Rough Notes:

Rough Notes:

Sacred architecture

From Wikipedia, the free encyclopedia
 
 

 

Example of early Islamic sacred architecture : the Great Mosque of Kairouan, founded in 670, dates in its present state from the 9th century;[1] it is the architectural ancestor of all the mosques in the eastern Islamic world and represents one of the best preserved and most significant examples of early great mosques.[2] The Great Mosque of Kairouan (also called the Mosque of Uqba) is located in the UNESCO World Heritage city of Kairouan in Tunisia.

 

Larger-than-life structures remain at the ancient Egyptian Luxor Temple approximately 3400 years after it was built.

Sacred architecture (also known as religious architecture) is a religious architectural practice concerned with the design and construction of places of worship or sacred or intentional space, such as churchesmosquesstupassynagogues, and temples. Many cultures devoted considerable resources to their sacred architecture and places of worship. Religious and sacred spaces are amongst the most impressive and permanent monolithic buildings created by humanity. Conversely, sacred architecture as a locale for meta-intimacy may also be non-monolithic, ephemeral and intensely private, personal and non-public.

Sacred, religious and holy structures often evolved over centuries and were the largest buildings in the world, prior to the modern skyscraper. While the various styles employed in sacred architecture sometimes reflected trends in other structures, these styles also remained unique from the contemporary architecture used in other structures. With the rise of Abrahamic monotheisms (particularly Christianity and Islam), religious buildings increasingly became centres of worshipprayer and meditation.

The Western scholarly discipline of the history of architecture itself closely follows the history of religious architecture from ancient times until the Baroque period, at least. Sacred geometryiconography, and the use of sophisticated semiotics such as signs, symbols and religious motifs are endemic to sacred architecture.

Spiritual aspects of religious architecture[edit]

 

An illustrated layout of the traditional interior of a ChristianOrthodox church.

Sacred or religious architecture is sometimes called sacred space.

Architect Norman L. Koonce has suggested that the goal of sacred architecture is to make "transparent the boundary between matter and mind, flesh and the spirit." In discussing sacred architecture, Protestant minister Robert Schuller suggested that "to be psychologically healthy, human beings need to experience their natural setting—the setting we were designed for, which is the garden." Meanwhile, Richard Kieckhefer suggests that entering into a religious building is a metaphor for entering into spiritual relationship. Kieckhefer suggests that sacred space can be analyzed by three factors affecting spiritual process: longitudinal space emphasizes the procession and return of sacramental acts, auditorium space is suggestive of proclamation and response, and new forms of communal space designed for gathering and return depend to a great degree on minimized scale to enhance intimacy and participation in worship.

Ancient architecture[edit]

 

The interior of the ancient Egyptian Karnak Temple.

Sacred architecture spans a number of ancient architectural styles including Neolithic architectureancient Egyptian architecture and Sumerian architecture. Ancient religious buildings, particularly temples, were often viewed as the dwelling place, the temenos, of the gods and were used as the site of various kinds of sacrifice. Ancient tombs and burial structures are also examples of architectural structures reflecting religious beliefs of their various societies. The Temple of Karnak at Thebes, Egypt was constructed across a period of 1300 years and its numerous temples comprise what may be the largest religious structure ever built. Ancient Egyptian religious architecture has fascinated archaeologists and captured the public imagination for millennia.

Classical architecture[edit]

 

The Parthenon in AthensGreece.

Around 600 BCE the wooden columns of the Temple of Hera at Olympia were replaced by stone columns. With the spread of this process to other sanctuary structures a few stone buildings have survived through the ages. Greek architecture preceded Hellenistic and Roman periods (Roman architecture heavily copied Greek). Since temples are the only buildings which survive in numbers, most of our concept of classical architecture is based on religious structures. The Parthenon which served as a treasury building as well as a place for veneration of deity, is widely regarded as the greatest example of classical architecture.

Indian architecture[edit]

 

Vishvakarman, Divine Architect of Vedas in a modern Hindu representation: note Chhatra.

Indian architecture is related to the history and religions of the time periods as well as to the geography and geology of the Indian subcontinent. India was crisscrossed by trading routes of merchants from as far away as Siraf and China as well as weathering invasions by foreigners, resulting in multiple influences of foreign elements on native styles. The diversity of Indian culture is represented in its architecture. Indian architecture comprises a blend of ancient and varied native traditions, with building types, forms and technologies from WestCentral Asia, and Europe.

Buddhism[edit]

Buddhist architecture developed in South Asia beginning in the third century BCE. Two types of structures are associated with early Buddhismviharas and stupas. Originally, Viharas were temporary shelters used by wandering monks during the rainy season, but these structures later developed to accommodate the growing and increasingly formalized Buddhist monasticism. An existing example is at Nalanda (Bihar).

The initial function of the stupa was the veneration and safe-guarding of the relics of the Buddha. The earliest existing example of a stupa is in Sanchi (Madhya Pradesh). In accordance with changes in religious practice, stupas were gradually incorporated into chaitya-grihas (stupa halls). These reached their highpoint in the first century BCE, exemplified by the cave complexes of Ajanta and Ellora (Maharashtra).

The pagoda is an evolution of the Indian stupa that is marked by a tiered tower with multiple eaves common in China, Japan, Korea, Nepal and other parts of Asia. Buddhist temples were developed rather later and outside South Asia, where Buddhism gradually declined from the early centuries CE onwards, though an early example is that of the Mahabodhi Temple at Bodh Gaya in Bihar. The architectural structure of the stupa spread across Asia, taking on many diverse forms as details specific to different regions were incorporated into the overall design. It was spread to China and the Asian region by Araniko, a Nepali architect in the early 13th century for Kublai Khan.

Hinduism[edit]

 

The Shikhara of the Raghunath Temple at JammuIndia is built in the Nagara style of temple architecture.

 

The Brihadeeswarar Temple, Tanjavur, built in the Dravida style

Hindu temple architecture is based on Sthapatya Veda and many other ancient religious texts like the Brihat SamhitaVastu Shastra and Shilpa Shastras in accordance to the design principles and guidelines believed to have been laid by the divine architect Vishvakarma. It evolved over a period of more than 2000 years. The Hindu architecture conforms to strict religious models that incorporate elements of astronomy and sacred geometry. In Hindu belief, the temple represents the macrocosm of the universe as well as the microcosm of inner space. While the underlying form of Hindu temple architecture follows strict traditions, considerable variation occurs with the often intense decorative embellishments and ornamentation.

A basic Hindu temple consists of an inner sanctum, the garbhagriha or womb-chamber, a congregation hall, and possibly an antechamber and porch. The sanctum is crowned by a tower-like shikhara. The Hindu temple represents Mount Meru, the axis of the universe. There are strict rules which describe the themes and sculptures on the outer walls of the temple buildings.

The two primary styles that have developed are the Nagara style of Northern India and the Dravida style of Southern India. A prominent difference between the two styles are the elaborate gateways employed in the South. They are also easily distinguishable by the shape and decoration of their shikharas. The Nagara style is beehive-shaped while the Dravida style is pyramid-shaped.

