Diciadain sa chaidh dh’fhàg sinn an clas tràth airson a dhol gu taisbeanadh mu Ro-aithris na Luingeis aig a’ Chrichton, le còrr is ceud neach. Tha an neach-labhairt, Charlie Connelly, air a bhith dèidheil air a’ chraoladh o chionn fhada, mar a tha gu follaiseach air mòran dhaoine eile. Bidh seòladairean agus iasgairean ga chleachdadh gu mòr ach tha a’ mhòr-chuid de luchd-èisteachd nan leabaidh agus tha e gan socrachadh agus gan cuideachadh a’ cadal. Gu neònach, tha e math airson seo a chluinntinn gu bheil uisge trom agus gaothan a-muigh air a’ mhuir, agus tha e dha-rìribh! Tha e ag obair dhòmhsa.
Bha Mgr Connelly glè mhath air gun a bhith sgìth ach chan eil mise agus mar sin cha bhith mi a’ bruidhinn airson ùine mhòr. Thòisich an t-Àrd-mharaiche Fitzroy, caiptean a’ Bheagle, air ro-aithris na sìde airson shoithichean san naoidheamh linn deug agus b’ e còd Morse agus teileagrafaireachd a’ chiad ro-aithris luingeis. Chaidh an cleachdadh an toiseach air an rèidio ann an 1924 agus bha iad cho cudromach is gun do rinn iad an ro-aithris fhathast nuair a bhàsaich an Rìgh agus dhùin an rèidio airson caoidh.
Bha Connelly airson tadhal air tìr anns a h-uile sgìre mara far an robh tìr. Chan eil fhios agam an do shoirbhich leis leotha uile, ach chaidh e gu eilean Danmhairgeach ann am Bàgh na Gearmailt far an robh ionad bhùthan an rud as inntinniche. Air Utsire bha sgioba ball-coise glè shoirbheachail a bha an urra ri an luchd-dùbhlain a bhith tinn mara nuair a chluicheadh iad nan aghaidh. Tha iad a-nis air an toirt gu bhith a’ cluich air tìr-mòr Nirribhidh agus chan eil iad air geama a bhuannachadh bhon uair sin. Thadhail e cuideachd air Sealand, ann an Linne na Tamais, a tha na àrd-ùrlar a tha ag ràdh gur e dùthaich neo-eisimeileach fhèin a th’ ann. Is dòcha gun deach e gu Lundy cuideachd, far a bheil càl ro-eachdraidheil. Cha b’ urrainn dha faighinn gu Rockall. Tha Rockall trì cheud cilemeatair bho Shòaigh anns na h-Eileanan Siar agus bha e na phàirt de Siorrachd Inbhir Nis. Tha e còig ceud seasgad seachd cilemeatair bho Inbhir Nis, agus tha sin na shlighe fhada ri thighinn gus na bionaichean fhalamhachadh. Chuir an t-Arm dithis shaighdear agus bogsa geàrd air airson beagan mhionaidean. Tha Èirinn ag ràdh gur ann leotha a tha e, agus tha Innis Tìle agus na h-Eileanan Fàro ga iarraidh.
Is e ‘Sailing By’ le Ronald Binge an ceòl airson an Shipping Forecast agus chaidh a sgrìobhadh airson bailiùnaichean èadhair theth.
Mu dheireadh, chleachd an nobhail agam ‘Unspeakable’ na sgìrean mara Shipping Forecast mar shiorrachdan, ach is e sin sgeulachd eile gu litireil.
Bigger On The Inside
“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:
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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.
The Self-Similarity Of Land Or Pareidolia?
If you’re British, have good eyesight and are more than about fifty, the chances are you will remember this station ident. Back then of course, nobody knew the words “station ident” even though there were about eighteen of them, mainly for ITV companies, in this country. They’re an interesting topic in themselves, but not one I want to cover today.
What I want to illustrate probably works best with a short video clip:
As humans, we have a predeliction for discerning patterns everywhere, even where those patterns are not significant. This clip used to trigger that in me, but what I’m not clear about is whether they’re important. The black and white image emphasised the contrast for me, with black oceans clearly outlining the white continents, and it seemed to me that the Atlantic Ocean looked like a kind of triangular face looking west, which was also quite similar to how the Indian Ocean west of the Indian subcontinent looked. But of course, this could just be nothing. Maybe younger children are more likely to pick up on patterns compared to adults, but it’s still with me whenever I look at a map or globe of the world.
