Aristotle would have said that a geometer considers the shapes of things in the natural world, not insofar as shapes are physical, but rather by abstracting the qualities of figure from the things themselves. In a philosophical sense this book by Mr. Arthur is thus one about geometry. The original illustrations go artistically beyond what a reader might expect to find in a textbook, and yet there is still a useful discussion revealing the important secrets of the creative process from a mathematical perspective. As an exploration of symmetry and smoothness, the work exhibits the precision of a draftsman unfettered by the practical constraints of real construction and engineering. Mr. Arthur develops a style of a visual art that is less dependent on the physical coordination of drawing with a pencil or a digital stylus and an art that is more of a purely intellectual nature. Clearly for the mathematical reader at a level of study somewhat exceeding vector calculus, the book presents new possibilities of exotic shapes (by their torsion or concavity) still having desirable qualities such as differentiability, self-similarity or compactness. Included in this book are many full-color pictures of tessellations, polyhedrons, unusual curves and surfaces, and fractals, along with their generating equations, coordinates and diagrams. (Note: the present edition does not publish the source code.)