Byzantine architecture[edit]

 

Hagia Sophia, the Church of Holy Wisdom

 

The 6th-century Kariye Camii, Istanbul, is now a museum.

 

The ruins of Menüçehr Camii near Kars, Turkey, believed to be the oldest Seljuk mosque in Anatolia

Byzantine architecture evolved from Roman architecture. Eventually, a style emerged incorporating Near East influences and the Greek cross plan for church design. In addition, brick replaced stone, classical order was less strictly observed, mosaics replaced carved decoration, and complex domes were erected. One of the great breakthroughs in the history of Western architecture occurred when Justinian's architects invented a complex system providing for a smooth transition from a square plan of the church to a circular dome (or domes) by means of squinches or pendentives. The prime example of early Byzantine religious architecture is the Hagia Sophia in Istanbul.

Islam[edit]

Byzantine architecture had a great influence on early Islamic architecture with its characteristic round arches, vaults and domes. Many forms of mosques have evolved in different regions of the Islamic world. Notable mosque types include the early Abbasid mosques, T-type mosques, and the central-dome mosques of Anatolia.

 

Interior of the Mezquita, a hypostyle mosque with columns arranged in grid pattern, in CórdobaSpain

The earliest styles in Islamic architecture produced Arab-plan or hypostyle mosques during the Umayyad Dynasty. These mosques follow a square or rectangular plan with enclosed courtyard and covered prayer hall. Most early hypostyle mosques had flat prayer hall roofs, which required numerous columns and supports.[3]The Mezquita in CórdobaSpain was constructed as a hypostyle mosque supported by over 850 columns.[4] Arab-plan mosques continued under the Abbasiddynasty.

The Ottomans introduced central dome mosques in the 15th century that have a large dome centered over the prayer hall. In addition to having one large dome at the center, there are often smaller domes that exist off-center over the prayer hall or throughout the rest of the mosque, in areas where prayer is not performed.[5] The Dome of the Rock mosque in Jerusalem is perhaps the best known example of a central dome mosque.

Iwan mosques are most notable for their domed chambers and iwans, which are vaulted spaces open out on one end. In iwan mosques, one or more iwans face a central courtyard that serves as the prayer hall. The style represents a borrowing from pre-Islamic Iranian architecture and has been used almost exclusively for mosques in Iran. Many iwan mosques are converted Zoroastrian fire temples where the courtyard was used to house the sacred fire.[3] Today, iwan mosques are no longer built.[5] The Shah Mosque in IsfahanIran is a classic example of an iwan mosque.

 

View of the square three-tiered minaret of the Mosque of Uqba (Great Mosque of Kairouan); this mosque, founded in 670, is one of the most impressive mosques in North Africa, situated in KairouanTunisia

A common feature in mosques is the minaret, the tall, slender tower that usually is situated at one of the corners of the mosque structure. The top of the minaret is always the highest point in mosques that have one, and often the highest point in the immediate area. The first mosques had no minarets, and even nowadays the most conservative Islamic movements, like Wahhabis, avoid building minarets, seeing them as ostentatious and unnecessary. The first minaret was constructed in 665 in Basra during the reign of the Umayyad caliph Muawiyah I. Muawiyah encouraged the construction of minarets, as they were supposed to bring mosques on par with Christian churches with their bell towers. Consequently, mosque architects borrowed the shape of the bell tower for their minarets, which were used for essentially the same purpose — calling the faithful to prayer.[6]

Domes have been a hallmark of Islamic architecture since the 7th century. As time progressed, the sizes of mosque domes grew, from occupying only a small part of the roof near the mihrabto encompassing all of the roof above the prayer hall. Although domes normally took on the shape of a hemisphere, the Mughals in India popularized onion-shaped domes in South Asia and Persia.[7]

 

The prayer hall, or musalla, in a Turkish mosque, with a minbar

 

People washing before prayer at the Badshahi mosque in LahorePakistan

The prayer hall, also known as the musalla, has no furniture; chairs and pews are absent from the prayer hall.[8] Prayer halls contain no images of people, animals, and spiritual figures although they may be decorated with Arabic calligraphy and verses from the Qur'an on the walls.

Usually opposite the entrance to the prayer hall is the qibla wall, which is the visually emphasized area inside the prayer hall. The qibla wall is normally set perpendicular to a line leading to Mecca.[9] Congregants pray in rows parallel to the qibla wall and thus arrange themselves so they face Mecca. In the qiblawall, usually at its center, is the mihrab, a niche or depression indicating the qibla wall. Usually the mihrab is not occupied by furniture either. Sometimes, especially during Friday prayers, a raised minbar or pulpit is located to the side of the mihrab for a khatib or some other speaker to offer a sermon (khutbah). The mihrab serves as the location where the imam leads the five daily prayers on a regular basis.[10]

Mosques often have ablution fountains or other facilities for washing in their entryways or courtyards. However, worshippers at much smaller mosques often have to use restrooms to perform their ablutions. In traditional mosques, this function is often elaborated into a freestanding building in the center of a courtyard.[4] Modern mosques may have a variety of amenities available to their congregants and the community, such as health clinicslibraries and gymnasiums.

Medieval architecture[edit]

 

Norwegian stave church

The religious architecture of Christian churches in the Middle Ages featured the Latin cross plan, which takes the Roman Basilica as its primary model with subsequent developments. It consists of a navetransepts, and the altar stands at the east end (see Cathedral diagram). Also, cathedralsinfluenced or commissioned by Justinian employed the Byzantine style of domes and a Greek cross (resembling a plus sign), centering attention on the altar at the center of the church. The Church of the Intercession on the Nerl is an excellent example of Russian orthodox architecture in the Middle Ages. The Urnes stave church (Urnes stavkyrkje) in Norway is a superb example of a medieval stave church.

Gothic architecture[edit]

Gothic architecture was particularly associated with cathedrals and other churches, which flourished in Europe during the high and late medieval period. Beginning in 12th century France, it was known as "the French Style" during the period. The style originated at the abbey church of Saint-Denis in Saint-Denis, near Paris.[11] Other notable gothic religious structures include Notre Dame de Paris, the Cathedral of Our Lady of Amiens, and the Chartres Cathedral.

Renaissance architecture[edit]

 

The Basilica of Saint Peter, Rome

The Renaissance brought a return of classical influence and a new emphasis on rational clarity. Renaissance architecture represents a conscious revival of Roman Architecture with its symmetry, mathematical proportions, and geometric order. Filippo Brunelleschi's plan for the Santa Maria del Fiore as the dome of the Florence Cathedral in 1418 was one of the first important religious architectural designs of the Italian renaissance.

