I discuss the symmetry of fullerenes, viruses and geodesic domes within a unified framework of icosadeltahedral representation of these objects. The icosadeltahedral symmetry is explained in details by examination of all of these structures. Using Euler’s theorem on polyhedra, it is shown how to calculate the number of vertices, edges, and faces in domes, and number of atoms, bonds and pentagon...

A pentagonal geometry PENT( k , r) is a partial linear space, where every line incident with points, point r lines, and for each x, there precisely those points that are not collinear x. Here we generalize the concept by allowing x to form set of Steiner system S ( 2 w ) whose blocks lines geometry.

Large carbon molecules, discovered at the end of the last century, are called fullerenes. The most famous of these, C60 has the structure of the soccer ball: the seams represent chemical bonds and the points where three seams come together represent the carbon atoms. We use the term fullerene to represent all mathematically possible structures for the chemical fullerenes: trivalent plane graphs...

In a recent paper, I began with four classical results due to Jacobi, Dirichlet and Lorenz which give the number of representations of a number by various forms involving squares in terms of divisor functions. I deduced another sixteen similar representation theorems involving triangular numbers and/or squares, including celebrated results of Legendre and Ramanujan (and omitted a seventeenth, g...