We define a cluster to be characterized by regions of high density separated by regions that are sparse. By observing the downward closure property of density, the search for interesting structure in a high dimensional space can be reduced to a search for structure in lower dimensional subspaces. We present a Hierarchical Projection Pursuit Clustering (HPPC) algorithm that repeatedly bi-partitions the dataset based on the discovered properties of interesting 1-dimensional projections. We describe a projection search procedure and a projection pursuit index function based on Cho, Haralick and Yi's improvement of the Kittler and Illingworth optimal threshold technique. The output of the algorithm is a decision tree whose nodes store a projection and threshold and whose leaves represent the clusters (classes). Experiments with various real and synthetic datasets show the effectiveness of the approach.