Textbook

# Squaring the Circle: Geometry in Art and Architecture

For Instructors
Geometry is a dynamic branch of mathematics that also serves as a creative tool for engineers, artists, and architects. Squaring the Circle: Geometry in Art and Architecture includes all the topics necessary for a solid foundation in geometry and explores the timeless influence of geometry on art and architecture.  The text offers wide-ranging exercise sets and related projects that allow students to practice and master the mathematics presented.  Each chapter introduces mathematical concepts geometrically and illustrates their nontraditional applications in art and architecture throughout the centuries.  Appropriate for both basic mathematics courses and cross-discipline courses in mathematics and art, Squaring the Circle requires no previous mathematics.
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Preface.

To the Student.

To the Instructor.

Acknowledgments.

1. Music of the Spheres.

Pythagoras and the Pythagoreans.

PROFILE: PYTHAGORAS.

Ratio and Proportion.

Pythagoras and the Musical Ratios.

Sequences, Series, and Means.

Plato and the Musical Ratios.

PROFILE: PLATO.

The Musical Ratios in Architecture.

PROFILE: VITRUVIUS.

PROFILE: LEON BATTISTA ALBERTI.

SUMMARY.

2. The Golden Ratio.

The Golden Ratio.

PROFILE: EUCLID.

The Fibonacci Numbers and the Golden Ratio.

PROFILE: LEONARDO FIBONACCI.

The Golden Ratio in Architecture.

PROFILE: LE CORBUSIER.

SUMMARY.

3. The Triangle.

Angles Between Lines.

Triangles.

Right Triangles. Congruence, Symmetry, Isometry, and Similarity.

SUMMARY.

The Square.

The Rectangle.

The Rectangular Format in Painting.

PROFILE: GUSTAV THEODOR FECHNER.

PROFILE: JAY HAMBIDGE.

Architectural Proportions Based on the Square and the Rectangle.

The Earthly and the Mundane.

SUMMARY.

5. Polygons, Tilings, and Sacred Geometry.

Polygons.

Pentagons and Pentagrams.

PROFILE: ALBRECHT DÜRER.

Other Polygons.

Tilings with Polygons.

SUMMARY.

6. The Circle.

Geometry of the Circle.

Lines Intersecting a Circle.

The Round Format in Art.

PROFILE: THE DELLA ROBBIA FAMILY.

The Circle as a Symbol in Art.

SUMMARY.

7. Circular Designs in Architecture.

More Circle Geometry.

The Circle in Gothic Architecture.

Round Windows.

Circular Ornamentation in Walls and Pavements.

Arches and Vaults.

The Vesica.

SUMMARY.

8. Squaring the Circle.

The Geometry of Squaring the Circle.

Squaring the Circle in the Great Pyramid.

Other Examples from Art and Architecture.

SUMMARY.

9. The Ellipse and the Spiral.

The Ellipse.

PROFILE: APOLLONIUS OF PERGA.

Graphing the Ellipse.

PROFILE: RENÉ DESCARTES.

The Spiral.

The Spiral in Art and Architecture.

SUMMARY.

10. The Solids.

Surfaces and Solids.

The Polyhedra.

Cylinders and Cones.

Similar Solids.

The Platonic Solids.

PROFILE: ARCHIMEDES.

The Creation—by Geometry?

Polyhedra in the Renaissance.

PROFILE: PIERO DELLA FRANCESCA.

PROFILE: LUCA PACIOLI.

Polyhedra as Art Motifs.

SUMMARY.

11. The Sphere and Celestial Themes in Art and Architecture.

The Sphere.

The Dome.

The Terrestrial Sphere.

The Celestial Sphere.

PROFILE: JOHANNES KEPLER.

Time.

The Sphere and the Heavens as Art Motifs.

PROFILE: MAURITS CORNELIUS ESCHER.

PROFILE: DANTE ALIGHIERI.

PROFILE: GALILEO GALILEI.

SUMMARY.

12. Brunelleschi's Peepshow and the Origins of Perspective.

What Is Perspective?

A Perspective Primer.

Paintings Before Perspective.

Setting the Stage.

Brunelleschi and Alberti.

PROFILE: FILIPPO BRUNELLESCHI.

Three Renaissance Artists.

PROFILE: DONATELLO.

PROFILE: MASACCIO.

PROFILE: ANDREA MANTEGNA.

Later Developments.

PROFILE: LEONARDO DA VINCI.

SUMMARY.

13. Fractals.

Dimension.

PROFILE: BENOIT B. MANDELBROT.

Self-Similarity.

How to Create a Fractal.

Fractal-Related Art.

SUMMARY.

Appendices.

A The Art–Math Tourist.

B Biographical Profiles.

C Listings of Constructions by Chapter.

D Index to Geometric Signs and Symbols.

E Number Symbolism and Famous Groupings in Art.

F Summary of Facts and Formulas.

G Bibliography.

Index.

Photo Credits.

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Paul A. Calter is a Visiting Scholar at Dartmouth College and Professor Emeritus of Mathematics at Vermont Technical College. He is a book review editor of the Nexus Network Journal and has interests in both the fields of mathematics and art. He received his B.S. from Cooper Union and his M.S. from Columbia University, both in engineering, and his Masters of Fine Arts Degree from Norwich University. Calter has taught mathematics for over twenty-five years and is the author of ten mathematics textbooks and a mystery novel. He has been an active painter and sculptor since 1968, has had many solo shows and participated in dozens of group art shows, and has permanent outdoor sculptures at a number of locations. Calter developed a course called “Geometry in Art & Architecture,” which he has taught at Dartmouth College and Vermont Technical College, and he has taught at Dartmouth College and Vermont Technical College, and he has given workshops and lectures on the subject. Calter's own art is concerned with astronomical and geometric themes; he searches for a link between the organic and geometric basis of beauty, what has been called the philosopher's stone of aesthetics.
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• Fully worked Examples and accompanying practice Exercises included in each chapter
• End of chapter Exercises and Projects containing a wide variety of problems, ranging from skill and drill to collaborative learning to mathematical challenges
• Emphasizes constructions, encouraging students to develop geometric skills using Euclid’s tools—the compass and straightedge
• Illustrations engage students who are visual learners while extensive definitions and examples appeal to students who are more mathematically oriented
• Since progress has been made since Euclid’s time, constructions can also be made using The Geometer’s Sketchpad® or other commonly used geometry software*
• Biographical profiles spotlight historical figures in mathematics, art, and architecture
• Rich appendix that contains indexes to historical figures and locations, geometric constructions and formulas, and a complete bibliography
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Instructors Resources
Wiley Instructor Companion Site
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Students Resources
Wiley Student Companion Site
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Purchase Options
Wiley E-Text
Squaring the Circle: Geometry in Art and Architecture
ISBN : 978-0-470-53034-4
\$39.50

Paperback
Squaring the Circle: Geometry in Art and Architecture
ISBN : 978-0-470-41212-1
554 pages