Victor Alexandrov
In the previous version of the paper it was announced that ``sphere homeomorphic flexible polyhedra (with self intersections) do really exist in n-dimensional Euclidean, Lobachevskij and spherical spaces for each $n\geq 3$.'' Now the paper has been withdrawn by the author, due a crucial error in the proof of the main theorem.
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Constructing flexible polyhedra by twinning
本文基于Bricard方法扩展，构造了具有自交的无限多面体家族，通过移除边得到crinkle，并展示如何用crinkle构建无自交的灵活多面体。
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 equivalent to $P$. This implies that the set of flexible nondegenerate polyhedra combinatorially equivalent to $P$ is not algebraic.
We demonstrate the existence of four types of flexible prismatic polyhedra that can be derived or inferred from a consideration of Bricard octahedra and generalizations of Bricard octahedra. These flexible polyhedra are of genus 0 and 1, have dihedral angles all of which are non-constant under flexion, have faces that exhibit self-intersections, have an indefinite number of faces and have vertex motion that can treated as a function of a single variable.
A new method for generalizing non-self-intersecting flexible polyhedra
本文提出“基底+褶皱”构造方法，用于生成非自交、非三角剖分、多自由度且拓扑超出球面的柔性多面体，并为折纸几何与密封腔机器人、变胞抓持器等工程机构设计提供新见解。
Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: R. Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove the both theorems. As a by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space is not an algebraic function of its edge lengths.
We discuss some recent results on flexible polyhedra and the bellows conjecture, which claims that the volume of any flexible polyhedron is constant during the flexion. Also, we survey main methods and several open problems in this area.
We demonstrate the construction of several families of flexible polyhedra by extending Bricard octahedra to form larger composite flexible polyhedra. These flexible polyhedra are of genus 0 and 1, have dihedral angles that are non-constant under flexion, exhibit self-intersections and are of indefinite size, the smallest of which is a decahedron with seven vertexes.
In the end of the 19th century Bricard discovered a phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit non-trivial flexes. One of the most important results in this field is a theorem of Sabitov asserting that the volume of a flexible polyhedron is constant during the flexion. In this paper we study flexible polyhedral surfaces in the 3-space two-periodic with respect to translations by two non-colinear vectors that can vary continuously during the flexion. The main result is that the period lattice of a flexible two-periodic surface homeomorph…
The bellows conjecture for small flexible polyhedra in non-Euclidean spaces
针对非欧空间中柔性多面体体积是否恒定的问题，本文证明在维数不低于3的球面和Lobachevsky空间中，棱长足够小的柔性多面体均满足风箱猜想。
Given a polyhedral surface, assume that it is prohibited to change the shape and size of any face but it is permissible to change the dihedral angles between the faces. A polyhedral surface is said to be flexible if it is possible to change its shape under the above restrictions. We prove that flexible polyhedral surfaces without boundary do exist in the Minkowski 3-space and each of them preserves the (inclosed) volume and the (total) mean curvature during a flex. To prove the latter result, we introduce the notion of the angle between two arbitrary non-null nonzero vectors in the Minkowski p…
Finding necessary conditions for the geometry of flexible polyhedra is a classical problem that has received attention also in recent times. For flexible polyhedra with triangular faces, we showed in a previous work the existence of cycles with a sign assignment for their edges, such that the signed sum of the edge lengths along the cycle is zero. In this work, we extend this result to flexible non-triangular polyhedra.
A Kokotsakis polyhedron with quadrangular base is a neighborhood of a quadrilateral in a quad surface. Generically, a Kokotsakis polyhedron is rigid. Up to now, several flexible classes were known, but a complete classification was missing. In the present paper, we provide such a classification. The analysis is based on the fact that the dihedral angles of a Kokotsakis polyhedron are related by an Euler-Chasles correspondence. It results in a diagram of elliptic curves covering complex projective planes. A polyhedron is flexible if and only if this diagram commutes.
We study the existence problem for a local implicit function determined by a system of nonlinear algebraic equations in the particular case when the determinant of its Jacobian matrix vanishes at the point under consideration. We present a system of sufficient conditions that implies existence of a local implicit function as well as another system of sufficient conditions that guarantees absence of a local implicit function. The results obtained are applied to proving new and classical results on flexibility and rigidity of polyhedra and frameworks.
