Geometry and the Imagination Copyleft* 2009 ---------------------------- Math 462 at the University of Massachusetts, Amherst (Spring 2009) MW 1-2:15 in Lederle 1033 (with a few exceptions) Prof. Rob Kusner NOTE: Math 461 is NOT a prerequisite! (However, Math 233, 235 and 300 are recommended.) ======================================================================== Office hours this semester are after class (till 3ish) and by appointment - please contact me by email (kusner@math.umass.edu) to let me know if you plan to come to my office hours, or if you wish to make an appointment for another time. My office is LGRT 1435G, but often you may find me one floor above in the GANG Lab (1535 LGRT). ===================================================================== Bicycle chain magic? Negatively curved vegetables?! Polydrons!?! What on earth is "Geometry and the Imagination" all about?!?! Learn more by reading below or by emailing me (kusner@math.umass.edu) or by signing up and coming to very first class!!!!! ===================================================================== This is an experimental course in which undergraduates will learn in a real laboratory setting how to explore basic and profound ideas about low-dimensional geometry and topology. It is an updated version of a popular course we tried here four years ago (and a decade earlier), based on similarly-named courses taught at Princeton, Dartmouth, Minnesota and elsewhere. The name borrows - indeed steals! - from the title of a classic book by David Hilbert and Stefan Cohn-Vossen that described the state of modern geometry in the 1930s. In contrast to their wonderfully readable book, however, this course is hands-on (active-learning, discovery-based), so not only will students learn some interesting mathematics, they also can learn about *how* to learn mathematics! There is no textbook - students will write their own.... ===================================================================== The instructor is indebted to all those geometers who have preceded him in this effort; Peter Doyle and John Sullivan in particular gave the instructor his first hands-on experience in how this hands-on course might be taught. And since the course is completely hands-on, CLASS ATTENDANCE - and full participation - IS MANDATORY! To repeat: this is not a course for the absentee student; YOU - and your open mind - NEED TO BE PRESENT for the course to be successful - for you, and for everyone!! Students - and the instructor - will often need to procure the right materials for some of the projects to work. Among the most important things to get (immediately, please) are a lab notebook (bound, not looseleaf or spiral) and some colored pens. The details of the notebook are not so important, but it should be big enough for me to read comfortably on a regular basis, perhaps 100 pages (200 sides), and not too big to feel uncomfortable packing around with you everywhere to write ideas down in. (If that's not enough space for the whole semester, get another when it becomes obvious....) Bring your lab notebook and pens to class all the time. We will work together in various groups, as well as individually, so the notes you take - your rough ideas and false starts, as well as your new (and perhaps great) insights - will form the basis for evaluation. You, the student, will be expected to reflect, work out, and expand upon the ideas germinated together in class, develop these ideas into coherent, mathematically precise thoughts, and rigorous arguments, during the rest of the week, and record this ALL in that very same notebook. Think of this as a lab course, except it's for a different kind of experimental science - mathematics - where the role of experiment is played by proofs, including the formulation of definitions, lemmas and illustrative examples. Indeed, just as experiment is the ultimate test for reality in other sciences, so mathematical reality is ultimately tested by working out proofs. We'll do this as rigorously as possible, recognizing that ultimate rigor, in something as rich as mathematics, is itself logically problematic (thanks to Kurt G\"odel and his successors). So our emphasis will be to approach rigor "from the top down" (like birds and real mathematicians) rather than "from the bottom up" (like worms and pedagogues). At this point, you may have reasons to be fainthearted - DON'T BE! We'll have fun in this class, and whatever your future plans - doing more math, teaching, doing other sciences or writing or carpentry or whatever - there'll be "relevant" stuff to take away with you. Just be prepared to work hard - and "smart" - to expand your mind.... And as far as evaluation: I already mentioned that I look forward to perusing your notebooks from time to time. But