken on 21 April 2014 with Hubble’s Wide Field Camera 3 (WFC3).

The planet with the most obvious deviation from spherical in this Solar System is Jupiter, which has a polar diameter of 133 708 kilometres but an equatorial one of 139 820, which is a variation of 4.5%. This is because Jupiter is a substantially fluid body consisting of liquids and gases, and because it spins very fast. In terms of velocity the planet’s equator is moving over thirty times faster than Earth’s and it’s also over three hundred times our mass. However, Jupiter happens not to be the least spherical planet in our neighbourhood. That honour goes to Saturn, whose rings may disguise the fact. Saturn has an equatorial diameter of 116 460 kilometres and a polar one of 108 728, which is a variation of over seven percent. This may be connected to the fact that it’s also the least dense planet.

It’s established, then, that a planet can be tangerine-shaped rather than spherical if it’s sufficiently fluid. These two examples are also large and rotate fast. Earth is not like that, but it’s theorised that there are planets out there in the Universe which are mainly made of water, or which have extremely deep oceans. These could presumably assume such a shape, which on a planet the size of Earth would be like having a stationary wave almost a thousand kilometres high circling the equator, and in fact this would even be noticeable from the surface as it’s close to a gradient of one in ten. This could be described as a tangerine-shaped planet, but I have to say I don’t find that idea very interesting. The shape is officially called an “oblate spheroid”. There are stars which are markèdly flattened in this way, such as VTFS 102 in the Large Magellanic Cloud’s Tarantula Nebula, which is three times wider at the equator than at the poles, but stars are not made of solids, liquids or gases.

Another well-known variation of a spheroid is the rugby-ball shape, or prolate spheroid, and there are also stars of this shape. Some binary stars orbit each other so closely that they are mutually distorted into elongated shapes of this kind, and I don’t know this but it seems possible to me that this is also because of how fast they’re whizzing round. The question arises of whether a planet could have such a shape. Larry Niven, again, imagined such a planet in the Sirius system he called Jinx, whose “poles”, so to speak, were effectively vast plateaux rising out of the atmosphere at each end, and the “equator” was a high gravity area. The humans living near the equator needed to be very strong and muscular to cope. I don’t feel convinced that this is possible for a largely solid planet, but just as Saturn and Jupiter can get squished by their rotation I can see that if there is a system somewhere with a double gas giant, this might be what shape those planets would assume. The same might even apply to double deep ocean planets.

Other possibilities are very limited. For instance, an egg-shaped planet flat at one end and pointed at the other is difficult to envisage. However, there is one possibility which, oddly, is very far from being spherical but is still possible.

I’ve mentioned the periodical ‘Manifold’ on here a couple of times. This was a mathematical magazine published by the University of Warwick, one of my almæ matres, from 1968 to 1980, one of whose claims to fame is that it invented the game ‘Mornington Crescent’. I used to read it back then, and one of its many whimsies was a fictional toroidal planet whose name escapes me, with six cities all joined together by an underground railway. This is a reference to a well-known mathematical puzzle involving three houses all of which need a water, gas and electricity supply but none of the pipes could cross each other. This is impossible to arrange on a flat surface but works fine on a torus.

When I read this, I mused that it was a shame that such a planet could never exist, and I started working out things like I did above, with the likes of pyramids as very high mountains and various irregularities in its surface ruling it out. I then realised that I couldn’t actually find a reason for such a planet not to exist, and just assumed I didn’t have the mathematical prowess to work out why it couldn’t.

Well, it turns out that it can, in the sense that if a toroidal object of Earth’s volume, mass and composition could be formed in the first place, it wouldn’t be susceptible to collapsing into a spheroidal form. The above shape, surprisingly, is gravitationally stable. Incidentally this would also apply to water in free fall, so a spinning doughnut shaped swimming pool in space made entirely of water is completely feasible, under a pressurised atmosphere of course. There’s a fairly easy way of understanding this. It’s already been shown that tangerine-shaped planets exist, which are largely fluid and flattened by spinning. This is a kind of limiting case of that situation. If a fluid planet ended up spinning fast enough, not only would it become flattened but its matter would be completely pulled away from its axis of rotation. Most planets can be considered either to start out as fluid, i.e. they are either actually liquid, such as made of magma, or of sufficiently small lumps of matter that they behave on a planetary scale as if they were, just as an actual liquid consists of molecules or a heap of sand or dust can flow like water, have surface waves and even drown people. The difficulty is in imagining a scenario where this would actually happen on its own. The alternative is simply to say it’s being done by intelligent life but then the imagination falters a little as well, because how powerful would a civilisation have to be to have the resources to make its own planets? Also, why?

