5,086 research outputs found
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
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
More indecomposable polyhedra
We apply combinatorial methods to a geometric problem: the classification of
polytopes, in terms of Minkowski decomposability. Various properties of
skeletons of polytopes are exhibited, each sufficient to guarantee
indecomposability of a significant class of polytopes. We illustrate further
the power of these techniques, compared with the traditional method of
examining triangular faces, with several applications. In any dimension , we show that of all the polytopes with or fewer edges,
only one is decomposable. In 3 dimensions, we complete the classification, in
terms of decomposability, of the 260 combinatorial types of polyhedra with 15
or fewer edges.Comment: PDFLaTeX, 21 pages, 6 figure
Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes
A quadrangulation is a graph embedded on the sphere such that each face is
bounded by a walk of length 4, parallel edges allowed. All quadrangulations can
be generated by a sequence of graph operations called vertex splitting,
starting from the path P_2 of length 2. We define the degree D of a splitting S
and consider restricted splittings S_{i,j} with i <= D <= j. It is known that
S_{2,3} generate all simple quadrangulations.
Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}.
First we show that the splittings S_{1,2} are exactly the monotone ones in the
sense that the resulting graph contains the original as a subgraph. Then we
show that they define a set of nontrivial ancestors beyond P_2 and each
quadrangulation has a unique ancestor.
Our results have a direct geometric interpretation in the context of
mechanical equilibria of convex bodies. The topology of the equilibria
corresponds to a 2-coloured quadrangulation with independent set sizes s, u.
The numbers s, u identify the primary equilibrium class associated with the
body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate
all primary classes from a finite set of ancestors which is closely related to
their geometric results.
If, beyond s and u, the full topology of the quadrangulation is considered,
we arrive at the more refined secondary equilibrium classes. As Domokos,
L\'angi and Szab\'o showed recently, one can create the geometric counterparts
of unrestricted splittings to generate all secondary classes. Our results show
that S_{1,2} can only generate a limited range of secondary classes from the
same ancestor. The geometric interpretation of the additional ancestors defined
by monotone splittings shows that minimal polyhedra play a key role in this
process. We also present computational results on the number of secondary
classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
The Rigidity of Spherical Frameworks: Swapping Blocks and Holes
A significant range of geometric structures whose rigidity is explored for
both practical and theoretical purposes are formed by modifying generically
isostatic triangulated spheres. In the block and hole structures (P, p), some
edges are removed to make holes, and other edges are added to create rigid
sub-structures called blocks. Previous work noted a combinatorial analogy in
which blocks and holes played equivalent roles. In this paper, we connect
stresses in such a structure (P, p) to first-order motions in a swapped
structure (P', p), where holes become blocks and blocks become holes. When the
initial structure is geometrically isostatic, this shows that the swapped
structure is also geometrically isostatic, giving the strongest possible
correspondence. We use a projective geometric presentation of the statics and
the motions, to make the key underlying correspondences transparent.Comment: 36 pages, 9 figure
Null twisted geometries
We define and investigate a quantisation of null hypersurfaces in the context
of loop quantum gravity on a fixed graph. The main tool we use is the
parametrisation of the theory in terms of twistors, which has already proved
useful in discussing the interpretation of spin networks as the quantization of
twisted geometries. The classical formalism can be extended in a natural way to
null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra
with space-like faces, and SU(2) by the little group ISO(2). The main
difference is that the simplicity constraints present in the formalims are all
first class, and the symplectic reduction selects only the helicity subgroup of
the little group. As a consequence, information on the shapes of the polyhedra
is lost, and the result is a much simpler, abelian geometric picture. It can be
described by an Euclidean singular structure on the 2-dimensional space-like
surface defined by a foliation of space-time by null hypersurfaces. This
geometric structure is naturally decomposed into a conformal metric and scale
factors, forming locally conjugate pairs. Proper action-angle variables on the
gauge-invariant phase space are described by the eigenvectors of the Laplacian
of the dual graph. We also identify the variables of the phase space amenable
to characterize the extrinsic geometry of the foliation. Finally, we quantise
the phase space and its algebra using Dirac's algorithm, obtaining a notion of
spin networks for null hypersurfaces. Such spin networks are labelled by SO(2)
quantum numbers, and are embedded non-trivially in the unitary,
infinite-dimensional irreducible representations of the Lorentz group.Comment: 22 pages, 3 figures. v2: minor corrections, improved presentation in
section 4, references update
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
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