Mathematical Underpinnings

 

BACKGROUND

 

Factors Affecting Mathematics

The development of mathematics in the Renaissance was influenced by several other developments.

 

Humanism.  The search for ancient texts by humanists turned up Greek, Arabic, and Hebrew manuscripts containing advanced mathematical knowledge that had been unknown since ancient times.  

 

Printing.  The invention of printing facilitated the distribution of Latin editions of previously unknown works to mathematicians, as well as of vernacular editions of previously known works to a less scholarly audience.  Euclid's Elements of Geometry, for instance, became available to individuals involved in a wide variety of practical fields.

 

Applications of Mathematics

Since ancient times mathematics has been used for both practical and mystical purposes.

 

Practical. Practical applications of mathematics in the Renaissance included such fields as cartography, engineering, surveying, and navigation.  

 

Mystical.  Mystical applications of mathematics in the Renaissance included such practices as astrology and divination.  Because suggestions of mysticism could be found in the works of ancient authors like Pythagoras and Plato, some Neo-Platonists searched for mystical significance in the relationships of numbers.

 

Symbolic Meanings of Numbers

As Christianity evolved, many numbers acquired special significance.

 

Three.  Three refers to the Trinity.  

 

Four.  Four is the number of the Evangelists, the writers of the Gospels. The evangelists were often depicted in places where there are four similar surfaces such as the inner faces of pendentives.  

 

Seven. Seven refers to the days of Creation, which are described in Genesis, the first book of the Old Testament. The days of Creation are often depicted as a cycle of scenes. Michelangelo's Creation of the Sun, Moon, and Plants, which is one of the nine central panels of the Sistine Chapel ceiling, combines the day that God created the sun and moon (the fourth day) with the day that He created the earth and plants (the third day).

 

Eight. Eight refers to the seven days of Creation plus Resurrection Day.  The use of an octagonal shape for baptisteries reflects the parallel between the beginning of the universe and the beginning of an individual's salvation through initiation into the Church and resurrection on Judgment Day.  The ceiling of the baptistery of Florence Cathedral depicts the seven days of creation plus the life of John the Baptist, to whom that baptistery is dedicated.

 

Twelve.  Twelve, which was the number of tribes in ancient Israel, refers foremost to the twelve Apostles.  The depiction of individual images of so many figures often required that they be placed on different walls, as they are in the Pazzi Chapel, where there are four on the two long walls and two on the two short walls of the rectangular area.  

 

Forms of Mathematics Affecting Design

Several forms of mathematics were especially important in the development of Renaissance design.

 

Geometry.  Geometry (discussed below), which is a separate branch of mathematics from number theory, deals with the properties of space in terms of points, lines, planes, and solids.  

 

Proportion. Proportion (discussed below) is a form of number theory, which is the area of mathematics that deals with the relationships of quantities.  Number theory was based on the work of the sixth-century BC mathematician Pythagoras. 

 

Golden section.  The golden section (discussed below) is a particular proportion.

 

 

GEOMETRY

 

Background

The properties of space in terms of points, lines, planes, and solids are the province of geometry.

 

In the Renaissance, geometry was at the root of two important aspects of art and architecture:  linear perspective and geometric composition.

 

History

From the early days of civilization, geometry had been an important tool for laying out buildings and surveying land.

 

Pythagoras founded a methodological basis for geometry using theorems (laws) that were derived from a series of seemingly self-evident premises.  His most famous theorem, which bears his name, states that, "The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides."

 

Later, Euclid, a fourth-century BC mathematician who lived in Alexandria (Egypt), expanded Pythagoras' foundations into a detailed body of postulates and proofs.  He wrote Elements of Geometry, the first text on the subject, which continues to be relevant to the study of geometry.

 

In the Middle Ages, geometry was part of the secular curriculum called the Liberal Arts.  This continued into the Renaissance, as geometry was a standard subject in a humanist education.

