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Proportion

Homage to Pythagoras: Rediscovering Sacred Science

Homage to Pythagoras: Rediscovering Sacred Science

By ed. by Christopher Bamford

This group of essays emerged from several ‘Schools of Sacred Architecture’ held under the auspices of the Lindisfarne Association in the early 1980’s. Keith Critchlow’s essays cover the intellectual, philosophical background of the architectural tradition while Robert Lawlor’s view into the mindset of ancient sacred tradition is profound and challenging.

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The Painter's Secret Geometry: A Study of Composition in Art

The Painter's Secret Geometry: A Study of Composition in Art

By Charles Bouleau

In this fascinating book, Charles Bouleau extracts the mathematical and geometric forms from images of classical traditions. The book seeks to explain that the two-dimensional arts were heavily influenced by math in a time when the visual, musical, and scientific arts were more intertwined. The book includes many preprints of the works of artistic masters with overlays of their mathematical means.

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Beautiful Necessity: Seven Essays on Theosophy and Architecture

Beautiful Necessity: Seven Essays on Theosophy and Architecture

By Claude Bragdon

This often cited reference is an early 20th century statement of an esoteric theory of beauty, and of what might be called the anthropic analogy in architecture. The book connects these spiritual ideas with proportional ideas in design. Many of the author’s personal line drawings are ‘classics’ that appear in later works on design.

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The Secrets of Ancient Geometry

The Secrets of Ancient Geometry

By Tons Brunes

The Secrets of Ancient Geometry reveals the standard geometrical patterns behind many of the ancient buildings of the world, from the Great Pyramids to the Cathedral of Notre Dame. In addition to the geometries in architecture, the geometries of other art and writing are explored. This is an interesting read for anyone that likes to find patterns and secrets in ancient things.

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Le Modulor & Le Modulor 2

Le Modulor & Le Modulor 2

By Le Corbusier

Architect Le Corbusier (1897-1965) published two books, the first in 1948 and second in 1955, in which he developed a system of proportion based on the Golden Section, Fibonacci numbers and the proportion of the human body. Corbusier believed that these measurements and patterns governed the rhythm of the universe and as such they should govern architecture. This idea put Corbusier in line with a long tradition of architects stretching back to Leon Battista Alberti, Leonardo da Vinci, and Vitruvius. Both of the books are printed as they originally were, in English, and are placed in a book jacket displaying Le Modulor’s dimensions.

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Islamic Patterns

Islamic Patterns

By Keith Critchlow

Due to prohibition of the portrayal of human and animal forms in Islamic law, Muslim artist had to find a different way to express themselves and inspire through art. This was done through intricate patterns based on simple geometries that were then made complicated. Behind this mathematical kind of art, however, there lied symbolic, cosmological, and religious meaning. In this book Keith Critchlow, director of Visual Islamic and Traditional Art at the Prince of Wales Institute of Architecture in England, explains and displays the complex geometries of Islamic art.

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The Power of Limits

The Power of Limits

By Gyorgy Doczy

In everyday life, little things are observed that suggest an underlying order – patterns in insect wings, shells, fruits and vegetables, and other things. In The Power of Limits, Gyorgy Doczy investigates and records the various patterns he encountered in the natural and built environment. Doczy takes this idea of order further to show how we as human beings are part of this celestial order and how it can be extended to our psychological and social orders. Gyorgy Doczy was an architect who practiced in Hungary, Sweden, Iran, and the United States.

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The Geometry of Art and Life

The Geometry of Art and Life

By Matila Ghyka

In The Geometry of Art and Life Matila Ghyka explains how nature is not random. All things are guided by concepts of ratio, proportion, geometries, the Golden Section, and other mathematical forms which give all things regularity and harmony. Ghyka transforms aesthetics into mathematics and in turn explains the reasons why things are deemed esthetically pleasing.

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Harmonies of Heaven and Earth

Harmonies of Heaven and Earth

By Joscelyn Godwin

Joscelyn Godwin, professor of the Colgate University Music Department, puts forth his thesis that the universe we inhabit is a harmonious one guided by divine and scientific laws of proportion. To support this idea Godwin presents musical theories starting with the harmony of the spheres of Pythagoras and continues to examine musical theory till the beginning of the 20th century. Godwin ends the book in a chapter with his own speculations on harmonies and musical theory. These high-minded theories may seem complicated to understand, but Godwin does a good job of breaking them down so that they are easy enough to comprehend.

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The Pythagorean Sourcebook

The Pythagorean Sourcebook

By Ken Guthrie

Pythagoras was one of the first great Greek philosophers. In this book writings about Pythagoras by his biographers as well as his students and contemporaries relate much of what is known about the ancient philosopher. This anthology is the largest collection of Pythagorean writing and deals with topics such as ethics, cosmology, geometry, mathematics, music, architecture, arts, logic, and others.

