A PARTICULAR CLASS OF PENTI-HEXAGONAL POLYHEDRA.
NAVAL RESEARCH LAB WASHINGTON D C
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In accordance with the Navys interest in developing an underwater spherical vessel capable of human occupancy, studies have been made of a particular class of convex polyhedra as a possible approximation to a sphere. The class referred to is a category of penti-hexagonal polyhedra, formed from the regular dodecahedron by inserting sucessive layers of equilateral convex hexagons within the basic pentagonal structure in a manner which maximizes the congruence and the symmetry of the polyhedral surfaces. The structure of a particular polyhedron in this class is determined by computing the vertex angles of all the distinct hexagons composing its surface, i.e., the polyhedron face angles. With this end in mind, trigonometric equations are derived for all hexagons having less than a certain degree of symmetry, and all dihedral angles whose edges are not perpendicular to a plane of symmetry of the polyhedron. These equations are then solved using a Newton-Raphson-type process. The structure of all polyhedra in this class up to 3242 faces has been determined, and the number of trigonometric equations in each case exactly equals the number of unknown angles. It is suspected that this equality is maintained for higher order polyhedra in this class. Author
- Numerical Mathematics
- Marine Engineering