2,094 research outputs found
Locked and Unlocked Chains of Planar Shapes
We extend linkage unfolding results from the well-studied case of polygonal
linkages to the more general case of linkages of polygons. More precisely, we
consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are
hinged together sequentially at rotatable joints. Our goal is to characterize
the families of planar shapes that admit locked chains, where some
configurations cannot be reached by continuous reconfiguration without
self-intersection, and which families of planar shapes guarantee universal
foldability, where every chain is guaranteed to have a connected configuration
space. Previously, only obtuse triangles were known to admit locked shapes, and
only line segments were known to guarantee universal foldability. We show that
a surprisingly general family of planar shapes, called slender adornments,
guarantees universal foldability: roughly, the distance from each edge along
the path along the boundary of the slender adornment to each hinge should be
monotone. In contrast, we show that isosceles triangles with any desired apex
angle less than 90 degrees admit locked chains, which is precisely the
threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof
details. (Fixed crash-induced bugs in the abstract.
Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes
We investigate how to make the surface of a convex polyhedron (a polytope) by
folding up a polygon and gluing its perimeter shut, and the reverse process of
cutting open a polytope and unfolding it to a polygon. We explore basic
enumeration questions in both directions: Given a polygon, how many foldings
are there? Given a polytope, how many unfoldings are there to simple polygons?
Throughout we give special attention to convex polygons, and to regular
polygons. We show that every convex polygon folds to an infinite number of
distinct polytopes, but that their number of combinatorially distinct gluings
is polynomial. There are, however, simple polygons with an exponential number
of distinct gluings.
In the reverse direction, we show that there are polytopes with an
exponential number of distinct cuttings that lead to simple unfoldings. We
establish necessary conditions for a polytope to have convex unfoldings,
implying, for example, that among the Platonic solids, only the tetrahedron has
a convex unfolding. We provide an inventory of the polytopes that may unfold to
regular polygons, showing that, for n>6, there is essentially only one class of
such polytopes.Comment: 54 pages, 33 figure
Self-Replicating Strands that Self-Assemble into User-Specified Meshes
It has been argued that a central objective of nanotechnology is to make
products inexpensively, and that self-replication is an effective approach to
very low-cost manufacturing. The research presented here is intended to be a
step towards this vision. In previous work (JohnnyVon 1.0), we simulated
machines that bonded together to form self-replicating strands. There were two
types of machines (called types 0 and 1), which enabled strands to encode
arbitrary bit strings. However, the information encoded in the strands had no
functional role in the simulation. The information was replicated without being
interpreted, which was a significant limitation for potential manufacturing
applications. In the current work (JohnnyVon 2.0), the information in a strand
is interpreted as instructions for assembling a polygonal mesh. There are now
four types of machines and the information encoded in a strand determines how
it folds. A strand may be in an unfolded state, in which the bonds are straight
(although they flex slightly due to virtual forces acting on the machines), or
in a folded state, in which the bond angles depend on the types of machines. By
choosing the sequence of machine types in a strand, the user can specify a
variety of polygonal shapes. A simulation typically begins with an initial
unfolded seed strand in a soup of unbonded machines. The seed strand replicates
by bonding with free machines in the soup. The child strands fold into the
encoded polygonal shape, and then the polygons drift together and bond to form
a mesh. We demonstrate that a variety of polygonal meshes can be manufactured
in the simulation, by simply changing the sequence of machine types in the
seed
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
Quantum Gravity in Large Dimensions
Quantum gravity is investigated in the limit of a large number of space-time
dimensions, using as an ultraviolet regularization the simplicial lattice path
integral formulation. In the weak field limit the appropriate expansion
parameter is determined to be . For the case of a simplicial lattice dual
to a hypercube, the critical point is found at (with ) separating a weak coupling from a strong coupling phase, and with degenerate zero modes at . The strong coupling, large , phase is
then investigated by analyzing the general structure of the strong coupling
expansion in the large limit. Dominant contributions to the curvature
correlation functions are described by large closed random polygonal surfaces,
for which excluded volume effects can be neglected at large , and whose
geometry we argue can be approximated by unconstrained random surfaces in this
limit. In large dimensions the gravitational correlation length is then found
to behave as , implying for the universal
gravitational critical exponent the value at .Comment: 47 pages, 2 figure
Polygonal path simplification with angle constraints
We present efficient geometric algorithms for simplifying polygonal paths in R2 and R3 that have angle constraints, improving by nearly a linear factor over the graph-theoretic solutions based on known techniques. The algorithms we present match the time bounds for their unconstrained counterparts. As a key step in our solutions, we formulate and solve an off-line ball exclusion search problem, which may be of interest in its own right
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