In this paper I describe a method – based on the projective interpretation of the hyperbolic geometry – that determines the data and the density of the optimal ball and horoball packings of each… (More)

We describe three hexacontahedra in which the faces are rectangles, all equivalent under symmetries of the icosahedral group and having all edges in the mirror planes of the symmetry group. Under the… (More)

A. Cayley and F. Klein discovered in the nineteenth century that euclidean and non-euclidean geometries can be considered as mathematical structures living inside projective-metric spaces. They… (More)

This investigation was inspired by a letter from Edward Pervin of Carnegie-Mellon University. He had noticed an unexpected property of the fundamental regions for the infinite Euclidean reflection… (More)

The 45 diagonal triangles of the six-dimensional polytope 2 21 (representing the 45 tritangent planes of the cubic surface) are the vertex figures of 45 cubes {4, 3} inscribed in the… (More)

Tony Bomford made six hooked rugs based on hyperbolic geometry, having received inspiration from the Canadian mathematician H.S.M. Coxeter. All rugs except one exhibit color symmetry to some degree.… (More)

Data Mining has many applications in the real world. One of the most important and widely found problems is that of classification. Recently, distance preserving data perturbation has gained… (More)

A 3 x 3 block of squares, with opposite sides identified in the usual way, yields a “map” of nine quadrangles covering a torus. The solid figure bounded by this surface may be called a ‘toroid’. The… (More)

Artin groups are easily defined but most of them are poorly understood. In this survey I try to highlight precisely where the problems begin. The first part reviews the close connection between… (More)