Mathematics and art

mathematical artmathematics of artartistic and imaginative pursuit
Leonardo da Vinci (1452–1519) illustrated the text with woodcuts of regular solids while he studied under Pacioli in the 1490s. Leonardo's drawings are probably the first illustrations of skeletonic solids. These, such as the rhombicuboctahedron, were among the first to be drawn to demonstrate perspective by being overlaid on top of each other. The work discusses perspective in the works of Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano. Da Vinci studied Pacioli's Summa, from which he copied tables of proportions. In Mona Lisa and The Last Supper, Da Vinci's work incorporated linear perspective with a vanishing point to provide apparent depth.

Piero della Francesca

Pierodella Francesca, PieroFrescoes
Piero's work on solid geometry was translated in Pacioli’s De divina proportione, a work illustrated by Leonardo da Vinci. Biographers of his patron Federico da Montefeltro of Urbino record that he was encouraged to pursue the interest in perspective which was shared by the Duke. In the late 1450s, Piero copied and illustrated the following works of Archimedes: On the Sphere and Cylinder, Measurement of a Circle, On Conoids and Spheroids, On Spirals, On the Equilibrium of Planes, The Quadrature of the Parabola, and The Sand Reckoner.

Florence Baptistery

BaptisteryBattistero di San GiovanniBaptistery of San Giovanni
Rustici may have been aided in his design by Leonardo da Vinci, who assisted him in the choice of his tools. Ghiberti was now widely recognized as a celebrity and the top artist in this field. He was showered with commissions, even from the pope. In 1425 he got a second commission, this time for the east doors of the baptistery, on which he and his workshop (including Michelozzo and Benozzo Gozzoli) toiled for 27 years, excelling themselves. These had ten panels depicting scenes from the Old Testament, and were in turn installed on the east side. The panels are large rectangles and are no longer embedded in the traditional Gothic quatrefoil, as in the previous doors.


FlorentineFirenzeFlorence, Italy
Medieval art was abstract, formulaic, and largely produced by monks whereas Renaissance art was rational, mathematical, individualistic, consisted of linear perspective and shading (Chiaroscuro) and produced by specialists (Leonardo da Vinci, Donatello, Michelangelo, and Raphael). Religion was important, but with this new age came the humanization of religious figures in art, such as Expulsion from the Garden of Eden, Ecce Homo (Bosch, 1470s), and Madonna Della Seggiola; People of this age began to understand themselves as human beings, which reflected in art.

Italian Renaissance

RenaissanceRenaissance ItalyItalian
Italian Renaissance art exercised a dominant influence on subsequent European painting and sculpture for centuries afterwards, with artists such as Leonardo da Vinci, Michelangelo, Raphael, Donatello, Giotto di Bondone, Masaccio, Fra Angelico, Piero della Francesca, Domenico Ghirlandaio, Perugino, Botticelli, and Titian. The same is true for architecture, as practiced by Brunelleschi, Leon Battista Alberti, Andrea Palladio, and Bramante. Their works include, to name only a few, the Florence Cathedral, St. Peter's Basilica in Rome, and the Tempio Malatestiano in Rimini, as well as several private residences.

Albrecht Dürer

DürerDürer, AlbrechtAlbrecht Durer
Finally, Dürer discusses the Delian Problem and moves on to the 'construzione legittima', a method of depicting a cube in two dimensions through linear perspective. It was in Bologna that Dürer was taught (possibly by Luca Pacioli or Bramante) the principles of linear perspective, and evidently became familiar with the 'costruzione legittima' in a written description of these principles found only, at this time, in the unpublished treatise of Piero della Francesca. He was also familiar with the 'abbreviated construction' as described by Alberti and the geometrical construction of shadows, a technique of Leonardo da Vinci.

De pictura

Della pittura
His treatment of perspective was the most influential of his recommendations, being powerfully implemented by Leonardo da Vinci, and through him to the whole Italian renaissance. Alberti made at least 29 uses of Pliny the Elder's Natural History, deriving his key themes of simplicity and seriousness directly from Pliny. For example, Alberti advised artists to use colour with restraint, and to paint in the effect of gold rather than using actual gold in their paintings. Gold did indeed vanish from Italian paintings of the second part of the 15th century. Artists however found their own ways to paint with restraint, rather than following Alberti's actual instructions directly.

Giorgio Vasari

VasariVasari, GiorgioVasari, G
He was the first to use the term "Renaissance" (rinascita) in print, though an awareness of the ongoing "rebirth" in the arts had been in the air since the time of Alberti, and he was responsible for our use of the term Gothic Art, though he only used the word Goth which he associated with the "barbaric" German style. The Lives also included a novel treatise on the technical methods employed in the arts. The book was partly rewritten and enlarged in 1568, with the addition of woodcut portraits of artists (some conjectural).