 

Homage in Art

Geometry, along with the other Liberal Arts, was honored in art by Pinturicchio's personification of the Liberal Arts in the Borgia Apartments of the Vatican Palace.  In his School of Athens in the Vatican Stanze, Raphael included both Pythagoras, who holds a slate, and Euclid, who draws with a compass. Raphael accentuated the link between architecture and geometry by giving Euclid the features of Bramante, the most important and respected architect in Italy at the time this fresco was painted.

 

Preference for Perfect Figures

Geometric figures that embody symmetry, especially radial or bi-axial symmetry, were thought to be inherently beautiful because of their regularity.

 

The circle was regarded as the most perfect figure because all points are equidistant from the center, and it has neither beginning nor end. Circular plans fascinated many architects.

 

Because of practical considerations, straight-sided figures, especially the square, were much used in architecture. Circle-in-square designs combined both.

 

In the first century BC, the Roman architect Vitruvius suggested a correspondence between human proportions and both the circle and the square (separately).

See more about geometric figures in Italian Renaissance architecture.

 

 

PROPORTION

 

Definition

Proportion refers to the relationships of groups of numbers, especially the relationship between two ratios.  An example is 2 is to 3 as 8 is to 12, which is expressed mathematically as 2:3::8:12.

 

Mathematics and Musical Harmony

The concept that mathematics is the language of nature was developed by the sixth-century BC. Greek mathematician Pythagoras, who demonstrated a connection between musical harmony and mathematics.

 

Pythagoras found that when tones are produced by stretched strings that have been made to vibrate, the tones that will be harmonious together are produced by strings whose lengths are mathematically related.  For instance, if a base tone is created by a string of a given length, a tone one octave higher can be produced using a string of half its length.

 

Linking Proportion to Visual Harmony

Harmony in the world of visual design was believed to be related to proportion, as it was in the world of sound.

 

Because harmony was thought to be a prerequisite to beauty in the Renaissance, the study of mathematics was included in the training of artists and architects.

 

Role of Proportion in Representation

In art, correct proportions were important in making an image correspond to its real-life counterpart and for indicating spatial position in two-dimensional representation.  The new system of linear perspective gave artists a scientific method with which to determine correct sizes and spatial relationships.

 

Use of Proportion in Architecture

In Renaissance architecture, the calculation of the relative dimensions of the orders enabled classically oriented architects to design buildings that suited contemporary needs but embodied classical principles and forms.  Using classical models, they examined ratios like a column's height to a number of other measurements such as its thickness, the space between columns, and the ceiling height.

 

The proportions of plans like those of Brunelleschi's San Lorenzo and Santo Spirito and Giuliano da Sangallo's Santa Maria delle Carceri are based on simple ratios like one to two and one to four.

 

In emulation of Vitruvius' ideas on ideal proportions, many Renaissance architects aimed to utilize the proportions of the human body in their designs, specifically for churches, where the nave of the church represented the body, the transept represented the arms, and the apse represented the head.

 

A specific proportion that was often used by architects in determining relative dimensions was the "golden section," referred to in the Renaissance as the "divina proportione."

 

 

THE GOLDEN SECTION

 

Description

The golden section refers to a proportion of two dimensions in which the ratio of the smaller one to the larger is the same as the ratio of the larger one to the total of both.

 

Phi

When the larger dimension of a golden section is divided by the smaller one, the result is a constant numerical value of 1.618033989....  This proportion is symbolized by the Greek letter phi, which was chosen because Phidias used this proportion in designing the sculpture program at the Parthenon.

 

The Greek letter phi, is pronounced "fi" or "fe" and is written as Φ.

 

Drawing a Golden Rectangle

To draw a golden rectangle, one whose length-to-width ratio constitutes a golden section, start by constructing a square, ABCD.

 

Establish a midpoint, E, along side CD.

 

Using E as a center and EB as a radius, draw an arc that crosses an extension of CD.

