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The Geometry of Art and Life


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CHAPTER I PROPORTION IN SPACE AND TIME Ratio and proportion. "The simplest asymmetrical division of a measurable whole into two parts, and Ockham's principle of economy." Generalisation of the concept of proportion. "Arithmetical, geometrical and harmonic proportion." The ten types of proportion. "Proportion, symmetry, eurhythmy." Rhythm in space and proportion in time. CHAPTER II THE GOLDEN SECTION Algebraical and geometrical properties of the Golden Section or Number F. The Fibonacci Series and the Golden Section. The F Rectangle. "Phyllotaxis and "Ideal Angle" in botany." The Golden Section and pentagonal symmetry. CHAPTER III GEOMETRICAL SHAPES ON THE PLANE Polygons. Regular polygons and star-polygons. "Remarkable triangles: Triangle of Pythagoras, Triangle of Price, "Sublime" Triangle." Rectangles: the F Rectangle and the vF Rectangle. "Pentagon, pentagram, decagon, and Golden Section." Hexagon and oxtagon. CHAPTER IV GEOMETRICAL SHAPES IN SPACE The five regular polyhedra or Platonic bodies. The thirteen semiregular Archimedian bodies. Regular prisms and anti-prisms. The two continuous star-dodecahedra of Kepler. "The dodecahedron, the icosahedron and the Golden Section." Other remarkable volumes. "The "Chamber of the King" in the Great Pyramid." "The Great Pyramid, star-dodecahedron, and the human body." Regular hypersolids in the fourth dimension. CHAPTER V THE REGULAR PARTITIONS ON THE PLANE AND IN SPACE "Equipartitions and partitions of the plane, regular and semi-regular." Equipartitions and partitions of space. Crystal lattices. Hexagonal and cubic symmetries. The cuboctahedron and the close-packing of spheres. "The principle of least action, most general law for inorganic systems." CHAPTER VI THE GEOMETRY OF LIFE Harmonious growth and logarithmic spiral. Pentagonal symmetry in living organisms. "Pentagon, F Spiral and Golden Section." Flowers and shells. The human body and the F Progression. CHAPTER VII THE TRANSMISSION OF GEOMETRICAL SYMBOLS AND PLANS Life and Legend of Pythagoras. Pythagorean number-mystic. "The Pythagorean tradition, the pentagram, decad and tetrakytus." Neo-Pythagorism and Kabbala. From the antique builders' guilds to the masons' guilds of the Middle Ages. Masons' marks and fundamental design. Masonic traditions and symbols. CHAPTER VIII GREEK AND GOTHIC CANONS OF PROPORTION Rediscovery of the Greek and Gothic canons of architecture. Proportions and dynamic symmetry. The dynamic rectangles of Hambidge and the directing circles of Moessel. "Greek vases, Greek temples, and the human body." Gothic master plans. CHAPTER IX SYMPHONIC COMPOSITION Periodic rediscovery of the Golden Section. Seurat's divisionism. Revival of Pythagorean doctrine in science and art. "Modern applications of dynamic symmetry in architecture, painting, and decorative art." Symphonic composition.

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