# The Dehn invariants of the Bricard octahedra

@article{Alexandrov2010TheDI, title={The Dehn invariants of the Bricard octahedra}, author={Victor Alexandrov}, journal={Journal of Geometry}, year={2010}, volume={99}, pages={1-13} }

We prove that the Dehn invariants of any Bricard octahedron remain constant during the flex and that the Strong Bellows Conjecture holds true for the Steffen flexible polyhedron.

## 14 Citations

Combinatorics of Bricard’s octahedra

- Mathematics
- 2020

We re-prove the classification of flexible octahedra, obtained by Bricard at the beginning of the XX century, by means of combinatorial objects satisfying some elementary rules. The explanations of…

The set of nondegenerate flexible polyhedra of a prescribed combinatorial structure is not always algebraic

- Mathematics
- 2015

We construct some example of a closed nondegenerate nonflexible polyhedron P in Euclidean 3-space that is the limit of a sequence of nondegenerate flexible polyhedra each of which is combinatorially…

A sufficient condition for a polyhedron to be rigid

- MathematicsJournal of Geometry
- 2019

We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if…

Flexible Octahedra in the Projective Extension of the Euclidean 3-Space

- Mathematics
- 2010

In this paper we complete the classification of flexible octahedra in the projective extension of the Euclidean 3-space. If all vertices are Euclidean points then we get the well known Bricard…

Dehn invariant of flexible polyhedra

- Mathematics
- 2017

We prove that the Dehn invariant of any flexible polyhedron in Euclidean space of dimension greater than or equal to 3 is constant during the flexion. In dimensions 3 and 4 this implies that any…

A necessary flexibility condition for a nondegenerate suspension in Lobachevsky 3-space

- Mathematics
- 2013

We show that some combination of the lengths of all edges of the equator of a flexible suspension in Lobachevsky 3-space is equal to zero (each length is taken with a ‘plus’ or ‘minus’ sign in this…

A necessary flexibility condition for a nondegenerate suspension in Lobachevsky 3-space

- Mathematics
- 2013

We show that some combination of the lengths of all edges of the equator of a flexible suspension in Lobachevsky 3-space is equal to zero (each length is taken with a 'plus' or 'minus' sign in this…

Algebraic methods for solution of polyhedra

- Mathematics
- 2011

By analogy with the solution of triangles, the solution of polyhedra means a theory and methods for calculating some geometric parameters of polyhedra in terms of other parameters of them. The main…

Dehn Invariant and Scissors Congruence of Flexible Polyhedra

- MathematicsProceedings of the Steklov Institute of Mathematics
- 2018

We prove that the Dehn invariant of any flexible polyhedron in n-dimensional Euclidean space, where n ≥ 3, is constant during the flexion. For n = 3 and 4 this implies that any flexible polyhedron…

Flexible suspensions with a hexagonal equator

- Mathematics
- 2009

We construct a flexible (non-embedded) suspension with a hexagonal equator in Euclidean 3-space. It is known that the volume bounded by such a suspension is well defined and constant during the flex.…

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