MODIFYING GAUSS-BONNET FOR CONE POINTS AND CORNER POINTS -------------------------------------------------------- SUMMARY OF THE OVERALL STORY TO THIS POINT We begin with a summary of the overall strategy for classifying symmetry types of plane patterns: We call a polyhedron_flat_ if the angles around any vertex sum to 360 degrees. Clearly, dividing the infinite plane into polygons yields a flat polyhedron. (Here you should think particularly of the division of a symmetric plane pattern into templates.) Now if we think of a plane pattern as a_map_ of a surface, the flatness of the plane should imply the flatness of the surface (in some suitable sense, fully explained below). Now a flat surface has all its_local angle defects_ equal 0, so it has _total angle defect_ equal 0 also. But the Gauss-Bonnet theorem says _total angle defect_ = 360(V - E + F) and Euler's theorem tells us that V-E+F (the so-called ``Euler number'' or ``Euler characteristic'', which we write using the Greek letter chi) depends only on the topology of the surface (and not on the details of its combinatorial geometry, meaning the configuration of the polygons which make it up.) This suggests that we make a catalog of topologies with Euler number 0. Taking on faith the classic result that all surfaces come from the sphere by adding handles, making holes and adding Mobius strips, we aim to produce a finite list which corresponds to our as yet tentitive list of symmetry types. This summary, correct as far as it goes, nevertheless neglects some important details. In three steps we refine the meaning of BOTH sides of the Gauss-Bonnet formula, always making sure that the formula remains valid. The first modification allows us to consider polyhedra with ``holes''. Next we deal with cone points, and last with corner points. Gauss-Bonnet WITH ``HOLES'' When we derived our original Gauss-Bonnet formula, a key step involved assuming that each edge lies adjacent to exactly two faces. That assumption rules out the possibility of holes in the surface of our polyhedra, at least as far as Gauss-Bonnet goes. But the polyhedra that arise from plane symmetries often do possess holes. So we we'll adjust our definitions again and derive our first refinement of the Gauss-Bonnet formula. Observe first that some edges lie adjacent to two faces, but some, those that border a ``hole'', lie adjacent to just one. If an edge borders a hole, then by moving to one of its endpoints and going around the angles that meet there, we must eventually come to ANOTHER edge that meets only one face. Thus we may trace a cycle of edges and vertices going all the way around the ``hole'' (it can't go on forever since we have a finite number of edges and vertices, so eventually we come back to where we started). Let us call such a cycle a _boundary_. Obviously the total number of edges that belong to _boundaries_ equals the total number vertices along _boundaries_. Call this number L . >From now on, when we compute the _local angle defect_ of a vertex along the _boundary_ we'll use the new formula 180 - (sum of the angles at the vertex). (Intuitively, _local angle defect_ equal 0 here means we have a ``straight'' boundary at that vertex.) Now let's derive the Gauss-Bonnet from scratch. (We could also build on our old derivation, but in this case it wouldn't save any effort.) As before, we can assume all faces triangular since adding diagonals to a polygonal face won't change either side of the formula. First we again count ``sides of edges'' two different ways: 3F = 2(E-L)+L = 2E - L, since E-L equals the number of two-sided edges. Next we observe that _Total Angle Defect_ = 360(V-L) +180 L - 180 F degrees since we subtract_from_ 360 degrees for V-L vertices, 180 degrees for L vertices and finally we subtract _away_ the sum of all the angles which still equals 180 F degrees (because the angles in a triangle sum to 180 degrees). Dividing both sides of the first equation by 2 gives (3/2)F = E - L/2 and the subtracting F from both sides gives F/2 = E - F - L/2 . On the other hand, dividing the second equation by 360 degrees gives _Total Angle Defect_/(360 degrees) = V - L + L/2 - F/2 = V - L/2 - F/2 Substituting for F/2 in the last equation from the previous equation gives _Total Angle Defect_/(360 degrees) = = V - L/2 - (E - F - L/2) = V - E + F as desired. CHANGING THE DEFINITIONS: CONE POINTS If a symmetric plane pattern has a gyration point of order n , the angles at the corresponding point (a ``cone point'') in the associated surface will sum only to 360/n. Thus flatness for the surface, as previously formulated, will fail at the point. To fix this, WE WILL MODIFY THE DEFINITIONS OF_local angle defect_ AND _flatness_ FOR CONE POINTS OF ORDER n . Henceforth, the_local angle defect_ at a cone point of order n shall equal 360/n - (sum of all the angles at the point) and now we shall call a surface_flat_ when when all its _local angle defects_ (in this new sense) equal 0. If we stopped now, we would INVALIDATE the Gauss-Bonnet theorem. After all, we've changed the way we compute _total angle defect_, the left side of the equation, so we had better make a corresponding change to the right side of the equation. How does our new definition affect the _total angle defect_? As compared the the old definition, the _total angle defect_ GOES DOWN by exactly 360 - 360/n for each cone point of order n . So we just