Squaring the Circle: Geometry in Art and Architecture

Preface. To the Student. To the Instructor. Acknowledgments. About the Author. 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. PROFILE: ANDREA PALLADIO. 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. 4. Ad Quadratum and the Sacred Cut. The Square. The Rectangle. Other Quadrilaterals. 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 DURER. Other Polygons. Tilings with Polygons. SUMMARY. 6. The Circle. Geometry of the Circle. Radians. 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: RENE 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. H Selected Answers. Index. Photo Credits.