58 research outputs found
A Class of Convex Polyhedra with Few Edge Unfoldings
We construct a sequence of convex polyhedra on n vertices with the property
that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap
approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each
does have (several) nonoverlapping edge unfoldings.Comment: 12 pages, 9 figure
Flat Zipper-Unfolding Pairs for Platonic Solids
We show that four of the five Platonic solids' surfaces may be cut open with
a Hamiltonian path along edges and unfolded to a polygonal net each of which
can "zipper-refold" to a flat doubly covered parallelogram, forming a rather
compact representation of the surface. Thus these regular polyhedra have
particular flat "zipper pairs." No such zipper pair exists for a dodecahedron,
whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an
inventory of the possibilities, and raises more questions than it answers.Comment: 15 pages, 14 figures, 8 references. v2: Added one new figure. v3:
Replaced Fig. 13 to remove a duplicate unfolding, reducing from 21 to 20 the
distinct unfoldings. v4: Replaced Fig. 13 again, 18 distinct unfolding
Band Unfoldings and Prismatoids: A Counterexample
This note shows that the hope expressed in [ADL+07]--that the new algorithm
for edge-unfolding any polyhedral band without overlap might lead to an
algorithm for unfolding any prismatoid without overlap--cannot be realized. A
prismatoid is constructed whose sides constitute a nested polyhedral band, with
the property that every placement of the prismatoid top face overlaps with the
band unfolding.Comment: 5 pages, 3 figures. v2 replaced Fig.1(b) and Fig.3 to illustrate the
angles delta=(1/2)epsilon (rather than delta=epsilon
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
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
Spiral Unfoldings of Convex Polyhedra
The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution
Spiral Unfoldings of Convex Polyhedra
The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution
Unfolding Orthogrids with Constant Refinement
We define a new class of orthogonal polyhedra, called orthogrids, that can be
unfolded without overlap with constant refinement of the gridded surface.Comment: 19 pages, 12 figure
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