need to have the right side of the equation go down BY THE SAME QUANTITY. This simply amounts to counting a cone point of order n as 1/n of a vertex when we compute V . (Make sure you understand the very simple algebra that backs up this assertion!) (Intuitively speaking, it makes reasonable sense to think of an order n cone point as 1/n of a point, as follows: If you slightly fatten up a gyration point of order to a tiny disk, then as pass from the plane pattern to the template to associated surface, the template will contain only 1/n of that disk or one of its identical copies elsewhere on the plane.) Now we try to clarify in advance some potentially confusing points: WHAT EXACTLY DOES IT MEAN TO CALL A VERTEX A CONE POINT OF ORDER n ? ANSWER: It means that we use the new formula when we compute _local angle defect_ there, and count the point as 1/n when we compute the Euler number by the formula V - E + F . SO HOW CAN I RECOGNIZE THE CONE POINTS OF ORDER n FROM AMONG THE OTHER VERTICES? ANSWER: In GENERAL, someone has to tell you the order of each vertex. We've now left the category of POLYHEDRA and passed to the category of POLYHEDRA-WITH-CONE-POINTS. Describing a (mere) POLYHEDRON required just specifying the polygons that constitute the faces (with all their lengths and angles) and the way they connect up. But describing a POLYHEDRON-WITH-CONE-POINTS requires giving more data: you must specify the orders of all the vertices that you wish to count as cone points. (We have a choice about the terminology: we may consider ordinary vertices as cone points of order 1, or we may reserve the term ``cone point'' for cone points of order 2 or more. Everything comes out the same either way.) IF WE CAN JUST DECLARE ANY VERTEX TO BE A CONE POINT OF ANY ORDER HOWEVER WE PLEASE, HOW CAN THE NOTION OF ``CONE POINT'' HAVE ANY VALUE???? ANSWER: The value of a mathematical structure lies in its ability to model a phenomenon that we wish to understand. In this case we start with a plane symmetric pattern and pass to an associated polyhedron. If the pattern has gyration points, WE DON'T WANT TO LOSE THAT INFORMATION, so we preserve it by calling some vertices cone points. ON TO REFLECTIVE EDGES AND CORNER POINTS So far have modified our definitions to deal with gyration points and holes. Now we must account properly for reflective edges and corner points. Again we must change our category, passing from POLYHEDRA-WITH-CONE-POINTS to POLYHEDRA-WITH-CONE-POINTS-REFLECTIVE-EDGES-AND-CORNER-POINTS (PWCPREACP for short). To describe a PWCPREACP you must 1) specify a polyhedron-with-cone-points (with everything that entails) 2) indicate which_boundaries_should count as_reflective_. 3) specify the orders of the corner points, the vertices found along the reflective boundaries of holes. Given a PWCPREACP, we shall calculate its_local angle defect_ at a corner point of order n by 180/n - (sum of the angles at the vertex). When we compute V - E + F, reflective edges shall make a contribution of only 1/2 to E and corner points of order n shall contribute only 1/(2n) to V . If we've done everything right, the Gauss-Bonnet formula should remain valid. That it does follows from two simple observations: 1) Suppose that a_boundary_has k vertices and k edges. Declaring that_boundary_reflective reduces V by k/2 and E by k/2 , so V - E + F does not change. Nor does the _total angle defect_, so neither side of the Gauss-Bonnet formula changes. If valid before the declaration, the formula must remain valid afterwards too. 2) Declaring a point on a_reflective boundary_to be a corner point of order n means, first, that we'll compute the local angle defect there as 180/n - (sum of the angles at the point) instead of 180 - (sum of the angles at the point), and second that the point will contribute 1/(2n) to V instead of 1/2 when we compute the Euler number V - E + F. So both sides of the Gauss-Bonnet formula drop by 180(1-1/n) and the formula remains valid. SUMMARY A plane symmetry pattern gives rise to a PWCPREACP. By the way we define _local angle defect_ for PWCPREACPs, we made sure that when a PWCPREACP arises this way, all its_local angle defects_ equal 0. So its_total angle defect_ equals 0 too. The Gauss-Bonnet formula then guarantees that V - E + F = 0 for such a PWCPREACP. Euler's formula tells us that we can determine V - E + F just from the topology of a surface (knowing how many handles, holes, mobius strips, cone points and corner points we have to add to a sphere to manufacture the surface). Indeed V - E + F goes down by 2 when we add a handle, by 1 when we add a hole or a Mobius strip, by (n-1)/n when we declare a cone point of order n and by by (n-1)/(2n) when we declare a corner point of order n. Since a sphere has Euler number 2, and all these operations drive the Euler number DOWN, we can classifify the combinations of modifications that drive the Euler number down to EXACTLY 0. This last step involves solving a PURELY NUMERICAL problem. The numerical problem can seem a bit messy, but we feel relieved to know that we've finished all our geometrical and conceptual work. Is anyone reading these things?? E-mail me if you get this far. Also, I'm trying very hard to be clear, elementary and complete (because my eventual goal is the publish these notes as a textbook). So please point out passages you find difficult to understand. Also typos, mistakes, etc.