8,515 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
Polyhedral billiards, eigenfunction concentration and almost periodic control
We study dynamical properties of the billiard flow on convex polyhedra away
from a neighbourhood of the non-smooth part of the boundary, called
``pockets''. We prove there are only finitely many immersed periodic tubes
missing the pockets and moreover establish a new quantitative estimate for the
lengths of such tubes. This extends well-known results in dimension . We
then apply these dynamical results to prove a quantitative Laplace
eigenfunction mass concentration near the pockets of convex polyhedral
billiards. As a technical tool for proving our concentration results on
irrational polyhedra, we establish a control-theoretic estimate on a product
space with an almost-periodic boundary condition. This extends previously known
control estimates for periodic boundary conditions, and seems to be of
independent interest.Comment: 32 pages, a few sections reorganised and a few results adde
Periodic orbits from Δ-modulation of stable linear systems
The �-modulated control of a single input, discrete time, linear stable system is investigated. The modulation direction is given by cTx where c �Rn/{0} is a given, otherwise arbitrary, vector. We obtain necessary and sufficient conditions for the existence of periodic points of a finite order. Some concrete results about the existence of a certain order of periodic points are also derived. We also study the relationship between certain polyhedra and the periodicity of the �-modulated orbit
Complexity in surfaces of densest packings for families of polyhedra
Packings of hard polyhedra have been studied for centuries due to their
mathematical aesthetic and more recently for their applications in fields such
as nanoscience, granular and colloidal matter, and biology. In all these
fields, particle shape is important for structure and properties, especially
upon crowding. Here, we explore packing as a function of shape. By combining
simulations and analytic calculations, we study three 2-parameter families of
hard polyhedra and report an extensive and systematic analysis of the densest
packings of more than 55,000 convex shapes. The three families have the
symmetries of triangle groups (icosahedral, octahedral, tetrahedral) and
interpolate between various symmetric solids (Platonic, Archimedean, Catalan).
We find that optimal (maximum) packing density surfaces that reveal unexpected
richness and complexity, containing as many as 130 different structures within
a single family. Our results demonstrate the utility of thinking of shape not
as a static property of an object in the context of packings, but rather as but
one point in a higher dimensional shape space whose neighbors in that space may
have identical or markedly different packings. Finally, we present and
interpret our packing results in a consistent and generally applicable way by
proposing a method to distinguish regions of packings and classify types of
transitions between them.Comment: 16 pages, 8 figure
Cubic Polyhedra
A cubic polyhedron is a polyhedral surface whose edges are exactly all the
edges of the cubic lattice. Every such polyhedron is a discrete minimal
surface, and it appears that many (but not all) of them can be relaxed to
smooth minimal surfaces (under an appropriate smoothing flow, keeping their
symmetries). Here we give a complete classification of the cubic polyhedra.
Among these are five new infinite uniform polyhedra and an uncountable
collection of new infinite semi-regular polyhedra. We also consider the
somewhat larger class of all discrete minimal surfaces in the cubic lattice.Comment: 18 pages, many figure
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