THE FIFTH OF EUCLID
A 2-Pager by Ajit Chaudhuri – February 2012
Six months down the PhD road, and the only thing I can claim to have learnt is the unwelcome art of living alone. This is no minor matter for someone who is generally useless at all things that require an element of practicality and/or pragmatism. How have I survived? Well, I have someone coming in to clean and wash clothes, and I have outsourced all gastronomic requirements to the institution’s students’ mess. In fact, the only thing that I have had to actually learn is ironing clothes. And I have been spectacularly unsuccessful at this.
The reason for this is that I have yet to figure out two inter-related things – stuff that my semi-educated neighbourhood ironing-lady has probably known from childhood. One, how do you project a curved surface onto a flat space? And two, how can something that covers a three dimensional figure (like clothes for my upper body) be reduced into two dimensions?
Looking for answers has been a journey (conducted while supposedly studying public policy and local governance) through geometry and abstract mathematics that touches upon issues such as the philosophy of science, the possibility of a fourth, or fifth, or even an infinite number of dimensions, and the likely shape of the universe. And I have just begun! This note is an attempt to suss out some of this, and to share the wonder and bewilderment that I feel. And let me assure those of you on the verge of closing this, there are no mathematical equations, there are no formulas, there are no graphs, and this is in plain and simple English from beginning to end. So please don’t!
Let me begin with flat spaces, curvature, and the shape of the Earth. The ancient Greeks were the first to wonder about these matters without ascribing every mystery to the presence of God. They had figured out that the Earth’s surface was curved and had even worked out a possible circumference (on the assumption that it was a sphere) that has since proved stunningly accurate. They also drew maps, depicting the parts of the Earth’s surface that they knew on to two-dimensional charts. But there was no way of knowing the shape of the Earth – land and sea seemed to go on forever, in every direction. The Earth could have been shaped like a pear, or an idli (the equivalent would be a flying saucer), or an infinitely long dosa (cylinder) or a vada (doughnut) or even two vadas joined together as sometimes happens in the cooking pot before the chef cleaves them apart. And as for the maps, what happened when one went, say, beyond the top right hand corner? Did one fall off into infinity? Or come back on the bottom left hand (or another) corner?
Ferdinand Magellan’s expedition around the world, much later in 1519-22, showed that if one headed continuously in one direction one could ultimately return to where one started. So even if the Earth’s surface was continuous, it was also finite. But what did that say about the shape of the Earth? He could have made his journey around the narrow part of a pear-shaped object, or along the surface of the circular cross-section of a dosa, or around the inside ring of a vada. It was only with the advent of space travel, and therefore of man’s ability to move outside the boundaries of the Earth to look at it, that one could confidently say that it is a sphere with slightly flattened poles.
As my ironing lady knows, the simplicity of flat two-dimensional spaces does not apply in curved three-dimensional ones. There is nothing like a straight line connecting two points – lines curve as they move along the surface of the Earth. The shortest distance between two places is not as per the line connecting them on a map – it is along a curved line (called the geodesic) running along the circumference of an imaginary circle that touches the two places, whose centre is the centre of the Earth. Which is why, you would have noticed on planes in which one channel on your armchair TV displays route and location, inter-continental journeys take seemingly elliptical directions.
This leads to interesting controversies. For example, Islamic convention decrees that all mosques be built facing Mecca. Now, in which direction should a proposed mosque in New York be built? Would it be the southeast, the shortest distance on a map and the direction New Yorkers know Mecca to be? Or would it be the north, as per the geodesic connecting the two places?
Complicated? Let’s merely agree that the laws of geometry that we learnt in school tend not to apply when the two-dimensional plane we are working upon (such as a map, or my ironed clothes) is actually a depiction of the surface of a three-dimensional curved object (such as the Earth, or my body).
