ANCIENT GEOMETRY, 4. RENAISSANCE

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The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics as a relevant subject needed to understand nature and the arts. Two major reasons drove Renaissance artists towards the pursuit of mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas.

Second, philosophers and artists alike were convinced that mathematics was the true essence of the physical world and that the entire universe, including the arts, could be explained in geometric terms. In light of these factors, Renaissance artists became some of the best applied mathematicians of their times.

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The first printed illustration of a rhombicuboctahedron, byLeonardo da Vinci, published in De divina proportione.

Written by Luca Pacioli in Milan from 1496–98, published in Venice in 1509, De Divina Proportione was about mathematical and artistic proportionLeonardo da Vinci drew illustrations of regular solids in De divina proportione while he lived with and took mathematics lessons from Pacioli. Leonardo’s drawings are probably the first illustrations of skeletonic solids, which allowed an easy distinction between front and back. Skeletonic solids, such as the rhombicuboctahedron, were one of the first solids drawn to demonstrate perspective by being overlaid on top of each other. Additionally, the work also discusses the use of perspective by painters such as Piero della FrancescaMelozzo da Forlì, and Marco Palmezzano.

It is in De Divina Proportione that the golden ratio is defined as the divine proportion. Pacioli also details the use of the golden ratio as the mathematical definition of beauty when applied to the human face.

“The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.” from De Divina Proportione (1509)

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ANCIENT GEOMETRY, 3. Great Mosque of Kairouan

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The oldest mosque in North Africa is the Great Mosque of Kairouan (Tunisia), built by Uqba ibn Nafi in 670 A.D. Boussora and Mazouz’s study of the mosque dimensions reveals a very consistent application of the golden ratio in its design.

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Floor plan of the Great Mosque of Kairouan.

The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organization. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period.

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Because of urban constraints, the mosque floor plan is not a perfect rectangle. Even so, for example, the division of the courtyard and prayer hall is almost a perfect golden ratio.

ANCIENT GEOMETRY, 2. Parthenon

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The Parthenon is a temple dedicated the Greek goddess Athena, built in the 5th century BC on the Athenian Acropolis. It is contended that Phidias, the main Greek sculptor in charge of decorating the Parthenon, also knew about the golden ratio and its aesthetic properties. In fact, the Greek symbol for the Golden Ratio is named Phi (φ) because of  Phidias The golden rectangle, a rectangle whose length to width ratio is the Golden Ratio and considered the most pleasing to the eye, is almost omnipresent in the façade and floor plans of the Parthenon. The entire façade may be enclosed within a golden rectangle.

The ratio of the length of a metope and triglyph to the height of the frieze, as well as the height of the columns and stylobate to the entire height of the temple is also the golden ratio.

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Phidias himself constructed many Parthenon statues that meticulously embody the golden ratio. He is also notable for his contributions to the Athena Parthenos and the Statue of Zeus. As with the Pyramids however, more recent historians challenge the purposeful inclusion of the golden ratio in Greek temples, such as the Parthenon, contending that earlier studies have purposefully fitted in measurements of the temple until it conformed to a golden rectangle.

ANCIENT GEOMETRY

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The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, 427- 347 bC

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writing that “Plato said God geometrizes continually”. In modern times the mathematician Carl Friedrich Gauss adapted this quote, saying “God arithmetizes”.

At least as late as Johannes Kepler (1571–1630), a belief in the geometric underpinnings of the cosmos persisted among scientists.

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According to Stephen Skinner, (Sydney, Australia, 1948)  the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein.

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Many forms observed in nature can be related to geometry, for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape

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Also, honeybees construct hexagonal cells to hold their honey. These and other correspondences are sometimes interpreted in terms of sacred geometry and considered to be further proof of the natural significance of geometric forms

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Geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture. Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra.

Many of the sacred geometry principles of the human body and of ancient architecture have been compiled into the Vitruvian Man drawing by Leonardo da Vinci, itself based on the much older writings of the Roman architect Vitruvius.

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