A new rectangular columnar liquid crystalline phase with p2gg lattice is reported, which represents a polygonal cylinder array composed of cylinders with trapezoidal cross section. In these polygonal cylinders, one of the sides has a different length and is composed of a different material than the others. This tiling pattern was obtained in two series of T-shaped facial amphiphilic triblock molecules in which a rigid rod-like p-terphenyl core is substituted laterally by a polar and flexible oligoethylene glycol chain, terminated either by a hydrogen-bonding COOH group or by a Li carboxylate group, and having identical or different alkyl groups in the terminal positions. The trapezoidal cylinder phase provides an improved packing for relatively long and rigid alkyl chains at lower temperature and more space inside the polygonal cylinders than triangular cylinders. This combination of conformational and space-filling effects leads to different phase sequences. The trapezoidal cylinder phases pave the way to a new level of complexity in LC engineering and show the huge potential of the general concept of polyphilic tectons for the design of new complex soft matter structures.