flattening the Earth

Some thoughts on

Mercator: The man who mapped the planet
Nicholas Crane
Weidenfeld & Nicolson 2002 / Phoenix paperback 2003

The Mercator projection is everywhere these days, although some of us are aware that it is problematic because it inflates the size of countries further away from the equator, thus creating the wrong impression that South America is no bigger than Greenland, and Africa smaller than North America. Alternatives are available but rarely used.

The reason is, of course, Google maps. The Mercator projection is mathematically the simplest way of straightening a globe to make a flat map. You just use the longitude and latitude lines as an orthogonal coordinate system, as your x and y axes. So once you know coordinates of places, it is computationally extremely simple to create a flat map. Nothing wrong with that when you do that for a city map or even a small country, but when people use Mercator to (mis)represent the globe, as Royal Mail does, I come out in a rash.

Reading the biography of the man behind the projection, I learned that mathematical simplicity wasn’t his motivation. He had happily drawn heart-shaped maps of the world before, and he had produced much-coveted globes. The problem he set out to solve was based on the fact that sailors in his time navigated by compass bearing, i.e. the angle between their direction of travel and the measured direction of magnetic north. The simplest way to navigate would be to keep this bearing constant throughout. However, this isn’t the shortest route on a globe, and on most maps it isn’t a straight line. So, to Mercator, the attraction of the orthogonal projection was that the longitude lines will always be straight and vertical, so a route following compass bearing will always be at the same angle to them and thus also be a straight line. Simples.

Born in 1512 as Gerard Kremer, the son of a cobbler latinised his name in line with humanist contemporaries such as Erasmus. His life of just over 80 years neatly falls into two halves, split by a near death experience courtesy of the Spanish Inquisition, after which he decided the Spanish Netherlands weren’t safe for him and settled in the small town of Duisburg on the other side of the river Rhine. Having grown up not too far from that place, which in the 20th century became a major centre of heavy industry complete with Europe’s largest inland harbour, I found it cute to read about the peace and quiet Mercator found in the town, where he completed most of the work that he is remembered for, including the first maps using his famous projection and the book of maps that gave us the word “atlas”.

His move was linked to plans to launch a university at Duisburg, which then fell through. The town later did get a university, which was founded in 1655 but dissolved in 1815. The modern university was set up in 1972 as a Gesamthochschule. In 1994 it was named Gerhard-Mercator-Universität, but in 2003 Mercator lost this honour again, as his university merged with the University of Essen.

Another irony is that, while Mercator mapped all of the known world, and even some speculative geographies that turned out to be fictional, such as a ring of islands surrounding the North Pole, he never left the boundaries of a map of the Rhineland area shown at the front of the book, which covers around 400 km by 600 km. (It amused me that this map prefacing a book about a map maker shows Cologne on the wrong side of the river Rhine. As Colonia is a Roman foundation, it’s not hard to remember that it belongs on the left bank.)

Crane published the biography in 2002, before Google maps started flooding the world with Mercator projections, but I was still a bit disappointed that the epilogue doesn’t even mention the perception problems it creates and the alternative projections that others have produced to create a fairer representation of the planet. The biography is available as an e-book but the paper version seems to have gone out of print, unfortunately. Mercator's 500th birthday in 2012 would have been an opportunity for a new edition both to celebrate his work and re-assess it in the brave new world of Google maps.

PS: I just discovered his descendants are on GedBas. No obvious connection to my family history, but good to know.

can Pythagoras take a shower?

During a recent stay at a cosy little hotel in Germany, we had a shower cabin with a square footprint (ABCD) and a sliding door (ab) that operated such that its endpoints a and b slid along the sides of the square, so point a moved along AB, while point b moved along BC.

We wondered if a well-rounded hotel guest – let’s call him Pythagoras (hint, hint) - could fit into the shower cabin and close the door behind him. Specifically, if the side length of the square is 1 meter, what is the maximal radius of a rigid rotund guest that could fit in and close the door?

Answers on a postcard. (This really happened, and the young mathematician in the family worked it out. However, I suspect that somebody, eg Martin Gardner, must have come up with this riddle before.)

the world according to Royal Mail

in attempt to get my brain round the staggering price increases, I had a good look at Royal Mail's new price leaflet and found that according to them, Greenland is larger than South America:

Now I do know that applications like Google maps use the Mercator projection because it is easiest to calculate when you want to zoom in and out of maps. However, in my opinion, if you display a static map like this there is simply no excuse for using it. It distorts the relative importance of various parts of the world and is thus offensive to anybody who cares about such "small" places like Africa and South America.

