Body

Chapter 1

Straightness and Symmetry

-1-

Problem 1. When Do You Call a Line Straight?
-1-

How Do You Construct a Straight Line?
-2-

The Symmetries of a Line
-4-

Local (and Infinitesimal) Straightness
-8-

Chapter 2

Straightness on Sphere, Cylinder, and Cone

-10-

Problem 2.1. What Is Straight on a Sphere?
-10-

Symmetries of Great Circles
-12-

Every Geodesic is a Great Circle
-15-

Intrinsic Curvature
-15-

Problem 2.2. Straightness on Cylinders and Cones
-15-

Cones with Varying Cone Angles
-16-

Geodesics on Cylinders
-18-

Geodesics on Cones
-19-

Locally Isometric
-20-

Is "Shortest" Always "Straight"?
-21-

Euclid's Postulates and Differential Geometry
-22-

Chapter 3

Angles, ITT, and Circles

-23-

Problem 3.1. Vertical Angle Theorem (VAT)
-23-

Problem 3.2. What Is an Angle?
-23-

Hints for Three Different Proofs
-25-

Problem 3.3. Isosceles Triangle Theorem (ITT)
-26-

Problem 3.4. Circles and Constructions
-27-

Chapter 4

Circles in the Plane

-29-

Problem 4.1. Angles & Power Points of Planar Circles
-30-

Problem 4.2. Inversions in Circles
-31-

*Problem 4.3. Applications of Inversions
-33-

Chapter 5

Hyperbolic Planes

-36-

Constructions of Hyperbolic Planes
-37-

Hyperbolic Planes of Different Radii (Curvature)
-41-

Problem 5.1. What is Straight in a Hyperbolic Plane?
-43-

Problem 5.2. The Upper Half Plane Model.
-44-

Problem 5.3. Hyperbolic Isometries and Geodesics
-47-

Chapter 6

Transformations and Triangles

-50-

Geodesics are Locally Unique
-50-

Problem 6.1. Properties of Geodesics
-50-

Problem 6.2. Transformations
-51-

Problem 6.3. Side-Angle-Side (SAS)
-51-

Problem 6.4. Angle-Side-Angle (ASA)
-55-

Chapter 7

Area and Holonomy

-59-

Problem 7.1. The Area of a Triangle on a Sphere
-59-

Problem 7.2. Area of Hyperbolic Triangles
-59-

Introducing Parallel Transport and Holonomy
-62-

Problem 7.3. The Holonomy of a Small Triangle
-64-

The Gauss-Bonnet Formula for Triangles
-65-

Problem 7.4. Gauss-Bonnet Formula for Polygons
-66-

Gauss-Bonnet Formula for Polygons on Surfaces
-68-

Chapter 8

Parallel Transport

-70-

Problem 8.1. Euclid's Exterior Angle Theorem (EEAT)
-70-

Problem 8.2. Symmetries of Parallel Transported Lines
-71-

Problem 8.3. Transversals through a Midpoint
-73-

Problem 8.4. What is "Parallel"?
-74-

Chapter 9

SSS, ASS, SAA, and AAA

-75-

Problem 9.1. Side-Side-Side (SSS)
-75-

Problem 9.2. Angle-Side-Side (ASS)
-76-

Problem 9.3. Side-Angle-Angle (SAA)
-77-

Problem 9.4. Angle-Angle-Angle (AAA)
