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Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws Pairs of Pants and Congruence Laws of Geometry S. Allen Broughton - Rose-Hulman Institute of Technology Rose Math Seminar, January 30, 2013 Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws geometric surfaces What are geometric surfaces and why study them? Surfaces are the boundary of objects in our 3D world, such as the surface of a planet. Surfaces, especially when equipped with geometry and topology, are the simplest examples of manifolds, explained next. Manifolds form the configuration space or state space for systems in science and engineering, such as the configuration space of a robot arm or the configuration space of a solar system. Surfaces and Geometry Tori pictures of surfaces sphere sphere with tesselation Hyperbolic Surfaces and Pairs of Pants Congruence Laws Surfaces and Geometry Tori pictures of surfaces torus "flat" torus with tiling Hyperbolic Surfaces and Pairs of Pants Congruence Laws Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants pictures of surfaces genus 2 surface genus two surface formed by surgery the genus is the number of holes Congruence Laws Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws geometry for surfaces geometry of surfaces All surfaces can be given a geometry - lines and angles - which locally looks like spherical, Euclidean, or hyperbolic geometry. We show some examples. Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants geometry for surfaces spherical geometry dotted curves are the lines Congruence Laws Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws geometry for surfaces Euclidean plane geometry for the torus the curves are the lines of the geometry on a flat torus the pattern locally looks like the unfolded plane picture Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws geometry for surfaces hyperbolic geometry for higher genus surfaces hyperbolic surface with geometry is hard to picture - we will use the pants model instead unfolded hyperbolic surface geometry - circles orthogonal to the boundary are the lines again patterns locally look Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants construction of tori equilateral triangle torus show how the parallelogram formed from equilateral triangles may be glued together by translation to form a torus Congruence Laws Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws equivalence of tori definition and example Two tori are equivalent if there there is a 1-1 correspondence preserving lines and angles Consider T1 formed from the rectangle with corners (0, 0), (1, 0, (1, 1), (0, 1) and T2 formed from the rectangle with corners (0, 0), (1, 0), (1, 1), (2, 1) T1 and T2 are geometrically equivalent Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws equivalence of tori another example Two tori formed from rectangles are equivalent (similar) if the ratio of sides are equal. Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws equivalence of tori why the interest in tori? Tori are useful in elliptic curve cryptography and the crypto-system depends on the geometric type in some way. Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws pairs of pants construction of genus surface from two pairs of pants Actual show and tell with pants. Also the next slide. Surfaces and Geometry Tori pairs of pants terminology waist and cuffs seams and inseams Hyperbolic Surfaces and Pairs of Pants Congruence Laws Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws pairs of pants variation in the geometry geometry of the pants depends only on the size of the waist and the cuffs geometry of the surface depends on twisting the way the pants are put together - show and tell Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws pairs of pants different pairs of pants decompositions show picture of genus 2 decompositions show picture of genus 3 decompositions the number of pairs of pants equals 2g − 2 where g is the genus - number of holes Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants hexagon decomposition construction of the hexagon decomposition - 1 rip apart the seams of the pants to get two hexagons show hexagons the number of hexagons is 4g − 4 Congruence Laws Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws hexagon decomposition construction of the hexagon decomposition - 2 the waist, cuffs and the seams can be chosen so that each hexagon is a hyperbolic hexagon each hexagon has six right angles the two hexagons from a pair of pants have corresponding pairs of equal sides because they have the same sides Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws hexagon decomposition construction a surface from hexagons pick 4g − 4 right angled hexagons with suitably matched sides form pairs of pants by sewing the seams and inseams sew the pants together with twists different constructions my yield geometrically equivalent surfaces - identifying equivalent surfaces is not so easy. Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws hexagon decomposition question on congruence of hexagons The two hexagons from a pair of pants have 6 equal angles and three congruent sides. Is this enough information to make the two hexagons congruent? Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws triangle case congruence of triangles There are six degrees of freedom in choosing the point of a triangle. Translating a vertex to a given point and rotating a side about that point uses up three degrees of freedom. So he should only be three degrees of freedom for congruence classes of triangles We also have three sides and three angles, all of which are invariant under translation and rotation. So selecting three of the three sides and three angles should determine a triangle up to congruence Similar reasoning for hyperbolic and spherical geometry. Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws triangle case not so fast on triangles We have side-side-side and side-angle-side congruence theorems. We do not have an angle-angle-angle congruence theorem Euclidean geometry (similarity only). There is an an angle-angle-angle congruence theorem in hyperbolic and spherical geometry. The possible angle-side-side fails in a two to one fashion show picture. Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws hexagon case congruence of hexagons There are twelve degrees of freedom in choosing the point of a hexagon. Rigid motions use up three degrees of freedom. So he should only be nine degrees of freedom for congruence classes of triangles. We also have six sides and six angles, all of which are invariant under congruence. So specifying nine of the six sides and the six angles should determine a hexagons up to congruence. Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants Congruence Laws hexagon case question on congruence of hexagons Which subsets of nine sides/angles yield a unique congruence class of hexagons. If the number of classes is finite in number, how many are there? Surfaces and Geometry Tori Hyperbolic Surfaces and Pairs of Pants done Any Questions? Congruence Laws