Graphyne allotropes of carbon are fascinating materials, and their electronic properties are predicted to rival those of the "wonder material" graphene. One allotrope of graphyne, having rectangular symmetry rather than hexagonal, stands out as particularly attractive, namely 6,6,12-graphyne. It is currently an insurmountable challenge, however, to design and execute a synthesis of this material. Herein, we present synthesis and electronic properties of molecules that serve as model compounds. These oligomers, so-called radiaannulenes, are prepared by iterative acetylenic coupling reactions. Systematic optical and redox studies indicate the effective conjugation length of the radiaannulene oligomers is nearly met by the length of the trimer. The HOMO-LUMO gap suggested by the series of oligomers is still, however, higher than that expected for 6,6,12-graphyne from theory, which predicts two nonequivalent distorted Dirac cones (no band gap). Thus, the radiaannulene oligomers present a suitable length in one dimension of a sheet, but should be expanded in the second dimension to provide a unique representation of 6,6,12-graphyne.