Flexible polyhedra
Let's look at a polygon with hard sides, at the vertices of which hinges are placed. If it has more than three vertices, then it can bend - the lengths of the sides do not uniquely determine the polygon. And what happens to polyhedra in three-dimensional space? If you fix the shape of their faces, can they bend?
It turns out that sometimes they can, but this is a very rare property. We
must say right away that bending means continuous bending, and not just that the polyhedron is not uniquely defined by its faces. It is quite easy to come up with such an example:
However, back in 1813, Cauchy proved that even such a situation is impossible for a convex polyhedron: a convex polyhedron is uniquely determined by its faces.
In 1897, it was possible to construct examples of self-intersecting flexible polyhedra (this can be visualized as a wire frame; the absence of hard edges does not matter, since they are all triangular and are uniquely defined by edges) - Bricard's octahedra. Wolfram demonstration
Only in 1976, Conelli proposed the construction of a non-self-intersecting non-convex bendable polyhedron. Following his ideas, Steffen soon built an example of a flexible polyhedron with 9 vertices (it was later proved that this could not be done with fewer vertices). A video with this polyhedron is posted at the beginning of the post, there is also a Wolfram demonstration .
We make a reservation in advance that polyhedra with triangular faces are considered. This does not change the essence of the matter, since for any bendable polyhedron, edges can be added by cutting the edges into triangles, from which its bending will not disappear. However, this simplifies the calculations, since now all the information about the faces of the polyhedron is contained in the combinatorial structure and the lengths of the edges.
Let us now try to understand why it turned out to be so difficult to find flexible polyhedra, while for polygons this is a very typical property. Let's look at a polygon with n vertices. Its shape is determined by the coordinates of the vertices, of which 2n. These coordinates determine not only the shape of the polygon, but also its position on the plane. The position is specified by 3 coordinates (for example, the pair of coordinates of one vertex and the angle of rotation of the polygon around it). Thus, a system with 2n-3 degrees of freedom is obtained, while edge lengths impose only n conditions, and for n> 3, 2n-3> n is obtained. In mathematical language, there are n functions of 2n-3 variables that map to the vertex coordinate set the set of squares of edge lengths (squares are taken to make the functions polynomial) and for n> 3 the image of the function far from uniquely defines the inverse image.
We now carry out a similar calculation for polyhedra. The shape of a polyhedron with n vertices is determined by 3n-6 parameters (since the polyhedron position in space is determined by 6 parameters). We now calculate the number of edges. Let their number be equal to e. If f is the number of faces, then 3f = 2e, since two edges are adjacent to each edge, and each face contains 3 edges. Applying the Euler Formula , we obtain n-e + 2e / 3 = 2, that is, e = 3n-6. It turns out that the number of conditions imposed on the polyhedron is exactly equal to the number of degrees of freedom.
This does not mean that the lengths of the edges uniquely determine the shape of the polyhedron. It is possible that each set of edge lengths will have several prototypes among the polyhedron forms, but they will be isolated (as in the example at the beginning of the post), but locally the prototype is unique. Cm.Implicit function theorem . A whole family of preimages needed for bending can only be found if the set of functions is degenerate, see Jacobian . Thus, for the possibility of bending, the combinatorial structure of the polyhedron must specify a degenerate system of equations for the lengths of the edges and the coordinates of the vertices, which explains the rarity of flexible polyhedra.
After constructing examples of flexible polyhedra, mathematicians began to study their properties under bending. In 1996, Sabitov discovered an amazing fact - a flexible polyhedron retains volume during bending (more precisely, he proved a stronger statement - the volume of a polyhedron is the root of a polynomial whose coefficients are polynomially expressed in terms of squares of edge lengths). What is noteworthy, despite the recent result, the proof is not at all complicated and understandable for a mathematics student of 1-2 courses.
Further, mathematicians began to study polyhedra of higher dimensions. A. Gayfullin proved an analogue of the Sabitov theorem in all dimensions and constructed examples of flexible polyhedra of all dimensions.