also I think it will be fun and instructive for you all to do a final project for the course that will involve a presentation to the others in the class, as well as something written for me to keep. So as the semester progresses, you might start thinking about a possible topic, and we should certainly settle on one by Spring Break. ======================================================================== Mon 26 Jan How do we use a cloverleaf intersection to make a left turn? If the usual right turn ramps are missing (or if we forgot to use them), how do we make a right turn? (This illustrates the old adage: two wrongs don't make right - but three lefts do!) We made the convention that left-turning is positive and right-turning is negative (as in politics, at least half the time :) and found that going all the way around the cloverleaf amounts to -3 turns. Work out the total turning for all the possible turns in a cloverleaf. ======================================================================== Wed 28 Jan SNOW DAY!!! ======================================================================== Mon 2 Feb We explored the turning number for closed and open curves, using the circle (or "compass") of directions indicated by the unit tangent vector to the curve. We related the classical fact that the sum of the interior angles of a triangle is 180 degrees (1/2 turn) to turning number, and used it to generalize for N-gons (you guys gave some nice alternative arguments too)! What happens if the N-gon isn't simple (has edges which cross)? ======================================================================== Wed 4 Feb We argued a couple ways that the turning number of a figure "8" curve must be zero. We also described turning number using calculus ideas as total curvature (integral of curvature with respect to arclength). You should explore this in more detail using formulas. Find an example with infinite turning angle but finite length. We saw that for closed curves, turning number can be computed by looking at how many times the tangent vector points east (or any fixed direction) and counting these with a plus or minus sign depending on whether the curve is smiling (bending left) or frowning (bending right) at such points. A conjecture is that simple closed curves must have turning number 1 or -1 only - you guys are working on a proof of this for homework! ======================================================================== Mon 9 Feb We consider some open curves (spirals) with infinite turning number - some had infinite lenth and some (e.g. the self-similar ones) had finite length. For spirals of the form r(t)*(cos 2\pi t, sin 2\pi t), t > 1, what condition on the integral of r(t) from 1 to infinity will guarantee finite length? We also found a (self-similar) closed curve (continuous image of a circle) with infinite turning number and finite length. Using real bicycle chains we experimnetally explore which closed plane curves can or cannot be deformed to simple closed curves. The handout (courtesy of John Sullivan) indicated some examples to try: when the turning number was NOT 1 or -1, what did we discover? ======================================================================== Wed 11 Feb After seeing part of the _Outside In_ video, we continue exploring the deformations of immersed curves using real (and "thought") bike-chains. Keep in mind the issue of "bike-chain magic" - that is, because a chain is not really a plane curve, there is either an over- or under-crossing at each place where the corresponding plane curve would cross. Without "magic", what rules for over- and under-crossing also need to be satisfied to deform the chain to standard position? Finally, there is the issue of knotting: the example on the handout in the lower right of Figure 2 is an overhand (or trefoil) knot: the corresponding plane curve has turning number 2 (or -2); you can add a kink to make this 1 (or -1); with "magic", the resulting curve can be deformed to standard position (draw a sequence of figures - like a flip book - to illustrate this deformation); but without "magic" the knot persists.... ======================================================================== Mon 16 Feb [honor Washington and Lincoln and...] ======================================================================== Wed 18 Feb [emergency meeting in 1535=GANG] We explored how to assign a handedness to each crossing in analogy to a screw-thread. The writhe of the plane curve (with crossings) is the number of right-handed crossings minus the number of left-handed crossings. In other words, assign a "+" for each right-handed crossing and a "-" for each left-handed crossing and add them up (with signs). We discovered that these signs didn't depend on how the curve was rotated