Baroque architecture[edit]

 

Baroque façade of Santa Susanna, by Carlo Maderno

Evolving from the renaissance style, the baroque style was most notably experienced in religious art and architecture. Most architectural historians regard Michelangelo's design of St. Peter's Basilica in Rome as a precursor to the Baroque style. Baroque style can be recognized by broader interior spaces (replacing long narrow naves), more playful attention to light and shadow, extensive ornamentation, large frescoes, focus on interior art, and frequently, a dramatic central exterior projection. The most important early example of the baroque period was the Santa Susanna by Carlo MadernoSaint Paul's Cathedral in London by Christopher Wren is regarded as the prime example of the rather late influence of the Baroque style in England.

Mormon temples[edit]

Temples of The Church of Jesus Christ of Latter-day Saints offer a unique look at design as it has changed from the simple church like structure of the Kirtland Temple built in the 1830s, to the castellated Gothic styles of the early Utah temples, to the dozens of modern temples built today. Early temples, and some modern temples, have a priesthood assembly room with two sets of pulpits at each end of the room, with chairs or benches that can be altered to face either way. Most, but not all temples have the recognizable statue of the Angel Moroni atop a spire. The Nauvoo Temple and the Salt Lake Temple are adorned with symbolic stonework, representing various aspects of the faith.

Modern and post-modern architectures[edit]

 

Community of Christ Temple in Independence, Missouri, USA is postmodern in design.

Modern architecture spans several styles with similar characteristics resulting in simplification of form and the elimination of ornament. The most influential modernist architects in the early to mid 20th century include Dominikus BöhmRudolf Schwarz, and Auguste Perret.[12] While secular structures clearly had the greater influence on the development of modern architecture, several excellent examples of modern architecture can be found in religious buildings of the 20th century. For example, Unity Temple in Chicago is a Unitarian Universalist congregation designed by Frank Lloyd Wright. The United States Air Force Academy Cadet Chapel, started in 1954 and completed in 1962, was designed by Walter Netsch and is an excellent example of modern religious architecture. It has been described as a "phalanx of fighters" turned on their tails and pointing heavenward. In 1967, Architect Pietro Belluschi designed the strikingly modern Cathedral of St. Mary of the Assumption (San Francisco), the first Catholic cathedral in the United States intended to conform to Vatican II.

Post-modern architecture may be described by unapologetically diverse aesthetics where styles collide, form exists for its own sake, and new ways of viewing familiar styles and space abound. The Temple at Independence, Missouri was conceived by Japanese architect Gyo Obata after the concept of the chambered nautilus. The Catholic Cathedral of Our Lady of the Angels (Los Angeles) was designed in 1998 by Jose Rafael Moneo in a post-modern style. The structure evokes the area's Hispanic heritage through the use of adobe coloring while combining stark modern form with some traditional elements. The Basilica of Our Lady of Licheń on the other hand is a much more traditional edifice. Designed by Barbara Bielecka and built between 1994 and 2004, its form includes references to a number of Polish structures. The columns possess a slenderness and delicacy inspired by those of the Renaissance court of Wawel Castle in Kraków, while the huge 420-foot spire that will be erected next to the basilica bears more than an accidental resemblance to the Baroque spire that adorns the Jasna Gora monastery of Czestochowa, home of the Black Madonna.

Shaker communities[edit]

Shaker communities were semiotically architectured upon the crux of the compass rose.[citation needed]

See also[edit]

 

The Temple of All Religions in KazanRussia, combining various forms of religious architecture

Notes[edit]

  1. Jump up^ Great Mosque of Kairouan (Qantara Mediterranean Heritage)
  2. Jump up^ John Stothoff Badeau and John Richard Hayes, The Genius of Arab civilization: source of Renaissance. Taylor & Francis. 1983. p. 104
  3. Jump up to:a b Hillenbrand, R. "Masdjid. I. In the central Islamic lands". In P.J. Bearman; Th. Bianquis; C.E. Bosworth; E. van Donzel; W.P. Heinrichs. Encyclopaedia of Islam Online. Brill Academic Publishers. ISSN 1573-3912.
  4. Jump up to:a b "Religious Architecture and Islamic Cultures". Massachusetts Institute of Technology. Retrieved 2006-04-09.
  5. Jump up to:a b "Vocabulary of Islamic Architecture". Massachusetts Institute of Technology. Archived from the original on 2005-11-24. Retrieved 2006-04-09.
  6. Jump up^ Hillenbrand, R. "Manara, Manar". In P.J. Bearman; Th. Bianquis; C.E. Bosworth; E. van Donzel; W.P. Heinrichs. Encyclopaedia of Islam Online. Brill Academic Publishers.
  7. Jump up^ Asher, Catherine B. (1992-09-24). "Aurangzeb and the Islamization of the Mughal style". Architecture of Mughal India. Cambridge University Press. p. 256.
  8. Jump up^ "Mosque FAQ". The University of Tulsa. Archived from the original on December 30, 2004. Retrieved 2006-04-09.
  9. Jump up^ Bierman, Irene A. (1998-12-16). Writing Signs: Fatimid Public Text. University of California Press. p. 150.
  10. Jump up^ "Terms 1: Mosque". University of Tokyo Institute of Oriental Culture. Retrieved 2006-04-09.
  11. Jump up^ Andrzej Piotrowski (2011), Architecture of Thought, U of Minnesota Press, p. 23
  12. Jump up^ Steven J. SchloederArchitecture in Communion: Implementing the Second Vatican Council through Liturgy and Architecture. (Ignatius Press: 1998): 23-24 and 234-38. ISBN 0-89870-631-9.

References[edit]

  • Jeanne Halgren Kilde, When Church Became Theatre: The Transformation of Evangelical Church Architecture and Worship in Nineteenth-Century America. (Oxford University Press:2002). ISBN
  • Michael E. DeSanctis, Building from Belief: Advance, Retreat, and Compromise in the Remaking of Catholic Church Architecture.. (Liturgical Press:2002). ISBN
  • Richard Kieckhefer, Theology in Stone: Church Architecture from Byzantium to Berkeley. (Oxford University Press, USA: 2004). ISBN
  • Anne C. Loveland and Otis B. Wheeler, From Meetinghouse to Megachurch: A Material and Cultural History. (University of Missouri Press:2003). ISBN
  • Michael S. Rose, Ugly as Sin: Why They Changed Our Churches from Sacred Places to Meeting Spaces -- and How We Can Change Them Back Again. (Sophia Institute Press: 2001). ISBN
  • Steven J. SchloederArchitecture in Communion: Implementing the Second Vatican Council through Liturgy and Architecture. (Ignatius Press: 1998). ISBN 0-89870-631-9.
  • R. Kevin Seasoltz, A Sense Of The Sacred: Theological Foundations Of Christian Architecture And Art. (Continuum International Publishing Group: 2005) ISBN

Further reading[edit]

