Realizations, symmetry and monodromy groups of abstract polytopes

抽象多胞体的实现、对称性和单峰群

• 批准号: 4818-2011
• 负责人: Monson, Barry
• 金额: $ 0.73万
• 依托单位: University of New Brunswick
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568378
• 关键词: Realizations; symmetry; monodromy; groups; abstract

An ordinary cube is a familiar example of a regular polytope: it has various square faces (top, bottom, front, back, left, right), separated by several edges (twelve of them), which meet by threes in a particular way at the eight corners. Usually such a cube suggests to the imagination a very symmetrical and rigid object, i.e. a Euclidean realization; but if we consider instead a rubber copy, then we can imagine twisting and distorting the cube in fantastic ways which destroy its symmetry. Nevertheless, the `combinatorial features' of the cube would survive this attempt at destruction; there would still be 6 square faces, 12 edges and 8 corners, interconnected as before. An `abstract regular polytope' is a mathematical object of just this sort - it has the basic structural features suggested by a cube or any of its relatives in any dimension. However, these combinatorial properties do not of themselves always imply that there should be any sort of real model for the object, since there could be some subtle incompatibility between mere combinatorial symmetry (think `assembly instructions') and actual spatial symmetry (think `able to actually build a nice model'). The thrust of this research project is to explore the mathematical machinery needed to describe and classify abstract regular polytopes and their less symmetrical kin. In mathematics, any such interplay between disparate branches of geometry and algebra promises to be both fascinating and worthwhile.

一个普通的立方体是一个常见的正多面体的例子：它有各种正方形面（顶，底，前，后，左，右），由几条边（其中12条）分开，这些边以特定的方式在八个角上相遇。通常，这样一个立方体会让人联想到一个非常对称和刚性的物体，即欧几里得的实现;但如果我们考虑一个橡胶复制品，那么我们可以想象以奇妙的方式扭曲和扭曲立方体，破坏它的对称性。尽管如此，立方体的“组合特征”将在这种破坏尝试中幸存下来;仍然有6个正方形面，12条边和8个角，像以前一样相互连接。 一个“抽象的正则多面体”就是这样一种数学对象--它具有立方体或其在任何维度上的任何亲戚所暗示的基本结构特征。然而，这些组合属性本身并不总是意味着应该存在对象的任何种类的真实的模型，因为在单纯的组合对称性（想想“组装对称性”）和实际的空间对称性之间可能存在一些微妙的不相容性。 对称性（认为“能够实际构建一个很好的模型”）。 这个研究项目的主旨是探索描述和分类抽象规则多面体及其不对称的亲属所需的数学机器。在数学中，几何学和代数学的不同分支之间的任何这种相互作用都将是既迷人又有价值的。