Research

446 KiB

Open AccessFeature PaperArticle

by Tomaž Pisanski, Gordon Williams and Leah Wrenn Berman

Symmetry 2017, 9(11), 274; https://doi.org/10.3390/sym9110274 - 14 Nov 2017

Viewed by 3683

A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as [...] Read more.

A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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5900 KiB

Open AccessArticle

Computer-Aided Panoramic Images Enriched by Shadow Construction on a Prism and Pyramid Polyhedral Surface

by Jolanta Dzwierzynska

Symmetry 2017, 9(10), 214; https://doi.org/10.3390/sym9100214 - 03 Oct 2017

Cited by 6 | Viewed by 4408

Abstract

The aim of this study is to develop an efficient and practical method of a direct mapping of a panoramic projection on an unfolded prism and pyramid polyhedral projection surface with the aid of a computer. Due to the fact that straight lines [...] Read more.

The aim of this study is to develop an efficient and practical method of a direct mapping of a panoramic projection on an unfolded prism and pyramid polyhedral projection surface with the aid of a computer. Due to the fact that straight lines very often appear in any architectural form we formulate algorithms which utilize data about lines and draw panoramas as plots of functions in Mathcad software. The ability to draw panoramic images of lines enables drawing a wireframe image of an architectural object. The application of the multicenter projection, as well as the idea of shadow construction in the panoramic representation, aims at achieving a panoramic image close to human perception. The algorithms are universal as the application of changeable base elements of panoramic projection—horizon height, station point location, number of polyhedral walls—enables drawing panoramic images from various viewing positions. However, for more efficient and easier drawing, the algorithms should be implemented in some graphical package. The representation presented in the paper and the method of its direct mapping on a flat unfolded projection surface can find application in the presentation of architectural spaces in advertising and art when drawings are displayed on polyhedral surfaces and can be observed from multiple viewing positions. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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13193 KiB

Open AccessFeature PaperArticle

On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters

by Jean E. Taylor, Erin G. Teich, Pablo F. Damasceno, Yoav Kallus and Marjorie Senechal

Symmetry 2017, 9(9), 188; https://doi.org/10.3390/sym9090188 - 11 Sep 2017

Cited by 8 | Viewed by 6290

Abstract

Where are the atoms in complex crystals such as quasicrystals or periodic crystals with one hundred or more atoms per unit cell? How did they get there? The first of these questions has been gradually answered for many materials over the quarter-century since [...] Read more.

Where are the atoms in complex crystals such as quasicrystals or periodic crystals with one hundred or more atoms per unit cell? How did they get there? The first of these questions has been gradually answered for many materials over the quarter-century since quasicrystals were discovered; in this paper we address the second. We briefly review a history of proposed models for describing atomic positions in crystal structures. We then present a revised description and possible growth model for one particular system of alloys, those containing Tsai-type clusters, that includes an important class of quasicrystals. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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4365 KiB

Open AccessArticle

Regular and Irregular Chiral Polyhedra from Coxeter Diagrams via Quaternions

by Nazife Ozdes Koca and Mehmet Koca

Symmetry 2017, 9(8), 148; https://doi.org/10.3390/sym9080148 - 07 Aug 2017

Cited by 1 | Viewed by 5586

Abstract

Vertices and symmetries of regular and irregular chiral polyhedra are represented by quaternions with the use of Coxeter graphs. A new technique is introduced to construct the chiral Archimedean solids, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal [...] Read more.

