Mathematical Perspective. Alex Jang, Shannon Jones, Anna Shapiro

Transcription

1 Mathematical Perspective Alex Jang, Shannon Jones, Anna Shapiro

2 Paintings During the Middle Ages -Often focusing on religion -Less attention to the body and detail -Sometimes very strange -Rarely, if ever, three dimensional

3 What Is Perspective?

4 Filippo Brunelleschi Not much was known about his early life, but apparently: - Born in 1377 Middle child of three children Became a master goldsmith in 1398 Not much was known about his transition from goldsmith to architect, or his training in art. Around 1400, a movement called Humanism became popular - - idealized the art of ancient Greece and Rome as opposed to those of the formal and lifeless Middle Ages Brunelleschi was inspired while studying ancient Roman Architecture Most of his life was dedicated to building the dome of the Florence Cathedral, a major architectural feat of the Renaissance

5 Brunelleschi s Early Application of Linear Perspective Brunelleschi s painting of the Florentine Baptistery is one of the earliest and most popular applications of linear perspective.. Curiously, Brunelleschi intended that the Baptistery painting only be observed by the viewer facing the Baptistery, looking through a hole in the painting, drilled through the central vanishing point, from the unpainted backside. As a mirror was moved into and out of view, the observer saw the striking similarity between the actual view of the Baptistery, and the reflected view of the painted Baptistery image. Brunelleschi wanted his new perspective "realism" to be tested not by comparing the painted image to the actual Baptistery but to its reflection in a mirror according to the Euclidean laws of geometric optics. This feat showed artists vividly how they might paint their images, not merely as flat two-dimensional shapes, but looking more like three-dimensional structures, just as mirrors reflect them. Both panels have since been lost. Around this time linear perspective spread not only in Italy but throughout Western Europe. It quickly became, and remains, standard studio practice.

6 Activity - For You to Try! With a partner, find the vanishing point(s) and orthogonal lines in the pictures that follow

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13 Leon Battista Alberti February 14, April 25, 1472 Italian artist, architect, linguist, and poet His mother is unknown and he was believed to be an illegitimate child His father s family was viewed as responsible for the success of Florence in the 14th century He went to boarding school in Padua and then studied Law at Bologna Travelled to Rome in 1431 Studied ancient ruins during that time 1435 his treatise Della pittura (On Painting) Inspired by pictorial art in Florence in the early 15th century First text on the geometry of perspective Believed that the first requirement of a painter was to know geometry Construction of a tiled floor in perspective He also later wrote treatises On Architecture and On the Art of Building

14 Construction of a Tiled Floor in Perspective Alberti gave no demonstration as to why this is a correct construction, but it has since been proved algebraically. 1) 2) 3) 4) 5) 6) 7) Draw a rectangle. Label equally spaced points along the bottom side of your rectangle. Label V as your vanishing point. Construct straight lines from each of your equally spaced points to V. Now draw straight lines from A, the corner of your triangle, to each of the equally spaced points. Now draw horizontal lines beginning from the intersection of your dashed lines and BV Continue doing this until you have a full tile floor and erase the dashed lines.

15 Construction of a Tiled Floor in Two Point Perspective

16 Projective Desargues Theorem: If two triangles are in perspective from a point, then their pairs of corresponding sides meet on a line. Desargues Theorem Desargues never published this theorem by itself, but it is included in Universal Method of M. Desargues for Using Perspective, published by his friend Abraham Bosse in 1648

17 Desargues Theorem - Coplanar Triangles

18 Albrecht Dürer Quick Timeline of Albrecht Dürer s Life Dürer born as the 3rd of 18 children Dürer produces many woodcut prints, which is what he is known for before his work on perspective, along with his paintings Dürer travels to Bologna to learn about the secret art of perspective. Probably obtains an untranslated version of Piero della Francesca s De Prospectiva Pingendi Maximilian I, Holy Roman Emperor, becomes Dürer s sponsor Earliest known draft of The Painter s manual Dürer produces many less works of art, focusing more on his work on perspective First edition of Painter s Manual Publishes Four Books on Measurement the first mathematics books for adults Dürer dies at the age of 56

19 Art of Perspective Since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art but the book is to be one for practical use, and not a treatise on pure mathematics. - Dürer Diagram for how to draw a cube using perspective, including the light source and shadows This figure shows how to accurately draw conic sections, using perspective and explaining why and how to scale each section For example, a circle when drawn realistically will look like an ellipse. Dürer s legacy is found mostly in his insistence that perspective is not only important for architects and painters, but for mathematicians as well.

20 Dürer s Devices for Drawing Using Perspective Dürer includes two apparatuses that can be used to aid in drawing using perspective, and explains how to build them. The first one builds off of taking a window panel, drawing a viewing hole through it, and tracing what you see through the hole The second one, pictured to the left, uses a large needle and thread to establish a vanishing point and the parallel lines of your object These can still be recreated today and allowed for painters to draw using perspective without knowing too much of the math behind it - sad, I know

21 Legacy of Perspective in Art During the Renaissance M.C. Escher s Views on Perspective

22 THE END

23 References Annalisa Crannell, Math and Art: the Good, the Bad, and the Pretty - (presentation) Albrecht Durer, Painter's Manual John Stillwell, The Four Pillars of Geometry Marc Frantz and Annalisa Crannell, Viewpoints: Mathematical Perspective and Fractal Geometry in Art Victor J. Katz, A History of Mathematics: An Introduction Dan Pedoe, Geometry and the Visual Arts