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   https://doi.org/https://doi.org/10.1007/978-3-030-22529-2_1  

   Hurd, H.; Pipiras, V. (January 2020, Cyclostationarity: Theory and Methods – IV. CSTA 2017)
2. Periodic Bootstrap Embedding

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   Meitei, Oinam Romesh; Van Voorhis, Troy (June 2023, Journal of Chemical Theory and Computation)
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4. Interactive 2D Periodic Graphs

   https://doi.org/10.4230/LIPIcs.SoCG.2023.63  

   Camero, Alexandra; Streinu, Ileana (June 2023, 39th International Symposium on Computational Geometry (SoCG 2023))

   Chambers, Erin W; Gudmundsson, Joachim (Ed.)

   We present an educational web app for interactively drawing and editing 2D periodic graphs. The user defines the unit cell and the finite set of vertex and edge representatives, from which a sufficiently large fragment of the periodic graph is created for the visualization. The periodic graph can also be modified by applying several transformations, including isometries and relaxations of the unit cell. A finite representation of the infinite periodic graph can be saved in an external file as a quotient graph with Z²-marked edges. Its geometry is recorded using fractional (crystallographic) coordinates. The facial structure of non-crossing periodic graphs can be revealed by the user interactively selecting face representatives. An accompanying video demonstrates the functionality of the web application. 

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5. How periodic surfaces bend

   https://doi.org/10.1098/rsta.2024.0016  

   Nassar, Hussein (October 2024, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   A periodic surface is one that is invariant by a two-dimensional lattice of translations. Deformation modes that stretch the lattice without stretching the surface are effective membrane modes. Deformation modes that bend the lattice without stretching the surface are effective bending modes. For periodic piecewise smooth simply connected surfaces, it is shown that the effective membrane modes are, in a sense, orthogonal to effective bending modes. This means that if a surface gains a membrane mode, it loses a bending mode, and conversely, in such a way that the total number of modes, membrane and bending combined, can never exceed 3. Various examples, inspired from curved-crease origami tessellations, illustrate the results. This article is part of the theme issue ‘Origami/Kirigami-inspired structures: from fundamentals to applications’. 

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