in 3-space, nor on the orientation of the curve (i.e. on which direction we trace the curve), and thus the same is true of the writhe. But what happens to the writhe if we take the mirror image curve in 3-space (i.e. if we reverse each of the crossings)? Conjecture: two bike chains (plane curves with crossings) can be deformed into each other without using "bike chain magic" if and only if they have the same turning number, writhe and knot type. We introduced the notion of the oriented connected sum C#D of closed curves C and D in R^2, and for homework you're going to figure out a relationship between their turning numbers T_C, T_D and T_D#D. ======================================================================== Mon 23 Feb We discussed oriented connected sum C#D of closed curves C and D in R^2, using a band whose boundary has turning number 1. We discovered that the turning number T_C#D = T_C + T_D - 1; in other words (T-1) is additive under connected sum. Please verify these for as many examples as you can, and see if you can make a proof (one suggestion from the class was to think of this operation as introducing a pair of U-turns). What happens to this above turning number formula if we use the "oppositely oriented" connected sum - that is, if we join C and D using a band whose boundary has turning number -1? How does writhe behave under #? If a curve bounds a surface in the plane, what is its writhe...? Try using the idea of connected sum - or rather something like its inverse operation, where the curve is cut apart, to show that simple closed curves must have turning number 1 or -1. Another way to do this was suggested in class: make a fine grid in the plane, then deform the curve a little to lie on the grid, arguing that the turning number doesn't change under the deformation; then deform one square at a time till all that is left is a single square.... We reviewed the Whitney-Graustein theorem: closed curves C and D in R^2 can be defomed (by regular homotopy) into each other if and only if T_C = T_D. Again, try to verify this with examples, or even see if you can prove the "only if" direction (the "if" direction is hard, and one proof due to Thurston was the subject one of the OutsideIn video). A consequence of the Whitney-Graustein theorem: a circle cannot be turned inside out (T=1 is not the same as T=-1)! We illustrated with bike chain why a naive attempt to do this leads to a pair of (illegal for regular homotopy) infinitely tight kinks. As we saw in the OutsideIn video, the analogous "eversion" can be done 1 dimension higher: a 2-sphere in R^3 can be turned inside out! ======================================================================== Wed 25 Feb [emergency meeting in GANG=1535] [Please try to get some POLYDRON pieces of your own: "A2Z" in Northampton may have some - call first - or you might find them on the web by googling "polydron"] We experimented with POLYDRONS to find turning numbers for closed curves on polyhedra. The turning number for a simple curve which surrounds a vertex differs from 1 by the "gap" angle at the vertex - what we see by flattening the polyhedron into the plane (here we orient curves to go counterclockwise around the vertex to avoid sign issues): Turn = 1 - Gap Please note: if the there is an overlap instead of a gap, we consider Gap to be negative! Using POLYDRONS, how close to 1 can you get for Turn without being exactly 1? (How small a Gap can you find - positive or negative?) Given the regular polygon pieces in POLYDRONS, please check that this amounts to determining small values of 60 - 18a - 15b - 10c, where a, b, c are the numbers of pentagons, squares, triangles, respectively, surrounding the vertex (a hexagon counts as 2 triangles - why?)! Does the turning number change if the curve is deformed through curves that don't pass through any vertices? What happens when the closed curve surrounds more than one vertex? ======================================================================== Mon 2 Mar [Another SNOW DAY!!] ======================================================================== Wed 4 Mar We continued experimenting with turning number of curves and gaps at vertices of polyhedra using POLYDRONS. In particular we found values of a, b, c which led to gaps of 1/30 and -1/60, and convinced ourselves that this is the best possible. (Note that although 10, 15 and 18 are relatively prime, the contraint that a, b, c are non-negative integers means we can't necessarily find a combination which gives 1/60 - but it also restricts the choices to a finite set of a, b, c that need to be considered.) John found an Archimedian polyhedron (using a dodecagon, a piece not available in POLYDRONS) with a 1/60 gap. I also poined out that a dihedron made from a pair of regular