  • Bain, George. Celtic Art: The Methods of Construction. Dover, 1973. ISBN 0-486-22923-8.
  • Bamford, Christopher, Homage to Pythagoras: Rediscovering Sacred Science, Lindisfarne Press, 1994, ISBN 0-940262-63-0
  • Critchlow, Keith (1970). Order In Space: A Design Source Book. Viking.
  • Critchlow, Keith (1976). Islamic Patterns: An Analytical and Cosmological Approach. Schocken Books. ISBN 0-8052-3627-9.
  • Hill, Marsha (2007). Gifts for the gods: images from Egyptian temples. New York: The Metropolitan Museum of Art. ISBN 9781588392312.
  • Iamblichus; Waterfield, Robin; Critchlow, Keith; Translated by Robin Waterfield (1988). The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers. Phanes Press. ISBN 0-933999-72-0.
  • Johnson, Anthony: Solving Stonehenge, the New Key to an Ancient Enigma. Thames & Hudson 2008 ISBN 978-0-500-05155-9
  • Lawlor, RobertSacred Geometry: Philosophy and practice (Art and Imagination). Thames & Hudson, 1989 (1st edition 1979, 1980, or 1982). ISBN 0-500-81030-3.
  • Lesser, George (1957–1964). Gothic cathedrals and sacred geometry. London: A. Tiranti.
  • Lippard, Lucy R.: Overlay: Contemporary Art and the Art of Prehistory. Pantheon Books New York 1983 ISBN 0-394-51812-8.
  • Michell, JohnCity of Revelation. Abacus, 1972. ISBN 0-349-12320-9.
  • Schloeder, Steven J.Architecture in Communion: Implementing the Second Vatican Council through Liturgy and Architecture. (Ignatius Press: 1998). ISBN 0-89870-631-9.
  • Steiner, Rudolf; Catherine Creeger (2001). The Fourth Dimension : Sacred Geometry, Alchemy, and Mathematics. Anthroposophic Press. ISBN 0-88010-472-4.
  • Schneider, Michael S.: A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science. Harper Paperbacks, 1995. ISBN 0-06-092671-6
  • Pennick, NigelBeginnings: Geomancy, Builders' Rites and Electional Astrology in the European Tradition
  • Pennick, Nigel: Sacred Geometry: Symbolism and Purpose in Religious Structures
  • Pennick, Nigel: The Ancient Science of Geomancy: Living in Harmony with the Earth
  • Pennick, Nigel: The Sacred Art of Geometry: Temples of the Phoenix
  • Pennick, Nigel: The Oracle of Geomancy
  • Pennick, Nigel: The Ancient Science of Geomancy: Man in Harmony with the Earth
  • West, John Anthony, Inaugural Lines: Sacred geometry at St. John the DivineParabola magazine, v.8, n.1, Spring 1983

External links[edit]

 

INTRODUCTION

Egyptian Rope Stretchers 
Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 22.

Welcome to Geometry in Art & Architecture. We'll be taking a long journey together, starting in Egypt, like the travellers in this picture. There, the story goes, geometry got its start when rope stretchers were sent out to put back the boundary markers washed away by the Nile.

In addition to looking at art and architecture, we'll cover any mathematics-related topics as we go along. The Math Topics for the first unit will be an introduction to the triangle in general, and the so-called Egyptian triangle, contained in the great pyramid. Since the Egyptian triangle contains the golden ratio, we'll introduce the ideas of ratio and proportion here, and for squaring of the circle, we must be able to find perimeters and areas of the square and the circle.

The plan is to go more or less chronologically, following threads of Art, Mathematics, and Architecture, from Egypt to the present.

 

Fractal Tetrahedron

We'll start our journey with a pyramid, and we'll also end with a very different pyramid, a Sierpinski tetrahedron, in our final unit on Chaos and Fractals.

We'll limit ourselves to Western art only, but even with that restriction the coverage is very wide. That means, of course, that we can't go too deeply into any one topic.

Things We'll Look For

 

Brunes, Tons, The Secrets of Ancient Geometry - and its use. Copenhagen. Rhodos, 1967.

Brune's Cover

As we go through the material we'll be looking for:

  1. Proportions between the parts of a building, a painting, or a sculpture. In particular, we'll look for the golden ratio and the musical ratios.
  2. Use of geometric symbols, such as the circle, mandala, triangle, square, pentagram, hexagon, or octagon, and their use in so-called Sacred Geometry.
  3. Geometric Constructions, like squaring the circle, the Gothic Master Diagram, the sacred cut, and constructions of the pentagon.
  4. Shapes of Frames; how they are chosen and how they affect the contents of a painting.
  5. Art Motifs, especially recurring themes that we see over and over in art.
  6. Math content; any geometry or other math that is closely related to the art or architecture we're studying.
  7. People, ones that played a key role in developing the ideas related to this course, and especially those that were both mathematicians and artists or architects.

Skeptical Attitude

We'll see that writers in this field sometimes make unsupported claims. Rudolf Wittkower, in his Architecture in the Age of Humanism says

". . . in trying to prove that a system of proportions has been deliberately applied . . . one is easily misled into finding . . . those ratios which one sets out to find. Compasses in the scholar's hand do not revolt."

In other words, we tend to find what we're looking for, whether its there or not. We will hope to avoid that pitfall by questioning everything.

Mathematics Across the Curriculum

MATC Logo

This course is one of several developed under a grant from the National Science Foundation to Dartmouth, called Mathematics Across the Curriculum. Some courses being developed at Dartmouth are ones that try to integrate math with:

physics and chemistry

textile design

psychology and medicine

music

earth sciences

Renaissance thought

and this one, combining math with art and architecture.

The Eternal Golden Braid

 

Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. NY: Vintage, 1979. 
Hofstadter Cover

In "Gödel, Escher, Bach:", Douglas Hostadter says

"I have sought to weave an eternal golden braid out of these three strands, Gödel, Escher, Bach, a mathematician, an artist, and a composer."

In other words, math, art, and music. In this course we hope to trace just two strands of his eternal golden braid, art (and architecture) and math, and sometimes connect them with strands from literature, mythology, and religion .

We've planned an exciting journey, to follow these strands over 5000 years and several continents, and we really hope that you'll join us for the trip!

The Golden Ratio &
Squaring the Circle
in the Great Pyramid

"Twenty years were spent in erecting the pyramid itself: of this, which is square, each face is eight plethra, and the height is the same; it is composed of polished stones, and jointed with the greatest exactness; none of the stones are less than thirty feet." -Herodotus, Chap. II, para. 124.

Slide 2-1: The Giza Pyramids and Sphinx as depicted in 1610, showing European travelers Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 22

 Outline:

 The Great Pyramid

 

 The Golden Ratio

 

 Egyptian Triangle

 

 Squaring the Circle

 

 Rope-Stretcher's Triangle

 

 Summary

Projects

Reading

The Great Pyramid

 

Slide 2-2: The Great Pyramid of Cheops

Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 205

We start our task of showing the connections between geometry, art, and architecture with what appears to be an obvious example; the pyramids, works of architecture that are also basic geometric figures.

The pyramids were built in the lifetime of a single king, and were to help him in become immortal. They were made mostly in 4th dynasty of the old kingdom, about 2800 B.C.

