Deformations of period lattices of flexible polyhedral surfaces

一种推广非自交柔性多面体的新方法

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。

具有柔性 $3 \ imes 3$ 子网的离散共轭网的刚性与柔性

Rigidity and flexibility of discrete conjugate nets with flexible $3 \times 3$-subnets

研究离散共轭网的刚性与柔性，通过反例驳斥了非退化离散共轭网在具有柔性 $3 \ imes 3$ 子网时必为柔性的错误论断，并在更强非退化条件下给出了修正版本。

Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional I: Polyhedral case

The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra, which deals with deformations that preserve face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volum…

Polyhedral Surfaces in Wedge Products

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially r…

所有弱凸且可分解的多面体表面是否都是无穷小刚性的？

Are all Weakly Convex and Decomposable Polyhedral Surfaces Infinitesimally Rigid?

本文研究了弱凸且可分解多面体的无穷小刚性猜想，通过构造非余可分解的 $T$-多面体族并利用离散 Hilbert-Einstein 泛函的 Hessian $M_T$ 进行特征值实验，提供了此类柔性多面体可能不存在的实验证据。

Auxetic deformations and elliptic curves

The problem of detecting auxetic behavior, originating in materials science and mathematical crystallography, refers to the property of a flexible periodic bar-and-joint framework to widen, rather than shrink, when stretched in some direction. The only known algorithmic solution for detecting infinitesimal auxeticity is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of 3D periodic bar-and-joint frameworks with 3 degrees-of-freedom. This class includes most zeo…

Flexible placements of graphs with rotational symmetry

We study the existence of an $n$-fold rotationally symmetric placement of a symmetric graph in the plane allowing a continuous deformation that preserves the symmetry and the distances between adjacent vertices. We show that such a flexible placement exists if and only if the graph has a NAC-colouring satisfying an additional property on the symmetry; a NAC-colouring is a surjective edge colouring by two colours such that every cycle is either monochromatic, or there are at least two edges of each colour.

The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in $\mathbb R^d$ does not always remain unaltered during the flex

Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in special bounded domains of Euclidean $d$-space. The boundary of such a domain is an embedded simplicial complex which allows a continuous deformation (a flex), under which each simplex of the complex moves as a solid body and the change in the spatial shape of the domain is achieved through a change of the dihedral angles only. The main result of this article is that both the Dirichlet and Neumann spectra of the Laplace operator in such a domain do not necessarily rem…