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Crystalline atomic layers on solid surfaces are composed of a single building block, unit cell, that is copied and stacked together to form the entire two-dimensional crystal structure. However, it appears that this is not an unique possibility. We report here on synthesis and characterization of the one-atomic-layer-thick Tl
x
Bi
1−
x
compounds which display quite a different arrangement. It represents a quasi-periodic tiling structures that are built by a set of tiling elements as building blocks. Though the layer is lacking strict periodicity, it shows up as an ideally-packed tiling of basic elements without any skips or halting. The two-dimensional Tl
x
Bi
1−
x
compounds were formed by depositing Bi onto the Tl-covered Si(111) surface where Bi atoms substitute appropriate amount of Tl atoms. Atomic structure of each tiling element as well as arrangement of Tl
x
Bi
1−
x
compounds were established in a detail. Electronic properties and spin texture of the selected compounds having periodic structures were characterized. The shown example demonstrates possibility for the formation of the exotic low-dimensional materials via unusual growth mechanisms.