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

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Problem 2.1. What Is Straight on a Sphere?

-10-

Symmetries of Great Circles

-12-

Every Geodesic is a Great Circle

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Intrinsic Curvature

-15-

Problem 2.2. Straightness on Cylinders and Cones

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Cones with Varying Cone Angles

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Geodesics on Cylinders

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Geodesics on Cones

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Locally Isometric

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Is "Shortest" Always "Straight"?

-21-

Euclid's Postulates and Differential Geometry

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Chapter 3

# Angles, ITT, and Circles

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Problem 3.1. Vertical Angle Theorem (VAT)

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Problem 3.2. What Is an Angle?

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Hints for Three Different Proofs

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Problem 3.3. Isosceles Triangle Theorem (ITT)

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Problem 3.4. Circles and Constructions

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Chapter 4

# Circles in the Plane

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Problem 4.1. Angles & Power Points of Planar Circles

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Problem 4.2. Inversions in Circles

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*Problem 4.3. Applications of Inversions

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Chapter 5

# Hyperbolic Planes

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Constructions of Hyperbolic Planes

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Hyperbolic Planes of Different Radii (Curvature)

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Problem 5.1. What is Straight in a Hyperbolic Plane?

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Problem 5.2. The Upper Half Plane Model.

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Problem 5.3. Hyperbolic Isometries and Geodesics

-47-

Chapter 6

# Transformations and Triangles

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Geodesics are Locally Unique

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Problem 6.1. Properties of Geodesics

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Problem 6.2. Transformations

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Problem 6.3. Side-Angle-Side (SAS)

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Problem 6.4. Angle-Side-Angle (ASA)

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Chapter 7

# Area and Holonomy

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Problem 7.1. The Area of a Triangle on a Sphere

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Problem 7.2. Area of Hyperbolic Triangles

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Introducing Parallel Transport and Holonomy

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Problem 7.3. The Holonomy of a Small Triangle

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The Gauss-Bonnet Formula for Triangles

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Problem 7.4. Gauss-Bonnet Formula for Polygons

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Gauss-Bonnet Formula for Polygons on Surfaces

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Chapter 8

# Parallel Transport

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

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Problem 8.2. Symmetries of Parallel Transported Lines

-71-

Problem 8.3. Transversals through a Midpoint

-73-

Problem 8.4. What is "Parallel"?

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Chapter 9

# SSS, ASS, SAA, and AAA

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Problem 9.1. Side-Side-Side (SSS)

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Problem 9.2. Angle-Side-Side (ASS)

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Problem 9.3. Side-Angle-Angle (SAA)

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Problem 9.4. Angle-Angle-Angle (AAA)

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Chapter 10

# Parallel Postulates

-81-

Parallel Lines on the Plane Are Special

-81-

Problem 10.1. Parallel Transport on the Plane

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Parallel Circles on a Sphere

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Parallel Postulates

-83-

Problem 10.2. Parallel Postulates on the Plane

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Problem 10.3. The P P on Sphere & Hyperbolic Plane

-84-

Parallelism in Spherical and Hyperbolic Geometry

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Problem 10.4. Sum of the Angles of a Planar Triangle

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Chapter 11

# Geometric 2-Manifolds and Coverings

-87-

*Problem 11.1. Geodesics on Cylinders and Cones

-87-

n-Sheeted Coverings of a Cylinder

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n-Sheeted (Branched) Coverings of a Cone

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Problem 11.2. Flat Torus and Klein Bottle

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*Problem 11.3. Universal Covering of Flat 2-Manifold

-93-

Problem 11.4. Spherical 2-Manifolds

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*Coverings of the Sphere

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Problem 11.5. Hyperbolic Manifolds

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Problem 11.6. Area, Euler Number and Gauss-Bonnet

-100-

Triangles on Geometric Manifolds

-101-

Problem 11.7. Can The Bug Tell Which Manifold?

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Chapter 12

# 3-Spheres, Hyperbolic 3-Spaces

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

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Problem 12.2. A 3-Sphere in 4-Space

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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

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Chapter 13

# Dissection Theory

-111-

Problem 13.1. Dissect Plane Triangle & Parallelogram

-112-

Dissection Theory on Spheres & Hyperbolic Planes

-113-

-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

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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-

-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.

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Problem 16.1. Gnomic Projection

-143-

Problem 16.2. Cylindrical Projection

-143-

Problem 16.3. Stereographic Projection

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Problem 16.4. Poincaré Disk Model

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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-

-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-

-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-