-79-

Chapter 10

Parallel Postulates

-81-

Parallel Lines on the Plane Are Special
-81-

Problem 10.1. Parallel Transport on the Plane
-81-

Parallel Circles on a Sphere
-83-

Parallel Postulates
-83-

Problem 10.2. Parallel Postulates on the Plane
-84-

Problem 10.3. The P P on Sphere & Hyperbolic Plane
-84-

Parallelism in Spherical and Hyperbolic Geometry
-85-

Problem 10.4. Sum of the Angles of a Planar Triangle
-86-

Chapter 11

Geometric 2-Manifolds and Coverings

-87-

*Problem 11.1. Geodesics on Cylinders and Cones
-87-

n-Sheeted Coverings of a Cylinder
-88-

n-Sheeted (Branched) Coverings of a Cone
-89-

Problem 11.2. Flat Torus and Klein Bottle
-90-

*Problem 11.3. Universal Covering of Flat 2-Manifold
-93-

Problem 11.4. Spherical 2-Manifolds
-94-

*Coverings of the Sphere
-96-

Problem 11.5. Hyperbolic Manifolds
-98-

Problem 11.6. Area, Euler Number and Gauss-Bonnet
-100-

Triangles on Geometric Manifolds
-101-

Problem 11.7. Can The Bug Tell Which Manifold?
-102-

Chapter 12

3-Spheres, Hyperbolic 3-Spaces

-103-

Problem 12.1. Explain 3-Space to 2-D Person
-103-

Problem 12.2. A 3-Sphere in 4-Space
-105-

Problem 12.3. Hyperbolic 3-Space (Upper Half Space)
-107-

*Problem 12.4. Disjoint Equidistant Great Circles
-108-

*Problem 12.5. Hyperbolic and Spherical Symmetries
-109-

Problem 12.6. Triangles in 3-Dimensions
-110-

Chapter 13

Dissection Theory

-111-

Problem 13.1. Dissect Plane Triangle & Parallelogram
-112-

Dissection Theory on Spheres & Hyperbolic Planes
-113-

Problem 13.2. Khayyam Quadrilaterals
-113-

Problem 13.3. Dissect Spherical & Hyperbolic Triangles and Khayyam Parallelograms
-114-

*Problem 13.4. Spherical Polygons Dissect to Biangles
-114-

Chapter 14

Square Roots, Pythagoras, and Similar Triangles

-116-

Square Roots
-116-

Problem 14.1. A Rectangle Dissects into a Square
-117-

Baudhayana's Sulbasutram
-119-

Problem 14.2. Equivalence of Squares
-122-

Any Polygon Can Be Dissected into a Square
-123-

Problem 14.3. Similar Triangles
-123-

Three-Dimensional Dissections and Hilbert's Third
-124-

Addendum: Numerical Approximations of Square Roots
-124-

Chapter 15

Geometric Solutions of Quadratic and Cubic Equations

-131-

Problem 15.1. Quadratic Equations
-131-

Problem 15.2. Conic Sections and Cube Roots
-134-

Problem 15.3. Roots of Cubic Equations
-136-

Problem 15.4. Algebraic Solution of Cubics
-138-

So What Does This All Point To?
-140-

Chapter 16

Projections of Spheres and Hyperbolic Planes.

-142-

Problem 16.1. Gnomic Projection
-143-

Problem 16.2. Cylindrical Projection
-143-

Problem 16.3. Stereographic Projection
-144-

Problem 16.4. Poincaré Disk Model
-145-

Problem 16.4. Projective Disk Model
-146-

Chapter 17

Duality and Trigonometry

-147-

Problem 17.1. Circumference of a Circle
-147-

Problem 17.2. Law of Cosines
-148-

Problem 17.3. Law of Sines
-150-

Duality on a Sphere
-151-

Problem 17.4. The Dual of a Small Triangle
-152-

*Problem 17.5. Trigonometry with Congruences
-153-

Duality on the Projective Plane
-153-

Problem 17.6. Properties on the Projective Plane
-154-

Perspective Drawings and Vision
-154-

Chapter 18

Isometries and Patterns

-156-

Definitions and Terminology
-156-

Problem 18.1. Examples of Patterns
-159-

Problem 18.2. Isometry Determined by Three Points
-159-

Problem 18.3. Classification of Isometries
-159-

Problem 18.4. Classification of Discrete Strip Patterns
-160-

Problem 18.5. Classification of Finite Plane Patterns
-160-

Problem 18.6. Regular Tilings with Polygons
-161-

Geometric Meaning of Abstract Group Terminology
-161-

Chapter 19

Polyhedra

-163-

Definitions and Terminology
-163-

Problem 19.1. Measure of a Solid Angle
-164-

Problem 19.2. Edges and Face Angles
-164-

Problem 19.3. Edges and Dihedral Angles
-165-

Problem 19.4. Other Tetrahedra Congruence Theorems
-166-

Problem 19.5. The Five Regular Polyhedra
-166-

Chapter 20

3-Manifolds — Shape of Space

-169-

Space as an Oriented Geometric 3-Manifold
-169-

Problem 20.1. Is Our Universe Non-Euclidean?
-170-

Problem 20.2. Euclidean 3-Manifolds
-171-

Problem 20.3. Dodecahedral 3-Manifolds
-173-

Problem 20.4. Some Other Geometric 3-Manifolds
-174-

Cosmic Background Radiation
-175-

Problem 20.5. Circle Patterns Show the Shape of Space
-176-

Appendices

A Geometric Introduction to Differential Geometry

-178-

The Universe — Zooms
-178-

Smooth Curves
-179-

Smooth Surfaces, Curvature, Geodesics, and Isometries
-179-

Theorems on Geodesics
-180-

Bibliography

-181-

AD. Art and Design
-181-

An. Analysis
-181-

AT. Ancient Texts
-181-

DG. Differential Geometry
-182-

DS. Dimensions and Scale
-184-

Fr. Fractals
-184-

GC. Geometry in Different Cultures
-184-

Hi. History
-185-

LA. Linear Algebra and Geometry
-186-

Mi. Minimal Surfaces
-186-

MP. Models, Polyhedra
-186-

Na. Nature
-187-

NE. Non-Euclidean Geometries (Mostly Hyperbolic)
-187-

Ph. Philosophy
-188-

RN. Real Numbers
-188-

SP. Spherical and Projective Geometry
-189-

SG. Symmetry and Groups
-189-

SE. Surveys and General Expositions
-190-

Tp. Topology
-191-

Tx. Geometry Texts
-191-

Z. Miscellaneous
-192-

Chapter 1
Straightness and Symmetry	-1-
Problem 1. When Do You Call a Line Straight?	-1-
How Do You Construct a Straight Line?	-2-
The Symmetries of a Line	-4-
Local (and Infinitesimal) Straightness	-8-
Chapter 2
Straightness on Sphere, Cylinder, and Cone	-10-
Problem 2.1. What Is Straight on a Sphere?	-10-
Symmetries of Great Circles	-12-
Every Geodesic is a Great Circle	-15-
Intrinsic Curvature	-15-
Problem 2.2. Straightness on Cylinders and Cones	-15-
Cones with Varying Cone Angles	-16-
Geodesics on Cylinders	-18-
Geodesics on Cones	-19-
Locally Isometric	-20-
Is "Shortest" Always "Straight"?	-21-
Euclid's Postulates and Differential Geometry	-22-
Chapter 3
Angles, ITT, and Circles	-23-
Problem 3.1. Vertical Angle Theorem (VAT)	-23-
Problem 3.2. What Is an Angle?	-23-
Hints for Three Different Proofs	-25-
Problem 3.3. Isosceles Triangle Theorem (ITT)	-26-
Problem 3.4. Circles and Constructions	-27-
Chapter 4
