Dear Colleagues,

The study of polyhedra and symmetry has a long and fascinating history tracing back to the early days of geometry. With the passage of time, various notions of polyhedra have attracted attention and have brought to light new exciting classes of regular or other symmetric polyhedra including well-known figures such as the Platonic solids, Kepler-Poinsot star polyhedra, and Petrie-Coxeter sponge polyhedra, as well as the more recently discovered new regular or chiral skeletal polyhedra. This flexibility of the concept proves an important point --- polyhedra and symmetry have shown an enormous potential for revival! One explanation for this is the appearance of symmetric polyhedra in many contexts that a priori seem to have little apparent relation to symmetry, such as the occurrence of many figures in nature as crystals. In addition, their internal beauty appeals to the artistic senses and sparks the desire for a rigorous mathematical analysis and understanding of the figures themselves, as well as of their relationships with other areas of science.

This Special Issue of Symmetry features articles about polyhedra and symmetry. We are soliciting contributions  covering a broad range of topics including:  convex and non-convex polyhedra in spherical, euclidean, hyperbolic, or other spaces; maps and polyhedra on surfaces of higher genus; abstract polyhedra; polyhedra and symmetry groups; classification of polyhedra by transitivity properties of symmetry groups; regular polyhedra; various classes of highly-symmetric polyhedra, such as vertex-, edge, or face-transitive polyhedra, regular-faced polyhedra, and equivelar maps or  polyhedra; space-filling polyhedra; polyhedra and crystallography; polyhedra in nature; polyhedra in art, design, ornament, and architecture; polyhedral models.

Prof. Dr. Egon Schulte
Guest Editor

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Research

2516 KiB  

Open AccessArticle

Non-Crystallographic Symmetry in Packing Spaces

by Valery G. Rau, Leonty A. Lomtev and Tamara F. Rau

Symmetry 2013, 5(1), 54-80; https://doi.org/10.3390/sym5010054 - 09 Jan 2013

Cited by 4 | Viewed by 6990

Abstract

In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings [...] Read more.

In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure. Full article

(This article belongs to the Special Issue Polyhedra)

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270 KiB  

Open AccessShort Note

A Note on Lower Bounds for Colourful Simplicial Depth

by Antoine Deza, Tamon Stephen and Feng Xie

Symmetry 2013, 5(1), 47-53; https://doi.org/10.3390/sym5010047 - 07 Jan 2013

Cited by 2 | Viewed by 4539

Abstract

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal [...] Read more.

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d² + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)² /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14. Full article

(This article belongs to the Special Issue Polyhedra)

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4194 KiB  

Open AccessArticle

Taylor–Socolar Hexagonal Tilings as Model Sets

by Jeong-Yup Lee and Robert V. Moody

Symmetry 2013, 5(1), 1-46; https://doi.org/10.3390/sym5010001 - 28 Dec 2012

Cited by 6 | Viewed by 8333

Abstract

The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model [...] Read more.

The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice Q in the plane. If X_Q denotes the set of all Taylor–Socolar tilings with centers on Q, then X_Q forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of Q.The Q-adic completion Q of Q is a natural factor of X_Q and the natural mapping X_Q → Q is bijective except at a dense set of points of measure 0 in /Q. We show that X_Q consists of three LI classes under translation. Two of these LI classes are very small, namely countable Q-orbits in X_Q. The other is a minimal dynamical system, which maps surjectively to /Q and which is variously 2 : 1, 6 : 1, and 12 : 1 at the singular points. We further develop the formula of what determines the parity of the tiles of a tiling in terms of the coordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull X_Q; more precisely the two hulls are mutually locally derivable. Full article

(This article belongs to the Special Issue Polyhedra)

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1042 KiB  

Open AccessArticle

A Peculiarly Cerebroid Convex Zygo-Dodecahedron is an Axiomatically Balanced “House of Blues”: The Circle of Fifths to the Circle of Willis to Cadherin Cadenzas

by David A. Becker

Symmetry 2012, 4(4), 644-666; https://doi.org/10.3390/sym4040644 - 15 Nov 2012

Viewed by 11108

Abstract

A bilaterally symmetrical convex dodecahedron consisting of twelve quadrilateral faces is derived from the icosahedron via a process akin to Fuller’s Jitterbug Transformation. The unusual zygomorphic dodecahedron so obtained is shown to harbor a bilaterally symmetrical jazz/blues harmonic code on its twelve faces [...] Read more.

