Polytopes with an Axis Of Symmetry

1970 ◽  
Vol 22 (2) ◽  
pp. 265-287 ◽  
Author(s):  
P. McMullen ◽  
G. C. Shephard
Keyword(s):  
Symmetry Group ◽  
Linear Subspace ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Simple Manner ◽  
Gale Transform ◽  
Enumeration Problems ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry ◽  
Gale Diagrams

During the last few years, Branko Grünbaum, Micha Perles, and others have made extensive use of Gale transforms and Gale diagrams in investigating the properties of convex polytopes. Up to the present, this technique has been applied almost entirely in connection with combinatorial and enumeration problems. In this paper we begin by showing that Gale transforms are also useful in investigating properties of an essentially metrical nature, namely the symmetries of a convex polytope. Our main result here (Theorem (10)) is that, in a manner that will be made precise later, the symmetry group of a polytope can be represented faithfully by the symmetry group of a Gale transform of its vertices. If a d-polytope P ⊂ Ed has an axis of symmetry A (that is, A is a linear subspace of Ed such that the reflection in A is a symmetry of P), then it is called axi-symmetric. Using Gale transforms we are able to determine, in a simple manner, the possible numbers and dimensions of axes of symmetry of axi-symmetric polytopes.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

scholarly journals Students’ understanding of axial and central symmetry

2021 ◽  
Vol 14 (1) ◽  
pp. 28-40
Author(s):  
Vlasta Moravcová ◽  
Jarmila Robová ◽  
Jana Hromadová ◽  
Zdeněk Halas
Keyword(s):  
Gender Differences ◽  
Quantitative Analysis ◽  
Mathematics Teachers ◽  
Infinite Number ◽  
Line Segment ◽  
Central Symmetry ◽  
Thought Processes ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry ◽  
Type Of School

The paper focuses on students’ understanding of the concepts of axial and central symmetries in a plane. Attention is paid to whether students of various ages identify a non-model of an axially symmetrical figure, know that a line segment has two axes of symmetry and a circle has an infinite number of symmetry axes, and are able to construct an image of a given figure in central symmetry. The results presented here were obtained by a quantitative analysis of tests given to nearly 1,500 Czech students, including pre-service mathematics teachers. The paper presents the statistics of the students’ answers, discusses the students’ thought processes and presents some of the students’ original solutions. The data obtained are also analysed with regard to gender differences and to the type of school that students attend. The results show that students have two principal misconceptions: that a rhomboid is an axially symmetrical figure and that a line segment has just one axis of symmetry. Moreover, many of the tested students confused axial and central symmetry. Finally, the possible causes of these errors are considered and recommendations for preventing these errors are given.

scholarly journals Computing Reeb dynamics on four-dimensional convex polytopes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Julian Chaidez ◽  
Michael Hutchings
Keyword(s):  
Convex Polytopes ◽  
Convex Polytope

<p style='text-indent:20px;'>We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>.</p>

Moveout approximations for P- and SV-waves in dip-constrained transversely isotropic media

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. C53-C59 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Transversely Isotropic ◽  
Arbitrary Orientation ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Dip Angle ◽  
P And Sv Waves ◽  
Axes Of Symmetry

We generalize the P- and SV-wave moveout formulas obtained for transversely isotropic media with vertical axes of symmetry (VTI), based on the weak-anisotropy approximation. We generalize them for 3D dip-constrained transversely isotropic (DTI) media. A DTI medium is a transversely isotropic medium whose axis of symmetry is perpendicular to a dipping reflector. The formulas are derived in the plane defined by the source-receiver line and the normal to the reflector. In this configuration, they can be easily obtained from the corresponding VTI formulas. It is only necessary to replace the expression for the normalized offset by the expression containing the apparent dip angle. The final results apply to general 3D situations, in which the plane reflector may have arbitrary orientation, and the source and the receiver may be situated arbitrarily in the DTI medium. The accuracy of the proposed formulas is tested on models with varying dip of the reflector, and for several orientations of the horizontal source-receiver line with respect to the dipping reflector.

The Salience of Vertical Symmetry

Perception ◽  
1994 ◽  
Vol 23 (2) ◽  
pp. 221-236 ◽  
Author(s):  
Peter Wenderoth
Keyword(s):  
Additional Data ◽  
Oblique Effect ◽  
Axis Of Symmetry ◽  
Random Dot Patterns ◽  
Symmetry Perception ◽  
Axes Of Symmetry ◽  
Other Orientation ◽  
Relative Salience ◽  
Sensitivity Performance ◽  
Better Than

It has long been accepted that amongst patterns which are bilaterally symmetrical, those which have their axis of symmetry vertical are more saliently symmetrical than patterns whose axis of symmetry is at some other orientation. The evidence regarding the relative salience of other orientations of axis of symmetry is somewhat more equivocal. In experiment 1, subjects were required to discriminate between symmetric or random-dot patterns when the axis of symmetry was at one of eighteen different orientations, spaced 10° apart, both clockwise and counterclockwise of vertical to horizontal. The data indicated that vertical was most salient, then horizontal but that, unlike in the classical oblique effect for contrast sensitivity, performance for precisely diagonal axes was better than that for surrounding axis orientations. Additional data (from experiments 2 and 3) also showed that the salience of vertical and horizontal axes of symmetry can be manipulated extensively by varying the range of stimuli presented, presumably by manipulating the scanning or attentional strategy adopted by the observer. Many previous studies of symmetry perception may have confounded hard-wired salience for vertical symmetry with scanning or attentional strategies.

