geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

(this is also true in higher dimensions).

Examples

The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by . They are self-intersecting surfaces isometric to an

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

. The first example of a flexible non-self-intersecting surface in

\mathbb^3

, the Connelly sphere, was discovered by . Steffen's polyhedron is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra.

Bellows conjecture

In the late 1970s Connelly and D. Sullivan formulated the bellows conjecture stating that the

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

by using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by . The proof extends

Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...

's formula for the volume of a tetrahedron to a formula for the volume of any polyhedron. The extended formula shows that the volume must be a root of a polynomial whose coefficients depend only on the lengths of the polyhedron's edges. Since the edge lengths cannot change as the polyhedron flexes, the volume must remain at one of the finitely many roots of the polynomial, rather than changing continuously.

Scissor congruence

Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. This was known as the strong bellows conjecture or (after it was proven in 2018) the strong bellows theorem. Because all configurations of a flexible polyhedron have both the same volume and the same Dehn invariant, they are scissors congruent to each other, meaning that for any two of these configurations it is possible to dissect one of them into polyhedral pieces that can be reassembled to form the other. The total mean curvature of a flexible polyhedron, defined as the sum of the products of edge lengths with exterior dihedral angles, is a function of the Dehn invariant that is also known to stay constant while a polyhedron flexes.

Generalizations

Flexible 4-polytopes in 4-dimensional Euclidean space and 3-dimensional hyperbolic space were studied by . In dimensions

n\geq 5

, flexible polytopes were constructed by .

References

Notes

Primary sources

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Secondary sources

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External links

* {{Mathematics of paper folding Nonconvex polyhedra Mathematics of rigidity Flexible polyhedra