Bricard octahedron

[3] The Bricard octahedra have six vertices, twelve edges, and eight triangular faces, connected in the same way as a regular octahedron.

His work in this area was later the subject of lectures by Henri Lebesgue at the Collège de France.

The resulting Bricard octahedron resembles one of the extreme configurations of the second animation, which has an equatorial antiparallelogram.

It is also possible to think of the Bricard octahedron as a mechanical linkage consisting of the twelve edges, connected by flexible joints at the vertices, without the faces.

But, by the symmetries of its construction, the flexing motions of these two open pyramids both move the equator along which they were cut in the same way.

[2][6] The property of having opposite sides of equal length is true of the rectangle, parallelogram, and antiparallelogram, and it is possible to construct Bricard octahedra having any of those flat shapes as their equators.

[9] However, there exist other self-crossing flexible polyhedra for which the Dehn invariant changes continuously as they flex.

[10] It is possible to modify the Bricard polyhedra by adding more faces, in order to move the self-crossing parts of the polyhedron away from each other while still allowing it to flex.