Introduction

   The Great Pyramid

   Music of the Spheres

   Number Symbolism

   Polygons and Tilings

   The Platonic Solids

   Roman Architecture

   Number Symbolism
      in the Middle Ages

   The Wheel of Fortune

   Celestial Themes in Art

   Origins of Perspective

   What Shape Frame?

   Piero della FrancescaArt

   LeonardoArt

   Façade measurement Art by Trigonometry

   Early Twentieth Century Art

   Dynamic symmetry &
      The Spiral

   The Geometric Art
      of M.C. Escher

   Later Twentieth Century
      Geometry Art

   Art and the Computer

   Chaos & Fractals

 Bibliography

  The Book

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• 1. Introduction

 

• 2. The Golden Ratio and Squaring the Circle in the Great Pyramid

Great Pyramid of Cheops

Introduction to the Triangle

Pythagorean theorem

Golden Ratio

Secrets of the Great Pyramid

Squaring the Circle

Perimeters and areas of squares and circles

The Rhind Mathematical Papyrus

Rope stretchers Triangle

Summary: The Importance of Egyptian Art

 

• 3. Music of the Spheres

Pythagoras

The Pythagoreans

Music of the Spheres

 

• 4. Number Symbolism

Elementary number symbolism

Zodiac number symbolism

Number symbolism in the Old Testament

Pythagorean number symbolism

Modern number terminology

 

• 5. Polygons and Tilings, and Sacred Geometry

The Regular Polygons

The Equilateral Triangle

Tilings

Hexagon & Hexagram

Pentagram & Pentagon

Golden triangle

 

• 6. The Platonic Solids

Plato (427-347 B.C.)

The Timaeus

Polyhedra

Polyhedra in Art & Architecture

 

• 7. Ad Quadratum, The Sacred Cut, and Roman Architecture

Vitruvius

The Square

Ad quadratum

The Sacred Cut

Pompeii and Herculaneum

Garden Houses of Ostia

Roman Rectangle

 

• 8. Number, Weight, and Measure: the Medieval Cathedral

Biblical, classical, & Islamic influences: Augustine, Hildegard von Bingen , Aquinas, Cabala, Gematria

Symbolism of numbers and geometric figures

Four & the Square: The Cross, Tree of Life, Herms, pillars

Five & the Pentagon

Six & the Hexagon

Seven & the Heptagon

Eight & the Octagon: The sacred, S. Giovanni Castel Del Monte

 

• 9. The Wheel of Fortune and the Rose Window

Symbolism of the circle: Celestial, Ring, Wheel, Rainbow, Crescent, Calendar circles, Stonehenge

Dante's circles of Heaven and Hell

Geometry of the circle

The circle and Gothic architecture: The Gothic style, The Masons, The Gothic Master Plan,

Catherine Window, Wheel Window, Rose window, Rosettes, pavements, intarsia designs, Gothic arch, Gothic tracery

Mandorla, Symbolism of the fish,

Squaring the Circle: Neolithic Stone Circles, Mandala, Sacred cut, Hindu temple layout, Yin yang symbol

 

• 10. Celestial Themes in Art

The Creation, God the Geometer

The Ptloemaic universe, the Sphaera mundi, the structuring of Celestial Space

Nine Ranks of Angels

Dante's nine circles of heaven

Cigoli's Immacolata

Portrayals of Astronomers and allegories to astronomy

Models of the cosmos: Maps, globes, armillary spheres, orreries, domes

Astonomical instruments: Sundials, sun calendars, stone circles

Twentieth-Century astronomical art

Sun symbols

Temple orientation, Pyramid placement

Zodiac

Radian measure, arc length, rotation, latitude and longitude, The Conic Sections, Elliptical orbits

 

• 11. Brunelleschi's Peepshow and the Origins of Perspective

What is perspective?

Brunelleschi

Donatello

Masaccio

Mantegna

Alberti

From Euclid to Desargues

Significance of perspective

Later developments in perspective: Perspective Machines and Cameras

 

• 12. What Shape Frame?

Why Frame a Picture?

What Shape Frame?

The Quadro or Rectangular Format

The Square Format

The Tondo or Round Format

The Roundel

Ovato Tondo and the Elliptical Format

Frequency Distributions, Mean, Standard Deviation

 

• 13. Piero della Francesca (1416-1492)

Mathematical education in the Renaissance: Proportion and the Rule of Three, Gauging of solids

Piero's math books

Piero's paintings and geometry

 

• 14. Leonardo (1452-1519)

The Ultimate Renaissance Man

Leonardo's mathematics

Leonardo's Writings

Perspective

Chaos and Fractals

 

• 15. Façade measurement by Trigonometry

Historical background

The method

Trial at VTC

The Torre Bernarda

Right triangle trigonometry, law of sines, probability

 

• 16. Relativity, the Fourth Dimension, and Early Twentieth Century Art

Imact of the new geometries, the fourth dimension, and physics on art:

Lobachevski (1793-1856), Bolyai (1802-1860), Reimann (1826-1866), Einstein (1879-1955)

Picasso, Braque, Mondrian, Malevich

Abstraction, Post-impressionism, Cubism, Constructivism , Suprematism, De Stijl

 

• 17. Dynamic symmetry

Jay Hambidge, Irma Richter

The Golden Numberists

Irrational numbers, Logarithmic spiral, Ratio and proportion

 

• 18. The Geometric Art of M.C. Escher

Portrayal of Infinity

Recursion and feedback

Tilings

Reflective transformations

Symmetry

 

• 19. Geometry & Architecture in our Century

Buckminster Fuller, Le Corbusier,

Geodesic domes

ratio and proportion, polyhedra

 

• 20. Art and the Computer

The Artist-mathematician

The Artist-Mathematician-Hacker

The Computer in Art

 

• 21. Chaos & Fractals

Real landscapes

Mandelbrot

Self-similarity

Fractal Art

Randomness in Art