In this paper we seek a Gaussian mixture model (GMM) of an n-variate probability density function. Usually the parameters of GMMs are determined by a maximum likelihood (ML) criterion. A practical deficiency of ML fitting of GMMs is poor performance when dealing with high-dimensional data since a large sample size is needed to match the accuracy that is possible in low dimensions. We propose a method to fit the GMM to multivariate data which is based on the two-dimensional projection pursuit (PP) method. By means of simulations we compare the proposed method with a one-dimensional PP method for GMM. We conclude that a combination of one- and twodimensional PP methods could be useful in some applications.