On the Number of Vertices of a Convex Polytope

1964 ◽  
Vol 16 ◽  
pp. 701-720 ◽  
Author(s):  
Victor Klee
Keyword(s):  
Sharp Estimate ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Bounded Subset ◽  
Empty Interior ◽  
System Of Linear Inequalities ◽  
System Of Inequalities ◽  
Basic Solutions ◽  
Upper Estimate

As is well known, the theory of linear inequalities is closely related to the study of convex polytopes. If the bounded subset P of euclidean d-space has a non-empty interior and is determined by i linear inequalities in d variables, then P is a d-dimensional convex polytope (here called a d-polytope) which may have as many as i faces of dimension d — 1, and the vertices of this polytope are exactly the basic solutions of the system of inequalities. Thus, to obtain an upper estimate of the size of the computation problem which must be faced in solving a system of linear inequalities, it suffices to find an upper bound for the number f0(P) of vertices of a d-polytope P which has a given number fd-1(P) of (d — l)-faces. A weak bound of this sort was found by Saaty (14), and several authors have posed the problem of finding a sharp estimate.

Estimate Volumes of Convex Polytopes: A New Method Based on Particle Swarm Optimization

2014 ◽  
Vol 1022 ◽  
pp. 357-360
Author(s):  
Sheng Xu ◽  
Dun Bo Cai
Keyword(s):  
Particle Swarm Optimization ◽  
Convex Polytopes ◽  
Particle Swarm ◽  
Convex Polytope ◽  
New Method ◽  
Monte Carlo Algorithm ◽  
Swarm Optimization ◽  
Uniform Sampling ◽  
Sampling Process ◽  
Sample Points

The volume of a convex polytope is important for many applications, and generally #P-hard to compute. In many scenarios, an approximate value of the volume is sufficed to utilize. Existing methods for estimating the volumes were mostly based on the Monte Carlo algorithm or its variants, which required a near uniform sampling process with a large number of sample points. In this paper, we propose a new method to estimate the volumes of convex polytopes that needs relative less sample points. Our method firstly searches for fringe points that are inside and near the border of a convex polytope, by a new way of utilizing the particle swarm optimization (PSO) technique. Then, the set of fringe points is input to a tool called Qhull whose output value serves as an estimate of the real volume. Experimental results show that our method is efficient and gives result with high accuracy for many instances of 10-dimension and beyond.

A Remark on Convex Polytopes

1965 ◽  
Vol 8 (6) ◽  
pp. 829-830
Author(s):  
A. S. Glass
Keyword(s):  
Dimensional Space ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Finite Family ◽  
Alternative Proof

In this note we wish to present an alternative proof for the following well-known theorem [1, Theorem 16]: every convex polytope X in Euclidean n-dimensional space Rn is the intersection of a finite family of closed half-spaces. It will be supposed that the converse of this theorem has been verified by conventional arguments, namely: every bounded intersection of a finite family of closed half-spaces in Rn is a convex polytope [cf. 1, Theorem 15].

Compressibility of two-dimensional pores having n -fold axes of symmetry

2006 ◽  
Vol 462 (2071) ◽  
pp. 1933-1947 ◽  
Author(s):  
Thushan C Ekneligoda ◽  
Robert W Zimmerman
Keyword(s):  
Scaling Law ◽  
Mapping Function ◽  
Closed Form Expression ◽  
Fold Axis ◽  
Two Dimensional ◽  
Form Expression ◽  
Pore Compressibility ◽  
Special Cases ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry

The complex variable method and conformal mapping are used to derive a closed-form expression for the compressibility of an isolated pore in an infinite two-dimensional, isotropic elastic body. The pore is assumed to have an n -fold axis of symmetry, and be represented by at most four terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. The results are validated against some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from three and four terms of the Schwarz–Christoffel mapping function for regular polygons, the compressibilities of a triangle, square, pentagon and hexagon are found (to at least three digits). Specific results for some other pore shapes, more general than the quasi-polygons obtained from the Schwarz–Christoffel mapping, are also presented. An approximate scaling law for the compressibility, in terms of the ratio of perimeter-squared to area, is also tested. This expression gives a reasonable approximation to the pore compressibility, but may overestimate it by as much as 20%.

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