## Viewpoints:Mathematical PerspectiveandFractal Geometryin Art

### by Marc Frantz & Annalisa Crannell

This is a textbook intended for undergraduated mathematics courses with orientation to fine arts majors.

The mathematical content revolves around two subjects: perspective projection and fractal geometry. The brick-and-mortar topics include 2d and 3d coordinates (Ch 1), similarity (Ch 2), central and inscribed angles (Ch 5), existence of orthocenter (Ch 6), self-similarity (Ch 8) and exponents and logarithms (Ch 9). On these, the authors build an edifice of perspective projection (Ch 1-7), with a spacious attic of fractal geometry (Ch 8-9).

Perspective projection is covered gently and at a considerable length. One-, two-, and three-point perspective drawings are treated in separate chapters and extend to anamorphic art (Ch 7) and painting where a sphere replaces the picture plane (Ch 6).

The treatment of perspective is rather unique. The authors derive all the standard formulas in Chapter 2, but then in Chapter 4 - independently - discuss tools for depicting a sequence of congruent rectangles in a perspective view. This is clearly designed with the student artists in mind. The techniques discussed in Chapter 4 are useful when painting long fences, electric poles on the side of a road, extended sidewalks, railroads and such. In addition, the book not only shows how to draw, but also how to view perspective paintings. This is quite uncommon. I do not believe I came across anything similar before.

The key is an observation that the viewpoint of a drawing that includes two vanishing points due to two perpendicular directions lies on a semircircle with a diameter joining the two vanishing points in the plane perpendicular to the picture plane (Ch 5). A practical construction of the best viewing distance and the viewpoint for a two-point perspective depends on identifying two pairs of vanishing points and finding the intersection of the corresponding semicirles. This works perfectly for the illustrations provided in the book and for several perspective paintings I found in my home library. For a one-point perspective the construction is even simpler; for a three-point perspective an engaging discussion leads to the question of the intersection of three spheres and the existence of the orthocenter.

Fractal geometry is introduced in Chapter 8 with make-belief photographs that convincingly illustrate the idea of many natural objects looking the same regardless of their size. That's a beautiful introduction to self-similarity. Self-similarity is pursued with several example and, towards the end of the chapter, is shown to arise via iteration function systems. The central topic of the last chapter is fractal dimension which is introduced as a means of measuring the size of a fractal. The chapter includes a review of exponents and logarithms making the material self-contained. The chapter draws attention to the photographs of Ansel Adams which juxtapose fractal mountain ridge skyline with the smooth river curves; to the drawings of Li Cheng and architecture of Frank Lloyd Write.

The textbook contains a wide variety of classroom-tested activities and problems and a good deal of pedagogical and learning opportunities for instructors and students. Each chapter ends with a list of exercises; solutions to some of the exercises are collected at the end of the book.

Last but not least, the book features a series of essays by contemporary artists written especially for the book. These fine artists - among which several are professional mathematicians and scientists - examine how mathematics influences their art. The essays are accompanied by images of their work. There are also several color inserts.

The book goes a long way trying to convey to its audience - through both theory and practice - professional techniques that could not fail but empower students to make accurate, sophisticated drawings. The book presents an elegant fusion of mathematical ideas and practical aspects of fine art.

Viewpoints: Mathematical Perspective and Fractal Geometry in Art by Marc Frantz & Annalisa Crannell. Princeton University Press. Hardcover, 258 pp, \$45.00. ISBN 9780691125923.