There are other examples. For instance, Afrika and Madagascar and the Indian subcontinent and Sri Lanka are both roughly triangular landmasses tapering towards the south with a somewhat rhomboid island to their southeast. Corsica and Sardinia are also somewhat similar islands. South America and Afrika, as well as fitting together due to having both been part of Gondwana in prehistoric times, are again both roughly triangular landmasses tapering towards the south. The same patterns repeat over and over again on the map. Another one I noticed recently which probably is spurious is the apparent similarity between the coastline of Europe and that of southwestern Great Britain. In this case, South Wales corresponds to Scandinavia, Land’s End to Iberia, the Lizard to Italy and the Isle of Wight to Crete. I’m pretty sure this one is nothing though, particularly when one considers the proportions.
All that said, one of the sources of fractal mathematics was the “Coastline Problem”. This was based on the realisation that measuring a coastline would vary according to the length of the ruler you were using. Great Britain on the crudest scale is roughly triangular, with a very approximate perimeter of 2 800 kilometres. A “ruler” able to produce a very crude but recognisable map of this island, with a length of about a hundred kilometres, yields a coastline of about 3 500 kilometres. According to the ‘CIA Factbook’, the perimeter is around 12 429 kilometres. At this point, one might find that if a ruler a mile, nautical mile or kilometre in length were to be used, different figures would be arrived at, so whereas it says 12 429 kilometres or 7 723 miles, which is correct as far as converting units is concerned, if the coastline had actually been measured twice, once in miles and the other time in kilometres, the results would not be convertible and the length in kilometres would’ve been greater. The same applies to millimetres, only much more so. One of the results of this is that it’s entirely possible to come up with similar figures for the length of the coastline from Aberystwyth to Hayling Island and Vingsand in Norway to Θεσσαλονικη by judicious choice of the right measuring sticks. Alternatively, even with the same yardstick the lengths could in theory be the same, and this is particularly plausible when comparing a fjord-rich coastline with a particularly smooth one. Also, if one considers the Mandelbrot Set, mini-versions of itself are found all over which are somewhat “morphed”, with larger regions minimised and smaller ones maximised, and this could happen to a coastline, but if that kind of thing’s allowed, it appears that anything could be made to fit.
The processes leading to the formation of the islands of Sri Lanka and Madagascar are entirely different. Madagascar is the result of rifting in the Afrikan continental plate causing it to calve off to the side like an iceberg, though one attached to the ocean bed of course. Sri Lanka has existed since Precambrian times in that position relative to the subcontinent and was joined to the land by an isthmus until recently, and is also in the centre of its plate. Nonetheless they both look superficially similar. Is it possible that there is another factor involved which leads to this kind of similarity?
As far as I can tell, although there are other continental archipelagos such as Indonesia and Japan, none are very similar to the islands I’m currently sitting in. That said, comparisons have been made between Japan and Great Britain in other ways, comparing Hokkaido and Scotland on the one hand and Wales and Shikoku on the other. Kyushu is also compared to the Six Counties on this map, but there isn’t an extra bit of Kyushu to be taken into consideration, and Scotland is no longer a different island to the one with England on it, although it was in the geologically distant past. Moreover, Sakhalin could be thought of as part of Japan geologically although there is no similar large landscape north of Great Britain which fills that rôle. That said, there are many similarities between Japan and Britain. Both have languages which are written non-phonetically under the influence of a powerful continental neighbour, both have a sense of reserve as part of their national character and both have an official state religion. To what extent, though, is this cherry-picking? This is what I’d like to get to the bottom of in all of these.
