A bolapolyphile, derived from a linear p-terphenyl core with glycerol groups at each end and two octadecyl chains fixed to the same side of the π-conjugated rod, is synthesized and is found to form a new type of self-assembled cubic LC phase with the Iad lattice. Each unit cell involves 16 polar aggregates interconnected by 24 coaxial rod-bundles at the single-molecule length, forming two interwoven skeletal networks with three-way junctions, being separated by a continuum of the lateral alkyl chains located around the gyroid minimal surface.