1. Slicing up hyperbolic tetrahedra: from the infinite to the finite
Yana Mohanty
University of California, San Diego
Yana Mohanty, University of California, San Diego,

2. Overview Problem statement:
Spherical 2-D geometry
Hyperbolic 2-D geometry
Hyperbolic 3-D geometry
Hyperbolic tetrahedra
Problem statement:
Construct a finite tetrahadron out of ideal tetrahedra
Overview
Outline of method
Motivation
Scissors congruence problems
Study of hyperbolic 3-manifolds
Yana Mohanty, University of California, San Diego,

3. Spherical geometry “Lines” are great circles
Triangles are “plump”
Any 2-dimensional map distorts angles and/or lengths
“Lines” are great circles
Each pair of lines intersects in two points
Yana Mohanty, University of California, San Diego,

4. 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,

5. the “opposite” of spherical geometry
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
Yana Mohanty, University of California, San Diego,

6. The Poincare model of the hyperbolic plane
Escher’s Circle Limit I
lines
Preserves angles
(conformal)
Distorts lengths
Yana Mohanty, University of California, San Diego,

7. Hyperbolic space LINES PLANES
Yana Mohanty, University of California, San Diego,

8. The Poincare and upper half-space models (obtained by inversion)
metric:
z>0
metric:
Inversion:
z=0
Yana Mohanty, University of California, San Diego,

9. H3: The upper halfspace model (obtained by inversion)
metric:
“point at infinity”
z>0
z=0
Yana Mohanty, University of California, San Diego,

10. Lines and planes in the half-plane model of hyperbolic space
Contains point at infinity
lines
PLANES
Yana Mohanty, University of California, San Diego,

11. Ideal tetrahedron in H3 (Poincare model)
Convex hull of 4 points at the sphere at infinity
Yana Mohanty, University of California, San Diego,

12. Ideal tetrahedron in H3 (half-space model)B b B b g a a g C A C A
Determined by triangle ABC
View from above
Yana Mohanty, University of California, San Diego,

13. 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,

14. 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,

15. 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,

16. Euclidean 3-manifold Example: 3-Torus Glue together opposite faces
Yana Mohanty, University of California, San Diego,

17. Hyperbolic 3-manifold Example: the Seifert-Weber space
Drawing from Jeff Weeks’
Shape of Space
Image by Matthias Weber
Glue together opposite faces
Yana Mohanty, University of California, San Diego,

18. A strange and amazing fact:
The volume of a hyperbolic 3-manifold is a topological invariant
(There is a continuous 1-1 map from X to Y with a continuous inverse)
Homeomorphic
3-manifold X
3-manifold Y
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,

19. 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,

20. 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”
Yana Mohanty, University of California, San Diego,

21. An expression for volume that also gives a canonical decompositon?
Exists for ideal tetrahedra:
volume=L(a)
volume=L(b)
volume=L(g)
(hidden) b a V
=L(a)+L(b)+L(g), g
finite!
where
is the Lobachevsky function.
Yana Mohanty, University of California, San Diego,

22. Construction of a 3/4-ideal tetrahedron out of ideal tetrahedra:
extends “volume formula as a decomposition” idea
to tetrahedra with finite vertex
Algebraic
Proved in 1982 by Dupont and Sah using homology.
History:
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,

23. Main idea behind proof
Make a a certain type of ¾-ideal tetrahedron first d d d p p
rotated
{a,b,c,p} = + - p c c c a a b b
Inspiration for choosing ideal tetrahedra: another proof of Dupont and Sah
Yana Mohanty, University of California, San Diego,

24. Remainder of the proof
Making a finite tetrahedron out of ¾-ideal tetrahedra
Step 1: finite out of 1-ideal A B C D
Step 2: 1-ideal out of ¾-ideal A B C E E C’ B’ E A B C D A’
ABCE=A’B’C’E-A’B’BE-B’C’CE-C’A’AE
1-ideal
ideal
¾-ideal
finite
1-ideal
ABCD=ABCE-ABDE
Yana Mohanty, University of California, San Diego,

25. Summary Comparison of spherical and hyperbolic geometries
Examples of conformal models
Spherical: Mercator projection
Hyperbolic: Poincare ball
Introduced hyperbolic tetrahedra
Yana Mohanty, University of California, San Diego,

26. 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,