An octahedral equal area partition of the sphere and near optimal configurations of points

The Diamond ensemble: A constructive set of spherical points with small logarithmic energy

“…In [12] an area preserving map from the unit sphere to the regular octahedron is defined. Considering some hierarchical triangular grids on the facets of the octahedron a grid can be mapped into the sphere obtaining two different sets of points: those coming from the vertex of the grid Ω N and the centers of the triangles Λ N .…”

“…Since the parallel at height z j is a circumference of radius 1 − z 2 j , quantity B equals B = − p j=1 r j log r j + r j (r j − 1) 2 log(1 − z 2 j ). In order to compute the logarithmic energy associated to the set Ω(p, r j , z j ) it only rest to sum the quantities (11), (12) and (13).…”

Distributing many points on spheres: Minimal energy and designs

“…also [142]. A. Holhoş and D. Roşca [130] study Riesz energy of points derived from an area-preserving map from the 2-sphere to the octahedron. 11 For d = 2 and d = 4, the critical distance is the golden ratio and the plastic number.…”

General Method for Extending Discrete Global Grid Systems to Three Dimensions

“…One way to create such a grid is by extending an existing DGGS to the third dimension to operate as a 3D global grid system. However, there are many different DGGSs with various desirable properties [16][17][18][19], and as such, the ideal DGGS to be extended depends on the application. Instead of extending a single DGGS to 3D, a more versatile approach is to create a general method that works for any arbitrary DGGS.…”