History and philosophy of maths · Solid · Dissertation

The parallel postulate and new geometries: a maths EPQ idea

Why it works as an EPQ

Two thousand years of attempted proofs end with consistent alternative geometries, which raises a real question about truth in mathematics.

Scope and difficulty

Solid. Solid. Spherical geometry is accessible; hyperbolic geometry at the level of one model.

The maths

Builds on these A Level topics: Trigonometry · Proof.

You would learn:

• Euclid's postulates
• Spherical triangles and angle sums
• A model of hyperbolic geometry

Learning maths beyond the course is expected in a maths EPQ, but you must understand and explain everything you use.

One possible plan

1. Set out Euclid's postulates and the parallel postulate.
2. Show on a sphere that triangle angle sums exceed 180°.
3. Describe one model of hyperbolic geometry.
4. Judge what 'true' means for a geometry, with the role of physics.

Pitfalls

• Claims about relativity you cannot support.
• Pictures without mathematics.

Where to start reading

• Euclid's Elements (any good edition with commentary)
• Search for: spherical excess area of spherical triangle

Prefer books, lecture notes, journal articles and official data to a single website, and record every source as you read (how to reference).

Similar ideas

• Ramanujan and mathematical intuitionHow did Ramanujan reach results without proofs, and what does his work say about intuition in mathematics?History and philosophySolidDissertation
• Why you cannot trisect an angleWhy could the ancient Greeks not trisect an angle with straight edge and compass, and how was it finally proved impossible?History and philosophyAmbitiousDissertation
• Florence Nightingale's statisticsHow did Florence Nightingale use statistics and diagrams to change policy, and would her diagrams pass today's standards?History and philosophyAccessibleDissertation
• Is maths discovered or invented?Is mathematics discovered or invented? Weighing Platonism against formalismHistory and philosophySolidDissertation

All history and philosophy of maths ideas · all 93 ideas

Your EPQ must be your own work. These pages coach: ideas, structure, checklists and planning. Submitting text, proofs, code or analysis written by someone else or by an AI tool as your own is malpractice. How to use help and AI honestly.