Convex polytopes with convex nets

1975 ◽  
Vol 78 (3) ◽  
pp. 389-403 ◽  
Author(s):  
G. C. Shephard
Keyword(s):  
Convex Polytopes ◽  
Three Dimensional ◽  
Convex Polytope ◽  
Regular Tetrahedron ◽  
Similar Material ◽  
Flat Sheet

The idea of a net will be familiar to anyone who has made a model of a three-dimensional convex polytope (3-polytope) out of a flat sheet of card or similar material. To begin with, one cuts out a polygon, and then the model is formed by folding this and joining its edges in an appropriate manner. For example, Fig. 1 illustrates two nets for a regular tetrahedron. The folds are indicated by broken lines and lettering (either of the edges or of the vertices) indicates how the edges are to be joined.

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.

Numerical Analysis for Process Parameter Effect in Electromagnetic Impact Welding of Aluminum Alloy Sheet

2014 ◽  
Vol 548-549 ◽  
pp. 297-300
Author(s):  
Dae Yong Kim ◽  
Hyeon Il Park ◽  
Ji Hoon Kim ◽  
Sang Woo Kim ◽  
Young Seon Lee
Keyword(s):  
Aluminum Alloy ◽  
Numerical Analysis ◽  
Deformation Behavior ◽  
Three Dimensional ◽  
Alloy Sheet ◽  
Aluminum Alloy Sheet ◽  
Sheet Metals ◽  
Charge Voltage ◽  
Flat Sheet ◽  
Impact Welding

Studies on electromagnetic impact welding between similar or dissimilar flat sheet metals using the flat one turn coil have been recently achieved. In this study, three dimensional electromagnetic-mechanical coupled numerical simulations are performed for the electromagnetic impact welding of aluminum alloy sheets with flat rectangular one turn coil. The deformation behavior during impact welding was examined. The effect of process parameters such as charge voltage, standoff distance and gap distance were investigated.

scholarly journals Numerical Study of the Zero Velocity Surface and Transfer Trajectory of a Circular Restricted Five-Body Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
R. F. Wang ◽  
F. B. Gao
Keyword(s):  
Body Problem ◽  
Numerical Study ◽  
Dimensional Space ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Regular Tetrahedron ◽  
Transfer Trajectory ◽  
Zero Velocity ◽  
Zero Velocity Surfaces ◽  
Zero Velocity Surface

We focus on a type of circular restricted five-body problem in which four primaries with equal masses form a regular tetrahedron configuration and circulate uniformly around the center of mass of the system. The fifth particle, which can be regarded as a small celestial body or probe, obeys the law of gravity determined by the four primaries. The geometric configuration of zero-velocity surfaces of the fifth particle in the three-dimensional space is numerically simulated and addressed. Furthermore, a transfer trajectory of the fifth particle skimming over four primaries then is designed.

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 Additive lattice kirigami

2016 ◽  
Vol 2 (9) ◽  
pp. e1601258 ◽  
Author(s):  
Toen Castle ◽  
Daniel M. Sussman ◽  
Michael Tanis ◽  
Randall D. Kamien
Keyword(s):  
Gaussian Curvature ◽  
Three Dimensional ◽  
Complex Structures ◽  
3D Structures ◽  
Flat Sheet ◽  
Simple Rules ◽  
New Material ◽  
3D Shapes ◽  
Huge Variety

Kirigami uses bending, folding, cutting, and pasting to create complex three-dimensional (3D) structures from a flat sheet. In the case of lattice kirigami, this cutting and rejoining introduces defects into an underlying 2D lattice in the form of points of nonzero Gaussian curvature. A set of simple rules was previously used to generate a wide variety of stepped structures; we now pare back these rules to their minimum. This allows us to describe a set of techniques that unify a wide variety of cut-and-paste actions under the rubric of lattice kirigami, including adding new material and rejoining material across arbitrary cuts in the sheet. We also explore the use of more complex lattices and the different structures that consequently arise. Regardless of the choice of lattice, creating complex structures may require multiple overlapping kirigami cuts, where subsequent cuts are not performed on a locally flat lattice. Our additive kirigami method describes such cuts, providing a simple methodology and a set of techniques to build a huge variety of complex 3D shapes.

scholarly journals An Euler-type volume identity

1999 ◽  
Vol 59 (3) ◽  
pp. 495-508
Author(s):  
Kevin Callahan ◽  
Kathy Hann
Keyword(s):  
Elementary Proof ◽  
Three Dimensional ◽  
Convex Polytope ◽  
Image Position

In this paper we present an elementary proof of a congruence by subtraction relation. In order to prove congruence by subtraction, we produce a dissection relating equal sub-polytopes. An immediate consequence of this relation is an Euler-type volume identity in ℝ3 which appeared in the Unsolved Problems section of the December 1996 MAA Monthly.This Euler-type volume identity relates the volumes of subsets of a polytope called wedges that correspond to its faces, edges, and vertices. A wedge consists of the inward normal chords of the polytope emanating from a face, vertex, or edge. This identity is stated in the theorem below.Euler Volume Theorem. For any three dimensional convex polytope PThis identity follows immediately from

