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arXiv · 2110.01854

Flexing infinite frameworks with applications to braced Penrose tilings

Sean Dewar·Jan Legerský
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Abstract

A planar framework -- a graph together with a map of its vertices to the plane -- is flexible if it allows a continuous deformation preserving the distances between adjacent vertices. Extending a recent previous result, we prove that a connected graph with a countable vertex set can be realized as a flexible framework if and only if it has a so-called NAC-coloring. The tools developed to prove this result are then applied to frameworks where every 4-cycle is a parallelogram, and countably infinite graphs with $n$-fold rotational symmetry. With this, we determine a simple combinatorial characterization that determines whether the 1-skeleton of a Penrose rhombus tiling with a given set of braced rhombi will have a flexible motion, and also whether the motion will preserve 5-fold rotational symmetry.

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Sean Dewar, Jan Legerský. 2022-10-05. Flexing infinite frameworks with applications to braced Penrose tilings. https://doi.org/10.1016/j.dam.2022.09.002

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