Paolo dal Pozzo Toscanelli

ToscanelliPaolo ToscanelliToscanelli, Paolo
His circle of friends included the architect of the Duomo, Filippo Brunelleschi, and the philosopher Marsilio Ficino; he knew Leon Battista Alberti, mathematician, writer and architect; and his closest friend was Cardinal Nicholas of Cusa, himself a wide-ranging intellect and early humanist, who dedicated two short mathematical works, both written in 1445, to Toscanelli, and made himself and Toscanelli the interlocutors in a dialogue entitled ‘On Squaring the Circle (De quadratura circuli) written in 1458.

Andrea Mantegna

MantegnaA. MantegnaMantegna, Andrea
Leonardo da Vinci took from Mantegna the use of decorations with festoons and fruit. Mantegna's main legacy in considered the introduction of spatial illusionism, both in frescoes and in sacra conversazione paintings: his tradition of ceiling decoration was followed for almost three centuries. Starting from the faint cupola of the Camera degli Sposi, Correggio built on the research of his master and collaborator into perspective constructions, producing eventually a masterwork like the dome of Cathedral of Parma. Mantegna's only known sculpture is a "Sant'Eufemia" in the Cathedral of Irsina, Basilicata. * Berger, John and Katya, Lying Down to Sleep. Corraini Edizioni. 2010.

Aerial perspective

atmospheric perspectiveaerialatmospheric effects
Aerial perspective was used in paintings from the Netherlands in the 15th century, and explanations of its effects were with varying degrees of accuracy written by polymaths such as Leon Battista Alberti and Leonardo da Vinci. The latter used aerial perspective in many of his paintings such as the Mona Lisa and The Last Supper. Atmospheric perspective was used in Pompeian Second Style paintings, one of the Pompeian Styles, dating as early as 30 BCE. A notable example is the Gardenscape from the Villa of Livia in Primaporta, Italy. The major component affecting the appearance of objects during daylight is scattering of light, called skylight, into the line of sight of the viewer.

Science and inventions of Leonardo da Vinci

conceptda Vinci Bridgeda Vinci's flying machine
List of works by Leonardo da Vinci. Leonardo da Vinci's personal life. The Art of War: Leonardo da Vinci's War Machines. Complete text & images of Richter's translation of the Notebooks. Leonardo da Vinci: Experience, Experiment, Design (review). Some digitized notebook pages with explanations from the British Library (Non HTML5 Available). Digital and animated compendium of anatomy notebook pages. BBC Leonardo homepage. Leonardo da Vinci: The Leicester Codex. Leonardo's Letter to Ludovico Sforza. Animations of anamorphosis of Leonardo and other artists. The Invention of the Parachute. Da Vinci - The Genius: A comprehensive traveling exhibition about Leonardo da Vinci.


the Renaissanceearly RenaissanceRenaissance Europe
One of the distinguishing features of Renaissance art was its development of highly realistic linear perspective. Giotto di Bondone (1267–1337) is credited with first treating a painting as a window into space, but it was not until the demonstrations of architect Filippo Brunelleschi (1377–1446) and the subsequent writings of Leon Battista Alberti (1404–1472) that perspective was formalized as an artistic technique. The development of perspective was part of a wider trend towards realism in the arts. Painters developed other techniques, studying light, shadow, and, famously in the case of Leonardo da Vinci, human anatomy.

M. C. Escher

EscherM.C. EscherMaurits Cornelis Escher
Escher's interest in curvilinear perspective was encouraged by his friend and "kindred spirit", the art historian and artist Albert Flocon, in another example of constructive mutual influence. Flocon identified Escher as a "thinking artist" alongside Piero della Francesca, Leonardo da Vinci, Albrecht Dürer, Wenzel Jamnitzer, Abraham Bosse, Girard Desargues, and Père Nicon. Flocon was delighted by Escher's Grafiek en tekeningen ("Graphics in Drawing"), which he read in 1959. This stimulated Flocon and André Barre to correspond with Escher and to write the book La Perspective curviligne ("Curvilinear perspective").

Italian Renaissance painting

Italian RenaissanceEarly RenaissanceItalian Renaissance painter
Masaccio's work became a source of inspiration to many later painters, including Leonardo da Vinci and Michelangelo. During the first half of the 15th century, the achieving of the effect of realistic space in a painting by the employment of linear perspective was a major preoccupation of many painters, as well as the architects Brunelleschi and Alberti who both theorised about the subject. Brunelleschi is known to have done a number of careful studies of the piazza and octagonal baptistery outside Florence Cathedral and it is thought he aided Masaccio in the creation of his famous trompe l'oeil niche around the Holy Trinity he painted at Santa Maria Novella.