n-gons has vertex gap 2/n, which can be made as small as one likes; I also observed that if we allow any shape polygons (like skinny triagles) we can make closed (irregular) polyhedra with all gaps arbitrarily small. Notice that if we rewrite the equation above as: Gap = 1 - Turn this should remind you of the quantity (up to a sign) that was additive under connected sums. This quantity - the Gap - clearly adds: when a curve surrounds more than a single vertex, the total Gap along the curve is the sum of the Gaps at each vertex. (Convince yourself of this with POLYDRONS!) We call this total Gap the Curvature of the region enclosed by the curve: The Curvature of regions is additive, and for polyhedra, equals the sum of the Gaps at the vertices contained in the region! Careful: if the curve bounding region passes through a vertex, we have a problem defining the Curvature of the region - we'll assume for now that this doesn't happen :-) ======================================================================== Mon 9 Mar We began by discussing possible project topics - please take a look at the sheet or use your own imagination to come up with some tentative topics which we can begin to discuss next class individually - the spring break would be be a good time to get started on this and to decide whether each project "fits" each student. Then we went back to discussing total curvature of a region R on a polyhedral surface, and we reformulated the above as: TotalCurvature(R) = \Sum_{v \in R) Gap (v) and observed that as the boundary curve \dR (oriented, as usual, so that R lies to the left) is deformed to pass across a vertex v, the turning number of \dR changes by -Gap(v), proving: TotalCurvature(R) = 1 - Turn(\dR) at least for regions R which are disk-like. We made some polyhedra that close up, like a tetrahedron, or cube or whatever: what is the Total Curvature for your examples? We found in all examples so far that the total Curvature of a sphere-like polyhedron is 2. This jibes with the above formula since if R is a disk-like region R which covers all vertices of your polyhedron and whose complement is a little disk on a face, then we checked that Turn(\dR) = -1 since \dR must go clockwise in order to keep R on its left! (This proof was discovered by students in my 2005 class :-) What happens if your polyhedron is like an inner-tube (a torus)? What if is like the surface of a pretzel withseveral holes? ======================================================================== Wed 11 Mar Curvature of vegetables (and fruits): We had some fun with oranges and kale in class. In fact, we found the total curvature of the whole orange was (pretty close to) 2. Why do you suppose that an orange (or cabbage) is positively curved and kale is negatively curved? One idea has to do with growth rate of cells along the edge of the leaf. Here's some stuff to contemplate over the break: It turns out that if a surface has constant curvature K, then the circumference C(r) of a circle of radius r grows like c(K) sin \sqrt{K} r for K > 0 2\pi r for K = 0 c(K) sinh \sqrt{-K} for K < 0 where c(K) = 2\pi/\sqrt{|K|}. Use L'Hospital's rule to check that as K approaches 0 both expressions give the familiar formula for the euclidean plane. Also check that the formula is correct for K > 0 - this jibes with the geometry of a sphere of radius R, since the curvature density of such a sphere is exactly K = 1/R^2, since the total curvature is K times Area = 4\pi radians = 2 turns! The "hyperbolic" case (K < 0) needs more theory than we can develop in this course to prove, but it makes sense that C(r) would grow like an exponential function of r (recall sinh t grows like e^t), since around each point of space, negative curvature ("negative gap") means excess surface.... ======================================================================== Mon 23 Mar Curvature of cabbage (is that a fruit, or a vegetable - or a former German Chancellor?! ;-): What did you get for the curvature of your various cabbage leaves? Were you able to "develop" strips along curves on a sphere (the red or yellow 4-square, 3-ball, 2-sphere) to their corresponding strips in the plane? Did you make any observations about how the total curvature is related to the area (or area fraction) of the region on the sphere bounded by your curve? (Laura, Liz and Katie discussed details of their potential projects with Rob as well. Hyperbolic sleeves and trousers; Platonic kaleidoscopes or panoramas; and delta-square-hedra, respectively, were proposed so far.) ======================================================================== Wed 25 Mar Spherical geometry: We figured out that the area fraction a of a spherical triangle with interior angles x, y, z are related by x + y + z = 1/2 + 2a which agrees