The full leaflet with the offending picture is also online as a PDF file.

what the world doesn't look like

I find the use of the Mercator projection to portray the world offensive in any context (as it hugely distorts the sizes of countries and continents, Greenland isn't really larger than South America, it's a lot smaller!), but in the context of trying to help developing countries (which all appear small and insignificant in this projection) it really drives me up the wall.

Apparently, applications that allow you to zoom in and out of maps, like Google maps routinely use the Mercator projection, as this is the easiest one to calculate (it’s all in squares), but that’s no excuse for displaying the world in a blatantly misleading way, as, for instance, Oxfam does here:

Food price volatility map

I am sure the clever people working at Google and other internet companies can come up with a way to program interactive maps without making tropical countries shrink to insignificance?

If one has to have a cylindrical projection (where all the longitudes and latitudes are straight lines), one could use the Gall-Peters projection:



(Wikipedia)

which with its very unfamiliar look reminds us how wrong the maps are that we see more often. Personally, I prefer the round or oval projections, as they also remind us that our planet isn't rectangular, e.g. the Lambert azimuthal equal-area projection.

PS checking up on Google I realised that even their satellite view is in Mercator projection. Surely the satellites don't observe a cylindrical planet from space?

in praise of squares

As cropping is the main (and usually the only) thing I do to my photos, I've been thinking of the merits of various formats, and at the moment I have a thing for squares, especially to capture things that normally aren't square, like petals:



Accordingly, the six photos I've added to my flickr photostream today are all square, and I also started a new set with squares. Obviously, if this 3D craze catches on, I'll have to move up to cubes :)

How to circle the square

-- updated 13.1.2011 --

Some of the simplest geometric shapes are really difficult to recreate in organic molecules. Carbon loves the flat hexagon that we find in benzene and its numerous derivatives. It is also happy with the five-membered analogues (such as the cyclopentadienyl anion or the nitrogen-containing pyrrole), but when it comes down to building squares, things are getting complicated. Firstly, the angles are all wrong, as carbon with a double bond wants 120 degree angles between its binding partners, not 90. Secondly, theoretical organic chemistry predicts that a ring with 6 (or, more generally, 4n+2) free-floating electrons (on top of the ones involved in the direct bonds) will be stabilised by these electrons. A ring with 4 (or any other 4n) will be destabilised.

Both these considerations suggest that a ring of four carbons with two double bonds should not exist. Chemists have, however, with a whole arsenal of tricks, managed to make such a molecule, but it could only exist at temperatures near absolute zero and only trapped inside a cavity within larger molecules. No structural information about this elusive molecular prisoner could be obtained – until now.

The group of Mihail Barboiu at the University of Montpellier, France, has now developed a new kind of cage to trap and study the elusive carbon square in. They made a crystalline matrix out of hollow molecules (calixarenes), which enabled them to perform crystallographic structure determination on several compounds along the reaction path that leads to the desired product, thus also clarifying open questions regarding its mechanism of formation.

As for the final ring of four carbons, there are two structural options: a square with the four (rather unhappy, anti-aromatic) electrons smeared out around the ring – the structure one could write as a square with a circle inside, in analogy to the benzene structure; or a rectangle with two shorter (double-bonded) and two longer (single-bonded) sides. It is known that the equally anti-aromatic ring of 8 carbon atoms prefers the uneven solution, with localised double bonds. For the 4-ring, chemists had to await the study by Barbou’s group to learn that the answer is … Well, in fact, the crystal structure shows both versions, the rectangular and the square shape. Faced with two equally uncomfortable solutions to its structural problems, the molecule just can’t make up its mind.

Reference:
Y-M Legrand et al. Science 2010, 329, 299.

Fig. 4 from the paper shows the structures obtained:



PS (Oct.2010): I've also written a one-page piece about this in German, which is now out in Chemie in unserer Zeit

PPS (Jan. 2011): The results of this work have been disputed by researchers using computer simulations, but defended by the original authors. See the technical comments in Science online, and a detailed discussion here

beauty of fractals

... time to break up the series of text posts with a picture, here's in praise of natural fractals:



Oddly enough this tree stands in Oxford city centre (Jowett Walk), on a small patch of land that is probably a College garden and really to small for it.

a mathematician's tree

If mathematicians were to design trees, the result would probably look a bit like this sculpture:



which sits in the parks of Magdalen college, Oxford, and which I snapped from the River Cherwell on one of our canoe trips.