Vertices and symmetries of regular and irregular chiral polyhedra are represented by quaternions with the use of Coxeter graphs. A new technique is introduced to construct the chiral Archimedean solids, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexecontahedron. Starting with the proper subgroups of the Coxeter groups

W (A_{1} \oplus A_{1} \oplus A_{1})

,

W (A_{3})

,

W (B_{3})

and

W (H_{3})

, we derive the orbits representing the respective solids, the regular and irregular forms of a tetrahedron, icosahedron, snub cube, and snub dodecahedron. Since the families of tetrahedra, icosahedra and their dual solids can be transformed to their mirror images by the proper rotational octahedral group, they are not considered as chiral solids. Regular structures are obtained from irregular solids depending on the choice of two parameters. We point out that the regular and irregular solids whose vertices are at the edge mid-points of the irregular icosahedron, irregular snub cube and irregular snub dodecahedron can be constructed. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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2281 KiB

Open AccessArticle

The Roundest Polyhedra with Symmetry Constraints

by András Lengyel, Zsolt Gáspár and Tibor Tarnai

Symmetry 2017, 9(3), 41; https://doi.org/10.3390/sym9030041 - 15 Mar 2017

Cited by 5 | Viewed by 7482

Abstract

Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the [...] Read more.

Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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2270 KiB

Open AccessArticle

Aesthetic Patterns with Symmetries of the Regular Polyhedron

by Peichang Ouyang, Liying Wang, Tao Yu and Xuan Huang

Symmetry 2017, 9(2), 21; https://doi.org/10.3390/sym9020021 - 03 Feb 2017

Cited by 5 | Viewed by 7786

Abstract

A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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783 KiB

Open AccessArticle

On Center, Periphery and Average Eccentricity for the Convex Polytopes

by Waqas Nazeer, Shin Min Kang, Saima Nazeer, Mobeen Munir, Imrana Kousar, Ammara Sehar and Young Chel Kwun

Symmetry 2016, 8(12), 145; https://doi.org/10.3390/sym8120145 - 02 Dec 2016

Cited by 5 | Viewed by 4028

Abstract

A vertex v is a peripheral vertex in G if its eccentricity is equal to its diameter, and periphery

P (G)

is a subgraph of G induced by its peripheral vertices. Further, a vertex v in G is a central vertex [...] Read more.

A vertex v is a peripheral vertex in G if its eccentricity is equal to its diameter, and periphery

P (G)

is a subgraph of G induced by its peripheral vertices. Further, a vertex v in G is a central vertex if

e (v) = r a d (G)

, and the subgraph of G induced by its central vertices is called center

C (G)

of

G .

Average eccentricity is the sum of eccentricities of all of the vertices in a graph divided by the total number of vertices, i.e.,

a v e c (G) = {\frac{1}{n} \sum e_{G} (u); u \in V (G)} .

If every vertex in G is central vertex, then

C (G) = G

, and hence, G is self-centered. In this report, we find the center, periphery and average eccentricity for the convex polytopes. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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307 KiB

Open AccessArticle

Regular and Chiral Polyhedra in Euclidean Nets

by Daniel Pellicer

Symmetry 2016, 8(11), 115; https://doi.org/10.3390/sym8110115 - 28 Oct 2016

Cited by 1 | Viewed by 4058

Abstract

We enumerate the regular and chiral polyhedra (in the sense of Grünbaum’s skeletal approach) whose vertex and edge sets are a subset of those of the primitive cubic lattice, the face-centred cubic lattice, or the body-centred cubic lattice. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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7222 KiB

Open AccessArticle

Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry

by Stan Schein, Alexander J. Yeh, Kris Coolsaet and James M. Gayed

Symmetry 2016, 8(8), 82; https://doi.org/10.3390/sym8080082 - 20 Aug 2016

Cited by 4 | Viewed by 8227

Abstract

The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a [...] Read more.

The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of hexagons and pasting them onto the full triangular faces of an icosahedron produces icosahedral fullerene cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the truncated tetrahedron, produces two series of tetrahedral (T_d) fullerene cages. Cages in the first series have 28n² vertices (n ≥ 1). Cages in the second (leapfrog) series have 3 × 28n². We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (T_d) symmetry, these new polyhedra constitute a new class of “convex equilateral polyhedra with polyhedral symmetry”. We also show that none of the other Archimedean polyhedra, six with octahedral symmetry and six with icosahedral, can host full-face cutouts from regular tilings to produce cages with the host’s polyhedral symmetry. Full article

(This article belongs to the Special Issue Polyhedral Structures)

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