engleski

Geometry of pentagons and volumes of fullerenes

We provide new proofs of some known facts from geometry of pentagons and hexagons and prove some new facts and formulas concerning areas. We reprove the Gauss pentagon formula, the hexagon analogue and show some consequences. We also give a new proof of the Robbins area formula for cyclic pentagons (and hexagons). This proof is intrinsic.We also prove formulas relating area, circumradius and side lengths of such polygons. The obtained results we apply to get an efficient algorithm for computing surface areas and volumes of inscribed "fullerenes", i.e. 3-polytopes (solids) inscribed in a sphere of given radius, whose faces are at most hexagons and knowing only the graph and edge lengths of the fullerene.

pentagons; volumes of fullerenes

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