9 research outputs found
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
Regular Polyhedra of Index Two, II
A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2
if it is combinatorially regular and its geometric symmetry group has index 2
in its combinatorial automorphism group; thus its automorphism group is
flag-transitive but its symmetry group has two flag orbits. The present paper
completes the classification of finite regular polyhedra of index 2 in 3-space.
In particular, this paper enumerates the regular polyhedra of index 2 with
vertices on one orbit under the symmetry group. There are ten such polyhedra.Comment: 33 pages; 5 figures; to appear in "Contributions to Algebra and
Geometry
Chiral polyhedra in ordinary space, II
A chiral polyhedron has a geometric symmetry group with two orbits on the
flags, such that adjacent flags are in distinct orbits. Part I of the paper
described the discrete chiral polyhedra in ordinary Euclidean 3-space with
finite skew faces and finite skew vertex-figures; they occur in infinite
families and are of types {4,6}, {6,4} and {6,6}. Part II completes the
enumeration of all discrete chiral polyhedra in 3-space. There exist several
families of chiral polyhedra with infinite, helical faces. In particular, there
are no discrete chiral polyhedra with finite faces in addition to those
described in Part I.Comment: 48 page
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Discrete Geometry
The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants