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The minimization principle of the second moment of the mass distribution ( M 2 ) is responsible for the unique structure of three-dimensional clusters by using emulsion droplet evaporation. Herein we study the structure of two-dimensional clusters of colloidal particles bound at the interface of liquid droplets in the plane. We found that, differently from the three-dimensional system, the two-dimensional clusters have multiple degenerate configurations (isomers). An interesting feature of such two-dimensional clusters is that they have the same packings as those belonging to a class of geometric figures known as polyiamonds. In particular, except for the six-particle cluster, many higher order clusters of polyiamond have not been reported previously. Using a simple geometrical approach, based on the number of ways to generate a packing, we calculated the occupation probabilities of distinct isomeric clusters. The level of agreement with the results of metropolis Monte Carlo simulations was good for clusters containing up to nine particles, suggesting that our two-dimensional cluster structures are not a result of the minimization of the second moment. In addition, the structure of these clusters is somewhat insensitive to the range and depth of the interparticle potential, in good agreement with the results in the literature.