标题： 四维超立方体中的曲面 显示英文标题

标题： Surfaces in The Tesseract

摘要： 如何可视化从超立方体的面可以制作的所有表面？ 在最近的工作中，Aveni、Govc 和 Roldán 表明，环面和球面是唯一可以通过超立方体的二维面子集实现的闭合表面。 他们还给出了这些两个表面的所有同构类型的嵌入的详尽列表。 在这里，我们生成所有这些表面的3D模型。 我们还借助一些超蚂蚁，展示了莫比乌斯带在超立方体上的最小实现。 显示英文摘要

摘要： How can we visualize all the surfaces that can be made from the faces of the tesseract? In recent work, Aveni, Govc, and Rold\'an showed that the torus and the sphere are the only closed surfaces that can be realized by a subset of two-dimensional faces of the tesseract. They also gave an exhaustive list of all the isomorphic types of embeddings of these two surfaces. Here, we generate 3D models of all these surfaces. We also exhibit, with the help of some hyper-ants, the minimum realization of the M\"obius strip on the tesseract.