72 research outputs found
Enumerating Foldings and Unfoldings between Polygons and Polytopes
We pose and answer several questions concerning the number of ways to fold a
polygon to a polytope, and how many polytopes can be obtained from one polygon;
and the analogous questions for unfolding polytopes to polygons. Our answers
are, roughly: exponentially many, or nondenumerably infinite.Comment: 12 pages; 10 figures; 10 references. Revision of version in
Proceedings of the Japan Conference on Discrete and Computational Geometry,
Tokyo, Nov. 2000, pp. 9-12. See also cs.CG/000701
Folding concave polygons into convex polyhedra: The L-Shape
Mathematicians have long been asking the question: Can a given convex polyhedron can be unfolded into a polygon and then refolded into any other convex polyhedron? One facet of this question investigates the space of polyhedra that can be realized from folding a given polygon. While convex polygons are relatively well understood, there are still many open questions regarding the foldings of non-convex polygons. We analyze these folded realizations and their volumes derived from the polygonal family of `L-shapes,\u27 parallelograms with another parallelogram removed from a corner. We investigate questions of maximal volume, diagonal flipping, and topological connectedness and discuss the family of polyhedra that share a L-shape polygonal net
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
Nonorthogonal Polyhedra Built from Rectangles
We prove that any polyhedron of genus zero or genus one built out of
rectangular faces must be an orthogonal polyhedron, but that there are
nonorthogonal polyhedra of genus seven all of whose faces are rectangles. This
leads to a resolution of a question posed by Biedl, Lubiw, and Sun [BLS99].Comment: 19 pages, 20 figures. Revised version makes two corrections: The
statement of the old Lemma 14 was incorrect. It has been corrected and merged
with Lemma 13 now. Second, Figure 19 (a skew quadrilateral) was incorrect,
and is now removed. It played no substantive role in the proof
Fun with Fonts: Algorithmic Typography
Over the past decade, we have designed six typefaces based on mathematical
theorems and open problems, specifically computational geometry. These
typefaces expose the general public in a unique way to intriguing results and
hard problems in hinged dissections, geometric tours, origami design,
computer-aided glass design, physical simulation, and protein folding. In
particular, most of these typefaces include puzzle fonts, where reading the
intended message requires solving a series of puzzles which illustrate the
challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font.
Original version in Proceedings of the 7th International Conference on Fun
with Algorithm
Locked and unlocked smooth embeddings of surfaces
We study the continuous motion of smooth isometric embeddings of a planar
surface in three-dimensional Euclidean space, and two related discrete
analogues of these embeddings, polygonal embeddings and flat foldings without
interior vertices, under continuous changes of the embedding or folding. We
show that every star-shaped or spiral-shaped domain is unlocked: a continuous
motion unfolds it to a flat embedding. However, disks with two holes can have
locked embeddings that are topologically equivalent to a flat embedding but
cannot reach a flat embedding by continuous motion.Comment: 8 pages, 8 figures. To appear in 34th Canadian Conference on
Computational Geometr
- …