Herodotus

 

Slide 2-4: Herodotus

Encarta ‘96 Encyclopedia. Funk and Wagnalls, 1995.

Herodotus (484?-425 BC), called the Father of History, was the first to write about the pyramids around 440 B.C.

In his History Herodotus says that the pyramids, already ancient, were covered with a mantle of highly polished stones joined with the greatest exactness.

Secrets of the Great Pyramid

The pyramids are claimed to have many "secrets;" that they are models of the earth, that they form part of an enormous star chart, that their shafts are aligned with certain stars, that they are part of par of a navigational system to help travelers in the desert find their way, and on and on.

In this unit we'll examine the claim that the Great Pyramid contains the Golden Ratio, whatever that is, and then look at the claim that the Great Pyramid squares the circle, whatever that is.

Golden Ratio

So what is this Golden Ratio that the Great Pyramid is supposed to contain?

ratio is the quotient of two quantities. The ratio of a to b is

a/b

The price/earnings ratio is the price of a share of stock divided by the earnings of that share.

Price/ Earnings

proportion results when two ratios are set equal to each other. Thus if the ratio of a to b equals the ratio of c to d, we have the proportion,

a/b = c/d

Systems of Proportions

Throughout much of art history, artists and architects were concerned with the proportions of the parts of their works. For example, if you were designing a temple, you might want to make the ratio of its height any old number, or perhaps, for some reason, a particular value. In fact, we'll see that there were not only particular ratios that were preferred, but sometimes entire systems of proportions.

Some systems of proportions were based on:

  1. The musical intervals
  2. The Human Body
  3. The Golden Ratio

We'll see as we go along that these systems of proportions will be recurring themes thoughout the course.

Definition of the Golden Ratio

The golden ratio is also called extreme and mean ratio. According to Euclid,

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

Derivation of the Golden Ratio

Let smaller part = 1, larger part = . Thus  is the golden ratio. It is often designated by the greek letter phi,  for Phideas, (fl. c. 490-430 BC), Athenian sculptor and artistic director of the construction of the Parthenon, who supposedly used the golden ratio in his work.

Then by the definition of the golden ratio,

 / 1 = (1 +   ) / 

so

 2 = 12 + 1

and we get the quadratic equation,

 2 -  - 1 = 0

As a project, solve this quadratic equation for the golden ratio . You should get,

 = 1/2 +  5 / 2   1.618

Project: Do this derivation.

Geometric Construction of the Golden Ratio

Subdivide a square of side 1 into two equal rectangles. Then lay out a distance equal to the diagonal of one of these half-squares, plus half the side of the original square. The ratio of this new distance to the original side, 1, is the golden ratio.

Project: Do this construction for the golden ratio.

Project: Mathematically show that this construction gives the golden ratio.

Egyptian Triangle

Let's now return to the pyramids. If we take a cross-section through a pyramid we get a triangle. If the pyramid is the Great Pyramid, we get the so-called Egyptian Triangle. It is also called the Triangle of Price, and the Kepler triangle.

This triangle is special because it supposedly contains the golden ratio. In particular,

the ratio of the slant height s to half the base b is said to be the golden ratio.

To verify this we have to find the slant height.

Computation of Slant Height s

The dimensions, to the nearest tenth of a meter, of the Great Pyramid of Cheops, determined by various expeditions.

height = 146.515 m,   and    base = 230.363 m

Half the base is

230.363   ÷   2 = 115.182 m

So,

2 = 146.515 + 115.182 2 = 34,733 m2

s = 18636.9 mm

Does the Great Pyramid contain the Golden Ratio?

Dividing slant height s by half base gives

186.369   ÷   115.182 = 1.61804

which differs from  (1.61803) by only one unit in the fifth decimal place.

The Egyptian triangle thus has a base of 1 and a hypotenuse equal to . Its height h, by the Pythagorean theorem, is given by

h2 =   2 - 12

Solving for h we get a value of .

Project: Compute the value for the height of the Egyptian triangle to verify that it is .

Thus the sides of the Egyptian triangle are in the ratio

Kepler Triangle

The astronomer Johannes Kepler (1571-1630) was very interested in the golden ratio. He wrote, "Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into mean and extreme ratios, that is, the Golden Mean. The first way may be compared to a measure of gold, the second to a precious jewel."

In a letter to a former professor he states the theorem, which I rephrase as:

If the sides of a right triangle are in geometric ratio, then the sides are

We recognize this as the sides of the Egyptian triangle, which is why its also called the Kepler triangle.

Project: Prove that If the sides of a right triangle are in geometric ratio, then the sides are

The Star Cheops

A British railway engineer, Robert Ballard, saw the pyramids on his way to Australia to become chief engineer of the Australian railways. He watched from a moving train how the relative appearance of the three pyramids on the Giza plateau changed. He concluded that they were used as sighting devices, and wrote a book with the grand title of The Solution of the Pyramid Problem in 1882.

He also noted that the cross-section of the Great Pyramid is two of what we have called Egyptian triangles. He then constructs what he called a Star Cheops, which, he says, "... is the geometric emblem of extreme and mean ratio and the symbol of the Egyptian Pyramid Cheops."

To draw a star Cheops:

  • Draw vertical and horizontal axes.
  • Using their intersection as center, draw two circles, radius 1, and radius 1 + .
  • Superscribe a square about the smaller circle. This will be the base of the pyramid,
  • From the point where an axis cuts the outer circle, draw two lines to the corners of the square. The triangle obtained will be one face of the pyramid.
  • Repeat the preceding step for the remaining three faces, getting a four-pointed star. Cut it out.
  • Fold each triangular face up from the base forming the pyramid.

Project Draw a star Cheops. Fold it to quickly make a model pyramid.

Squaring the Circle

 

Slide 2-3: The Great Pyramid

National Geographic. April '88

Now we'll look at his other claim, that the Great Pyramid's dimensions also show squaring of the circle. But just what is that?

The problem of squaring the circle is one of constructing, using only compass and straightedge;

(a) a square whose perimeter is exactly equal to the perimeter of a given circle, or

(b) a square whose area is exactly equal to the area of a given circle.

There were many attempts to square the circle over the centuries, and many approximate solutions, some of which we'll cover. However it was proved in the ninteenth century that an exact solution was impossible.

Squaring of the Circle in the Great Pyramid

The claim is:

The perimeter of the base of the Great Pyramid equals the circumference of a circle whose radius equal to the height of the pyramid.

Does it? Recall from the last unit that if we let the base of the Great pyramid be 2 units in length, then

pyramid height = 

So:

Perimeter of base = 4 x 2 = 8 units

Then for a circle with radius equal to pyramid height .

Circumference of circle = 2    7.992

So the perimeter of the square and the circumference of the circle agree to less than 0.1%.

An Approximate Value for  in Terms of 

Since the circumference of the circle (2  ) nearly equals the perimeter of the square (8)

2   8

we can get an approximate value for ,

  4 /  = 3.1446

which agrees with the true value to better than 0.1%.

