15 research outputs found

    Rigidity and volume preserving deformation on degenerate simplices

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    Given a degenerate (n+1)(n+1)-simplex in a dd-dimensional space MdM^d (Euclidean, spherical or hyperbolic space, and dβ‰₯nd\geq n), for each kk, 1≀k≀n1\leq k\leq n, Radon's theorem induces a partition of the set of kk-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in MdM^d for d=nd=n, and the volumes of kk-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all kk-faces; and this property still holds in MdM^d for dβ‰₯n+1d\geq n+1 if an invariant ckβˆ’1(Ξ±kβˆ’1)c_{k-1}(\alpha^{k-1}) of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant ck(Ο‰)c_k(\omega) we discovered for any kk-stress Ο‰\omega on a cell complex in MdM^d. We introduce a characteristic polynomial of the degenerate simplex by defining f(x)=βˆ‘i=0n+1(βˆ’1)ici(Ξ±i)xn+1βˆ’if(x)=\sum_{i=0}^{n+1}(-1)^{i}c_i(\alpha^i)x^{n+1-i}, and prove that the roots of f(x)f(x) are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.Comment: 27 pages, 2 figures. To appear in Discrete & Computational Geometr

    Simplices with fixed volumes of codimension 2 faces in a continuous deformation

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    For any nn-dimensional simplex in the Euclidean space Rn\mathbb{R}^n with nβ‰₯4n\ge 4, it is asked that if a continuous deformation preserves the volumes of all the codimension 2 faces, then is it necessarily a \emph{rigid} motion. While the question remains open and the general belief is that the answer is affirmative, for all nβ‰₯4n\ge 4, we provide counterexamples to a variant of the question where Rn\mathbb{R}^n is replaced by a pseudo-Euclidean space Rp,nβˆ’p\mathbb{R}^{p,n-p} for some unspecified pβ‰₯2p\ge 2.Comment: 16 pages, 1 figure. major revision, changed the title, added counterexample for n=4n=4 in R3,1\mathbb{R}^{3,1} as well, so we now have counterexamples for all $n\ge 4
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