Squaring the Circle: Geometry in Art and Architecture

Preface viii

To the Student xv

To the Instructor xvii

Acknowledgments xix

About the Author xx

1 Music of the Spheres 3

Pythagoras and the Pythagoreans 4

Profile: Pythagoras 4

Ratio and Proportion 8

Pythagoras and the Musical Ratios 14

Sequences, Series, and Means 16

Plato and the Musical Ratios 20

Profile: Plato 21

The Musical Ratios in Architecture 25

Profile: Vitruvius 26

Profile: Leon Battista Alberti 28

Profile: Andrea Palladio 30

Summary 32

2 The Golden Ratio 39

The Golden Ratio 40

Profile: Euclid 41

The Fibonacci Numbers and the Golden Ratio 46

Profile: Leonardo Fibonacci 47

The Golden Ratio in Architecture 49

Profile: Le Corbusier 55

Summary 57

3 The Triangle 63

Angles Between Lines 64

Triangles 67

Right Triangles 74

Congruence, Symmetry, Isometry, and Similarity 78

Summary 84

4 Ad Quadratum and the Sacred Cut 93

The Square 94

The Rectangle 98

Other Quadrilaterals 101

The Rectangular Format in Painting 103

Profile: Gustav Theodor Fechner 106

Profile: Jay Hambidge 109

Architectural Proportions Based on the Square and the Rectangle 110

The Earthly and the Mundane 120

Summary 124

5 Polygons, Tilings, and Sacred Geometry 131

Polygons 132

Pentagons and Pentagrams 136

Profile: Albrecht Dürer 136

Other Polygons 142

Tilings with Polygons 153

Summary 160

6 The Circle 167

Geometry of the Circle 168

Radians 172

Lines Intersecting a Circle 175

The Round Format in Art 178

Profile: The Della Robbia Family 179

The Circle as a Symbol in Art 181

Summary 187

7 Circular Designs in Architecture 195

More Circle Geometry 196

The Circle in Gothic Architecture 201

Round Windows 206

Circular Ornamentation in Walls and Pavements 208

Arches and Vaults 212

The Vesica 217

Summary 219

8 Squaring the Circle 229

The Geometry of Squaring the Circle 230

Squaring the Circle in the Great Pyramid 234

Other Examples from Art and Architecture 235

Summary 241

9 The Ellipse and the Spiral 247

The Ellipse 248

Profile: Apollonius of Perga 249

Graphing the Ellipse 254

Profile: René Descartes 255

The Spiral 260

The Spiral in Art and Architecture 266

Summary 270

10 The Solids 281

Surfaces and Solids 282

The Polyhedra 283

Cylinders and Cones 289

Similar Solids 292

The Platonic Solids 295

Profile: Archimedes 300

The Creation—by Geometry? 302

Polyhedra in the Renaissance 304

Profile: Piero Della Francesca 305

Profile: Luca Pacioli 307

Polyhedra as Art Motifs 308

Summary 312

11 The Sphere and Celestial Themes in Art and Architecture 319

The Sphere 320

The Dome 323

The Terrestrial Sphere 328

The Celestial Sphere 333

Profile: Johannes Kepler 336

Time 336

The Sphere and the Heavens as Art Motifs 343

Profile: Maurits Cornelius Escher 348

Profile: Dante Alighieri 350

Profile: Galileo Galilei 353

Summary 355

12 Brunelleschi’s Peepshow and the Origins of Perspective 363

What Is Perspective? 364

A Perspective Primer 366

Paintings Before Perspective 371

Setting the Stage 373

Brunelleschi and Alberti 375

Profile: Filippo Brunelleschi 376

Three Renaissance Artists 378

Profile: Donatello 379

Profile: Masaccio 380

Profile: Andrea Mantegna 382

Later Developments 384

Profile: Leonardo Da Vinci 386

Summary 388

13 Fractals 393

Dimension 394

Profile: Benoit B. Mandelbrot 396

Self-Similarity 400

How to Create a Fractal 402

Fractal-Related Art 412

Summary 415

Appendices 421

A The Art–Math Tourist 421

B Biographical Profiles 425

C Listings of Constructions by Chapter 427

D Index to Geometric Signs and Symbols 433

E Number Symbolism and Famous Groupings in Art 435

F Summary of Facts and Formulas 467

G Bibliography 481

H Selected Answers 491

Index 499

Photo Credits 523