Recent work presented a Monte Carlo algorithm based on Markov Chain operators for generating replicas of textile composite specimens that possess the same statistical characteristics as specimens imaged using high resolution x-ray computed tomography. That work represented the textile reinforcement by one-dimensional tow loci in three-dimensional space, suitable for use in the Binary Model of textile composites. Here analogous algorithms are used to generate solid, three-dimensional (3D) tow representations, to provide geometrical models for more detailed failure analyses. The algorithms for generating 3D models are divided into those that refer to the topology of the textile and those that deal with its geometry. The topological rules carry all the information that distinguishes textiles with different interlacing patterns (weaves, braids, etc.) and provide instructions for resolving interpenetrations or ordering errors among tows. They also simplify writing a single computer program that can accept input data for generic textile cases. The geometrical rules adjust the shape and smoothness of the generated virtual specimens to match data from imaged specimens. The virtual specimen generator is illustrated using data for an angle interlock weave, a common 3D textile architecture.