Download Pairs of Pants and Congruence Laws of Geometry - Rose

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