103 research outputs found
Regular Polygonal Complexes of Higher Ranks in E^3
The paper establishes that the rank of a regular polygonal complex in 3-space
E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in
3-space are the eight regular 4-apeirotopes
Chiral polyhedra in ordinary space, II
A chiral polyhedron has a geometric symmetry group with two orbits on the
flags, such that adjacent flags are in distinct orbits. Part I of the paper
described the discrete chiral polyhedra in ordinary Euclidean 3-space with
finite skew faces and finite skew vertex-figures; they occur in infinite
families and are of types {4,6}, {6,4} and {6,6}. Part II completes the
enumeration of all discrete chiral polyhedra in 3-space. There exist several
families of chiral polyhedra with infinite, helical faces. In particular, there
are no discrete chiral polyhedra with finite faces in addition to those
described in Part I.Comment: 48 page
Combinatorial Space Tiling
The present article studies combinatorial tilings of Euclidean or spherical
spaces by polytopes, serving two main purposes: first, to survey some of the
main developments in combinatorial space tiling; and second, to highlight some
new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science
Polyhedra, Complexes, Nets and Symmetry
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are
finite or infinite 3-periodic structures with interesting geometric,
combinatorial, and algebraic properties. They can be viewed as finite or
infinite 3-periodic graphs (nets) equipped with additional structure imposed by
the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is
"regular" if its geometric symmetry group is transitive on the flags (incident
vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra
and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all
infinite, which are not polyhedra. Their edge graphs are nets well-known to
crystallographers, and we identify them explicitly. There also are 6 infinite
families of "chiral" apeirohedra, which have two orbits on the flags such that
adjacent flags lie in different orbits.Comment: Acta Crystallographica Section A (to appear
Reflection Groups and Polytopes over Finite Fields, III
When the standard representation of a crystallographic Coxeter group is
reduced modulo an odd prime p, one obtains a finite group G^p acting on some
orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p
will often be the automorphism group of a finite abstract regular polytope. In
parts I and II we established the basics of this construction and enumerated
the polytopes associated to groups of rank at most 4, as well as all groups of
spherical or Euclidean type. Here we extend the range of our earlier criteria
for the polytopality of G^p . Building on this we investigate the class of
3-infinity groups of general rank, and then complete a survey of those locally
toroidal polytopes which can be described by our construction.Comment: Advances in Applied Mathematics (to appear); 19 page
Reflection groups and polytopes over finite fields, II
When the standard representation of a crystallographic Coxeter group
is reduced modulo an odd prime , a finite representation in some orthogonal
space over is obtained. If has a string diagram, the
latter group will often be the automorphism group of a finite regular polytope.
In Part I we described the basics of this construction and enumerated the
polytopes associated with the groups of rank 3 and the groups of spherical or
Euclidean type. In this paper, we investigate such families of polytopes for
more general choices of , including all groups of rank 4. In
particular, we study in depth the interplay between their geometric properties
and the algebraic structure of the corresponding finite orthogonal group.Comment: 30 pages (Advances in Applied Mathematics, to appear
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
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