You are here

Math and Art: An Introduction to Visual Mathematics

Sasho Kalajdzievski
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Paperback with CDROM
[Reviewed by
Barbara Reynolds
, on

This delightful book grew out of set of teaching notes for an interdisciplinary course called Math in Art that was co-taught by a mathematician and an artist or architect. As stated in the Introduction, the goals of this book are “to indicate the potential of mathematics for generating (visually) appealing objects” and “to reveal some of the visual beauty of mathematics.” This book does a remarkable job of meeting these goals.

The author tells us that topics (see the table of contents for details) were chosen for their accessibility, visual appeal, and potential for mathematical interconnections. This book could be used for an introductory course in mathematics for non-mathematics majors, for a mathematics course for art majors, or for an independent reading course. The mathematical ideas are presented visually in a way that seems quite natural, and it engages the reader through explorations with lots of hands-on exercises. The mathematical presentation is solid, and the choice of topics puts the focus on the visual presentation of mathematical concepts.

The illustrations are beautiful! Most of the illustrations in the text are in grey-scale, but there are 10 pages of color plates with 22 full-color figures. The accompanying CD includes all the illustrations of the text in full color, some additional color illustrations, and 21 animations, six of which are referred to in the text.

This text is very readable. The mathematics is accessible to those with little mathematical background, and yet the presentation is still engaging for those with more background. Good mathematical notes (called “A Bit of Math”) are interspersed throughout the text so that an instructor could make the course more or less mathematically rigorous depending in the mathematical background of the students.

There is enough material that this could be used as the text for a semester-long course; sections that are more mathematical are clearly identified, and while these add depth to the discussion, they could be skipped without sacrificing continuity or the integrity of the course.

Chapter 1 begins with familiar Euclidean geometry, Euclid’s axioms, ruler and compass constructions, the golden ratio, and Fibonacci numbers. Chapter 2 introduces plane transformations, symmetries in the plane symmetries, frieze patterns, wallpaper designs, and tilings of the plane. The discussion of groups of symmetries stops just short of talking about symmetry groups — so that the discussion is clear, and leads to some deep concepts without overwhelming a student who may be more visually motivated.

The chapter on Similarities, Fractals, and Cellular Automata includes an optional section on complex numbers, but presents core concepts of fractals in a way that would be appealing to less mathematically inclined student, so that the instructor could decide how deep to take the discussion depending on the backgrounds of the students. The chapter on hyperbolic geometry discusses the tiling of the hyperbolic plane, thus making a clear connection to the discussion of symmetries and tilings given in Chapter 2. The chapter on perspective includes a discussion of the mathematics of perspective drawing, but puts the focus on the interplay between 2-dimensional representations and 3-dimensional objects being represented.

One delightful unifying technique that bears mention is the Dadat, a toy-like character, circa “2 x 109 BC to present, Everybody’s contemporary.” Datat is introduced in Chapter 1, and appears throughout the text, often introducing visual concepts. For example, in Chapter 2, Dadat illustrates rotations and glide reflections, in Chapter 3 Dadat illustrates similarity and opposite orientation (a similar Dadat may be smaller or larger, and either left- or right-handed), and in Chapter 5 Dadat illustrates concepts in three dimensions. By the end of the book, Dadat has become something of a friend of the reader. This is done in a simple and unobtrusive way.

Sr. Barbara E. Reynolds, SDS, Ph.D., is Professor of Mathematics & Computer Science at Cardinal Stritch University, Milwaukee, WI.

Euclidean Geometry 
The Five Axioms of Euclidean Geometry
Ruler and Compass Constructions
The Golden Ratio
Fibonacci Numbers
Plane Transformations
Plane Symmetries
Plane Symmetries, Vectors, and Matrices
Groups of Symmetries of Planar Objects
Frieze Patterns
Wallpaper Designs and Tiling of the Plane
Tilings and Art
Similarities, Fractals, and Cellular Automata
Similarities and Some Other Planar Transformations
Complex Numbers
Fractals: Definition and Some Examples
Julia Sets
Cellular Automata
Hyperbolic Geometry 
Non-Euclidean Geometries: Background and Some History
Hyperbolic Geometry
Some Basic Constructions in the Poincaré Model
Tilings of the Hyperbolic Plane
Perspective and Some Three-Dimensional Objects
The Mathematics of Perspective Drawing: A Brief Overview
Regular and Other Polyhedra
Sphere, Cylinder, Cone, and Conic Sections
Geometry, Tilings, Fractals, and Cellular Automata in Three Dimensions
Homotopy of Spaces: An Informal Introduction
Two-Manifolds and the Euler Characteristic
Nonorientable Two-Manifolds and Three-Manifolds