Accurate Drawings of Three Dimensional Figures

1944 ◽  
Vol 37 (8) ◽  
pp. 350-353
Author(s):  
Margaret Joseph
Keyword(s):  
Three Dimensional ◽  
Regular Tetrahedron ◽  
Solid Geometry

Every teacher of solid geometry has no doubt experienced having a student fail to prove an original exercise just because his drawing of the figure required was so much out of proportion that he failed to establish the proper relationship between lines, angles, triangles, or planes necessary in the proof. If, for example, a student would draw a square for the base of an oblique prism or a circle for the base of a cone, instead of the parallelogram and ellipse as they appear in perspective, then the whole figure is distorted. Just recently I had a student come to me for help in deriving the formula for the altitude of a regular tetrahedron in terms of its edge e. As soon as I saw the rough sketch he was using, where the faces were scalene triangles, I knew why he had two unknowns in the equation that he was trying to solve. The altitude did not meet the base of the figure at its center. I made a better drawing for him and he immediately recognized his error and proceeded with the solution without further help from me.

scholarly journals Design and analysis of three-dimensional printing of a porous titanium scaffold

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Jiajie Yang ◽  
Yaqiang Li ◽  
Xiaojian Shi ◽  
Meihua Shen ◽  
Kaibing Shi ◽  
...  
Keyword(s):  
Compressive Strength ◽  
3D Printing ◽  
Pore Size ◽  
Three Dimensional ◽  
Structural Models ◽  
Porous Titanium ◽  
Knee Joints ◽  
Regular Tetrahedron ◽  
Diamond Structure ◽  
Pore Morphology

Abstract Objective Mechanic strength, pore morphology and size are key factors for the three-dimensional (3D) printing of porous titanium scaffolds, therefore, developing optimal structure for the 3D printed titanium scaffold to fill bone defects in knee joints is instructive and important. Methods Structural models of titanium scaffolds with fifteen different pore unit were designed with 3D printing computer software; five different scaffold shapes were designed: imitation diamond-60°, imitation diamond-90°, imitation diamond-120°, regular tetrahedron and regular hexahedron. Each structural shape was evaluated with three pore sizes (400, 600 and 800 μm), and fifteen types of cylindrical models (size: 20 mm; height: 20 mm). Autodesk Inventor software was used to determine the strength and safety of the models by simulating simple strength acting on the knee joints. We analyzed the data and found suitable models for the design of 3D printing of porous titanium scaffolds. Results Fifteen different types of pore unit structural models were evaluated under positive pressure and lateral pressure; the compressive strength reduced when the pore size increased. Under torsional pressure, the strengths of the imitation diamond structure were similar when the pore size increased, and the strengths of the regular tetrahedron and regular hexahedron structures reduced when the pore size increased. In each case, the compressive strength of the regular hexahedron structure was highest, that of the regular tetrahedron was second highest, and that of the imitation diamond structure was relatively low. Fifteen types of cylindrical models under a set force were evaluated, and the sequence of comprehensive compressive strength, from strong to weak was: regular hexahedron > regular tetrahedron > imitation diamond-120° > imitation diamond-90° > imitation diamond-60°. The compressive strength of cylinder models was higher when the pore size was smaller. Conclusion The pore size and pore morphology were important factors influencing the compressive strength. The strength of each structure reduced when the pore size (400, 600 and 800 μm) increased. The models of regular hexahedron, regular tetrahedron and imitation diamond-120°appeared to meet the conditions of large pore sizes and high compressive strength.

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>

scholarly journals Geometry of Multiprimary Display Colors I: Gamut and Color Control

2021 ◽  
Author(s):  
Gaurav Sharma ◽  
Carlos Rodriguez-Pardo
Keyword(s):  
Three Dimensional ◽  
Convex Polytope ◽  
Companion Paper ◽  
Light Sources ◽  
Color Management ◽  
Color Representation ◽  
Geometric Representations ◽  
Color Control ◽  
Surface Facets

<div>Displays that render colors using combinations of more than three lights are referred to as multiprimary displays. For multiprimary displays, the gamut, i.e., the range of colors that can be rendered using additive combinations of an arbitrary number of light sources (primaries) with modulated intensities, is known to be a zonotope, which is a specific type of convex polytope. Under the specific three-dimensional setting relevant for color representation and the constraint of physically meaningful nonnegative primaries, we develop a complete, cohesive, and directly usable mathematical characterization of the geometry of the multiprimary gamut zonotope that immediately identifies the surface facets, edges, and vertices and provides a parallelepiped tiling of the gamut. We relate the parallelepiped tilings of the gamut, that arise naturally in our characterization, to the flexibility in color control afforded by displays with more than four primaries, a relation that is further analyzed and completed in a Part II companion paper. We demonstrate several applications of the geometric representations we develop and highlight how the paper advances theory required for multiprimary display modeling, design, and color management and provides an integrated view of past work on on these topics. Additionally, we highlight how our work on gamut representations connects with and furthers the study of three-dimensional zonotopes in geometry.</div>

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