The Geometry of the Bees

When constructing a honeycomb, bees aim to minimize wax usage and honey consumption, using the least wax necessary for a comb with maximum honey storage. The wax cells are designed with interlocking opposing layers, sharing facets at closed ends while having open ends facing outwards (see fig. 1). Each cell is a ten-sided structure with a rhombic decahedron form – a hexagonal prism with three rhombi at its closed end (fig. 2). Mathematicians have extensively studied the highly efficient isoperimetric properties of these cells. The question remains: What angle alpha maximizes volume while minimizing surface area on each cell face when the hexagonal prism’s faces have a width of 1 unit?

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