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Topology
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Steve Huntsman, Jimmy Palladino, Michael Robinson
When describing the local structure of a simplicial complex, it is often useful to delineate which simplices are subsets of each other. If a and b are simplices of a simplicial complex X and a⊆b, we say that ‘a is a face of b’ or equivalently that ‘b is a coface of a.’ These relationships determine the topology of an abstract simplicial complex, in terms of its open and closed subsets. A closed setA of a simplicial complex contains every possible subset of every element of A. The star of a subset A of a simplicial complex consists of the set of all simplices containing an element of A. An open set of an abstract simplicial complex is one that can be written as a union of stars.
Metadesigning Customizable Houses
Published in Branko Kolarevic, José Pinto Duarte, Mass Customization and Design Democratization, 2018
According to its mathematical definition, topology is the study of intrinsic, qualitative properties of geometric forms that are not normally affected by changes in size or shape, i.e. which remain invariant through continuous one-to-one transformations or elastic deformations, such as stretching or twisting. A circle and an ellipse, for example, or a square and a rectangle, can be considered to be topologically equivalent, as both a circle and a square could be deformed by stretching them into an ellipsoid or rectangle, respectively. A square and a rectangle have the same number of edges and the same number of vertices, and are, therefore, topologically identical, or homeomorphic. This quality of homeomorphism is particularly interesting, as focus is on the relational structure of an object and not on its geometry – the same topological structure could be geometrically manifested in an infinite number of forms (figure 10.2).
Computer Networking
Published in Mohssen Mohammed, Al-Sakib Khan Pathan, Automatic Defense Against Zero-day Polymorphic Worms in Communication Networks, 2016
Mohssen Mohammed, Al-Sakib Khan Pathan
In broad terms, mainly four topologies are recognized: (1) bus topology, (2) ring topology, (3) star topology, and (4) mesh topology. However, in this book, we note a wide range of topologies that should be clearly distinguished without resorting to grouping them under the banner of any of the four mentioned categories. Instead of using the term hybrid topology, we prefer to allow the readers to clearly identify some topologies [1] that should be studied distinctively in any form of computer networking. It should be noted that both physical and logical topologies are considered here, and we do not classify the mentioned terms under physical or logical as some of them contain both notions in the structure and in the operational method.
Topological perception on attention to product shape
Published in International Journal of Design Creativity and Innovation, 2020
Topology is a major branch of mathematics concerned with spatial properties that are preserved under one-to-one, continuous transformations, such as stretching and bending, but not breaking or fusing. Important topological properties include connectedness and compactness. Hence, solid figures such a cube and a tetrahedron are topologically equivalent, because one can be transformed into the other through continuous transformations. An attribute of an object is called topologically invariant if it does not change under a continuous transformation. If two objects have different topological invariants, they must be different homeomorphisms. One of the most common topological invariants is the Euler characteristic, which is commonly denoted by χ and conventionally defined for the surfaces of a polyhedron as χ= V – E + F, where V, E, and F are the numbers of vertices (corners), edges, and faces in the polyhedron, respectively. For example, the surface of a convex polyhedron has the Euler characteristic χ = V – E + F = 2. On the other hand, several human-made goods are homeomorphic to either spheres or rings and can be molded forming tiles. The topological transformations can be pictured as distorting a rubber sheet without creating holes or fusing edges. The Euler characteristics of a sphere and a ring (i.e. a torus) are 2 and 0, respectively.
Geometry in Our Three-Dimensional World
Published in Technometrics, 2023
Topology is a branch of mathematics in which shapes obtained from each other by compression or stretching, or more technically by continuous deformations, are considered equivalent. For a topologist, a donut is not different from a coffee cup. One-sided surfaces such as the Mobius strip and nonorientable surfaces such as Klein’s bottle are some of the biggest surprises geometry offers us. Such is the subject matter of Chapter 10.