The Author of this paper has developed a family of Precious Tangrams based upon dissections of the first six regular polygons. Each set of tiles has similar properties to that of the regular tangram. In particular the property called Preciousness. It includes a discussion of some of the mathematical aspects of the dissections with examples of non- periodic tessellating patterns. It continues with examples of the unique way in which they can produce an infinite number of designs. It explains the iterative nature of the process as applied to designs for mosaics, quilts and animation.