DIVIDING A SPHERE HIERARCHICALLY INTO A LARGE NUMBER OF SPHERICAL PENTAGONS USING EQUAL AREA OR EQUAL LENGTH OPTIMIZATION

Dividing a 2-dimensional sphere uniformly into a large number of spherical polygons is a challenging mathematical problem, which has been studied across many disciplines due to its important practical applications. Most sphere subdivisions are achieved using spherical triangles, quadrangles, or a combination of hexagons and pentagons. However, spherical pentagons, which may create elegant configurations, remain underexplored. 

 

This study presents a new sphere subdivision method to generate a large number of spherical pentagons based on successively subdividing a module of an initial spherical dodecahedron. The new method can conveniently control the shapes of generated spherical pentagons through specified design parameters. Two optimization problems have been investigated: (I) dividing a sphere into spherical pentagons of equal area; (II) minimizing the number of different arc lengths used in the pentagonal subdivision. A variety of examples are presented to demonstrate the effectiveness of the new method. This study shows that treating the mathematical challenge of dividing a sphere uniformly into a large number of spherical polygons as an optimization problem can effectively obtain equal area or equal length sphere subdivisions. Furthermore, considering additional constraints on the optimization problem may achieve sphere subdivisions of specific characteristics.