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| Introduction The acute triangle theorem Discreteness theorem proof outline | |
| Preliminaries Triangle groups and their tilings Pentagons Hyperbolic formulae & geometry Extending Knapp & Poincare | |
| Geometric equivalence and the discreteness theorem The standard acute triangles Nielsen eq: $(2,3,n)t=3; k=3$ Nielsen eq: $(2,4,n)t=2; k=2$ Pentagon $t=9$ & 2-2 spectrum The seven & geometric eq $t=9$ Discreteness theorem proof | |
| The real number algorithm and the Turing machine algorithm Forms of the algorithm | |
| Appendix Verify Matelski-Beardon count A summary of notation | |
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