Flexible polyhedra
Let's look at the polygon with rigid sides, at the vertices of which the hinges are placed. If it has more than three vertices, then it can be bent - 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.
At once, we say that bending means continuous bending, and not just that a polyhedron is not uniquely defined by its faces. Such an example is pretty easy to come up with
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However, in 1813, Cauchy proved that even 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 (it can be visualized as a wire frame, the absence of rigid edges does not matter, since all of them are triangular and are uniquely defined by edges) - Bricard octahedra. Wolfram demonstration
Only in 1976, Conelli proposed the construction of a non-self-intersecting nonconvex flexible polyhedron. Following his ideas, Steffen soon built an example of a flexible polyhedron with 9 vertices (it was later proved that with a smaller number of vertices it was impossible to do this). A video with this polyhedron is placed at the beginning of the post, there is also a Wolfram demonstration .
Let's stipulate in advance that polyhedra with triangular faces are considered. This does not change the essence of the matter, since any bendable polyhedron can add edges by cutting faces into triangles, from which its bendability does 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 edge lengths.
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 given by the coordinates of the vertices, of which 2n. These coordinates define not only the shape of the polygon, but also its position on the plane. The position is defined by 3 coordinates (for example, a pair of coordinates of one vertex and the angle of rotation of a polygon around it). Thus, a system with 2n-3 degrees of freedom is obtained, while the lengths of the edges impose only n conditions, and for n> 3, 2n-3> n is obtained. Mathematically speaking, there are n functions in 2n-3 variables that associate a set of vertex coordinates with a set of squares of edge lengths (squares are taken so that the functions turn out to be polynomial) and for n> 3 the image of the function is far from uniquely defining the pre-image.
We now carry out a similar calculation for polyhedra. The shape of a polyhedron with n vertices is given by 3n-6 parameters (since the polyhedron's position in space is given by 6 parameters). Now calculate the number of edges. Let their number be e. If f is the number of faces, then 3f = 2e, since there are two faces 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 a polyhedron is exactly equal to the number of degrees of freedom.
This does not mean that the edge lengths uniquely define the shape of a polyhedron. It is quite possible that each set of edge lengths will have several preimages among the shapes of polyhedra, but they will be isolated (as in the example at the beginning of the post), but locally the type is unique. See Implicit Function Theorem . The whole family of preimages needed for bending can be found only if the set of functions is degenerate, see Jacobian . Thus, in order to bend, the combinatorial structure of a polyhedron must define a degenerate system of equations for the lengths of the edges and coordinates of the vertices, which explains the rarity of the flexible polyhedra.
After constructing examples of flexible polyhedra, mathematics began to study their properties in flexing. In 1996, Sabitov discovered an amazing fact - a flexible polyhedron preserves volume when bent (more precisely, he proved a stronger statement - the volume of a polyhedron is the root of a polynomial, whose coefficients polynomially expressed in terms of squares of edge lengths). What is remarkable, despite the recent results, the proof is not at all difficult and understandable for a 1-2 year math student.
Then mathematicians began to study polyhedra of higher dimensions. A. Gaifullin proved an analogue of Sabitov's 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.
At once, we say that bending means continuous bending, and not just that a polyhedron is not uniquely defined by its faces. Such an example is pretty easy to come up with
')
However, in 1813, Cauchy proved that even 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 (it can be visualized as a wire frame, the absence of rigid edges does not matter, since all of them are triangular and are uniquely defined by edges) - Bricard octahedra. Wolfram demonstration
Only in 1976, Conelli proposed the construction of a non-self-intersecting nonconvex flexible polyhedron. Following his ideas, Steffen soon built an example of a flexible polyhedron with 9 vertices (it was later proved that with a smaller number of vertices it was impossible to do this). A video with this polyhedron is placed at the beginning of the post, there is also a Wolfram demonstration .
Let's stipulate in advance that polyhedra with triangular faces are considered. This does not change the essence of the matter, since any bendable polyhedron can add edges by cutting faces into triangles, from which its bendability does 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 edge lengths.
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 given by the coordinates of the vertices, of which 2n. These coordinates define not only the shape of the polygon, but also its position on the plane. The position is defined by 3 coordinates (for example, a pair of coordinates of one vertex and the angle of rotation of a polygon around it). Thus, a system with 2n-3 degrees of freedom is obtained, while the lengths of the edges impose only n conditions, and for n> 3, 2n-3> n is obtained. Mathematically speaking, there are n functions in 2n-3 variables that associate a set of vertex coordinates with a set of squares of edge lengths (squares are taken so that the functions turn out to be polynomial) and for n> 3 the image of the function is far from uniquely defining the pre-image.
We now carry out a similar calculation for polyhedra. The shape of a polyhedron with n vertices is given by 3n-6 parameters (since the polyhedron's position in space is given by 6 parameters). Now calculate the number of edges. Let their number be e. If f is the number of faces, then 3f = 2e, since there are two faces 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 a polyhedron is exactly equal to the number of degrees of freedom.
This does not mean that the edge lengths uniquely define the shape of a polyhedron. It is quite possible that each set of edge lengths will have several preimages among the shapes of polyhedra, but they will be isolated (as in the example at the beginning of the post), but locally the type is unique. See Implicit Function Theorem . The whole family of preimages needed for bending can be found only if the set of functions is degenerate, see Jacobian . Thus, in order to bend, the combinatorial structure of a polyhedron must define a degenerate system of equations for the lengths of the edges and coordinates of the vertices, which explains the rarity of the flexible polyhedra.
After constructing examples of flexible polyhedra, mathematics began to study their properties in flexing. In 1996, Sabitov discovered an amazing fact - a flexible polyhedron preserves volume when bent (more precisely, he proved a stronger statement - the volume of a polyhedron is the root of a polynomial, whose coefficients polynomially expressed in terms of squares of edge lengths). What is remarkable, despite the recent results, the proof is not at all difficult and understandable for a 1-2 year math student.
Then mathematicians began to study polyhedra of higher dimensions. A. Gaifullin proved an analogue of Sabitov's theorem in all dimensions and constructed examples of flexible polyhedra of all dimensions.
Additional materials:
- Video on the website etudes.ru about bent polyhedra
- I. Maksimov 's article on flexible polyhedra with a small number of vertices
- Lecture A.Gayfullina
- Instructions for gluing the Steffen polyhedron
Source: https://habr.com/ru/post/234315/
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