Pairs of Pants and Congruence Laws of Geometry
Hyperbolic Surfaces and Pairs of Pants
Pairs of Pants and Congruence Laws of
Geometry
S. Allen Broughton - Rose-Hulman Institute of Technology
Rose Math Seminar, January 30, 2013
Hyperbolic Surfaces and Pairs of Pants
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.
sphere
Hyperbolic Surfaces and Pairs of Pants
sphere with tesselation
torus
Hyperbolic Surfaces and Pairs of Pants
"flat" torus with tiling
genus 2 surface
Hyperbolic Surfaces and Pairs of Pants
genus two surface formed by surgery the genus is the number of holes
geometry of surfaces
Hyperbolic Surfaces and Pairs of Pants
All surfaces can be given a geometry - lines and angles - which locally looks like spherical, Euclidean, or hyperbolic geometry.
We show some examples.
spherical geometry
Hyperbolic Surfaces and Pairs of Pants
dotted curves are the lines
Hyperbolic Surfaces and Pairs of Pants
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
Hyperbolic Surfaces and Pairs of Pants
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
equilateral triangle torus
Hyperbolic Surfaces and Pairs of Pants
show how the parallelogram formed from equilateral triangles
may be glued together by translation to form a torus
definition and example
Hyperbolic Surfaces and Pairs of Pants
Two tori are equivalent if there there is a 1-1 correspondence preserving lines and angles
Consider T
1 formed from the rectangle with corners
( 1 , 0, ( 1 , 1 ) , ( 0 , 1 )
( 0 , 0 ) , and T
2 formed from the rectangle with corners
( 1 , 0 ) , ( 1 , 1 ) , ( 2 , 1 )
( 0 , 0 ) ,
T
1 and T
2 are geometrically equivalent
another example
Hyperbolic Surfaces and Pairs of Pants
Two tori formed from rectangles are equivalent (similar) if the ratio of sides are equal.
why the interest in tori?
Hyperbolic Surfaces and Pairs of Pants
Tori are useful in elliptic curve cryptography and the crypto-system depends on the geometric type in some way.
Hyperbolic Surfaces and Pairs of Pants
construction of genus surface from two pairs of pants
Actual show and tell with pants. Also the next slide.
terminology
Hyperbolic Surfaces and Pairs of Pants
waist and cuffs seams and inseams
variation in the geometry
Hyperbolic Surfaces and Pairs of Pants
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
Hyperbolic Surfaces and 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 2 g − 2 where g is the genus - number of holes
Hyperbolic Surfaces and Pairs of Pants
construction of the hexagon decomposition - 1
rip apart the seams of the pants to get two hexagons show hexagons the number of hexagons is 4 g − 4
Hyperbolic Surfaces and Pairs of Pants
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
Hyperbolic Surfaces and Pairs of Pants
construction a surface from hexagons
pick 4 g − 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.
Hyperbolic Surfaces and Pairs of Pants
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?
congruence of triangles
Hyperbolic Surfaces and Pairs of Pants
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.
not so fast on triangles
Hyperbolic Surfaces and Pairs of Pants
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.
congruence of hexagons
Hyperbolic Surfaces and Pairs of Pants
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.
Hyperbolic Surfaces and Pairs of Pants
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?
Hyperbolic Surfaces and Pairs of Pants