Geometry

A book in 21 chapters, from points, lines and angles to the curved space of general relativity and the shape of the universe. Each chapter starts from a question, draws it, proves what it claims, works examples step by step, and ends with questions whose answers are folded underneath.

How to read it

• In order. Each chapter needs only the ones before it; the Next link at the end of every chapter walks you through.
• Levels inside a chapter. The box at the top of each chapter says which sections are beginner, intermediate, advanced or expert. On a first read, skip the harder sections and come back later.
• Slower help. The first chapters point to the matching lessons on the Topics shelf, which teach the same ideas more gently, for younger readers.
• Draw along. A ruler, a compass and paper make every proof easier to follow.

Contents

Part	Chapter	Level
I. Euclid's world	1. Points, lines and angles	beginner
	2. Triangles and congruence	beginner
	3. Parallel lines and the angle sum	beginner
	4. Area and Pythagoras	beginner to intermediate
	5. Similarity	beginner to intermediate
II. Measuring and moving	6. Circles	intermediate
	7. Trigonometry	intermediate
	8. Coordinate geometry	intermediate
	9. Vectors	intermediate
	10. Transformations and symmetry	intermediate to advanced
	11. Solid geometry	intermediate to advanced
III. Classical depth	12. Triangle centres and classical theorems	advanced
	13. Inversion and complex geometry	advanced
	14. Constructions and impossibility	advanced
IV. Other geometries	15. Axioms and the parallel postulate	advanced
	16. Spherical geometry	advanced
	17. Hyperbolic geometry	advanced to expert
	18. Projective geometry	advanced to expert
V. Curved space and the frontier	19. Curves and surfaces	expert
	20. Manifolds and Riemannian geometry	expert
	21. Where geometry goes	expert

The glossary defines every term, with a link to the chapter that teaches it.

Three ways in

• Starting out, or coming back after years away: read chapters 1 to 8. That is all of school geometry and trigonometry, with every reason written out.
• You know school geometry: start at chapter 9 or 10, then read chapters 12 to 18 for the classical depth and the geometries Euclid never imagined.
• You know calculus and want curved space: read chapters 15 to 17, then 19 to 21.

The Algebra book is the companion: chapter 14 here uses its chapter 19, and the symmetry groups of chapter 10 are studied there in chapters 13 to 16.