We show that metric trees and their products meet the octahedron comparison, which is a certain six-point comparison similar to Alexandrov’s CAT(0) comparison.

In this paper, we examine the octahedron as a parameterization scheme to represent environment maps in real-time rendering applications. We discuss two projection schemes and show two convenient unfold mappings to pack the platonic solid into quadratic or rectangular textures. We carry out an analysis and consider rendering performance for interactive applications as well as the quality of repr...

Icosahedral symmetry can not be allowed to exist in crystalline phases. However, the structures of some crystalline alloy phases are characterized by packing of atom clusters with icosahedral symmetry, and their structures are considered to be closely related to the structures of icosahedral quasicrystals. For examples, a cubic α-(AlMnSi) crystalline phase with a lattice constant of 1.264 nm is...

We show that every triangulation on a closed surface, except the sphere, with minimum degree at least 4 can be obtained from an irreducible triangulation by two kinds of local deformations, called 4-splitting and addition of an octahedron and that every triangulation on the sphere can be obtained from the octahedron by them. c © 2002 Elsevier Science B.V. All rights reserved.

We give a fairly simple proof of a result announced in [2] that there are only seven non-extendible finite (r, q)-polycycles: the tetrahedron without a face, the cube without a face, the octahedron without a face, the dodecahedron without a face, the icosahedron without a face, the split-vertex octahedron and the split-vertex icosahedron.

The topotactic connection style of oxygen octahedron/tetrahedron in transition metal oxides (TMOs) is an important feature that modulates their corresponding physical properties. Using a simple chemical doping technique, we obtained Sr1−xNdxCoO3-δ with crystal structure from face-shared octahedron to corner-shared octahedron/tetrahedron. Rietveld analyses the X-ray diffraction (XRD) patterns sh...

The structures and coalescence behavior of size-selected, matrix-isolated silicon clusters have been studied using surface-plasmon-polariton ~SPP! enhanced Raman spectroscopy. The cluster ions were produced in a laser vaporization source, mass selected then deposited into a co-condensed matrix of Ar, Kr or N2 on a liquid He cooled substrate. Raman spectra from monodisperse samples of the smalle...