Probably the largest example of this in Earth’s geography is the fairly minor similarity between Afrika and South America. This is of course helped by the fact that they both used to be joined along an entire coastline before the breakup of Gondwana, but there are other factors. Both are roughly triangular continents straddling the Equator with a bulge in the north and a consequential human-impenetrable region in that bulge. This last point is stretching it a bit because hot deserts and tropical rain forest are opposite ends of the spectrum regarding rainfall, and the picture is complicated by the presence of rain forest in the Congo. There’s a possibly rather fruitless question regarding which bits of Afrika and South America correspond. The Amazon is in a sense the Congo of South America and there is no corresponding huge desert, although there are deserts in southern South America, none of them are that similar to the Kalahari. The Atacama is much drier and Cabo Polonio is a cold desert. The Andes are also very important for South America and there is no corresponding range in Afrika. One thing they do have in common is a thin piece of land linking them to a larger continent to the north. One might expect all of these similarities would have led to similar histories and ecologies, but it isn’t clear that this has happened. For instance, the Sahara Pump, if it happened, led to a distribution of the Afro-Asiatic language family across the Sahara and to the south about to the level of the Horn of Afrika and also into Asia in the form of Arabic, but nothing similar happened in South America. There are anteater-like animals on both continents in the form of anteaters themselves in South America and aardvarks in Afrika, and pangolins and armadillos are somewhat similar, but I don’t think there are any Afrikan “sloths”. Both of them, though, have been subjected to colonialism, although only Afrika lost people to slavery abroad due to imperial powers. Both have Hispanophone, Lusophone, Francophone and Anglophone nations, but South America is dominated by Portuguese and Spanish to a much greater extent. A lot of the differences are to do with South America being situated further to the south than Afrika, although there is of course a big overlap.
It’s conceivably instructive to look at Venus and Mars with flooded lower altitudes to see if the same kind of land forms can be identified. This is Venus:
and this is Mars:
There is a fairly clear problem with comparing either of these with Earth. In fact there are several. Both of these maps are based on the idea that 71% of the surface is covered in water as it is on Earth at the moment. However, this doesn’t mean they’re proportionately similar. That is, given that Mars is half Earth’s diameter, a proportionate amount of water would be about an eighth of ours, but that wouldn’t provide 71% cover, and on Venus, which is slightly smaller than Earth, the cover is provided by far less water because there’s less variation in altitude, possibly due to melting mountains (that’s me, not science). Taking the Martian map first, the planet is fairly neatly divided into highland and lowland regions in the south and north respectively, and again this is not a scientific judgement but I think of Mars as consisting of a single continent plus a single ocean due to its size not allowing for anything more complex. Also, the terrain illustrated depends on the absence of plate tectonics. Tharsis in the west of that map is a complex of huge volcanoes caused by a hot spot which doesn’t move and has caused a build up of a massive plateau heavy enough to have cracked the land to its east, which is the blue channel referred to as Noctis Labyrinthis followed by the great canyon, here below sea level, called Velles Marineris or Mariner Valley. Also clear is the large depression Hellas, visible on the eastern side and possibly the antipodes of Tharsis. Having a thin atmosphere, Mars shows obvious craters in the highland “continent” which wouldn’t be there if it genuinely had large bodies of water and rain eroding its surface. Therefore there isn’t really anything similar to the kind of land shapes found here.
Turning to Venus, at first glance the planet looks much more like Earth. The eastern side of the northern continent even looks rather like Siberia. However, there are many more approximately circular “islands” than there are here and the oceans are very different, being much shallower and lacking the oceanic ridges characteristic of Earth with its tectonic plates and continental drift. There are also a lot of archipelagos including relatively large islands and intermediately-sized masses of land between large islands and small continents.