with the euclidean limit (as a tends to 0). Show that if instead we use turning angles, the total turning number satisfies t = 1 - 2a, where we surround an area to the left of the triangular curve. This makes sense for small triangles (a nearly 0). Check that this works not only for triangles, but for any simple polygon - and by a limiting argument, for any simple curve on the sphere! In fact, for an equator (or any curve surrounding area fraction a = 1/2) we see t = 0 as expected. (Lubna discussed with Rob a potential project about tiling hyperbolic surfaces using right-angled pentagons.....) ======================================================================== Mon 30 Mar We began to explore (in the case of regular polyhedra, like tetrahedron or cube) why the curvature (= gap) at a vertex equals the sherical image area (normalized so that the whole sphere has area 2, in units of turning). Work out the area of spherical image for any vertex of a regular polyhedron in two ways: by dividing 2 by the number of vertices, and using the detailed geometry - does the answer agree with the gap? Now try this for a general polyhedral vertex, using the fact that the perpendicular plane to any edge is what cuts out the great circle arc of the edge's spherical image. (Rob continued discussion with Lubna on approximating hyperbolic surfaces with squares meeting five to a vertex, and also discussed with Chris about moduli of elliptic curves and hyperbolic orbifolds in general....) ======================================================================== Wed 1 Apr (no foolin'!) Today we began to "put it all together" by working out that spherical image area (extrinsic) is really the same as (intrinsic) total curvature (as measured by gaps) for polyhedra. This leads, in effect, to two deep theorems of Gauss rolled into one: Gauss's Theorem Egregium and the "Gauss-Bonnet" Theorem! Gauss's Theorem Egregium asserts that the intrinsic and extrinsic curvature measures agree. For "Gauss Bonnet" we need to recall the notion of Euler number of a region R on a surface, denoted Euler(R) or \chi(R). On a polyhedron, by definition Euler(R) = V - E + F where V, E and F are the numbers of vertices, edges and faces, respectively. We explored why this is independent of the subdivision into polygons, and why it makes sense for surfaces with more general cellular subdivisions. We also saw that Euler number is additive under disjoint unions and used that to compute what happens under "connected sums" and "surgeries" of surfaces (where a pair of disks is cut out and an annulus is glued in, reducing Euler by 2). ======================================================================== Mon 6 Apr And now for the combined Theorem Egregium and Gauss-Bonnet: As we saw earlier when R is a disk-like region (where Euler(R) = 1) the formula for the curvature measure of a region on a surface in terms of turning angle of the boundary gives: TotalCurv(R) + Turn(\d R) = Euler(R) where \d R is the boundary of R. Try to do the general case using the disk case as a model! This can also be thought of as Gauss's Theorem Egregium (intrinsic curvature agrees with extrinsic curvature), since (in case R is a disk) we can compute Turn(\d R) in two ways: Turn(\d R) = 1 - TotalCurv(R) as we know from a long time ago (or see last time with Euler(R) = 1). In the special case of R being the neighborhood of a one vertex on a polyhedron, this is just Turn(\d R) = 1 - gap. On the other hand, in the case of a vertex with three faces meeting it, whose normal image N(R) is thus a spherical triangle with interior angles x, y, z (you can try the more general vertex, which invlolves not a spherical triangle, but spherical polygon as normal image) we have (by "developing" or rolling on the plane): Turn(\d R) = Turn(\d N(R)) = (1/2 - x) + (1/2 - y) + (1/2 - z) = 3/2 - (x + y + z) = 1 + (1/2 - (x + y + z)) = 1 - 2a = 1 - A where a is the area fraction (or A is the area) of N(R). Thus the intrinsic curvature equals the extrinsic curvature: gap = A. Try to do the same computation using spherical trigonometry directly instead of this developing/rolling trick. (Chris discussed his sundial for an ideal general planet project with Rob.) ======================================================================== Wed 10 Apr Here's another way to compute the Euler number of a surface in space. It uses height functions and should remind you how turning number of plane curves (1D in 2D) could be computed using "smiles" and "frowns" - this idea was mentioned in the _OutsideIn_ video. First, note that in addition to 2D-smiles (pits) and 2D-frowns (peaks), which can be thought of as critical points (local minima and maxima) of a height function, there is third version - a kind of 2D-frown/smile (pass) - of critical