标题： 四面体的平衡状态 显示英文标题

标题： On equilibria of tetrahedra

摘要： 多面体的单稳性（即仅具有一个稳定或不稳定静态平衡点的性质）自Conway和Guy\cite{Conway}发表了第一个此类物体存在性的证明以来，一直受到研究的关注。 在同一篇文章中，他们还证明了一个均匀的四面体至少有两个稳定的平衡点。 通过使用极对偶性，同样的想法已被用于\cite{balancing}证明一个均匀的四面体至少有两个不稳定的平衡点。 Conway\cite{Dawson}还声称在非均匀的四面体中可以找到单稳的四面体。 在这里，我们不仅给出了该陈述的正式证明，并表明单稳的四面体恰好有4个平衡点，而且还展示了这个问题的一个令人惊讶的新方面：单稳性意味着形状的某些\emph{可见的}特征，反之亦然。 我们的结果还表明，仅有一个稳定和一个不稳定平衡点的单-单稳四面体不存在。 相反，我们证明对于任何其他合法的面数、边数和顶点数，都存在具有该面向量的单-单稳多面体。 显示英文摘要

摘要： The monostatic property of polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in a focus of research ever since Conway and Guy \cite{Conway} published the proof of the existence of the first such object. In the same article they also proved that a homogeneous tetrahedron has at least two stable equilibrium points. By using polar duality, the same idea has been used \cite{balancing} to prove that a homogeneous tetrahedron has at least two unstable equilibria. Conway \cite{Dawson} also claimed that among inhomogeneous tetrahedra one can find monostable ones. Here we not only give a formal proof of this statement and show that monostatic tetrahedra have exactly 4 equilibria, but also demonstrate a startling new aspect of this problem: being monostatic implies certain \emph{visible} features of the shape and vice versa. Our results also imply that mono-monostatic tetrahedra (having just one stable and just one unstable equilibrium point) do not exist. In contrast, we show that for any other legal number of faces, edges, and vertices there is a mono-monostatic polyhedron with that face vector.