 

F, the point where the arc crosses an extension of CD, defines the smaller dimension, CF, of the ratio with CD.

 

Projecting a rectangle from F to A produces a large golden rectangle, AGFD.  The newly formed smaller rectangle, BGFC, is also a golden rectangle.

 

DC/CF = DF/DC = 1.618033989

 

Expanding a Golden Rectangle

A golden rectangle can be enlarged by locating the new corner along a line passing through diagonal corners.

phi

 

Subdividing a Golden Rectangle

By marking off a square in a golden rectangle, a smaller golden rectangle is created.

 

A golden rectangle can be subdivided into smaller and smaller golden rectangles by repeating the process of marking off squares in each newly created rectangle until the divisions are too small to see.

 

Creating a Golden Spiral

The subdivisions can be used to create a golden spiral by drawing an arc through the outer sides of the squares.  The resulting golden spiral approximates the spiral found in such natural phenomena as the galaxy, many sea shells, and many plant forms.

 

History

Knowledge of phi dates back to Old Kingdom Egypt (third millennium BC), where it was used in building the pyramids.  In ancient Greece, its calculation was described by Euclid.  Some scholars believe that golden section proportions were used on the Parthenon.

 

In the Renaissance, Luca Pacioli (1445-1514) wrote De divina proportione (1509), a book discussing the golden section.  He was acquainted with three of the most mathematically oriented Italian artists/architects of the day: Leon Battista Alberti in Rome, Piero della Francesca in Urbino, and Leonardo da Vinci in Milan, who made the book's illustrations.  A skilled mathematician in his own right, Leonardo regarded mathematics and quantitative measurement to be at the foundation of science and wrote "Let no one read my principles who is not a mathematician."

 

Application to Renaissance Arts

Because of its unique properties, the golden section was believed to be especially harmonious.

 

It was used by painters and sculptors in calculating proportions and poses. Michelangelo applied golden section proportions to David.

 

Leonardo made many studies of proportion using golden section proportions.  He made the golden section the basis of poses like that of St. Jerome.

 

In architecture, rectangles whose dimensions embodied golden-section proportions could be integrated with the design of both elevations and plans.

 

 

FIBONACCI NUMBERS

 

Relationship to Golden Section

Fibonacci numbers are a series of numbers whose adjacent pairs approximate the golden section.

 

Formation of the Series

The series is formed by adding the last two numbers to generate the next.  For instance, the total of 3 + 5 is 8, and the ratio of 5:8 is roughly equivalent to 3:5.

gsec1

 

The series begins with zero and one, and the numbers increase rapidly:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 . . . .  Any two adjacent numbers will approach the phi proportion of 1.618033989, and the higher the pair of numbers, the more exactly their ratio will approach it.

 

For instance, the mathematical value of 5/3 = 1.6666666666, whereas that of 233/144 = 1.6180555.

 

The Mathematician Fibonacci

This system is named for its inventor, Leonardo da Pisa, known as Fibonacci (1175-1250), who was the most advanced mathematician in Europe at the time.  He described the series in Liber abaci, which he wrote in 1202.  His development of the series was in response to a mathematical puzzle concerning rabbit reproduction over time.

 

Fibonacci is also important for having introduced the zero and the Arabic decimal system, which led the substitution of Arabic numerals for Roman.  

 

Significance in Nature

The Fibonacci numbers are significant in nature in many radial plant forms ranging from branches around tree trunks to flower petals around stigmas.  In forms involving clockwise and counterclockwise parallel spirals that radiate from centers, the number of spirals in each direction corresponds to pairs of Fibonacci numbers.  For instance, on pine cones, the number of clockwise spirals is five and the number of counterclockwise spirals is eight.

 

 


Introduction 4 of 7

Outline

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Francesco di Giorgio's treatise, Trattato di architettura civile e militare, included drawings that illustrate his notion of a relationship between the human body and architectural forms.  Such ideas were rooted in Vitruvius and Alberti.