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Elements of Dynamic Symmetry

Elements of Dynamic Symmetry

By Jay Hambridge

Jay Hambridge (1867-1924) was an artist who, through his study of classical art and architecture, became convinced that the study of arithmetic and geometry was the basis on which all art of a classical nature derived. Elements of Dynamic Symmetry describes his theory that the beauty of Greek design was the result of mathematical theories of growth in nature and man, such as the Golden Ratio and the Fibonacci series. Habmridge asserts that rather than formulating a new theory he has recovered a lost technique.

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The Divine Proportion

The Divine Proportion

By H. E. Huntley

In this book professor Huntley attempts to relate mathematics to everyday life in the complex society we live in. Geometry is related to aesthetics using simple but powerful formulas and ideas like Pythagoras’ theorem. Dr. J. Bronowski of the Salk Institute writes, “it wanders here and there through some of the most attractive byways of simple mathematics, returning always to the oddities and pleasures of the Golden Section.” Huntley asks “why?’ in this book on mathematics and the universe; it contains 59 diagrams.

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The Theology of Arithmetic

The Theology of Arithmetic

By Iamblichus

Iamblichus’ (245-325) The Theology of Arithmetic is the longest work on number theory from the ancient world that still exists today. A follower of Neo-Platonist thought, Iamblichus sought to place symbolic meaning on numbers and expound on earlier philosophy originating with Pythagoras. The complicated theories, translated to English for the first time and presented here, are accompanied by footnotes, an extensive bibliography, glossary, and forward by Keith Critchlow.

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Sacred Geometry: Philosophy & Practice

Sacred Geometry: Philosophy & Practice

By Robert Lawlor

This is perhaps ‘the’ statement for our generation of spiritual principles as a background for design, specifically of proportion in architecture. Through a workbook style presentation of geometry, Lawlor builds up an integration of mind and body skills essential to approaching Beauty and Unity in design. This is a key work in the revitalization of a spiritual architecture.

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The Manual of Harmonics

The Manual of Harmonics

By Nicomachus, trans. by Levin

Nicomachus (c.60-c.120) was strongly influenced by the philosophy of Pythagoras. In The Manual of Harmonics, Nicomachus provides the third major surviving treatise on the theory of music from the Classical era. It is from this that Pythagoras’ discovery of pitch’s relationship to numbers is first recorded. Additionally, Nicomachus provides a discussion on the relationship of music and the order of the universe known as the “music of spheres.” This translation includes an introduction and additional commentary by translator Flora Levine in which she explains the principles of Pythagorean theories on harmony to make the reading clearer.

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Introduction to Arithmetic

Introduction to Arithmetic

By Nicomachus

Nicomachus (c.60-c.120), Pythagorean theorist, writes about number theory in this book. He stresses the significance of prime and perfect numbers while at the same time argues that all other mathematics derives from arithmetic.

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Timaeus

Timaeus

By Plato, trans by Jowett

In this dialogue by Plato, Socrates, after describing his ideal states the day before, goes on to discuss the interactions between two states – Athens and Atlantis. But before this discussion ensues, Timaeus is directed to tell the story of the origin of the universe and man. Timaeus relies heavily on geometry to explain cosmic realities, the creation of the universe, elements – based on five geometric shapes, the world, and the soul.

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Commentary on the First Book of Euclid's Elements

Commentary on the First Book of Euclid's Elements

By Proclus, trans. by G Morrow, Princeton U.

Proclus Lycaeus (412-485) was one of the last philosophers of the Neo-Platonist thought of the classical era, and his largely developed body of work went on to influence later philosophy of the Western and Islamic world. In this book, Proclus comments and expands on the theories of geometry set forth in Elements, by earlier philosopher and mathematician Euclid (fl. 300 BC). Through his commentary on Elements, Proclus sets forth his own theories on the science and mysticism of geometry. Additionally the book provides a complete overview of the changes in the philosophy of geometry during the Classical period.

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A Beginner’s Guide to Constructing the Universe

A Beginner’s Guide to Constructing the Universe

By Michael S. Schneider

This book makes available in the most direct and disarming way the sometimes difficult ideas of the ancient Platonic Quadrivium. The author has a background in primary education and the book is presented with the clarity of a teacher’s lesson plan. Each section is integrated with analysis of architectural and art objects across the historical and cultural spectrum.

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Theoretic Arithmetic of the Pythagoreans

Theoretic Arithmetic of the Pythagoreans

By Thomas Taylor

The arithmetic of Pythagorean thought is examined in this work originally published in 1816. When released, this book expounded upon Pythagorean thought in ways that have not been explored before.

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Math Useful for Understanding Plato

Math Useful for Understanding Plato

By Theon, trans by Lawlor

Theon of Smyrna explains the great influence mathematics played in Plato’s philosophy, Pythagoras’ philosophy, and other philosophical knowledge available in the first century. This book is more of a compilation of philosophical and mathematical thought at the time as opposed to the presentation of new ideas. In this interesting mélange of topics, Theon describes numbers, instrumental music, harmony – or music of numbers or proportion, and astronomy.

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