6,796 research outputs found
Regular Polygonal Complexes of Higher Ranks in E^3
The paper establishes that the rank of a regular polygonal complex in 3-space
E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in
3-space are the eight regular 4-apeirotopes
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
Combinatorial Space Tiling
The present article studies combinatorial tilings of Euclidean or spherical
spaces by polytopes, serving two main purposes: first, to survey some of the
main developments in combinatorial space tiling; and second, to highlight some
new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science
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
Intelligent Network Management and Functional Cerebellum Synthesis
Transdisciplinary modeling of the cerebellum across histology, physiology, and network engineering provides preliminary results at three organization levels: input/output links to central nervous system networks; links between the six neuron populations in the cerebellum; and computation among the neurons of the populations. Older models probably underestimated the importance and role of climbing fiber input which seems to supply write as well as read signals, not just to Purkinje but also to basket and stellate neurons. The well-known mossy fiber-granule cell-Golgi cell system should also respond to inputs originating from climbing fibers. Corticonuclear microcomplexing might be aided by stellate and basket computation and associate processing. Technological and scientific implications of the proposed cerebellum model are discussed
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