Area Squaring of the Circle

The claim here is:

The area of that same circle, with radius equal to the pyramid height equals that of a rectangle whose length is twice the pyramid height() and whose width is the width (2) of the pyramid.

Area of rectangle = 2 () ( 2 ) = 5.088

Area of circle of radius  =  r 2   () 2    = 5.083

an agreement withing 0.1%

The Pizza-Cutter Theory

Suppose that the Egyptians didn't know anything about but laid out the pyramid using a measuring wheel, such as those used today to measure distances along the ground.

Take a wheel of any diameter and lay out a square base one revolution on a side. Then make the pyramid height equal to two diameters

By this simple means you get a pyramid having the exact shape of the Great Pyramid containing perimeter-squaring of the circle and area squaring of the circle and, for no extra cost, the golden ratio!

Project: Use a pizza cutter or a similar disk to construct a pyramid similar to the Great Pyramid.

Project: Show, by calculation, that using a measuring wheel as described will give a pyramid of the same shape as the Great Pyramid.

Project: Find the diameter of the measuring wheel required so that:

100 revolutions = the base of the Great Pyramid

200 diameters = the height of the Great Pyramid

We'll see that this idea of squaring the circle will be a recurring theme throughout most of this course. But lets leave it for now and get back to triangles.

Rope-Stretcher's Triangle

One practical value of any triangle is its rigidity. A triangular frame is rigid, while a four-sided one will collapse.

Another imortant use is for triangulation, for locating things as in surveying and navigation, and this property takes us back to the very origins of geometry, in ancient Egypt.

The Origins of Geometry

 

Slide 2-5: Hardenonaptai: Rope stretchers or engineers

Tompkins, Peter. Secrets of the Great Pyramid.NY: Harper, 1971. p. 22

Geometry means earth measure. Geo + Metry. According to the Herodotus the Nile flooded its banks each year, obliterating the markings for fields.

He wrote, "This king divided the land . . . so as to give each one a quadrangle of equal size and . . . on each imposing a tax. But everyone from whose part the river tore anything away . . . he sent overseers to measure out how much the land had become smaller, in order that the owner might pay on what was left . . . In this way, it appears to me, geometry originated, which passed thence to Greece.

The Rope-stretcher's Triangle

One tool they may have used is a rope knotted into 12 sections stretched out to form a 3-4-5 triangle. Does it Produce a Right Angle?

According to the Pythagorean theorem,

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

The converse of is also true,

If the square of one side of a triangle equals the sum of the squares of the other two sides, then we have a right triangle.

For the 3-4-5 triangle;

52 = 32 + 42

25 = 9 + 16

It checks, showing a rope knotted like this will give a right angle.

The rope-stretcher's triangle is also called the 3-4-5 right triangle, the Rope-Knotter's triangle, and the Pythagorean triangle.

Project: Use a long knotted rope to make a rope-stretcher's triangle. Use it outdoors to lay out a right angle on some field. Then continue, making three more right angles to form a square. How accurate is your work? Did you come back to the starting point?

Summary

Was the golden ratio intentionally built into the Great Pyramid of Cheops? Why would anyone intentionally build the golden ratio into a pyramid, or other structure? What was the significance of to the Egyptians? And did the ancient Egyptians intentionally design the Great Pyramid to square the circle?

Its hard to know, but at any rate, we've introduced the golden ratio and squaring the circle themes which we'll encounter many times again in this study.

We've also some symbolism here:

If flooding of the Nile symbolized the annual return of watery chaos, then geometry, used to reestablish the boundaries, was perhaps seen as restoring law and order on earth. We'll see this notion again of geometry being sacred because it represents order, especially in the Middle Ages.

The rope stretchers triangle when opened out gives a zodiac circle, with the number of knots the most important of the astrological numbers

The square, with its four corners like the corners of a house, represents earthly things, while the circle, perfect, endless, infinite, has often been taken to represent the divine or godly. So squaring the circle is a universal symbol of bringing the earthly and mundane into a proper relationship with the divine.

And the Golden Ratio reverberates with the idea of the Golden Mean, the principle of moderation, defined by Aristotle as the mean between the two extremes of excess and insufficiency, as generosity is the mean between prodigality and stinginess, and by Horace, called the philosopher of the golden mean, advocated moderation even in the pursuit of virtue.

Remember that the pyramids were tombs, and that much of Egyptian art is funarary art. One Egyptian word for sculptor literally means He who keeps alive. To help the king acheive immortality, it was important that he didn't rot, hence the elaborate embalming. But embalming was not enough. The likeness of the king must also be preserved in gold or granite. So the tomb was seen as a sort of life insurance policy. Thus sculpture evolved.

But there is another angle to sculptor ... he who keeps alive. Once, the servants and slaves were buried with the king to help him in the other world. Then art came to the rescue, providing carved and painted substitutes for the real people. So the sculptor not only kept alive the memory of the dead king but literally kept alive all these people that would have been buried with the king.

Who says art isn't important?

 

Slide 2-7: King Tutankhamun

Metropolitan Museum of Art Gift Catalog, The Treasures of Tutahkhamun. NY: Met 1978.

Finally, in these units on Egypt we've started down the road that we'll follow right to the present time. The art historian Ernst Gombrich writes,

". . . the story of art as a continuous effort does not begin in the caves of southern France or among the North American Indians. . . there is no direct tradition which links these strange beginnings with our own days . . .But there isa direct tradition, handed down from master to pupil . . . which links the art of our own days with the art of the Nile valley some 5000 years ago. . .".. the Greek masters went to school with the Egyptians, and we are all the pupils of the Greeks.

In our next unit we'll cross the Mediteranean Sea where we too will be pupils of the Greeks.

Pythagoras
&
Music of the Spheres

There is geometry in the humming of the strings
... there is music in the spacing of the spheres.

 

Pythagoras.

The History of Philosophy (c.1660) by Thomas Stanley.

 Outline:

  Pythagoras

 

  The Pythagoreans

 

  Pythagorean Number Symbolism

 

  Music of the Spheres

 

 Summary

 Reading

 

Pythagoras

From Egypt we move across the Mediterranean Sea to the Greek island of Samos, the birthplace of Pythagoras, whose ideas dominate most of the material in this course. We'll introduce Pythagoras and his secret society of the Pythagoreans.

We'll look at the Pythagoreans' ideas about numbers, as a prelude to our next unit on number symbolism. Finally, we'll introduce a new idea that will be recurring theme throughout this course, the musical ratios, which will reappear in discussions of the architecture of the Renaissance.

Our main link between Egypt and Greece seems to be Thales c 640-550 BC, father of Greek mathematics, astronomy, and Philosophy, and was one of the Seven Sages of Greece. A rich merchant, his duties as a merchant took him to Egypt, and so became one of the main sources of Egyptian mathematical information in Greece. It was Thales advised his student to visit Egypt, and that student was Pythagoras.

Raphael's School of Athens

Slide 3-1: Raphael's School of Athens 1510-11.

Janson, H. W. History of Art. Fifth Edition. NY: Abrams, 1995. p.497

Pythagoras is shown in this famous painting, done by Raphael in 1510-11, which also shows most of the Greek philosophers.

Socrates sprawls on the steps at their feet, the hemlock cup nearby.

His student Plato the idealist is on the left, pointing upwards to divine inspiration. He holds his Timaeus, a book we'll talk about soon.

Plato's student Aristotle, the man of good sense, stands next to him. He is holding his Ethics in one hand and holding out the other in a gesture of moderation, the golden mean.

Euclid is shown with compass, lower right. He is the Greek mathematician whose Elements we'll mention often.

 

Slide 3-2: Pythagoras in Raphael's School of Athens
Janson, H. W. History of Art. Fifth Edition. NY: Abrams, 1995. p.497

Finally, we see Pythagoras (582?-500? BC), Greek philosopher and mathematician, in the lower-left corner.

The Pythagoreans

Pythagoras was born in Ionia on the island of Sámos, and eventually settled in Crotone, a Dorian Greek colony in southern Italy, in 529 B.C.E. There he lectured in philosophy and mathematics.

He started an academy which gradually formed into a society or brotherhood called the Order of the Pythagoreans.

Disciplines of the Pythagoreans included:

silence

music

incenses

physical and moral purifications

rigid cleanliness

a mild ascetisicm

utter loyalty

common possessions

secrecy

daily self-examinations
(whatever that means)

   

pure linen clothes

     

We see here the roots of later monastic orders.

For badges and symbols, the Pythagoreans had the Sacred Tetractys and the Star Pentagram, both of which we'll talk about later.

There were three degrees of membership:

  1. novices or "Politics"
    2. Nomothets, or first degree of initiation
    3. Mathematicians

The Pythagoreans relied on oral teaching, perhaps due to their pledge of secrecy, but their ideas were eventually committed to writing. Pythagoras' philosophy is known only through the work of his disciples, and it's impossible to know how much of the "Pythagorean" discoveries were made by Pythagoras himself. It was the tradition of later Pythagoreans to ascribe everything to the Master himself.

Pythagorean Number Symbolism

The Pythagoreans adored numbers. Aristotle, in his Metaphysica, sums up the Pythagorean's attitude towards numbers.

"The (Pythagoreans were) ... the first to take up mathematics ... (and) thought its principles were the principles of all things. Since, of these principles, numbers ... are the first, ... in numbers they seemed to see many resemblances to things that exist ... more than [just] air, fire and earth and water, (but things such as) justice, soul, reason, opportunity ..."

The Pythagoreans knew just the positive whole numbers. Zero, negative numbers, and irrational numbers didn't exist in their system. Here are some Pythagorean ideas about numbers.

Masculine and Feminine Numbers

Odd numbers were considered masculine; even numbers feminine because they are weaker than the odd. When divided they have, unlike the odd, nothing in the center. Further, the odds are the master, because odd + even always give odd. And two evens can never produce an odd, while two odds produce an even.

Since the birth of a son was considered more fortunate than birth of a daughter, odd numbers became associated with good luck. "The gods delight in odd numbers," wrote Virgil.

1   Monad. Point. The source of all numbers. Good, desirable, essential, indivisible.

2   Dyad. Line. Diversity, a loss of unity, the number of excess and defect. The first feminine number. Duality.

3   Triad. Plane. By virtue of the triad, unity and diversity of which it is composed are restored to harmony. The first odd, masculine number.

4   Tetrad. Solid. The first feminine square. Justice, steadfast and square. The number of the square, the elements, the seasons, ages of man, lunar phases, virtues.

5   Pentad. The masculine marriage number, uniting the first female number and the first male number by addition.

  • The number of fingers or toes on each limb.
  • The number of regular solids or polyhedra.

Incorruptible: Multiples of 5 end in 5.

6   The first feminine marriage number, uniting 2 and 3 by multiplication.
The first perfect number (One equal to the sum of its aliquot parts, IE, exact divisors or factors, except itself. Thus, (1 + 2 + 3 = 6).
The area of a 3-4-5 triangle

7   Heptad. The maiden goddess Athene, the virgin number, because 7 alone has neither factors or product. Also, a circle cannot be divided into seven parts by any known construction).

8   The first cube.

9   The first masculine square. 
Incorruptible - however often multiplied, reproduces itself.

10   Decad. Number of fingers or toes.
Contains all the numbers, because after 10 the numbers merely repeat themselves.
The sum of the archetypal numbers (1 + 2 + 3 + 4 = 10)

27   The first masculine cube.

28   Astrologically significant as the lunar cycle. 
It's the second perfect number (1 + 2 + 4 + 7 + 14 = 28). 
It's also the sum of the first 7 numbers (1 + 2 + 3 + 4 + 5 + 6 + 7 = 28)!

35   Sum of the first feminine and masculine cubes (8+27)

36   Product of the first square numbers (4 x 9) 
Sum of the first three cubes (1 + 8 + 27) 
Sum of the first 8 numbers (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)

Figured Numbers

The Pythagoreans represented numbers by patterns of dots, probably a result of arranging pebbles into patterns. The resulting figures have given us the present word figures.

Thus 9 pebbles can be arranged into 3 rows with 3 pebbles per row, forming a square.

Similarly, 10 pebbles can be arranged into four rows, containing 1, 2, 3, and 4 pebbles per row, forming a triangle.

From these they derived relationships between numbers. For example, noting that a square number can be subdivided by a diagonal line into two triangular numbers, we can say that a square number is always the sum of two triangular numbers.

Thus the square number 25 is the sum of the triangular number 10 and the triangular number 15.

Sacred Tetractys

One particular triangular number that they especially liked was the number ten. It was called a Tetractys, meaning a set of four things, a word attributed to the Greek Mathematician and astronomer Theon (c. 100 CE). The Pythagoreans identified ten such sets.

Ten Sets of Four Things

Numbers

1

2

3

4

Magnitudes

point

line

surface

solid

Elements

fire

air

water

earth

Figures

pyramid

octahedron

icosahedron

cube

Living Things

seed

growth in length

in breadth

in thickness

Societies

man

village

city

nation

Faculties

reason

knowledge

opinion

sensation

Seasons

spring

summer

autumn

winter

Ages of a Person

infancy

youth

adulthood

old age

Parts of living things

body

 

three parts of the soul

 

Gnomons

Gnomon means carpenter's square in Greek. Its the name given to the upright stick on a sundial. For the Pythagoreans, the gnomons were the odd integers, the masculine numbers. Starting with the monad, a square number could be obtained by adding an L-shaped border, called a gnomon.

Thus, the sum of the monad and any consecutive number of gnomons is a square number.

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

and so on.

The Quadrivium

While speaking of groups of four, we owe another one to the Pythagoreans, the division of mathematics into four groups,

giving the famous Quadrivium of knowledge, the four subjects needed for a bachelor's degree in the Middle Ages.

