simplices

简明释义

英[ˈsɪmplɪˌsiːz]美[ˈsɪmplɪˌsiːz]

n. 单形；单层公寓（simplex 的复数形式）

英英释义

单词用法

同义词

反义词

例句

1.Sperner's lemma concerns the decomposition of a simplex into smaller simplices.

斯波纳引理涉及到把一个单纯形分解为较小的单纯形的问题。

2.Let G be a graph and let V(G) be the vertex set of G. Define the neighborhood complex N(G) as the simplicial complex whose simplices are those subsets of V(G) which have a common neighbor.

一个图G的邻域复形是以G的顶点为顶点，以G的具有公共邻接顶点的顶点子集为单形的抽象复形。

3.Let G be a graph and let V(G) be the vertex set of G. Define the neighborhood complex N(G) as the simplicial complex whose simplices are those subsets of V(G) which have a common neighbor.

一个图G的邻域复形是以G的顶点为顶点，以G的具有公共邻接顶点的顶点子集为单形的抽象复形。

4.In geometry, a triangle is one of the simplest forms of a simplex.

在几何学中，三角形是最简单的单纯形之一。

5.In algebraic topology, we often use simplices to build complex structures.

在代数拓扑中，我们常常使用单纯形来构建复杂的结构。

6.The concept of simplices can be extended to higher dimensions, such as tetrahedrons and beyond.

单纯形的概念可以扩展到更高的维度，比如四面体及更高的形状。

7.The study of simplices is fundamental in understanding multi-dimensional spaces.

对单纯形的研究对于理解多维空间至关重要。

8.A single edge and two vertices form a one-dimensional simplex.

一条边和两个顶点形成一个一维的单纯形。

作文

In the realm of mathematics, particularly in topology and geometry, the term simplices refers to the simplest form of a geometric object. A simplex can be defined as a generalization of a triangle or tetrahedron to arbitrary dimensions. For instance, in two dimensions, a triangle is a simplex, while in three dimensions, a tetrahedron serves as a simplex. This concept extends to higher dimensions, where an n-dimensional simplex consists of n + 1 vertices, which are the points that define the shape. Understanding simplices is crucial for various fields, including computer graphics, data analysis, and even in the study of complex systems.The significance of simplices can be observed in their application to simplicial complexes, which are built by connecting multiple simplices. These complexes serve as a foundational structure in algebraic topology, allowing mathematicians to study the properties of spaces in a more manageable way. By breaking down complex shapes into simplices, researchers can analyze topological features such as holes and connectivity. This decomposition is akin to simplifying a problem into smaller, more solvable parts, making the analysis of high-dimensional data sets much more feasible.Moreover, the concept of simplices is not limited to pure mathematics; it has practical applications in various scientific domains. For example, in data science, simplices can be used to represent relationships between different data points in a multi-dimensional space. By visualizing data in terms of simplices, analysts can uncover patterns and insights that might not be apparent when looking at raw data alone. This visualization helps in clustering algorithms, where data points are grouped based on their proximity in a simplex-like structure.In the context of computer graphics, simplices play a vital role in rendering 3D models. By using triangles as the basic building blocks, graphics engines can efficiently create complex surfaces and shapes. The process involves breaking down intricate models into a mesh of simplices, enabling smoother rendering and better performance in real-time applications like video games and simulations. This technique highlights the importance of simplices in bridging the gap between theoretical mathematics and practical applications in technology.Furthermore, the study of simplices leads to fascinating connections with other mathematical concepts, such as homology and cohomology theories, which explore the properties of spaces through algebraic means. By examining how simplices fit together, mathematicians can derive essential information about the overall structure and behavior of a given space. This interplay between geometry and algebra illustrates the beauty of mathematics, where simple elements combine to form complex and rich structures.In conclusion, the term simplices encompasses a fundamental concept in mathematics that has far-reaching implications across various fields. Whether in theoretical studies or practical applications, simplices provide a framework for understanding and analyzing complex systems. Their ability to simplify and clarify multidimensional relationships makes them indispensable tools in both mathematics and its applications in the real world. As we continue to explore the complexities of our universe, the role of simplices will undoubtedly remain significant, guiding us toward deeper insights and innovations.

在数学领域，特别是在拓扑学和几何学中，术语simplices指的是几何对象的最简单形式。Simplex可以定义为三角形或四面体向任意维度的推广。例如，在二维中，三角形是一个simplex，而在三维中，四面体则作为一个simplex。这个概念扩展到更高的维度，其中n维simplex由n + 1个顶点组成，这些顶点是定义形状的点。理解simplices对于多个领域至关重要，包括计算机图形学、数据分析，甚至复杂系统的研究。simplices的重要性可以在它们对单纯复形的应用中观察到，单纯复形是通过连接多个simplices构建的。这些复形作为代数拓扑中的基础结构，使数学家能够以更可控的方式研究空间的性质。通过将复杂形状分解为simplices，研究人员可以分析拓扑特征，如孔和连通性。这种分解类似于将问题简化为更小、更易解决的部分，使得对高维数据集的分析变得更加可行。此外，simplices的概念并不限于纯数学；它在多个科学领域中具有实际应用。例如，在数据科学中，simplices可以用来表示多维空间中不同数据点之间的关系。通过以simplices的形式可视化数据，分析师可以发现原始数据中可能不明显的模式和见解。这种可视化有助于聚类算法，其中数据点根据在simplex样结构中的接近程度进行分组。在计算机图形学的背景下，simplices在渲染3D模型中发挥着重要作用。通过使用三角形作为基本构建块，图形引擎可以有效地创建复杂的表面和形状。该过程涉及将复杂模型分解为一系列simplices的网格，从而实现更平滑的渲染和在视频游戏和模拟等实时应用中的更好性能。这种技术突显了simplices在理论数学与技术实际应用之间架起桥梁的重要性。此外，对simplices的研究与其他数学概念之间产生了迷人的联系，例如同调和上同调理论，它们通过代数手段探索空间的属性。通过检查simplices如何组合，数学家可以推导出关于给定空间的整体结构和行为的重要信息。这种几何与代数之间的相互作用展示了数学的美丽，其中简单元素结合形成复杂而丰富的结构。总之，术语simplices包含了数学中的一个基本概念，该概念在各个领域具有深远的影响。无论是在理论研究还是实际应用中，simplices提供了一个理解和分析复杂系统的框架。它们简化和澄清多维关系的能力使其成为数学及其在现实世界应用中不可或缺的工具。随着我们继续探索宇宙的复杂性，simplices的角色无疑将保持重要，引导我们走向更深刻的见解和创新。