Circles in the Plane	-29-
Problem 4.1. Angles & Power Points of Planar Circles	-30-
Problem 4.2. Inversions in Circles	-31-
*Problem 4.3. Applications of Inversions	-33-
Chapter 5
Hyperbolic Planes	-36-
Constructions of Hyperbolic Planes	-37-
Hyperbolic Planes of Different Radii (Curvature)	-41-
Problem 5.1. What is Straight in a Hyperbolic Plane?	-43-
Problem 5.2. The Upper Half Plane Model.	-44-
Problem 5.3. Hyperbolic Isometries and Geodesics	-47-
Chapter 6
Transformations and Triangles	-50-
Geodesics are Locally Unique	-50-
Problem 6.1. Properties of Geodesics	-50-
Problem 6.2. Transformations	-51-
Problem 6.3. Side-Angle-Side (SAS)	-51-
Problem 6.4. Angle-Side-Angle (ASA)	-55-
Chapter 7
Area and Holonomy	-59-
Problem 7.1. The Area of a Triangle on a Sphere	-59-
Problem 7.2. Area of Hyperbolic Triangles	-59-
Introducing Parallel Transport and Holonomy	-62-
Problem 7.3. The Holonomy of a Small Triangle	-64-
The Gauss-Bonnet Formula for Triangles	-65-
Problem 7.4. Gauss-Bonnet Formula for Polygons	-66-
Gauss-Bonnet Formula for Polygons on Surfaces	-68-
Chapter 8
Parallel Transport	-70-
Problem 8.1. Euclid's Exterior Angle Theorem (EEAT)	-70-
Problem 8.2. Symmetries of Parallel Transported Lines	-71-
Problem 8.3. Transversals through a Midpoint	-73-
Problem 8.4. What is "Parallel"?	-74-
Chapter 9
SSS, ASS, SAA, and AAA	-75-
Problem 9.1. Side-Side-Side (SSS)	-75-
Problem 9.2. Angle-Side-Side (ASS)	-76-
Problem 9.3. Side-Angle-Angle (SAA)	-77-
Problem 9.4. Angle-Angle-Angle (AAA)	-79-
Chapter 10
Parallel Postulates	-81-
Parallel Lines on the Plane Are Special	-81-
Problem 10.1. Parallel Transport on the Plane	-81-
Parallel Circles on a Sphere	-83-
Parallel Postulates	-83-
Problem 10.2. Parallel Postulates on the Plane	-84-
Problem 10.3. The P P on Sphere & Hyperbolic Plane	-84-
Parallelism in Spherical and Hyperbolic Geometry	-85-
Problem 10.4. Sum of the Angles of a Planar Triangle	-86-
Chapter 11
Geometric 2-Manifolds and Coverings	-87-
*Problem 11.1. Geodesics on Cylinders and Cones	-87-
n-Sheeted Coverings of a Cylinder	-88-
n-Sheeted (Branched) Coverings of a Cone	-89-
Problem 11.2. Flat Torus and Klein Bottle	-90-
*Problem 11.3. Universal Covering of Flat 2-Manifold	-93-
Problem 11.4. Spherical 2-Manifolds	-94-
*Coverings of the Sphere	-96-
Problem 11.5. Hyperbolic Manifolds	-98-
Problem 11.6. Area, Euler Number and Gauss-Bonnet	-100-
Triangles on Geometric Manifolds	-101-
Problem 11.7. Can The Bug Tell Which Manifold?	-102-
Chapter 12
3-Spheres, Hyperbolic 3-Spaces	-103-
Problem 12.1. Explain 3-Space to 2-D Person	-103-
Problem 12.2. A 3-Sphere in 4-Space	-105-
Problem 12.3. Hyperbolic 3-Space (Upper Half Space)	-107-
*Problem 12.4. Disjoint Equidistant Great Circles	-108-
*Problem 12.5. Hyperbolic and Spherical Symmetries	-109-
Problem 12.6. Triangles in 3-Dimensions	-110-
Chapter 13
Dissection Theory	-111-
Problem 13.1. Dissect Plane Triangle & Parallelogram	-112-
Dissection Theory on Spheres & Hyperbolic Planes	-113-
Problem 13.2. Khayyam Quadrilaterals	-113-
Problem 13.3. Dissect Spherical & Hyperbolic Triangles and Khayyam Parallelograms	-114-
*Problem 13.4. Spherical Polygons Dissect to Biangles	-114-
Chapter 14
Square Roots, Pythagoras, and Similar Triangles	-116-
Square Roots	-116-
Problem 14.1. A Rectangle Dissects into a Square	-117-
Baudhayana's Sulbasutram	-119-
Problem 14.2. Equivalence of Squares	-122-
Any Polygon Can Be Dissected into a Square	-123-
Problem 14.3. Similar Triangles	-123-
Three-Dimensional Dissections and Hilbert's Third	-124-
Addendum: Numerical Approximations of Square Roots	-124-

