Geometry and its Applications in Arts, Nature and Technology

Geometrical form has fascinated humans for ages. In this book, Professor Glaeser of the University of Applied Arts Vienna rekindles that fascination through a multitude of examples drawn from astronomy, architecture, botany, computer graphics, mechanical designs, photography, zoology and more. More than a mere compendium, I liken this book to a fine museum providing a curated selection of exhibits that both informs and delights. To promote accessibility and the exercise of geometric intuition, the author has given preference to synthetic methods over algebraic and analytic methods, and has limited his examples to those visualizable in (low-dimensional) Euclidean space. At least one diagram or photo accompanies each example, and as in a museum, some exhibits receive brief comment while others are the subject of extensive discussion. The cumulative effect is that of wonder at the richness of geometrical form around us and an invitation to participate in its exploration.

 

Continuing the museum analogy, on the ground floor, we have a gallery illustrating basic geometric notions such as congruence, duality, and various types of projections. From there we are invited to explore the museum's other thematic galleries. For example, there is one on polyhedra including Platonic, Archimedean and rhombic solids, and another on tilings both periodic and non-periodic. Even non-Euclidean tilings make an appearance. Another gallery focuses on geometric questions related to perspective and image perception. Questions related to computer graphics and computational geometry receive particular attention.

 

One floor of this museum is devoted entirely to a menagerie of curves and surfaces: spheres, cylinders, conics, quadric surfaces, surfaces of revolution, developable and ruled surfaces, helical surfaces, B-spline surfaces, and more. Gauss curvature and the Gauss map make appearances, but those desirous of a more analytical treatment can seek Professor Glaeser's separate coauthored volumes The Universe of Conics and The Universe of Quadrics, both of which contain numerous illustrations.

 

In another wing we find exhibits on kinematics and spatial motions. Linkages and ellipse and trochoid motions are exemplified by various mechanical inventions, and there is also a detailed, but elementary discussion of the motions of the sun and moon as viewed from earth. There are further galleries, and the reader is invited to check the table of contents for a full listing. The two appendices provide geometric advice on freehand drawing and photography and are yet another way in which the author invites us to be attentive to geometric forms around us. The index is adequate, though the bibliography is somewhat short given the breadth of the content. Nonetheless, this is minor, since more on any topic can be readily obtained via an internet search.

 

The book's most distinctive and attractive aspects-- its wealth of examples and illustrations--make it a good supplement for geometry courses, providing inspiration for projects and reports. If used as a primary course text, the terse exposition and required geometric reasoning may be challenging. There are no problem sets, but the geometric arguments needed in the examples make them tantamount to a collection of open-ended problems. As with a fine museum, there is something for everyone: young readers can delight in the many beautiful illustrations or exercise their geometric imagination to check the author's claims while those with technical training can supplement this reasoning with analytical arguments. I would encourage readers to read just a few sections at a time, and then to search for similar geometric structures in their daily surroundings. They may be pleasantly surprised at how much they discover.

 

In summary, this book is a celebration of geometric form, and its cornucopia of examples an excellent resource for sharpening geometric reasoning and nurturing fascination with geometric form in all its guises. 

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Jer-Chin Chuang is an instructor in mathematics at the University of Illinois Urbana-Champaign.