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Slicing up hyperbolic tetrahedra:
from the infinite to the finite
Yana Mohanty
University of California, San Diego
[email protected]
Yana Mohanty, University of California, San Diego, [email protected]
p. 1
Overview
Spherical 2-D geometry
Hyperbolic 2-D geometry
Hyperbolic 3-D geometry
Hyperbolic tetrahedra
Problem statement:
Construct a finite tetrahadron out of ideal tetrahedra
Motivation
Study of hyperbolic 3-manifolds Scissors congruence problems
Outline of method
Yana Mohanty, University of California, San Diego, [email protected]
p. 2
Spherical geometry
•“Lines” are great circles
•Each pair of lines intersects in two points
•Triangles are “plump”
•Any 2-dimensional map distorts angles
and/or lengths
Yana Mohanty, University of California, San Diego, [email protected]
p. 3
The Mercator projection:
a conformal map
of the sphere
Angles shown are
the true angles!
(conformal)
Areas near poles
are greatly distorted
Yana Mohanty, University of California, San Diego, [email protected]
p. 4
Hyperbolic geometry:
the “opposite” of spherical geometry
•Triangles are “skinny”
•Given a point P and a line L there are
many lines through P that do not
intersect L.
•Any 2-dimensional map distorts angles
and/or lengths.
A piece of a hyperbolic surface in space
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p. 5
The Poincare model of the hyperbolic plane
lines
•Preserves angles
(conformal)
•Distorts lengths
Escher’s Circle Limit I
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p. 6
Hyperbolic space

LINES
PLANES
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p. 7
The Poincare and upper half-space models
x 2  y 2  z 2  1;
Inversion: d 
metric: ds 2  4
dx  dy  dz
[1  ( x 2  y 2  z 2 )]2
2
2
(obtained by inversion)
2

1
d
metric:
2
2
2
dx

dy

dz
ds 2 
z2


z=0
z>0
Yana Mohanty, University of California, San Diego, [email protected]
p. 8
H3: The upper halfspace model

(obtained by inversion)

metric:
2
2
2
dx

dy

dz
ds 2 
z2
“point at infinity”
z>0

z=0
Yana Mohanty, University of California, San Diego, [email protected]
p. 9
Lines and planes in the half-plane model of
hyperbolic space
Contains point at infinity
lines

PLANES
Yana Mohanty, University of California, San Diego, [email protected]
p. 10
Ideal tetrahedron in H3 (Poincare model)
Convex hull of 4 points at the sphere at infinity
Yana Mohanty, University of California, San Diego, [email protected]
p. 11
Ideal tetrahedron in H3
(half-space model)

B
b
B
b
a
A
a
g
C
Determined by triangle ABC
g
C
A
View from above
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p. 12
Hyperbolic tetrahedra
ideal: 2 parameters
¾-ideal: 3 parameters
finite: 6 parameters
1 or 2 ideal vertices also possible
Yana Mohanty, University of California, San Diego, [email protected]
p. 13
Problem statement:
How do you make

out of finitely many of these?



The rules:
•an ideal tetrahedron may count as + or –
•use finitely many planar cuts
Yana Mohanty, University of California, San Diego, [email protected]
p. 14
What is this needed for: part I
Study of hyperbolic 3-manifolds
2-Manifold: An object which is homeomorphic to a plane near every
one of its points.
can be stretched into without tearing
Example of a Euclidean 2-manifold
A 2 manifold may NOT
contain
Can’t be stretched into
a plane near this point
Yana Mohanty, University of California, San Diego, [email protected]
p. 15
Euclidean 3-manifold
Example: 3-Torus
Glue together opposite faces
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p. 16
Hyperbolic 3-manifold
Example: the Seifert-Weber space
Drawing from Jeff Weeks’
Shape of Space
Image by Matthias Weber
Glue together opposite faces
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p. 17
A strange and amazing fact:
The volume of a hyperbolic 3-manifold is a topological invariant
3-manifold X
3-manifold Y
Homeomorphic
(There is a continuous 1-1 map from X to Y with a continuous inverse)
X and Y have the same volume
Volume computation generally requires triangulating,
that is, cutting up the manifold into tetrahedra.
Yana Mohanty, University of California, San Diego, [email protected]
p. 18
Triangulating a hyperbolic 2-manifold
Finite hyperbolic
octagon
2-holed torus
glue
Drawing by Tadao Ito
In hyperbolic space triangulation involves finite tetrahedra (6-parameters)
Better: express in terms of ideal tetrahedra (2-parameters)
Yana Mohanty, University of California, San Diego, [email protected]
p. 19
What is this needed for: part II
Solving scissors congruence problems in hyperbolic space:
Given 2 polyhedra of equal volume, can one be cut up into a finite
number of pieces that can be reassembled into the other one?
Example in Euclidean space:
“Hill’s tetrahedron”
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p. 20
An expression for volume that also gives a canonical decompositon?
Exists for ideal tetrahedra:
volume=L(g)
(hidden)
b
a
V
g
volume=L(b)
L(a)L(b)L(g),
finite!
volume=L(a)

where
L ( )    log 2 sin u du
is the Lobachevsky function.
0
Yana Mohanty, University of California, San Diego, [email protected]
p. 21
Construction of a 3/4-ideal tetrahedron out of ideal tetrahedra:
extends “volume formula as a decomposition” idea
to tetrahedra with 1  finite vertex
History:
Algebraic
Proved in 1982 by Dupont and Sah using homology.
Geometric
•Mentioned as unknown by W. Neumann in 1998 survey
article on 3-manifolds.
•Indications of construction given by Sah in 1981, but
these were not well known.
Yana Mohanty, University of California, San Diego, [email protected]
p. 22
Main idea behind proof
Make a a certain type of ¾-ideal tetrahedron first
d
d
d
rotated
{a,b,c,p}
+
=
p
p
p
c
a
b
c
c
a
b
Inspiration for choosing ideal tetrahedra:
another proof of Dupont and Sah
Yana Mohanty, University of California, San Diego, [email protected]
p. 23
Remainder of the proof
Making a finite tetrahedron out of ¾-ideal tetrahedra
Step 2: 1-ideal out of ¾-ideal
Step 1: finite out of 1-ideal
D
E
E
C
B
A
C’
C
B
E
A’
D
B’
A
ABCE=A’B’C’E-A’B’BE-B’C’CE-C’A’AE
C
B finite
A
1-ideal
1-ideal
ideal
¾-ideal
ABCD=ABCE-ABDE
Yana Mohanty, University of California, San Diego, [email protected]
p. 24
Summary
• Comparison of spherical and hyperbolic geometries
• Examples of conformal models
Spherical: Mercator projection
Hyperbolic: Poincare ball
•Introduced hyperbolic tetrahedra
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p. 25
Summary, continued
•Constructing a finite tetrahedron out of ideal ones is helpful for studying
-hyperbolic 3-manifolds
volume is an invariant, so construction is helpful in the
3-dimensional equivalent of
-scissors congruences
want volume formula that is also a decomposition
•Main ingredient: constructing a certain ¾-ideal tetrahedron
out of ideal tetrahedra. Idea comes from a proof by
Dupont and Sah.
Yana Mohanty, University of California, San Diego, [email protected]
p. 26