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1996 | Book

Introduction Read chapter Read first chapter

Symmetry Orbits

Author: Dr. Hugo F. Verheyen

Publisher: Birkhäuser Boston

Book Series : Design Science Collection

Included in: Professional Book Archive

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About this book

In a broad sense design science is the grammar of a language of images rather than of words. Modem communication techniques enable us to transmit and reconstitute images without needing to know a specific verbal sequence language such as the Morse code or Hungarian. Inter­ national traffic signs use international image symbols which are not An image language differs specific to any particular verbal language. from a verbal one in that the latter uses a linear string of symbols, whereas the former is multidimensional. Architectural renderings commonly show projections onto three mutually perpendicular planes, or consist of cross sections at different altitudes capable of being stacked and representing different floor plans. Such renderings make it difficult to imagine buildings compris­ ing ramps and other features which disguise the separation between and consequently limit the creative process of the architect. floors, Analogously, we tend to analyze natural structures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures.

Table of Contents

Frontmatter

Introduction

Introduction
Abstract
In the realm of symmetry various disciplines meet, resulting in a combination of knowledge that extends the abilities of each separate discipline. Publications on symmetry-related subjects are now appearing from different angles of scientific research, and conferences are increasingly being held in places all over the world. Topics such as polyhedral shapes, domes, expandability, membranes, and lightweight structures in architecture all attract specialists of a wide variety; among these are architects, engineers, mathematicians, astronomers, crystallographers, chemists, biologists, physicists, artists, designers, industrialists, and more. This blending of information is occurring with a common goal: to achieve a functional application of geometry. In such a sense, this volume basically investigates the applicatory qualities of an orbit in group-theory, for Design Science.
Hugo F. Verheyen

Realization of Symmetry Groups

Frontmatter
Chapter 1. Groups of Isometries
Abstract
The purpose of this chapter is to offer a compendium of things worth knowing about isometries, in general, and especially about their finite groups. Often in the literature, this knowledge is taken for granted, whereas in practice the composition, angles between axes, and other data are seldom listed. Most proofs of properties belong to elementary geometry and can be found here and there in such treatments, as, e.g., with Coxeter [3, 4]. Definitions and a number of properties will be merely mentioned here, but a full list of finite groups of isometries will be presented, together with numerical and geometric data, and all provided with clear illustrations.
Hugo F. Verheyen
Chapter 2. Symmetry Action
Abstract
The gravitational force of the earth remains only an abstract idea as long as there is no object which it is attracting. Similarly, the sun is shining, but when there is no object reflecting the sunrays, it remains dark and cold out in space.
Hugo F. Verheyen
Chapter 3. Orbit Systems
Abstract
In this chapter, some geometrical application possibilities of the realization theory in the previous chapter will be explained. It will be shown how different sorts of polyhedral symmetric shapes are actually associated with realizations of groups of isometries. According to the given definitions, a group realization implies a homogenous space isomorphism between a right G-set and the orbit of the descriptive. The right G-set is determined by the stabilizer of the positioned descriptive. Examples of descriptives will be restricted to space-(l) bodies, whereas in Chapter 6, some attention will be also dedicated to higher-level descriptives. Two distinct categories occur on level (1): point singletons and sets of more points.
Hugo F. Verheyen

Compounds of Cubes

Frontmatter
Chapter 4. Classification of the Finite Compounds of Cubes
Abstract
A compound of cubes has been defined as the orbit of a centered cube for an action group G of isometries. As such, it is the expression of an isomorphic right G-set G/H, where H is the stabilizer of the descriptive position of the cube. In this part, such orbits will be constructed and classified for the finite groups of isometries, and discussed afterward. As has been explained, however, these groups may be restricted to those, containing I, which will be further referred to as I-groups (and I-subgroups).
Hugo F. Verheyen
Chapter 5. Stability of Subcompounds
Abstract
A next subject in the discussion of finite cube compounds concerns the relationship of a compound with its orientative subcompounds. Distinct versions contain subcompound versions that may be distinct, yet holding their number of constituents. Hence, with respect to the maximal orientative stability, the stability of a subcompound version is either basic or superior. Therefore, a special version of the compound occurs when the basic stability of a subcompound becomes superior.
Hugo F. Verheyen
Chapter 6. Higher Descriptives
Abstract
Our last subject to be discussed is the orbit system of a higher body whose system on level (2) is composed of congruent, centered cubes. Let p (and specifically p i ) denote centered positions of a cube for an I-group G.
Hugo F. Verheyen
Chapter 7. Assembling Models
Abstract
Three-dimensional images have always been a highly desirable source of inspiration since the previous century. Who has never seen one of these wooden “belle époque” 3D-viewers, in which pairs of pictures had to be inserted? Or colored printed figures—and even movies—that have to be viewed with red-and-green spectacles?
Hugo F. Verheyen
Backmatter
Metadata
Title
Symmetry Orbits
Author
Dr. Hugo F. Verheyen
Copyright Year
1996
Publisher
Birkhäuser Boston
Electronic ISBN
978-1-4612-4074-7
Print ISBN
978-1-4612-8639-4
DOI
https://doi.org/10.1007/978-1-4612-4074-7