In this paper, the generation of fractal curves and surfaces along with their properties, using ternary 4-point interpolatory subdivision scheme with one parameter, are analyzed. The relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples. The specific range of the tension parameter has also been depicted, which provides a clear way to generate fractal curves. Since the fractal scheme introduces, in the paper, have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve.