Simpler foundations for the hyperbolic plane
-
John Bamberg
and
Tim Penttila
Abstract
H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger’s axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala’s formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.
Funding source: Australian Research Council
Award Identifier / Grant number: FT120100036
Funding statement: The first author acknowledges the support of the Australian Research Council Future Fellowship FT120100036.
Acknowledgements
The first author would like to give special thanks to Sue Barwick for accommodating the first author’s research visit to the University of Adelaide where much of this work was done. We are especially grateful to an anonymous referee for their comments on a draft of this paper.
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Articles in the same Issue
- Frontmatter
- On the arithmetic of polynomial semidomains
- Extra-special Leibniz superalgebras
- Restricted iso-minimum condition
- On the spectral large sieve inequality for symmetric-squares
- Simple 𝔰𝔩(V)-modules which are free over an abelian subalgebra
- Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
- Non-weight modules over N = 1 Lie superalgebras of Block type
- Simpler foundations for the hyperbolic plane
- Characterizations of the mixed radial-angular central Campanato space via the commutators of Hardy type
- The fourth moment of Dirichlet L-functions along the critical line
- Diagonal restriction of Eisenstein series and Kudla–Millson theta lift
- Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term
- Jantzen filtration of Weyl modules for general linear supergroups
- Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems
