Saturday, November 08, 2014

A balanced world map: more than cute arts?

Frank Jacobs blogging for Big Think posted (and Sean Carroll reposted) a fascinating distorted map of the world drawn by Gaicomo Faiella:

What's so fascinating about it, Carroll asks? If you want to think yourself, please don't open the rest of this blog post, and/or don't read anything beneath the ads.

In fact, I can tell you that the map contains infinitely many coincidences or fine-tunings relatively to a generic set of continents you could draw according to the same template. Don't you still know what is special? Try to think harder and really don't read anything below the other ad. ;-)

What are those coincidences? The first one that many people notice is that the ocean has been suppressed and the continents were expanded – in such a way that their ratio is 50:50. You might think it's just approximately true. But it's actually exactly half.

And this exact coincidence is just one among infinitely many similar coincidences. The truth is that the map is actually completely symmetric – or antisymmetric, if you wish. Rotate the distribution of the continents and oceans, the image, by 180 degrees, and you will obtain this image. It isn't just a "rotated Earth": it is the same Earth with continents and oceans interchanged!

How does the symmetry work? Look at the picture once again:

Obviously, the boundaries have some kind of a \(\ZZ_2\) symmetry. But how does the symmetry act on the surface of the Earth? Look at Caracas, Venezuela. You probably don't know exactly where it is. OK, so find the Panama Canal – the thinnest point separating the Americas – and the nearby point where South America and Africa nearly touch. Find the center of this line interval. That's Caracas.

Caracas is a center of the world! Hugo Chávez would have been thrilled to read such a comment on a conservative blog.

If you rotate the map around Caracas by 180 degrees, you get the same (or "opposite") map. In particular, now you see that the South America is the rotated version of the North Atlantic Ocean! And Africa is the rotated image of the North Pacific Ocean! If you still haven't seen the pattern with your eyes, look at South America; at North Atlantic; at South America; at North Atlantic; at South America... do you already see it? ;-) Similarly, do it with Africa and the North Pacific.

And you may continue like that. The Mediterranean Sea has the same shape as the 180-degree-rotated Australia. And the North and Baltic seas between Europe and Scandinavia are the image of New Zealand. And so on. Feel free to find the image of the place where you are right now. But be careful: Chances are that the image will be in the ocean. :-) Central Europe ends up being some boring place of the sea South of Australia.

If you get familiar with these maps, you will notice that the 180-degree rotation has another center beyond Caracas, of course. It's Vishakhapatnam, India. OK, that word sounds complicated so what I mean is the center of the Eastern Coast of India. If you rotate the map around this place by 180 degrees, you get the "anti-same" map, too.

I have some rough idea how I could generate this "best \(\ZZ_2\)-symmetric distortion of the Earth" using a computer, in Mathematica. (The artist has probably created it manually.) But it could be somewhat difficult for me who has no experience with similar manipulations. I think that Mathematica can do many of the steps concisely, by a single command, but I don't know these commands.

Fine. So because this "slightly distorted Earth" has a \(\ZZ_2\) symmetry and it rotates the Earth by 180 degrees around an axis, it is useful to redefine the polar coordinates so that this axis plays a special role. In particular, let's use the term "New North Pole" for Caracas and "New South Pole" for the Visha-city. (I apologize to my dear Indian readers for shortening the name for the sake of the convenience of us, the barbarian non-Indians. My special apologies go to Balasubramaniam Sethuramanathan Sreejahannathan and his four brothers.)

Let's introduce the new latitude and the new longitude. Caracas has \(\theta=0\) and the V-city has \(\theta=\pi\). Note that on the (new) poles, the longitude is ill-defined. However, the \(\ZZ_2\) symmetry is generated simply by\[

(\theta,\varphi) \mapsto (\theta, \varphi+\pi)

\] So if the point \((\theta,\varphi)\) sits in the water, the point \((\theta, \varphi+\pi)\) is a point on the land, and vice versa! The existence of Feiella's image – and its relative proximity to the actual map of the world – suggests that this rule is "approximately" true for our mostly blue, not green planet (it is exactly true for the distorted half-blue, half-green planet).

Now, I think that a good question (OK, at least my question!) is whether there is any reason to think that this statement should be true?

My model will be weirder than an average science-fiction movie but of course I can offer you a mechanism that would produce a planet with this property. ;-) To prepare you, let's try something simpler. From the Earth, the Moon and the Sun look almost equally large. That's also a shocking coincidence, isn't it? (But in this case, it's just one fine-tuning, not infinitely many.) How could you explain this one?

Well, it's easy. Millions of years ago, the Moon was significantly larger. Or smaller. But there was a civilization on Earth based on a two-party system. One party wanted to preserve the full solar eclipses because these eclipses were a pillar of their religion. The other party wanted to sell the material from the Moon because the extraterrestrials paid a significant amount of cash for Earth to afford some cute things. What was the outcome? Of course, the second party sold as much as it could and the Moon was reduced to the minimum size allowing full solar eclipses.