Nonetheless, however it came into being, once it was there it could continue as long as any other planet in its current shape, and this is a little surprising because the deviation from a sphere in this case is extreme. I also have to admit to a little confusion and have to insert an explanatory note. I can’t honestly tell whether this shape is sustainable simply due to the planet’s gravity or whether it would also need to be rotating fast with the axis passing “vertically” through the hole. I suspect the former is the case, but even if it isn’t, the second case would guarantee that it’s possible, although it’s not clear how fast it would have to be spinning. That would also depend on the proportions of the torus. Now for the explanatory note. Thus far I’ve assiduously avoided using the words “centrifugal force” because that doesn’t exist as such, as is well-known, but it can be quite awkward to express oneself without using those words. What is in fact happening in this situation is that the mass of the planet is constantly “trying” to move in an infinite number of straight lines, all tangent to its surface at the outer equator, but is pulled away from that path owing to the electromagnetic and gravitational forces holding it together.

It’s also very unclear how big this planet would be. According to the second picture at the top of this post, the north-south distance across Afrika is rougly equivalent to the width of the “tube” of the torus, the same distance for Australia is its thickness and the hole is about the same size as Australia again. Hence that version of a toroidal Earth is 3 000 kilometres thick, 7 000 kilometres wide on either side and has a hole 3 000 kilometres in diameter. This raises two questions for me about how to calculate the volume and surface area of a torus and also what to call the different features of the shape. Strictly speaking, the shape in the picture is not a torus because it’s not circular but elliptical in cross-section. The distance from the centre of the hole to the outer edge is called the “major radius”, R, and that from the centre to the inner edge is the “minor radius”, r. There’s also the aspect ratio, which is R/r. Strictly speaking, not only is the above not a torus (although the blue image is), but even if it was, it would only be a particular kind, namely the ring torus. There are also horn and spindle tori. A horn torus has its circular cross-sections touch at the centre, so strictly speaking has no hole, and a spindle torus has the circles overlapping. Both of these shapes are slightly more achievable for a planet than the ring in terms of events happening without intelligent intervention.

The formulæ for surface area and volume are respectively:

where p=R and q=r. This suggests several “equivalences”. One is the size of a torus with the same surface area as Earth’s, another the volume of such a torus, another the size of a torus with the same volume as Earth’s and another the surface area of that torus. All of these are also dependent on R and r, and thus the aspect ratio. I’m not going to address these immediately.

The torus in the second picture has basically the same continents as the real world, but Antarctica seems to be missing. In fact it can be concluded that the polar regions are missing altogether. However, there are two circles corresponding to the poles and of course two further circles corresponding to the Equator. Assuming the planet is held together by its rotation and doesn’t have constant daylight anywhere on its surface, i.e. not rotating with the hole axis facing the Sun, the “polar” regions are pretty close to lands which are equatorial on the real Earth in that image, although the other side has another circle which in the Arctic. Meanwhile there are inner and outer equators, and the outer passes through the Mediterranean. Assuming no axial tilt the inner equator is in eternal darkness and therefore colder than Antarctica, which would take a lot of water out of circulation and probably cool the whole planet. If it’s tilted at the same angle as we are, on the other hand, it would be exposed to sunlight some of the time and the “polar” circles would also have seasons, half a year of night and half of day and so forth, as they have here.

If this planet maintains its shape through rotation, there will probably be strong winds and ocean currents everywhere. There’s also an important topological difference between a spheroidal and a toroidal Earth. Topologically, considering the troposphere (the bit with the weather in it) as a single layer, there must always be at least two locations on Earth where there is no wind. This is not so on a toroidal planet because the hypothetical still spots could be lined up to be in the hole. Ocean currents are like this in the real world, because the land punches holes in the ocean in which potential still points could be located. If you go high enough in Earth’s atmosphere, the air is no longer dragged along by our rotation, so perhaps a toroidal Earth could have a relatively calm troposphere like ours is.

