86 research outputs found
A porism for cyclic quadrilaterals, butterfly theorems, and hyperbolic geometry
If there exists a cyclic quadrilateral whose sides go through the given four
collinear points, then there are infinitely many such quadrilaterals inscribed
in the same circle. We give two proofs of this porism; one based on
cross-ratios, the other on compositions of hyperbolic isometries.Comment: 9 pages, 10 figures, to appear in Amer. Math. Monthl
Variational properties of the discrete Hilbert-Einstein functional
This is a survey on rigidity and geometrization results obtained with the
help of the discrete Hilbert-Einstein functional, written for the proceedings
of the "Discrete Curvature" colloquium in Luminy.Comment: 10 page
Branched Coverings, Triangulations, and 3-Manifolds
A canonical branched covering over each sufficiently good simplicial complex
is constructed. Its structure depends on the combinatorial type of the complex.
In this way, each closed orientable 3-manifold arises as a branched covering
over the 3-sphere from some triangulation of S^3. This result is related to a
theorem of Hilden and Montesinos. The branched coverings introduced admit a
rich theory in which the group of projectivities plays a central role.Comment: v2: several changes to the text body; minor correction
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