Chapter 15
Geometric Solutions of Quadratic and Cubic Equations	-131-
Problem 15.1. Quadratic Equations	-131-
Problem 15.2. Conic Sections and Cube Roots	-134-
Problem 15.3. Roots of Cubic Equations	-136-
Problem 15.4. Algebraic Solution of Cubics	-138-
So What Does This All Point To?	-140-
Chapter 16
Projections of Spheres and Hyperbolic Planes.	-142-
Problem 16.1. Gnomic Projection	-143-
Problem 16.2. Cylindrical Projection	-143-
Problem 16.3. Stereographic Projection	-144-
Problem 16.4. Poincaré Disk Model	-145-
Problem 16.4. Projective Disk Model	-146-
Chapter 17
Duality and Trigonometry	-147-
Problem 17.1. Circumference of a Circle	-147-
Problem 17.2. Law of Cosines	-148-
Problem 17.3. Law of Sines	-150-
Duality on a Sphere	-151-
Problem 17.4. The Dual of a Small Triangle	-152-
*Problem 17.5. Trigonometry with Congruences	-153-
Duality on the Projective Plane	-153-
Problem 17.6. Properties on the Projective Plane	-154-
Perspective Drawings and Vision	-154-
Chapter 18
Isometries and Patterns	-156-
Definitions and Terminology	-156-
Problem 18.1. Examples of Patterns	-159-
Problem 18.2. Isometry Determined by Three Points	-159-
Problem 18.3. Classification of Isometries	-159-
Problem 18.4. Classification of Discrete Strip Patterns	-160-
Problem 18.5. Classification of Finite Plane Patterns	-160-
Problem 18.6. Regular Tilings with Polygons	-161-
Geometric Meaning of Abstract Group Terminology	-161-
Chapter 19
Polyhedra	-163-
Definitions and Terminology	-163-
Problem 19.1. Measure of a Solid Angle	-164-
Problem 19.2. Edges and Face Angles	-164-
Problem 19.3. Edges and Dihedral Angles	-165-
Problem 19.4. Other Tetrahedra Congruence Theorems	-166-
Problem 19.5. The Five Regular Polyhedra	-166-
Chapter 20
3-Manifolds — Shape of Space	-169-
Space as an Oriented Geometric 3-Manifold	-169-
Problem 20.1. Is Our Universe Non-Euclidean?	-170-
Problem 20.2. Euclidean 3-Manifolds	-171-
Problem 20.3. Dodecahedral 3-Manifolds	-173-
Problem 20.4. Some Other Geometric 3-Manifolds	-174-
Cosmic Background Radiation	-175-
Problem 20.5. Circle Patterns Show the Shape of Space	-176-

Appendices
A Geometric Introduction to Differential Geometry	-178-
The Universe — Zooms	-178-
Smooth Curves	-179-
Smooth Surfaces, Curvature, Geodesics, and Isometries	-179-
Theorems on Geodesics	-180-

Bibliography	-181-
AD. Art and Design	-181-
An. Analysis	-181-
AT. Ancient Texts	-181-
DG. Differential Geometry	-182-
DS. Dimensions and Scale	-184-
Fr. Fractals	-184-
GC. Geometry in Different Cultures	-184-
Hi. History	-185-
LA. Linear Algebra and Geometry	-186-
Mi. Minimal Surfaces	-186-
MP. Models, Polyhedra	-186-
Na. Nature	-187-
NE. Non-Euclidean Geometries (Mostly Hyperbolic)	-187-
Ph. Philosophy	-188-
RN. Real Numbers	-188-
SP. Spherical and Projective Geometry	-189-
SG. Symmetry and Groups	-189-
SE. Surveys and General Expositions	-190-
Tp. Topology	-191-
Tx. Geometry Texts	-191-
Z. Miscellaneous	-192-

Contents

Straightness and Symmetry

Straightness on Sphere, Cylinder, and Cone

Angles, ITT, and Circles

Circles in the Plane

Hyperbolic Planes

Transformations and Triangles

Area and Holonomy

Parallel Transport

SSS, ASS, SAA, and AAA

Parallel Postulates

Geometric 2-Manifolds and Coverings

3-Spheres, Hyperbolic 3-Spaces

Dissection Theory

Square Roots, Pythagoras, and Similar Triangles

Geometric Solutions of Quadratic and Cubic Equations

Projections of Spheres and Hyperbolic Planes.

Duality and Trigonometry

Isometries and Patterns

Polyhedra

3-Manifolds — Shape of Space

A Geometric Introduction to Differential Geometry

Bibliography