Additional materials:
It turns out that sometimes they can, but this is a very rare property. We
must say right away that bending means continuous bending, and not just that the polyhedron is not uniquely defined by its faces. It is quite easy to come up with such an example:
However, back in 1813, Cauchy proved that even such a situation is impossible for a convex polyhedron: a convex polyhedron is uniquely determined by its faces.
In 1897, it was possible to construct examples of self-intersecting flexible polyhedra (this can be visualized as a wire frame; the absence of hard edges does not matter, since they are all triangular and are uniquely defined by edges) - Bricard's octahedra. Wolfram demonstration
Only in 1976, Conelli proposed the construction of a non-self-intersecting non-convex bendable polyhedron. Following his ideas, Steffen soon built an example of a flexible polyhedron with 9 vertices (it was later proved that this could not be done with fewer vertices). A video with this polyhedron is posted at the beginning of the post, there is also a Wolfram demonstration .
We make a reservation in advance that polyhedra with triangular faces are considered. This does not change the essence of the matter, since for any bendable polyhedron, edges can be added by cutting the edges into triangles, from which its bending will not disappear. However, this simplifies the calculations, since now all the information about the faces of the polyhedron is contained in the combinatorial structure and the lengths of the edges.
Let us now try to understand why it turned out to be so difficult to find flexible polyhedra, while for polygons this is a very typical property. Let's look at a polygon with n vertices. Its shape is determined by the coordinates of the vertices, of which 2n. These coordinates determine not only the shape of the polygon, but also its position on the plane. The position is specified by 3 coordinates (for example, the pair of coordinates of one vertex and the angle of rotation of the polygon around it). Thus, a system with 2n-3 degrees of freedom is obtained, while edge lengths impose only n conditions, and for n> 3, 2n-3> n is obtained. In mathematical language, there are n functions of 2n-3 variables that map to the vertex coordinate set the set of squares of edge lengths (squares are taken to make the functions polynomial) and for n> 3 the image of the function far from uniquely defines the inverse image.
We now carry out a similar calculation for polyhedra. The shape of a polyhedron with n vertices is determined by 3n-6 parameters (since the polyhedron position in space is determined by 6 parameters). We now calculate the number of edges. Let their number be equal to e. If f is the number of faces, then 3f = 2e, since two edges are adjacent to each edge, and each face contains 3 edges. Applying the Euler Formula , we obtain n-e + 2e / 3 = 2, that is, e = 3n-6. It turns out that the number of conditions imposed on the polyhedron is exactly equal to the number of degrees of freedom.
This does not mean that the lengths of the edges uniquely determine the shape of the polyhedron. It is possible that each set of edge lengths will have several prototypes among the polyhedron forms, but they will be isolated (as in the example at the beginning of the post), but locally the prototype is unique. Cm.Implicit function theorem . A whole family of preimages needed for bending can only be found if the set of functions is degenerate, see Jacobian . Thus, for the possibility of bending, the combinatorial structure of the polyhedron must specify a degenerate system of equations for the lengths of the edges and the coordinates of the vertices, which explains the rarity of flexible polyhedra.
After constructing examples of flexible polyhedra, mathematicians began to study their properties under bending. In 1996, Sabitov discovered an amazing fact - a flexible polyhedron retains volume during bending (more precisely, he proved a stronger statement - the volume of a polyhedron is the root of a polynomial whose coefficients are polynomially expressed in terms of squares of edge lengths). What is noteworthy, despite the recent result, the proof is not at all complicated and understandable for a mathematics student of 1-2 courses.
Further, mathematicians began to study polyhedra of higher dimensions. A. Gayfullin proved an analogue of the Sabitov theorem in all dimensions and constructed examples of flexible polyhedra of all dimensions.
Additional materials:
- Video on etudes.ru about flexible polyhedra
- An article by I. Maximov on flexible polyhedra with a small number of vertices
- Lecture by A. Gaifullin
- Instructions for gluing the Steffen polyhedron