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Compact hyperbolic tetrahedra with non-obtuse dihedral angles.

Roland K.W. Roeder (2006)

Publicacions Matemàtiques

Given a combinatorial description C of a polyhedron having E edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize C is generally not a convex subset of RE. If C has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angles AC obtained by restricting to non-obtuse angles is a convex polytope. In this paper we explain why Andreev did not consider tetrahedra, the only polyhedra having fewer than five faces, by demonstrating that the...

Hyperideal polyhedra in hyperbolic 3-space

Xiliang Bao, Francis Bonahon (2002)

Bulletin de la Société Mathématique de France

A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic 3 -space ℍ 3 which, in the projective model for ℍ 3 ⊂ ℝℙ 3 , is just the intersection of ℍ 3 with a projective polyhedron whose vertices are all outside ℍ 3 and whose edges all meet ℍ 3 . We classify hyperideal polyhedra, up to isometries of ℍ 3 , in terms of their combinatorial type and of their dihedral angles.

La conjecture des soufflets

Jean-Marc Schlenker (2002/2003)

Séminaire Bourbaki

On sait depuis les travaux de Bricard et de Connelly qu’il existe dans l’espace euclidien des polyèdres (non convexes) qui sont flexibles : on peut les déformer continûment sans changer la forme de leurs faces. La conjecture des soufflets affirme que le volume interieur de ces polyèdres est constant au cours de la déformation. Elle a été démontrée récemment par I. Sabitov, qui a pour cela utilisé des outils algébriques inattendus dans ce contexte.

Properties of triangulations obtained by the longest-edge bisection

Francisco Perdomo, Ángel Plaza (2014)

Open Mathematics

The Longest-Edge (LE) bisection of a triangle is obtained by joining the midpoint of its longest edge with the opposite vertex. Here two properties of the longest-edge bisection scheme for triangles are proved. For any triangle, the number of distinct triangles (up to similarity) generated by longest-edge bisection is finite. In addition, if LE-bisection is iteratively applied to an initial triangle, then minimum angle of the resulting triangles is greater or equal than a half of the minimum angle...

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