During the Renaissance Italian polymaths such as Leonardo da Vinci (1452–1519), Michelangelo (1475–1564) and Leon Battista Alberti (1404–1472) made important contributions to a variety of fields, including biology, architecture, and engineering. Galileo Galilei (1564–1642), a physicist, mathematician and astronomer, played a major role in the Scientific Revolution. His achievements include key improvements to the telescope and consequent astronomical observations, and ultimately the triumph of Copernicanism over the Ptolemaic model. Other astronomers suchs as Giovanni Domenico Cassini (1625–1712) and Giovanni Schiaparelli (1835–1910) made many important discoveries about the Solar System.

Ibn al-Haytham

This translation was read by and greatly influenced a number of scholars in Christian Europe including: Roger Bacon, Robert Grosseteste, Witelo, Giambattista della Porta, Leonardo Da Vinci, Galileo Galilei, Christiaan Huygens, René Descartes, and Johannes Kepler. His research in catoptrics (the study of optical systems using mirrors) centred on spherical and parabolic mirrors and spherical aberration. He made the observation that the ratio between the angle of incidence and refraction does not remain constant, and investigated the magnifying power of a lens. His work on catoptrics also contains the problem known as "Alhazen's problem".

Fra Angelico

AngelicoBeato AngelicoBlessed Fra Angelico
Masaccio ventured into perspective with his creation of a realistically painted niche at Santa Maria Novella. Subsequently, Fra Angelico demonstrated an understanding of linear perspective particularly in his Annunciation paintings set inside the sort of arcades that Michelozzo and Brunelleschi created at San’ Marco's and the square in front of it. When Fra Angelico and his assistants went to the Vatican to decorate the chapel of Pope Nicholas, the artist was again confronted with the need to please the very wealthiest of clients. In consequence, walking into the small chapel is like stepping into a jewel box.

Perspective projection distortion

distortion of perspectiveperspective distortion
One of the first uses of perspective was in Giotto’s Jesus Before the Caïf, more than 100 years before Filippo Brunelleschi’s perspectival demonstrations galvanized the proper widespread use of convergent perspective of the Renaissance. "Artificial perspective projection" was the name given by Leonardo da Vinci to what today is called "classical perspective projection" and, as noted above, is the result of a geometric protocol. 'Artificial perspective projection' is used here rather than 'classical perspective projection' in order to recognize Leonardo's priority in developing the concept.

Donato Bramante

Leon Battista Alberti. Giorgio Vasari.

Culture of Italy

ItalianItalian cultureculture
To him we also owe a scientific discovery of the first importance in the history of art: the rules of perspective. In painting, Leonardo da Vinci and other Italian painters used a technique called sfumato that created softness in their portraits. At the same time, Italy witnessed the revival of the fresco. In music, both the small-scale madrigal and the large-scale opera were inventions of the period with a long future. Italian cities invented the modern conservatory to train professional musicians, as they invented the art academy as a place to master the techniques and the theory of painting, sculpture, and architecture.


Italian Renaissance painters and sculptors, such as Leonardo da Vinci and Michelangelo Buonarroti, and scores of others of the first rank, were greatly admired and acclaimed, and had a widespread influence on artistic concepts and esthetic standards throughout Europe. Italian artists, beginning with Giotto, mastered the use of perspective and chiaroscuro (light and shadow), which had a widespread influence on Western art. Brunelleschi, an architect, was the first to explain perspective in terms of a well-defined set of geometric rules.

Renaissance architecture

RenaissanceRenaissance styleItalian Renaissance
The influence of Renaissance architecture can still be seen in many of the modern styles and rules of architecture today. * Renaissance Architecture in Great Buildings Online The first treatise on architecture was De re aedificatoria ("On the Subject of Building") by Leon Battista Alberti in 1450. It was to some degree dependent on Vitruvius's De architectura, a manuscript of which was discovered in 1414 in a library in Switzerland. De re aedificatoria in 1485 became the first printed book on architecture.

Florentine painting

FlorentineFlorentine SchoolFlorentine art
Masaccio's work became a source of inspiration to many later painters, including Leonardo da Vinci and Michelangelo. During the first half of the 15th century, the achieving of the effect of realistic space in a painting by the employment of linear perspective was a major preoccupation of many painters, as well as the architects Brunelleschi and Alberti who both theorised about the subject. Brunelleschi is known to have done a number of careful studies of the piazza and octagonal baptistery outside Florence Cathedral and it is thought he aided Masaccio in the creation of his famous trompe l'oeil niche around the Holy Trinity he painted at Santa Maria Novella.

List of Italian inventions

Paddle boat, first designed by Leonardo da Vinci in the 1490s. Pantelegraph, a device for telegraphic transmission of writing and drawing invented by Giovanni Caselli. Commercial service started in 1865. It was the first functional Fax Machine to enter commercial service. Parachute, dates back to the Renaissance Italy. Personal Computer (in a broad sense, not referring to the modern IBM PC compatible architecture), due to the pioneering work of Pier Giorgio Perotto. Perspective Linear perspective was first invented by Renaissance architect Filippo Brunelleschi, in Florence, who created a system that helped show how objects shrink in size according to their distance from the eye.