A bilaterally symmetrical convex dodecahedron consisting of twelve quadrilateral faces is derived from the icosahedron via a process akin to Fuller’s Jitterbug Transformation. The unusual zygomorphic dodecahedron so obtained is shown to harbor a bilaterally symmetrical jazz/blues harmonic code on its twelve faces that is related to such fundamental music theoretical constructs as the Circle of Fifths and Euler’s tonnetz. Curiously, the patterning within the aforementioned zygo-dodecahedron is discernibly similar to that observed in a ventral view of the human brain. Moreover, this same pattern is arguably evident during development of the embryonic pharynx. A possible role for the featured zygo-dodecahedron in cephalogenesis is considered. Recent studies concerning type II cadherins, an important class of proteins that promote cell adhesion, have generated data that is demonstrated to conform to this zygo-dodecahedral brain model in a substantially congruous manner. Full article

(This article belongs to the Special Issue Polyhedra)

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452 KiB  

Open AccessArticle

Hexagonal Inflation Tilings and Planar Monotiles

by Michael Baake, Franz Gähler and Uwe Grimm

Symmetry 2012, 4(4), 581-602; https://doi.org/10.3390/sym4040581 - 22 Oct 2012

Cited by 11 | Viewed by 8576

Abstract

Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which [...] Read more.

Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces. Full article

(This article belongs to the Special Issue Polyhedra)

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355 KiB  

Open AccessArticle

Barrel Pseudotilings

by Undine Leopold

Symmetry 2012, 4(3), 545-565; https://doi.org/10.3390/sym4030545 - 30 Aug 2012

Viewed by 5884

Abstract

This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel [...] Read more.

This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel tiles or congruent apeirogonal pseudotiles. No physical space is occupied by the apeirogonal pseudotiles. Many interesting pseudotilings arise from plane tilings with high symmetry. As combinatorial structures, these are abstract polytopes of rank 4 with both finite and infinite 2-faces and facets. Full article

(This article belongs to the Special Issue Polyhedra)

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786 KiB  

Open AccessArticle

Superspheres: Intermediate Shapes between Spheres and Polyhedra

by Susumu Onaka

Symmetry 2012, 4(3), 336-343; https://doi.org/10.3390/sym4030336 - 03 Jul 2012

Cited by 12 | Viewed by 8067

Abstract

Using an x-y-z coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of {100}, {111} and {110} surfaces with round edges are treated in the present study, where [...] Read more.

Using an x-y-z coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of {100}, {111} and {110} surfaces with round edges are treated in the present study, where {100}, {111} and {110} are the Miller indices of crystals with cubic structures. The three parameters p, a and b are included to describe the {100}-{111}-{110} near-polyhedral shapes, where p describes the degree to which the shape is a polyhedron and a and b determine the ratios of the {100}, {111} and {110} surfaces. Full article

(This article belongs to the Special Issue Polyhedra)

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262 KiB  

Open AccessArticle

Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators

by Steve Wilson

Symmetry 2012, 4(2), 265-275; https://doi.org/10.3390/sym4020265 - 16 Apr 2012

Cited by 17 | Viewed by 5149

Abstract

This paper introduces the idea of a maniplex, a common generalization of map and of polytope. The paper then discusses operators, orientability, symmetry and the action of the symmetry group. Full article

(This article belongs to the Special Issue Polyhedra)

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234 KiB  

Open AccessArticle

Self-Dual, Self-Petrie Covers of Regular Polyhedra

by Gabe Cunningham

Symmetry 2012, 4(1), 208-218; https://doi.org/10.3390/sym4010208 - 27 Feb 2012

Cited by 4 | Viewed by 4165

Abstract

The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build the [...] Read more.

The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build the minimal common cover of two polyhedra, and by mixing a regular polyhedron with its five other images under the duality operations, we are able to construct the minimal self-dual, self-Petrie cover of a regular polyhedron. Determining the full structure of these covers is challenging and generally requires that we use some of the standard algorithms in combinatorial group theory. However, we are able to develop criteria that sometimes yield the full structure without explicit calculations. Using these criteria and other interesting methods, we then calculate the size of the self-dual, self-Petrie covers of several polyhedra, including the regular convex polyhedra. Full article

(This article belongs to the Special Issue Polyhedra)

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271 KiB  

Open AccessArticle

Convex-Faced Combinatorially Regular Polyhedra of Small Genus

by Egon Schulte and Jörg M. Wills

Symmetry 2012, 4(1), 1-14; https://doi.org/10.3390/sym4010001 - 28 Dec 2011

Cited by 7 | Viewed by 4732

Abstract

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E³ of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally [...] Read more.

Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E³ of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E³. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E³. Full article

(This article belongs to the Special Issue Polyhedra)

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