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A new class of $32n^{2}$ boron cages which are made closed by six squares is proposed and a procedure to build such cages using an α-boron sheet is described. Each member from this infinite set of boron cages has a structure that is compatible with the most stable α-boron sheet that maintains an optimal balance of the two-center and three-center bonds. Accurate density functional calculations with a large polarized Gaussian basis set show that B32, B96, B128, and B288 are energetically stable structures. The smallest B32 cage from this class has the HOMO-LUMO gap of 1.32 eV, the largest amongst the boron cages and boron fullerenes studied so far