Packing of equal regular pentagons on a sphere (Q2748096)

In this paper the authors discuss how \(n\) equal non-overlapping regular spherical pentagons can be packed on a sphere so that the angular radius of the circumcircles of the pentagons is the largest possible. Locally extremal results are conjectured for the cases when \(n=1, 2, 3, 4, 7, 9, 11, 12\). Locally non-extremal results are provided for \(n=10, 32\). These conjectural results were obtained by using a mechanical algorithm similar to the ``heating technique'' developed for spherical circle packings. Also, locally extremal configurations under octahedral and icosahedral symmetry constraints are presented for \(n=24, 72\). A detailed table of the known best arrangements is given, too. NEWLINENEWLINENEWLINEThe paper is illustrated with a large number of figures and some pictures showing examples of arrangements of pentagons on spherical surfaces in architecture.