Minimum Surface Area of a Capped Hexagonal Prism

A hexagonal prism is capped with three rhombi at one of its ends. The surface area of , designated , is minimized when , the ratio of the diagonals of the rhombi, equals , which also pertains to the faces of the rhombic dodecahedron. Honeycombs are likewise characterized by this geometry.


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The points that are movable are the top vertex and three of the vertices of the top hexagon, in the opposite direction. Let be the magnitude of this vertical motion, be the height of the prism and be the ratio of diagonals of the rhombus. Then the surface area is determined by:
The surface area has a minimum value when its derivatives with respect to equals zero.
[1] Archimedes Lab. "Puzzle #117: The Geometry of the Bees." (Aug 16, 2016) 72.html.
[2] F. Tóth, "What the Bees Know and What They Do Not Know," Bulletin of the American Mathematical Society, 70, 1964 pp. 468–481.
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