EULER'S THEOREM: TOPOLOGY AND V - E + F --------------------------------------- INTRODUCTION TO TOPOLOGY ------------------------ One may speak of polyhedra either as a metric geometer or as a combinatorial geometer. For a metric geometer, describing a polyhedron involves specifying the side lengths and angles of all the polygons which will form the faces of the polyhedron and then specifying which edges of which polygons get ``taped together'' in order to assemble the unit. For example, a metric geometer might describe a certain square pyramid as follows: The set of faces shall consist of 5 polygons: 1. ABCD, a 90-90-90-90 quadrilateral with sides of 1 ft.; 2. ABE, a 60-60-60 triangle with sides of 1 ft.; 3. BCE, a 60-60-60 triangle with sides of 1 ft.; 4. CDE, a 60-60-60 triangle with sides of 1 ft.; 5. DAE, a 60-60-60 triangle with sides of 1 ft. Then: side AB of ABCD gets taped to side AB of ABE; side BC of ABCD gets taped to side BC of BCE; side CD of ABCD gets taped to side CD of CDE; side DA of ABCD gets taped to side DA of DAE; side AE of ABE gets taped to side AE of DAE; side BE of BCE gets taped to side BE of ABE; side CE of CDE gets taped to side CE of BCE; side DE of DAE gets taped to side DE of CDE. Observe how the names of the vertices and edges already determine what gets taped to what. This means that we could shorten the description down to just the first part, The set of faces shall consist of 5 polygons: 1. ABCD, a 90-90-90-90 quadrilateral with sides of 1 ft.; 2. ABE, a 60-60-60 triangle with sides of 1 ft.; 3. BCE, a 60-60-60 triangle with sides of 1 ft.; 4. CDE, a 60-60-60 triangle with sides of 1 ft.; 5. DAE, a 60-60-60 triangle with sides of 1 ft. and anyone could figure out the rest. Now combinatorial geometry doesn't deal with lengths or angles, so from that optic, the describing the pyramid only requires us to say: The set of faces shall consist of 5 polygons: ABCD, ABE, BCE, CDE, DAE. Of course we've lost a lot of information when we cut the metric geometer's description down to the combinatorial geometer's description. This exemplifies the process of ABSTRACTION, throwing away inessential information to concentrate on the core of a phenomenon. (Whether information counts as essential or inessential depends on the phenomenon under study, of course.) Now a topologist would specify a polyhedron the same way a combinatorial geometer would, but with this difference: where a combinatorial geometer counts two polyhedra as structurally identical only when their faces, edges and vertices match up, the topologist REGARDS CERTAIN MODIFICATIONS OF THE COMBINATORIAL STRUCTURE AS INESSENTIAL, namely, given a polyhedron: i) adding a vertex in the middle of an edge: Before: o--------------o After: o-------o------o ; i') the reverse of i); ii) Adding a diagonal to a face. (DIAGONAL means a line that connects any two nonadjacent vertices.) o---------o o---------o / \ / \ \ / \ / \ \ / \ / \ \ / \ / \ \ Before: o o After: o \ o ; \ / \ \ / \ / \ \ / \ / \ \ / \ / \ \ / o---------o o---------o ii') the reverse of ii); iii) Subdividing a face by adding a vertex within it and connecting it to all the corners. o---------o o---------o / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ Before: o o After: o---------o---------o ; \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / o---------o o---------o iii') the reverse of iii). Comments: a) Since one may accomplish a modification of type iii) by a sequence of modifications of types i) and ii), we could have omitted type iii) from this list, but type iii) modifications prove useful, so we call explicit attention to them. b) Though these diagrams show hexagonal faces, the modifications apply to faces with any number of sides, of course. c) Let's understand the philosophy behind the list of modifications the topologist considers inessential. Suppose you had a square pyramid such as we described above, but made out of flexible rubber. Suppose it has WHITE faces, but BLACK edges. Suppose now that you inflate it like a balloon. Inflation eventually rounds out the corners and the edges and the object becomes more and more spherical. The black edges on the surface of the balloon remind us of the combinatorial geometry. The modifications above change the markings on the surface, but they obviously don't change the underlying spherical shape of the surface. THE TOPOLOGIST REGARDS ONLY THIS UNDERLYING SHAPE AS ESSENTIAL. The topologists considers the specification of a specific combinatorial structure only as A NECESSARY EVIL, a nuisance one suffers in order to bring a topological structure under discussion. Compare this to the following real world situation. We measure heights in feet, yards, meters or some other unit, but WE MUST ALWAYS CHOOSE A UNIT. To say a man measures 6 feet tall means EXACTLY THE SAME as saying that he measures 2 yards tall. The notions of foot and yard have no intrinsic meaning but rather exist as conventions that facilitate communication. You see nothing ``sixish'' when you see a man 6 feet tall the way you do when you look at a snowflake. Choosing particular units often turns into a nuisance because often we must translate measurements into other units to get some job done. ``Six feet'' and ``two yards'' represent the same length much the way ``a square pyramid'' and ``a cube'' represent the same topology. Exercise: Use the modifications above to turn a square pyramid into a cube. EULER'S THEOREM --------------- The works of Swiss mathematician Leonhard Euler (1707-1783) (pronounced ``oiler'') fill 70 volumes making him history's most prolific mathematician. After more than 200 years, his ideas continue to spur modern research; mathematics students see his name everywhere. A simple but profound observation of Euler interests us here. Basically, Euler observed that the modifications which don't alter topology also don't alter the quantity V - E + F . Indeed, modification i) raises both V and E by 1; modification ii) raises both E and F by 1 and modification iii) raises V by 1, E by n (where n equals the number of sides on the given face) and F by n-1 (since we create n small triangular faces but lose the original face). What makes this so important? Suppose we had two polyhedra and we wanted to know whether they had the same topology. If yes, we could see this by exhibiting (once we found it) the sequence of modification which transform the combinatorial geometry of one into the combinatorial geometry of the other. But if no, then WE COULD NEVER FINISH EXAMINING ALL THE MODIFICATIONS of one polyhedron to ensure we never could never produce the other. But if calculating V - E + F for one gives a different result than calculating V - E + F for the other, then we can conclude they have different topologies. After all, no single modification changes V - E + F, so no sequence, no matter how long, can either. One calls a quantity like V - E + F a topological invariant - it stays the same when the combinatorial geometry changes because it depends only on the topology. Of course we MIGHT have two polyhedra with different topologies and V - E + F equal, and then we need a new idea. Fortunately it turns out that Euler's theorem tells almost the whole story. The results of Wednesday's lecture show (implicitly) that two polyhedra have the same topology if 1) they have V - E + F equal; 2) they have the same number of boundary cycles; 3) they either both contain or do not contain Mobius strips. Of course for our story the full signficance of Euler's theorem emerges when we combine it with Gauss Bonnet theorem. Then we see that the _Total Angle Defect_ depends only on topology. Once we list all the topologies that have V - E + F equal 0 (which we've already done), we have our list of plane symmetry types. If you read this far and you haven't done so previously, send me an e-mail to say so. (So far I've had just 3 responses from a class of more than 70!) Also I'd like to know about any passage in the notes you find unclear or obscure.