Both these maps show no sign of water erosion, and there having been no continental drift the land is not like it is on this planet. Water also propels plate tectonics. It might be more informative to turn to Titan, since that alone in the solar system has both land and bodies of liquid on its surface like Earth, and it looks like this:
Like Mars, Titan’s size must be borne in mind. It’s a little smaller than Mars but its composition is very different, having substantial quantities of water ice in its make up. It’s difficult to work the consequences of this out because we’re used to water ice near its melting point, whereas Titanian water ice would be mixed with rock and therefore kind of “muddy”, and also much harder due to being well below the temperature of our South Pole in midwinter. Nonetheless, the solid surface material on Titan is eroded in a similar manner to how water erodes our rocks:
Those are pebbles, possibly of water ice (frozen water) which have been rubbed smooth by liquid ethane and methane. Titan always faces Saturn and takes just over a fortnight to orbit the planet. It’s also a lot colder than Earth and its atmosphere is thicker and therefore carries heat around the moon more effectively. Therefore, even though the surface temperature of Titan is below -180°C, it behaves as if its climate is tropical, bearing in mind also that the boiling point of methane on Titan, with its higher atmospheric pressure, is -155°C. Moreover, although we tend to think of temperature as a linear scale, it often makes more sense to see it as exponentially colder and to scale it relative to the temperatures we’re used to, because absolute zero, -273.15°C, is actually infinitely cold as it can never be reached and it always takes the same amount of energy to halve the temperature. Titan is only 5°C warmer than the melting point of methane but since its surface temperature is around a third of Earth’s, that’s equivalent to our whole planet being at 15°C. However, water is also a highly unusual substance because it expands when it freezes, which is not unique as bismuth and gallium, for example, also do it, but there are unlikely to be any celestial bodies with oceans of bismuth or gallium. Water also turns white when it freezes, which reflects heat and therefore tends to produce a feedback effect, cooling it further. Erosion and weathering from ice are therefore different from erosion would be from frozen methane. For instance, water can seep into rocks, expand on freezing and force rocks apart, and naturally glaciers can carve out U-shaped valleys and fjords. Titan would not have fjords or that kind of erosion. Nor does it have continental drift because that too requires water, meaning that there is more opportunity for erosion to smooth the surface without it being replaced by volcanism, which is also stimulated by continental drift. There are faults on Titan, but they don’t follow continental plates and there may also be volcanoes, but nothing like the “Ring Of Fire” around the Pacific Basin on Earth. Consequently something like the island of Madagascar cleaving away from Afrika won’t happen, although that doesn’t mean there wouldn’t be islands off larger landmasses on such a world.
I don’t know about you, but I find that map of Titan difficult to read. I presume that the darker patches are exposed liquid and the lighter patches land, but I’m not sure. This set of pictures, however, may make things clearer:
Despite what I said, that looks quite fjordy to me. There are also rivers running into it. This is Ligeia, a lake about a third the size of the Caspian and is the second largest body of liquid on the moon. This should be put in perspective in that in terms of proportion of coverage it’s more than four times larger and therefore bigger than the Caspian by scale. The largest body is Kraken Mare:
Kraken Mare has an approximate area of half a million square kilometres, making it almost twice the size of the Caspian in absolute terms but still only a small fraction of the size of the smallest ocean, the Arctic, even in proportion to the size of the moon. The essential difference between Titan and Earth is therefore that whereas Earth has land surrounded by continuous liquid, Titan has liquid surrounded by continuous land, meaning that comparisons of land forms are only meaningful for islands in its lakes and seas, although the shapes of the lakes and rivers are more meaningfully compared between the two worlds. The rivers of Titan seem to be much more subject to tributaries than those of Earth, which may be what gives the moon’s lakes and islands their fjordier appearance. The peninsula jutting out into the lake near the bottom left does have a larger headland to it rather than just being a finger of land, which is superficially like Iberia except that in that case the land used to be an island that collided with the rest of Europe. On the whole, it doesn’t look that familiar though. It also occurs to me that the density and viscosity of a mixture of liquid methane and ethane could be somewhat different, although the molecular weight of methane is very close to that of water. Gravity must also be a factor.
Without another Earth-like world to compare it to, it’s difficult to say what’s happening and what forms are likely or unlikely, but the considerations I’ve had to make here might narrow down the conditions somewhat. They seem to include the following:
- Relative quantity and depth of liquid.
- Coverage of surface. I’m guessing that more than fifty percent of the surface must be covered for there to be reliable continents and extensive islands.
- Plate tectonics. A lot of what’s visible on the surface of this planet is a manifestation of continental drift, which allows land to be rebuilt via volcanism and islands to surface, submerge and so forth.
- Density of liquid.
- Weight of liquid. This is not the same as density because it’s related to gravity.
- Coefficient of expansion. There may or may not be any fjords on worlds whose oceans are not made of water.
- Difference in density of liquid and solid components of the surface.
- Hardness of solid components.
- Thickness of the atmosphere – a thinner atmosphere would lead to more craters which would probably flood, although they would also be eroded quite quickly.
I haven’t been able to come to any conclusions yet about this. I can see that a supercontinent might fracture into roughly triangular continents, that island chains form from both continental drift and volcanic hot spots moving around relative to plates and various other things, but to be honest I’m no closer to being able to decide whether the apparently similar land I saw on the BBC globe in the early 1970s was mere pareidolia or a pattern which exists independently from human perception.
A Box Of Chocolates 