Music of the Spheres

Jubal and Pythagoras

 

Slide 3-4: Theorica Musica
F. Gaffurio, Milan, 1492
Lawlor, Robert. Sacred Geometry. NY: Thames & Hudson, 1982. p.7

So the Pythagoreans in their love of numbers built up this elaborate number lore, but it may be that the numbers that impressed them most were those found in the musical ratios.

Lets start with this frontispiece from a 1492 book on music theory.

The upper left frame shows Lubal or Jubal, from the Old Testament, "father of all who play the lyre and the pipe" and 6 guys whacking on an anvil with hammers numbered 4, 6, 8, 9, 12, 16.

The frames in the upper right and lower left show Pithagoras hitting bells, plucking strings under different tensions, tapping glasses filled to different lengths with water, all marked 4, 6, 8, 9, 12, 16. In each frame he sounds the ones marked 8 and 16, an interval of 1:2 called the octave, or diapason.

In the lower right, he and Philolaos, another Pythagorean, blow pipes of lengths 8 and 16, again giving the octave, but Pythagoras holds pipes 9 and 12, giving the ratio 3:4, called the fourth or diatesseron while Philolaos holds 4 and 6, giving the ratio 2:3, called the fifth or diapente.

They are:

8 : 16 or   1 : 2

Octave

diapason

4 : 6   or   2 : 3

Fifth

diapente

9 : 12 or   3 : 4

Fourth

diatesseron

These were the only intervals considered harmonious by the Greeks. The Pythagoreans supposedly found them by experimenting with a single string with a moveable bridge, and found these pleasant intervals could be expressed as the ratio of whole numbers.

Pythagoras in the School of Athens

 

Slide 3-3: Closeup of Tablet

Bouleau

Janson, H. W. History of Art. Fifth Edition. NY: Abrams, 1995. p.497

Raphael's School of Athens shows Pythagoras is explaining the musical ratios to a pupil.

Notice the tablet. It shows:

The words diatessaron, diapente, diapason
The roman numerals for 6, 8, 9, and 12, showing the ratio of the intervals, same as in the music book frontispiece.

The word for the tone, EPOGLOWN, at the top.

Under the tablet is a triangular number 10 called the sacred tetractys, that we mentioned earlier.

The Harmonic Scale

 

Slide 3-5: Gafurio Lecturing 
F. Gafurio, De Harmonia musicorum instrumentorum, 1518, Wittkower, Rudolf.Architectural Principles in the Age of Humanism.NY: Random, 1965. 43a.

This diagram from a book written in 1518 shows the famous Renaissance musical theorist Franchino Gafurio with three organ pipes and 3 strings marked 3 , 4, 6. This indicates the octave, 3 : 6 divided by the harmonic mean 4, into the fourth, 3 : 4, and the fifth, 4 : 6 or 2 : 3.

The banner reads, "Harmonia est discordia concors" or Harmony is discordant concord, propounding the thesis that harmony results from two unequal intervals drawn from dissimilar proportions. The diagram shows compasses, suggesting a link between geometry and music.

So What?

So after experimenting with plucked strings the Pythagoreans discovered that the intervals that pleased people's ears were

octave

1 : 2

fifth

2 : 3

fourth

3 : 4

and we can add the two Greek composite consonances, not mentioned before . . .

octave plus fifth

1 : 2 : 3

double octave

1 : 2 : 4

Now bear in mind that we're dealing with people that were so nuts about numbers that they made up little stories about them and arranged pebbles to make little pictures of them. Then they discovered that all the musical intervals they felt was beautiful, these five sets of ratios, were all contained in the simple numbers

1, 2, 3, 4

and that these were the very numbers in their beloved sacred tetractys that added up to the number of fingers. They must have felt they had discovered some basic laws of the universe.

Quoting Aristotle again ... "[the Pythagoreans] saw that the ... ratios of musical scales were expressible in numbers [and that] .. all things seemed to be modeled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of number to be the elements of all things, and the whole heaven to be a musical scale and a number."

Music of the Spheres

 

Slide 3-6: Kepler's Model of the Universe 
Lawlor, Robert. Sacred Geometry. NY: Thames & Hudson, 1982. p. 106

"... and the whole heaven to be a musical scale and a number... "

It seemed clear to the Pythagoreans that the distances between the planets would have the same ratios as produced harmonious sounds in a plucked string. To them, the solar system consisted of ten spheres revolving in circles about a central fire, each sphere giving off a sound the way a projectile makes a sound as it swished through the air; the closer spheres gave lower tones while the farther moved faster and gave higher pitched sounds. All combined into a beautiful harmony, the music of the spheres.

This idea was picked up by Plato, who in his Republic says of the cosmos; ". . . Upon each of its circles stood a siren who was carried round with its movements, uttering the concords of a single scale," and who, in his Timaeus, describes the circles of heaven subdivided according to the musical ratios.

Kepler, 20 centuries later, wrote in his Harmonice Munde (1619) says that he wishes "to erect the magnificent edifice of the harmonic system of the musical scale . . . as God, the Creator Himself, has expressed it in harmonizing the heavenly motions."

And later, "I grant you that no sounds are given forth, but I affirm . . . that the movements of the planets are modulated according to harmonic proportions."

Systems of Proportions based on the Musical Ratios

Slide 17-1: Villa Capra Rotunda
citatation

What does this have to do with art or architecture? The idea that the same ratios that are pleasing to the ear would also be pleasing to the eye appears in the writings of Plato, Plotinus, St. Augustine, and St. Aquinas. But the most direct statement comes from the renaissance architect Leone Battista Alberti (1404-1472), "[I am] convinced of the truth of Pythagoras' saying, that Nature is sure to act consistently . . . I conclude that the same numbers by means of which the agreement of sounds affect our ears with delight are the very same which please our eyes and our minds."

Alberti then gives a list of ratios permissible, which include those found by Pythagoras. We'll encounter Alberti again for he is a central figure in the development of perspective in painting.

We'll also discuss another architect who used musical ratios, Andrea Palladio (1518-1580), who designed the Villa Capra Rotunda shown here.

Summary

 

Slide 3-7: Correspondence School in Crotone

W. S. Anglin. Mathematical Intelligence V19, No. 1, 1997

I always wanted to make a pilgrimage to Crotone, site of the Pythagorean cult, but this is all that's there to mark their presence. Pythagoras and his followers died when their meetinghouse was torched. We'll have more on the Pythagoreans later, in particular their fondness for the star pentagram.

<p" style="color: rgb(0, 0, 0); font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration-style: initial; text-decoration-color: initial;">In this unit we've had some Pythagorean number lore and soon we'll add to it by talking about number symbolism in general, especially numbers in astrology and the Old Testament.<p">Somewhere I had read that one answer to the question, Why study history? was To keep Pythagoras alive! I've forgotten where I read that, but anyway, it makes a nice goal for this course.</p"></p">