Alternatively, the Moon materialized in the cone connecting a point P on Earth with the Sun out of the special miraculous solar photons that arrive to P. I don't know how it could happen but the Moon is the largest ball at the same distance that fits into this shining cone. :-)

Now you've been calibrated to the "generalized physics" we allow here. Fine. What about the symmetric distorted Earth? You may start with an exactly spherical (or ellipsoidal) Earth. But it is bombarded by meteors whose trajectories are always perpendicular to the axis of the Earth's rotation and they actually intersect this axis, too. When they hit the surface, they cause a depression at the point \((\theta,\varphi)\) where they hit the surface – this point becomes more ocean-like – but they add matter on the opposite side, \((\theta,\varphi+\pi)\). So the opposite point's altitude is increased – it becomes more continent-like.

Of course, I can't tell you why meteoroids' paths should be perpendicular to (and intersecting with) the axis of the Earth's spin. And I can't explain many other things that my "explanation" apparently depends upon. But it's fun to think about these possible reasons. Perhaps, it could be more than a coincidence that this \(\ZZ_2\) symmetry is "almost" possible for Earth. Perhaps you might invent a somewhat more plausible mechanism that generates such a pattern on Earth?

(It would be slightly less awkward to invent mechanisms if the symmetry were a \(\ZZ_2\) that reflects all three Cartesian coordinates' signs. But note that the orbifold \(S^2/\ZZ_2\) would be an \({\mathbb R \mathbb P}^2\) in that case: on the surface, the continents and their wet images would look like mirror images of each other, not simple rotations.)

Even without a more plausible mechanism to generate the pattern, you may want to ask the question: Does the actual shape of the Earth's oceans and continents nontrivially support the claim that the \(\ZZ_2\) symmetry holds more accurately than it does for a random distribution of the continents? I roughly know how I would construct a quantity that measures the degree at which this symmetry is preserved. But I don't want to tell you. Maybe you can offer your own, simpler rule? And what is the answer? Is there something special about the distribution we have on the Earth? I have no idea!

Sometimes, arts is just arts but sometimes arts adds lots of interesting questions if you have some fantasy that is waiting to be liberated. I will add one more comment.

Note that in all the comments about mechanisms, I was neglecting continental drift. I was assuming the current distribution and locations of the continents to be scientifically relevant but the continents have actually been moving and reshaping due to plate tectonics. An actual viable explanation would have to address the distribution of the continents as they existed billions of years ago, and the current distribution would simply be evolved out of the old one. It's strange because it's the current distribution, and not necessarily the old one, that supports the symmetry.

But couldn't it be true that the symmetry is "approximately true" at every point of the continental drift? If true, the continental drift would sort of have to respect the symmetry axis connecting Caracas and the V-city. Isn't this axis going through the Earth – approximately (and, in the idealization, exactly) belonging to the equatorial plane – the axis of some internal circulation in the Earth that dominate the continental drift?

The distribution of continents and oceans as we know them from the world map may be dismissed as a "random picture". And when most people hear "random", they no longer ask any questions. They think that if chance (a random generator) decides about something, it's no longer our business. However, if you're scientific, you know that there isn't anything such as a universal "random" thing. When something is random, you want to know what are the odds, what are the probability distributions (and what are the correlations in these distributions etc.), and those may reveal or hide lots of additional patterns and explanations that wait to be discovered.

Note that quantum mechanics also says that the results of all observations are "random" but this single adjective isn't everything that quantum mechanics tells us. Instead, it tells us the probability of any possible outcome for any possible experimental setup. And those calculable numbers ultimately end up being "similarly constrained" as the outcomes are in a deterministic theory. So when something looks "seemingly random", it just doesn't mean that there is nothing else to be discovered and it isn't true that there can be no way to "localize the right answer" by pure thought almost exactly!


  1. That is a truly beautiful rendering! The areas of the land masses are well ratioed, too. Another planetary coincidence: Between the poles, little dry land is antipodal to other dry land. It is almost always land, through the center of the Earth, water or water, through the center of the Earth, land. Here is your antipodes utility. Click drag to move and mouse wheel to magnify,

  2. I wonder if the land arising as fragments of one original big island helps the probability? Obviously this implies the water can also fit together. It seems plausible this increases the chance of this sort of similarity.

  3. I don't quite see why you think that the fragmentation of a Pangaea should mean that the remaining ocean is being fragmented in a (nearly) symmetric way.

    If a hemispherical big continent splits into two, the topology of the two continents will be two disks which is something else than the topology of the ocean - a disk with a strip in the middle which is topologically equivalent to a disk with an extra hole in it.

    There is an important issue that our eyes only remember the shapes of some "sufficiently compact" objects. We don't really remember the shape of the Pacific Ocean as well as we remember the shape of the continents or the shape of the Mediterrenean Sea. So the "distortion" may cleverly build on the difference of patterns that we easily perceive and those we don't.