We show how a set of orthogonal coordinates is defined for a system having an arbitrary number of components. These coordinates, in conjunction with the limiting atom fractions of the components, define the equilateral triangle for a three component system and a regular tetrahedron for a four component system. Various manipulations of such figures in higher order space are carried out using linear algebra. Two dimensional sections are considered the most important construction. The utility of such sections is enhanced if the sections are taken parallel to important sets of tie lines. The present exercise is but one facet of the larger problem in which complex phase equilibria are calculated from thermodynamic descriptions of binary systems. We suggest that, eventually, problems involving phase equilibria for systems of many components will be addressed by calculating the required sections for the problem at hand.