Let’s go back to the ancient Greeks for a bit, and to their search for answers to questions regarding the Earth. Scientists today are asking similar questions of the Universe. Does it continue forever, or is it finite? What is its shape? What lies beyond it? If we move in one direction continuously, for a long time, where will we end up – in some form of infinity, or back where we are now?
We are also trying to map it, just as the ancient Greeks did of the Earth, using accounts of our travels, our powers of observation, and our mathematics. But, in addition to our telescopes being seriously powerful and our mathematical tools considerably more developed, there is a critical difference. The Greeks mapped out the surface of the Earth, a three dimensional object, on to a two dimensional chart. And we are working one dimension higher, using a three dimensional space (a cube, or a shoebox) to map the surface of the Universe.
We are able to say, so far, that the Universe is a finite space that exists in multiple dimensions, and that it possibly expanded from one single point. The space that we know is possibly a curved three-dimensional surface of a four-dimensional Universe, much as the Earth’s surface is a curved two-dimensional boundary of a three-dimensional sphere. Parameters such as time, distance and speed become much more complicated when one moves from the three dimensions of a cube or shoebox (such as our map of the Universe) to a four-dimensional space with a curved three-dimensional surface (such as the Universe itself), just as they do on similar switching between two and three dimensions (as my ironing lady and I can confirm).
Most sciences, like our senses, can function in up to three dimensions. Abstract mathematics has no such limitation, and mathematicians have shown that there are limited possible shapes for curvaceous four-dimensional objects with a single point of origin, all of them being spherical in some way.
What does all this mean? I hesitate here – I am out of my depth. But I do know that it means something from recent churnings in the field of mathematics. One of mathematics’ greatest and longest unsolved problems, and the subject of a Millennium Prize (immense prestige plus $ 1 million), was the Poincare conjecture about multidimensional spheres, which is critical to speculation on the shape of the Universe. The conjecture was solved in 2003 by a reclusive Russian mathematician, Grigori Perelman, who just put his proof up on a public website on the Internet and left it there. It took three years and much controversy before the proof became accepted (some Chinese mathematicians tried to claim credit). Perelman was awarded the Fields Medal, the mathematical equivalent of the Nobel Prize, in 2006 for his work despite the facts that the proof was not published in a prestigious peer-reviewed mathematical journal as per rules and that the authorities knew that he would refuse it, which he duly did. He also refused the Millenium Prize (and the money) in 2010. Perelman’s proof of the Poincare Conjecture was honoured by the journal Science in 2006 as the breakthrough of the year, the first time ever that it has been bestowed upon the field of mathematics.
Why is this so important? It took 23 centuries from when the Greeks posed questions to when humankind became able to move out of the Earth, look at it, and confirm its shape. The recent ability of mathematics to make sense of possible shapes of curved four-dimensional objects, of which Perelman’s proof is a critical component, may just be an important step in obtaining confirmatory answers to similar questions regarding the Universe.
For those of you still here, I would like to conclude with something on the joys of mathematics. Mathematics, like single malt whisky, is an acquired taste – few of us are born with mathematical ability, and even fewer with an innate distaste for it. It is mostly that some of us are fortunate with our middle school maths teachers, and others are not. Mathematics is the only science that does not require expensive laboratories and complicated equipment to be able to practise – a pencil and paper, and a mind, are all one needs. And as for whether mathematics makes you a boring dullard – let’s look at some mathematicians. Pythagoras of the eponymous theorem fame was also the head honcho of a secretive spiritualist cult. Descartes, who built the foundation for understanding physical objects as algebraic equations, was a mercenary soldier who often fought both sides of the same war (depending on who was paying more money). Poincare was from a famous political family (his cousin, with the same surname, was a French Prime Minister), born with a silver spoon and a love for the good things in life. And, last but not least, Perelman himself is an unemployed bum who stays with his mother in a run-down St. Petersburg apartment and lives on her Soviet-era pension. It is entirely possible that he, too, would have difficulty ironing his clothes.