Summary: The development and use of an approach to three-dimensional projection pursuit are discussed. The well-established Jones and Sibson moments index [\textit{M. C. Jones} and \textit{R. Sibson}, J. R. Stat. Soc., Ser. A 150, 1-36 (1987; Zbl 0632.62059)] is chosen as a computationally efficient projection index to extend to three dimensions. The three-dimensional index was initially developed to find interesting linear combinations of spectral bands in a multispectral image. Computer algebraic methods are extensively employed to handle the complex formulae that constitute the index; these methods are explained in detail. A discussion of important practical issues such as interpreting projection solutions, dealing with outliers and optimization techniques completes the description of the index. An artificial tetrahedral data set is used to demonstrate how three-dimensional projection pursuit can produce better clusters than those obtained by principal components analysis. The main example shows how three-dimensional projection pursuit can successfully combine bands to discover alternative clusters to those produced by, say, principal components.