point (some call this a saddle point because a neighborhood of the surface resembles a horse's saddle). To find an analogue of the turning number which is invariant under deformations, we looked at some examples: if a sphere has an indentation near its peak, a new pit and pass are created; if a "horn" grows from the side of a sphere, a new peak and pass are created. And if we rotate a sphere, what was a peak becomes a pit and vice versa. This suggests that the number peaks - passes + pits should be invarant under deformations. Compute this for lots of deformed shapes of spheres - do you always get 2? (What happens if there are infinitely many critical points?) Now try computing this number for a bagel, a pretzel, etc. If we keep track of the connections (gradient flow lines) between critical points, this gives a decomposition of the surface into cells of dimension 0, 1 or 2, analogous to the decomposition of a polyhedron into V, E or F. And how does all this compare with what we did with total curvature and Gauss normal spherical images? What is the *sign* of the area of the normal image near a pit or peak? What about near a pass?! All this should convince you that Euler = peaks - passes + pits for a closed surface. What about when we have a region R with nonempty boundary \dR? (Try to figure out a formula that corrects for critical points of height restricted to \dR!) ======================================================================== Mon 13 Apr We considered the Mobius classificication of surfaces. If we remove a disk from a sphere, what's left is a disk. Removing more disks is the same as attaching (untwisted) bands to a disk. Mobius' idea is to do something similar for a general surface: Start with disk, which has k=1 boundary components. Attaching bands to the boundary circle in an "unlinked" way increases k by one for each band. Suppose we want to increase the genus g (recall this is the maximum number of disjoint simple curves which fail to separate the surface) by one. We can do that by attaching a pair of bands to the boundary whose attaching regions are "linked" on the circle (that is, if we look at how the the core curves of the bands intersect, we get an odd number, unlike the case of unlinked bands above, whose core curves intersect an even number of times). Finally, we can attach m half-twisted (a.k.a. Mobius) bands to tbe boundary. Mobius's classification theorem is that every surface can be realized this way. See if you can decide when (g, k, m) and (g', k', m') yield equivalent (homeomorphic) surfaces. This involves sliding one band along another. Back to Morse height functions: We also saw how to associate an h and its gradient field to a (polyhedral) surface in such a way that pits, passes and peaks correspond to (midpoints of) vertices, edges and faces - that confirms our Euler number speculations from last time. We also discussed how critical points on the boundary might be counted (with 1/2 weighting). ======================================================================== Tax 15 Apr We discussed a little more about the Mobius classification of surfaces, and noted three things: 1) there more information if we consider the surface to be immersed on the plane, namely, the even or odd number of full twists in the (ordinary) bands (can you think what might be done with the Mobius bands along the same lines?); 2) if (g,k,m) is finite, then when m > 0 there a relation g+1+m=g'+3m' giving another surface of type (g',k',m') homeomorphic to the first; if this triple is infinite then g (and m) must be countable, but k may not be.... We gave an alternative proof of Euler number of a sphere = 2 using "charges" and a puff of west wind (a vector field w on the sphere) in order to blow all the charges a bit to the east (except for the two positive charges at the N and S poles. Because the number of edges (where we placed negative charges and were careful to not align them on the east-west axis) on the west side of any face is one more that the number vertices on the west side of the same face (we put positive charges at the vertices, as well as on each face), the windblown charges (now all on the interior of the faces, except at the poles, of course) sum to zero on each face, and thus to 2 althogether. On the other hand, the total charge just counts V - E + F = Euler! We also see this works on a torus (Euler = 0), since then we can find a vector field w which never vanishes. Check that whether this works (mutatis mutandis) on any closed surface of any genus.... (Something should puzzle you: can we get Euler < 0?) ======================================================================== Fri 24 Apr ...... [still being edited] Observe that Total curvature = Euler # by splitting charges in the obvious way: the edge charges split into -1/2 for each adjacent face; the vertex charges split into the fraction of angle associated to each face, leaving the "gap" or curvature of the vertex as the net charge their. The total charge in each face is again zero (this time by looking at the tunring number of the boundary of each face) and the total curvature is thus the total net charge (which is the Euler # by the above). Explore a little more about wind patterns on surfaces and discover that the total number of ZEROES (of the vector field, where the wind stops, where the "hair" has a "cowlick" ;) also equals the Euler #(something we alread saw for gradient vector fields. This number of zeroes is counted with an index that is +1 for a sink or source and -1 for a saddle - and more generally is the number of times the field's direction winds around 0 when we follow a little loop around the place where it vanishes. ======================================================================== Mon 27 Apr ...... [still being edited] Orbifolds and their (fractional) Euler numbers The tile patterns on the hall floor, the brick wall and the Campsu Center patio, and their orbifolds (17 of them). ======================================================================== Wed 29 Apr ...... [still being edited] The 2-orbifolds for various platonic polyhedra. Unifolds (1-orbifolds - we found 4 connected examples, including R with trivial symmetry; R/-=ray; R/Z = S^1 and R/D=S^1/-=interval). In considering the last of these, we need to clear up on confusion: the glide-reflection group is not cyclic (it's not even abelian - it's actually the infinite dihedral group) and so it is not generated by a single element (as I was trying to make it be); the two generators are reflections S(x)=-x and R(x)=1-x, and the their composition is the translation T(x)=R(S(x))=1+x. (I'm not quite sure what the hop-twists that I did across the stage were, but maybe you can figure that out? :-) ======================================================================== Mon 04 May and beyond.... [still being edited] We defined the orbifold Euler number using charges, and we explored the teardrop T_n and cigar C_n = T_n # T_n, which have orbifold Euler number 1 + 1/n and 2/n, respectively. What other orbifolds with local structure R^2/Rot have positive orbifold Euler number? (Hint: first decide how many cone points are possible!?!) We explored the orbifolds T_a # T_b # T_c associated with the (rotation) symmetries of the various platonic solids, including the infinite family of "dihedra" (plural of "dihedron", where [a,b,c] = [2,2,n]), the tetrahedron [2,3,3], the cube (or its dual, the octahedron [2,3,4]) and the dodecahedron (or its dual, the icosahedron [2,3,5]): these give rise to the dihedral groups D_n, the tetrahedral group A_4 (even permutations of 4 objects), the octahedral group S_4 (all permutations of 4 objects) and icosahedral group A_5 (even permutations of 5 objects). These orbifolds, together with the teardrops T_n, their connected sums (including the cigars C_n = T_n # T_n, which correspond to cyclic group of symmetries, also denoted C_n) are the only ones of this form with positive curvature, i.e. positive (orbifold) Euler number. All the rest give rise to infinite symmetry groups, such as the frieze or wallpaper groups (the orbifolds are flat, i.e. have Euler number 0, such as [2,2,\infinity], [2,3,6], [2,4,4], [3,3,3] and [2,2,2,2]), or the hyperbolic (Klein) groups corresponding to orbifolds like [2,3,7] or even the "ideal homentaschen" [\infinity, \infinity, \infinity]. ======================================================================== All material copyleft by Rob Kusner, 2005-2009 ======================================================================== Project Presentation Schedule for Spring 2009 6 May @ 1 ----- Katie M: Deltahedra Lubna M: Tiling Hyperbolic Surfaces 11 May @ 1 ------ Chris E: Geometry of Sundials Andrew R: Plane curves, Arnol'd invariants and Blank words 13 May @ 1 [meet at 1535] ------ Liz K: Panoramic polyhedra Christina B: P#P#P = T#P (and other amazing surface topology) Ian C: Schwarzian derivatives Spencer R: ????? 18 May @ 10:30 [meet at 1535] ------ James: All the way with Gauss-Bonnet (well, almost)! Laura W: Hyperbolic crocheting: strips, cusps and pseoudosphere Chris B: Hypercube model - see photo Jess R: Klein surface realizing the Hurwitz 84(g-1) bound *"Copyleft" means that you have permission to use this material for non-commercial purposes as long as you acknowledge its source. Any use, in whole or in part, must include the "Copyleft 2009 by Rob Kusner" message or its equivalent. Commercial use without prior approval of the copyleft holder is strictly forbidden, and is punishable by methods not even imagined by John Yoo!