This paper defines and studies a general algorithm for constructing new families of fractals in Euclidean space. This algorithm involves a sequences of linear interscale transformations that proceed from large to small scales. We find that the fractals obtained in this fashion decompose in intrinsicfashion into linear combinations of a variable number of “addend’ fractals. The addends’ relative weights and fractal dimensionalities are obtained explicitly through an interscale matrix, which we call the transfer mafrix of the fractal (TMF). We first demonstrate by a series of examples, then prove rigorously, that the eigenvalues of our TMFS are real and positive, and that the fractal dimensions of the addend fractals are the logarithms of the eigenvalues of our TMF. We say that these dimensions form the overall fractal’s eigendimensional sequence. The eigenvalues of our TMF are integers in the non-random variants of the construction, but are non-integer in the random variants. A geometrical interpretation of the eigenvalues and the eigenvectors is given. Our TMF have other striking and very special properties that deserve additional attention.