Apparent gravity would also vary across its surface. The rapid spin would act against gravity at the outer equator and in favour of it at the inner one. Some time ago, Alfred Wegener attributed continental drift to the centrifugal effect he called Pohlflucht. Arguably, as it depends on how rigid the planet is, Pohlflucht could be a reality on this world. Perhaps the continents actually would cluster around the outer equator. If they did, though, they would have to be quite mountainous to prevail over the water, which would be pulled into a belt in the same region. This, however, might actually be so because the lower gravity would favour higher mountains in that area. It seems to be shaping up into a situation where the inner region is a cold, flat desert, there are two strips of land either side of the outer equator along with a tendency for continents to move towards the outer equator where they form fold mountains which are, however, submerged under a deep ocean, which resembles the Tethys of our prehistoric past.

There needs to be a moon of some kind to generate a protective magnetic field. This could orbit at the outer equator. The toroidal magnetosphere thus formed would be a different shape than the real one.

From the surface, there are conventional horizons to the north and south, but to the east and west the vista depends on where you are. On the outer equator, the situation is pretty much as it is here. Near the “polar” circles, the planet is effectively flat along one circumference, and the landscape or seascape (snowscape more likely) disappears into the haze of the atmosphere. The inner equator offers the most spectacular view. During the day, the sides climb upwards into curved, hornlike shapes which gradually plunge into night, forming an overhead arc. At night, the situation is the same but there would be a visible daylit sector which would first recede up the horn, travel across the sky and then descend towards the observer until dawn. On this inner surface, gravity would be high, so the view might be nice but it would also be quite uncomfortable or even uninhabitable. I’m assuming here that there would be an axial tilt.

There’s a limit to the relative size of the hole. The narrower the ring, the less stable the planet. Both of the first two illustrations are viable, but a more traditional banded ring shape would be highly volcanic because it would tend to flex and crack under the forces maintaining its shape. Hence a doughnut shape is best. Even then the day would only last about three hours. A moon might move in a straight line in and out of the hole, or it could follow an ∞-shaped orbit.

The remarkable thing about this scenario is, of course, that it isn’t impossible. There could never be a tetrahedral or cube-shaped planet and the largest conceivable regular polyhedral planet would probably be something like a dodecahedron perhaps somewhat larger than Mimas but still much smaller than Earth or even Mercury, because the vertices would effectively be high mountains. In fact planets are in a sense polyhedral because they aren’t perfectly smooth spheroids on one scale, although on a smaller scale the jagged peaks and steep valleys would be rounded – this is a fractal issue, because on a smaller scale still they’d be jagged again, and so on. However, they are also very close to being smooth. As far as I can tell, the only possible shape a planet could be which is radically different from a sphere is a torus. What isn’t clear is whether it could ever happen on its own. I can easily believe that there are occasionally asteroids which have holes all the way through the middle, although as far as I know there are no known examples in this Solar System. A very rapidly spinning protoplanet could form into a torus, and the question then arises of what could cause it to spin so rapidly. Perhaps if it were high in iron and close to a neutron star this could happen, but it would be unlikely to be habitable. A non-habitable toroidal planet is unsurprisingly much easier to devise than a habitable one. However, given the will, the technology and the access to resources, nothing at all seems to stop an intelligent technological culture from making such a planet on a whim, or perhaps as a work of art. Isn’t that amazing?

Upside Down Maps

This map is not upside down. In fact, it retains a northern bias in a way I’ll soon mention. Down is obviously towards Earth’s centre around these parts, so an upside down map would be something like one of our core, or perhaps a mirrored projection of the surface. North is not at the top except on many maps. And as I said, this map retains a northern, or perhaps I should say Atlantic, bias, because it still puts Greenwich in the middle. To make it even fairer it needs to be rolled round a bit. This would be an example of a Pacific-centred map, with north at the top:

It’s surprisingly hard to find an example of a map which puts both south at the top and Australasia in the middle. One thing which I think this Pacific-centred map does is emphasise the impression that in terms of the human population, the Americas are in a sense the Far East, not the West. Humans spread out of Afrika, across Eurasia and then across Beringia (the land bridge between Siberia and Alaska) into North America, followed by South America, meaning that Native Americans are genetically relatively close to East Asians. It might even go further than that, because for example the abacus and paper existed in Mesoamerica in similar forms to the Chinese versions and there are cultural similarities around the Asian and American Pacific, such as the use of a two-pronged fishing spear and the cultivation of sweet potatoes on Pacific islands before European contact.

