Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 1

Slicing up hyperbolic tetrahedra: from the infinite to the finite

Yana Mohanty

University of California, San Diego

mohanty@math.ucsd.edu

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 2

Outline of method

Motivation

Scissors congruence problemsStudy of hyperbolic 3-manifolds

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

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 3

Spherical geometry

•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, mohanty@math.ucsd.edu p. 4

The Mercator projection: a conformal map

of the sphere

Angles shown arethe true angles!(conformal)

Areas near polesare greatly distorted

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 5

Hyperbolic geometry:

•Triangles are “skinny”

•Any 2-dimensional map distorts angles and/or lengths.

the “opposite” of spherical geometry

A piece of a hyperbolic surface in space

•Given a point P and a line L there are many lines through P that do not intersect L.

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 6

The Poincare model of the hyperbolic plane

•Preserves angles (conformal)

•Distorts lengths

Escher’s Circle Limit I

lines

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 7

Hyperbolic space

PLANES

LINES

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 8

The Poincare and upper half-space models (obtained by inversion)

z>0

metric:

2

2222

z

dzdydxds

dd

1Inversion:

metric:

2222

2222

)](1[4

zyx

dzdydxds

;1222 zyx

z=0

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 9

H3: The upper halfspace model

(obtained by inversion)

metric:

2

2222

z

dzdydxds

z=0

z>0

“point at infinity”

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 10

Lines and planes in the half-plane model of hyperbolic space

PLANES

lines

Contains point at infinity

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 11

Ideal tetrahedron in H3 (Poincare model)

Convex hull of 4 points at the sphere at infinity

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 12

Ideal tetrahedron in H3

(half-space model)

A

B

C

B

CA

View from above

Determined by triangle ABC

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 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, mohanty@math.ucsd.edu p. 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, mohanty@math.ucsd.edu p. 15

What is this needed for: part IStudy 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

A 2 manifold may NOTcontain

Can’t be stretched into a plane near this point

Example of a Euclidean 2-manifold

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 16

Euclidean 3-manifold

Example: 3-Torus

Glue together opposite faces

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 17

Hyperbolic 3-manifold

Example: the Seifert-Weber space

Glue together opposite faces

Image by Matthias Weber

Drawing from Jeff Weeks’Shape of Space

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 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, mohanty@math.ucsd.edu p. 19

Triangulating a hyperbolic 2-manifold

glue

In hyperbolic space triangulation involves finite tetrahedra (6-parameters)

Better: express in terms of ideal tetrahedra (2-parameters)

Drawing by Tadao Ito

Finite hyperbolic octagon 2-holed torus

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 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, mohanty@math.ucsd.edu p. 21

An expression for volume that also gives a canonical decompositon?

Exists for ideal tetrahedra:

V

where duu

0

sin2log)( is the Lobachevsky function.

finite!

volume=

volume=

volume=(hidden)

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 22

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.

Construction of a 3/4-ideal tetrahedron out of ideal tetrahedra:

extends “volume formula as a decomposition” ideato tetrahedra with finite vertex 1

Algebraic

Proved in 1982 by Dupont and Sah using homology.

History:

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 23

Make a a certain type of ¾-ideal tetrahedron first

Main idea behind proof

+ -=

a

b

c

d

a

d

c

p

b

c

d

pp

rotated{a,b,c,p}

Inspiration for choosing ideal tetrahedra: another proof of Dupont and Sah

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 24

Remainder of the proof

Making a finite tetrahedron out of ¾-ideal tetrahedra

A

BC

D

Step 1: finite out of 1-ideal

AB

C

DE

E

Step 2: 1-ideal out of ¾-ideal

AB

C

E

A’

B’C’

ABCE=A’B’C’E-A’B’BE-B’C’CE-C’A’AE

ABCD=ABCE-ABDE

finite 1-ideal1-ideal idea

l¾-ideal

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 25

Summary

•Introduced hyperbolic tetrahedra

• Comparison of spherical and hyperbolic geometries

• Examples of conformal modelsSpherical: Mercator projectionHyperbolic: Poincare ball

Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu p. 26

Summary, continued

•Main ingredient: constructing a certain ¾-ideal tetrahedron out of ideal tetrahedra. Idea comes from a proof by Dupont and Sah.

•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