We formulate and prove a periodic analog of Maxwell’s theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.

中文翻译：

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平面周期框架的升力和应力

我们制定并证明了麦克斯韦定理的周期性模拟，该定理将受力平面框架及其升力与具有球形拓扑的多面体表面相关联。我们使用我们的提升定理来证明平面周期性伪三角剖分的变形和刚度理论特性，概括以其有限对应项而闻名的特征。然后将这些特性应用于源自数学晶体学和材料科学的问题，涉及平面周期拉胀结构和超刚性周期框架。