It has also been suggested that the Solutrean people of Europe colonised North America during the last Ice Age by walking across the ice, but genetic and linguistic issues rule this out. Although it’s been a respectable hypothesis in itself, it tends to be adopted by White supermacists, which is ironic because although the Solutreans were Western European they were also dark-skinned.

Putting Europe at the top influences how we think of the planet. In particular, people tend to talk about “Sub-Saharan Africa”. Afrika south of the Sahara is of course not literally under the desert and “sub” carries with it notions of inferiority, so I would prefer to avoid this. A similar phenomenon exists in the use of “cis” and “trans” to refer to different parts of the world with respect to locations in Western Europe, so for example we talk about Transylvania and Cisalpine Gaul, or Transnistria, when in fact for the people born in these places they are cis. Getting back to the North-South divide, this actually works to the advantage of this country because it means we can talk about “coming up to see” someone in York from the South of England, for instance, and it means there’s a sense of superiority about Scotland and the North of England which is absent from the South of England, although Southerners already feel superior. In Kent, we used to talk about people being from “under London”, meaning they were “furreners” from Sussex or Surrey, or perhaps just from the other side of the Medway. Kent, incidentally, holds the distinction of being the home of Greenwich, the origin of the Prime Meridian, except that in administrative terms that location became part of Greater London quite some time ago:

This means, of course, that Kent is almost entirely in the Eastern Hemisphere, a distinction it shares with Essex, Cambridge, Norfolk and Suffolk. Since the division in Kent is between East and West according to which side of the Medway one is on, the creation of Greater London means that West Kent is now sadly reduced in terms of area and population. However, this division is more a reflection of our imperialist past than anything “real”, as is our tendency to put North at the top.

What kind of planet is this “upside down” world? Well, it’s bottom-heavy in terms of land. There’s a large uninhabited wasteland of a continent at the top and most of that side is ocean, or rather, there’s less land. I remember hearing that the Southern Hemisphere has 20% of the land and 80% of the population, but whereas I think the former is so, I don’t think the latter is because there’s Antarctica and the sparsely-populated Australia to be taken into consideration. Even so, most of the land is definitely at the bottom in this map. This contrasts with Pangæa, which was mainly in the South. Things haven’t always been this way. This clustering has led to an almost landlocked ocean in the north, which is therefore less saline due to rivers discharging fresh water into it, and consequently more frozen over than it would otherwise be. The Sun doesn’t care that we think of Earth as this way up, and carries on doing its thing with El Niño and the rest, which is not tucked away down there mainly on the southern side of the Equator where it can safely be ignored. I wouldn’t want to overemphasise it, but it certainly seems to contribute to our blasé attitude that we’re able to think of that, for example, as happening far away in a place we don’t really need to think about.

El Niño, of course, is also an oceanic phenomenon, and that’s another bias we have in these maps: they’re very much focussed on land rather than the water. This makes a limited amount of sense given that humans tend to live on land, although it does also mean that Pacific nations tend to be ignored. Here is a way of looking at this planet which puts Aotearoa/New Zealand at the centre:

This still isn’t quite the same as centring on the oceans, because for example the Arctic on this map is relegated to the top left hand corner, but it does show quite effectively how much of the planet is covered in water. In fact it even comes close to showing the hemispheres of land and water, which used to be believed in very firmly in mediæval times and whose literal existence is instrumental in the formation of the British Empire.

I started this post with the Gall-Peters Projection, which was popular in the last couple of decades of the twentieth century as a kind of counterpoint to the imperialist-seeming Mercator Projection. However, it actually dates from the mid-nineteenth century, when it was invented by the Scots cleric James Gall in 1855, meaning that there’s been something of an intellectual property dispute. The idea of it being a fair representation of the sizes of land in comparison to Mercator is kind of a marketing ploy, because although it may do this, it’s been overhyped as a projection which shows the true relative sizes of the land on different parts of the globe. There was a time when other map projections looked distorted to me compared to Gall-Peters, and Mercator is still like that. It looks very top-heavy and the fact that it could be infinitely tall is quite jarring and absurd.

