AI 中文总结
研究基于多面体的等周问题，即三种平面镶嵌中用多面体围出的最大面积，给出岛内证明及整数线性规划转换，提供网络应用程序和相关材料方便读者测试挑战与实际操作。
AI 中文摘要
我们给出了一个基于多面体的等周问题的结果：在三种平面镶嵌中，使用多面体能围成的最大封闭面积是多少？我们向读者提出挑战，并给出岛内关于由五连块制成的围栏能围成的最大面积的证明。还给出了使用整数线性规划对该实例的转换。有一个配套的网络应用程序可用于测试我们提出的一些挑战并进行相关活动，我们还提供了额外页面，有棋盘的剪裁讲义和拼图碎片，以便打印和剪裁后实际操作阅读本文。
英文摘要
We present results around an isoperimetric problem built on polyforms: What is the biggest enclosed area one can build using polyforms in each of the three plane tessellations? We give challenges to the readers and present Shimauchi's proof of the biggest area a fence made of pentominoes can enclose. A translation of the instance using integer linear programming is also given. A companion web app is available to test some of the challenges we propose and for activities, and we included extra pages with a cutout handout of the board and pieces of the puzzles, so that you can print and cut them to read this paper hands on.
Comments18p. 14 fig., many drawings. English version of Langlois-Rémillard, Müßig, Roldán, Mitt. Deutsch. Math. Ver. 33.3 (2025) 187-199 (in German). Solutions to the challenges will be added once the follow-up paper is published
Journal refA. Langlois-Rémillard, M.N. Müßig and É. Roldán, Maximale Zäune mit Polyformen, Mitt. Dtsch. Math.-Ver. 33.3 (2025), 187-199