We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, i.e., can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and -1. We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem t…
Polyhedra called Siamese dipyramids are known to be non-flexible, however their physical models behave like physical models of flexible polyhedra. We discuss a simple mathematical method for explaining the model flexibility of the Siamese dipyramids.
Since Gruenbaum and Shephard's investigation of self-intersection-free polyhedra with positive genus and vertex-transitive symmetry in 1984 the question of complete classification of such objects in Euclidean 3-space has been open. Due to a recent article by Gevay, Schulte, and Wills, we now know that the genus range g>=2 can only admit a finite number of vertex-transitive polyhedra, all with rotational Platonic symmetry. In this article, we show that the symmetry group must also act simply transitively on the vertices. Furthermore, the case of rotational tetrahedral symmetry is settled comple…
Dehn invariant of flexible polyhedra
本文证明高维 Euclid 空间柔性多面体的 Dehn 不变量在弯折中恒定，修正既往反例并证明强风箱猜想，同时刻画球面与 Lobachevsky 空间中的等价条件。
The action of the rotation group $SO(3)$ on systems of $n$ points in the $3$-dimensional Euclidean space $\mathbf{R}^3$ induces naturally an action of $SO(3)$ on $\mathbf{R}^{3n}$. In the present paper we consider the following question: do there exist $3$ polynomial functions $f_1$, $f_2$, $f_3$ on $\mathbf{R}^{3n}$ such that the intersection of the set of common zeros of $f_1$, $f_2$, and $f_3$ with each orbit of $SO(3)$ in $R^{3n}$ is nonempty and finite? Questions of this kind arise when one is interested in relative motions of a given set of $n$ points, i.e., when one wants to exclude the…
Quasi-symmetric nets: A constructive approach to the equimodular elliptic type of Kokotsakis polyhedra
本文提出拟对称网（QS-nets）来构造四边底等模椭圆型柔性Kokotsakis多面体，给出显式代数刻画和闭式变形参数化，并开发数值管道验证实例。
An orthant polyhedron is a polyhedron with $m$ hyperfaces, that could be realized as a section of the $m$-dimensional non-negative orthant. We classify all 2-dimensional orthant polyhedra and provide some partial results towards the classification of higher dimensional orthant polyhedra. As a consequence of our results we show that every polytope could be realized as a section of the $n$-dimensional non-negative orthant with $n$ being big enough.
In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Unti…
The theory of the isoptic curves is widely studied in the Euclidean plane $\bE^2$ (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic $\bH^2$ and elliptic $\cE^2$ planes (see \cite{CsSz1}, \cite{CsSz2}, \cite{CsSz5}), but in the higher dimensional spaces there are only a few result in this topic. In \cite{CsSz4} we gave a natural extension of the notion of the isoptic curves to the $n$-dimensional Euclidean space $\bE^n$ $(n\ge 3)$ which are called isoptic hypersurfaces. Now we develope an algorithm to determ…
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, the only known polytopes (besides polygons) that are edge-transitive without being vertex-transitive. We show that these polyhedra do not have higher-dimensional analogues, that is, that in dimension $d\ge 4$, edge-transitivity of convex polytopes implies vertex-transitivity. More generally, we give a classification of all convex polytopes which at the same time have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show that any such po…
Pentagonal bipyramids lead to the smallest flexible embedded polyhedron
针对无自相交可弯曲多面体的顶点数下限问题，本文通过Galois群分类可弯曲五角双锥，发现可扩展为8顶点嵌入可弯曲多面体的构造，解决了该未决问题。
The Simplest Flexible Cross-Polytopes
基于Bricard的柔性八面体分类，研究n维欧氏空间中允许两个平坦姿态的柔性交叉多胞体，构造其平坦姿态并证明其与Bricard第三类八面体的相似性质。
The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces
本文通过将 Lobachevsky 空间单纯形体积视为棱长双曲余弦的函数并研究其解析延拓，证明了奇数维 Lobachevsky 空间中有界柔性多面体满足 Bellows 猜想。
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