Arno Peters himself was not a cartographer, and professional cartographers have treated the projection he promoted with considerable disdain. Now the question here is how to balance the opinions of experts and the hostility towards the other. If an outsider to a profession devises something new and influential, it might well meet with a poor reception from within that profession, but on the other hand there’s the Dunning-Kruger Effect, that the more one knows about a subject, the more one realises there is to know and that one does not know. Not being a cartographer, all I can really do is report what real cartographers felt about his projection. Peters claimed that the projection preserved compass direction and distance when in fact it clearly doesn’t. Mercator’s projection actually does preserve compass direction, which made it very practically useful on long sea voyages when the European empires were being founded, and this does carry a lot of negative baggage with it in a similar way to institutions, streets and statues commemorating slave traders might, and it would be fair to reject Mercator simply for that reason – it’s a tainted projection in a way. However, it’s very obvious to anyone who can be bothered to look that a horizontal line on the Equator of that map of the same length as a horizontal line passing through Patagonia is not going to be the same distance, and that a 45° diagonal drawn between the Gulf Of Guinea and Buenos Aires might be close to a NW-SE compass bearing but the same angle between Sydney and the South Island of Aotearoa New Zealand is definitely not SW-NE.

From my experience in pressure groups of the 1980s, when this map became popular, there was a particular style of rhetoric which was quite unhelpful to the causes they tried to promote. I’m not expert in rhetoric by any means, but two aspects of it are λογος and παθος. The former uses fact to persuade and the latter attempts to elicit an emotional response. Particularly in the ’80s there was a tendency for groups such as Greenpeace (not the real group but the one which stole the name and is well-known) and CND to use fairly dubious factoids to present their case. One which comes to mind is a popular counterpoint to nuclear power focussing on the half-life of the waste as ” deadly” for however long that half-life is, which doesn’t make sense. It’s a big number used for rhetorical purposes, and yes it’s persistent in the environment and very harmful, but the half-life is not a measure of how dangerous it is. This is not an attempt to defend, or for that matter attack, nuclear power, but it seems unwise to make such claims when they only have a thin veneer of scientific respectability. It seems to me that the Gall-Peters Projection is of that ilk. It isn’t actually all that marvellous and is easily proven not to live up to its claims just by looking with a dispassionate eye. I don’t know to what extent this has changed since, but there often seems to be an inappropriate use of science, or in this case maths, following from an appropriate emotional response. For this reason I tend to argue from an emotive rather than a logical point of view, since this, not rationality, is the true root of our beliefs.

It’s said to be impossible to map a sphere onto a plane without distortion. This is technically untrue, although distance can’t be preserved in the case I’m thinking of. Imagine the globe divided into a large number of strips like a Chinese lantern and flattened out. There is no distortion along the midline of any of the strips by definition, but the distortion tends to increase towards the edges of all of the strips. However, the strips are also linked to each other along a great circle, along with there is again no distortion, and the strips taper towards their ends, reducing the proportionate length. There would still be distortion, but it would be smaller the narrower the strips become and the more of them there are. There is a limiting case with infinitely many infinitely thin strips which would be absolutely faithful. Another way of doing this is to draw a spiral beginning at one point on the surface, widening out to a great circle and diminishing again at the antipodes of the start, and again, the narrower this spiral becomes the less distortion there is until finally, and infinitely long straight line can be formed along which all the distances are exactly in proportion. Both of these maps are of course useless, and in fact even the projections which are not at the limits are not very useful because they constantly interrupt the surface of the globe. They might be useful for plotting out routes which are mainly linear, if a custom map for each route was generated anew.

Things get considerably easier if you abandon the aim of preserving direction and long distances. This can be done by projecting the globe onto a convex polyhedron in various ways. The role-playing system GURPS has used maps based on regular icosahedra, since it’s easier to work out distances in this way for dice throws. That produces a map which looks like this:

Other polyhedra are available, and if they consist of mixtures of faces they can approach the sphere more closely, but of course my personal favourite Platonic solid is the regular dodecahedron, whose net and map look like this:

It isn’t entirely clear which of these is closest to a sphere as it depends on what’s meant by that. Inscribed in a sphere, the dodecahedron occupies more of the sphere’s volume than a regular icosahedron at 66.49% to the icosahedral 60.55%. However, in relation to a sphere inscribed within an icosahedron that polyhedron is the most spherical as it occupies 89.635% of its volume. The midsphere of an icosahedron, which is the sphere touching each edge at exactly one point, is only 1.66% larger than the icosahedron itself, and the midsphere is important for map projections because it shares out distortions for areas whose diameter is smaller than the polyhedron and those whose are larger, thereby reducing the distortion. However, I always feel strongly drawn to the dodecahedron so that’s what I’m going to cover now.

I am not afraid of maths. That said, neither am I particularly good at it. I enjoy it but it isn’t my strongest point, and my opinion isn’t based on bad teaching, as it so often seems to be. I mention it here because I’m about to present a number of equations and do a bit of my own maths to address the question of a secant-based dodecahedral projection. This is probably best approached visually. The Mercator projection is what you’d get if you had a luminous globe wrapped in an infinitely long cylinder. You get this:

Actually that’s only part of what you get. What you actually get is an infinitely tall map, meaning that another very minor drawback of the Mercator is that you can never get the whole of the globe onto it. Equations germane to this projection include:

This is for angles expressed in degrees, hence the values 180 and 45. I’m not entirely enlightened as to the meaning of these equations, but the tan in the second must indicate that that’s the vertical component of the projection, as does the y, because tangent functions go to infinity. λ is longitude and φ latitude. The Gall-Peters projection uses these:

The difference between these two makes me wonder if I’ve got the first one right. A simpler version of these is x=Rλ and y=2R sin φ, which presumably means that the sine function squishes the latitudes and ensures they never get to infinity. I’m guessing here, but I imagine this is like putting a globe inside a cylinder and using horizontal knitting needles perpendicular to its axis to work out where to draw the map features, although if that’s true it surprises me that Afrika, for example, gets so spaghettified, so maybe there’s something else going on.

I would imagine that that attempt to set out the maths behind these has failed to inspire confidence. Nonetheless I will continue. Dodecahedra have the interesting and unique feature that an infinite number of lines can be drawn from any vertex back to itself without crossing any other vertex. This means that something approaching a great circle can be drawn in a large variety of directions. If the projection is from the midsphere, there will be twelve circles along whose circumference there will be no distortion of distance or direction. Assuming Earth to be a sphere with a diameter of exactly 40 000 kilometres, each pentagonal face would cover an area of 42 441 318.16 km². The total distance covered by the circumference of the circles where there is no distortion is something like 251 327 kilometres with spurious accuracy. They also extend to five edges on each face and are in contact with five other circles on adjacent faces, meaning that there is also no distortion at the five midpoints of the edges. Maximum distortion is at the vertices and the centres of each face, a total of thirty-two points. Nonetheless, aligning this projection such that the poles are at the centres of two opposite faces would place them in what seemed to be the correct places.

I wish I had the νους to follow this through to the end. The main point here is that this comes very close to being an undistorted flat map of the world, and in thinking about it another issue has been raised in my mind. Earth is in fact not a perfect sphere, and therefore a perfectly spherical globe is not an accurate representation of its surface either. The equator is slightly elliptical and it’s 12 756 kilometres in diameter at the equator (on average, presumably) and 12 717 kilometres in diameter at the poles. Each hemisphere is also slightly off from even that difference, meaning that the South pole is slightly closer to the centre than the North. There are also some other irregularities. Mapping this shape onto a sphere will distort it very slightly, and as far as I know globes, although they aren’t perfect spheres, are not designed to be geoid in shape (Earth-shaped). Another question is how much map projections onto flat surfaces distort the shape not because they’re flat, but because they assume Earth is perfectly spherical.

I feel that to some extent I’ve let you down because my mathematical abilities as they stand, along with the time available to me, makes it implausible to do the calculations to work out exactly how distorted this map is. Also, I’ve concentrated on a dodecahedral projection (of which there could be several which distribute the distortion in various ways) when an icosahedral one is more promising. Nonetheless I hope I’ve made it clear that sometimes maps are adopted for apparently rational motives when in fact they’re more to do with rhetoric, and my preference for the dodecahedron is also like that. So I’m just like everybody else really.