Perspective

1. Perspective Part 2

2. Raphael, The School of Athens (1509) • An example of one-point perspective. • Can you find the vanishing point? • Raphael painted this using mathematical discoveries of perspective from the 1400s. • We’ll discuss this mathematics today. • Note the tiles!

3. Tintoretto, Finding of the body of St Mark (1562) • Where is the vanishing point? • Tintoretto used mathematical principles from the 1400s to achieve this perspective. • Tile floor! • Note the scale of the squares on the columns as they recede into the distance.

4. “Pavimento Pictures” • “Pavimento” is Italian for floor. • It is hard accurately to draw a tile floor in perspective by visual impression alone. • A mathematical discovery in 1413 made it possible finally to do this with precision. • Using this discovery, artists such as Raphael and Tintoretto (the two we just saw) were able to make remarkable use of perspective in their art. • This use of perspective required painters of the time to be very good at geometry and proportional reasoning. Some of the most famous artists of the Renaissance were excellent mathematicians also. • Recall a few more examples from last time…

5. Vermeer, The music lesson (1662)

6. Here’s a detail of the tile floor.

7. Here’s a detail of a tile floor from Dali’s Crucifixion (1950s), which we saw last time.

8. Brunelleschi’s Discovery • Brunelleschi gave the first correct formulation of linear perspective in 1413. • There should be a single vanishing point to which all parallel lines in a plane converge. • But also he understood scale, so figures in distance are correct proportion.

9. Brunelleschi’s Demonstration that his method worked • Brunelleschi made a small hole in his painting at the vanishing point. • A spectator looked through the hole from behind the painting at a mirror which reflected the painting. • In this way Brunelleschi controlled precisely the position of the spectator so that the geometry was guaranteed to be correct.

10. Leon Battista Alberti • Alberti was the first to write down the mathematical rules of perspective in 1435. • His (very mathematical) definition of painting: A painting is the intersection of a visual pyramid at a given distance, with a fixed center and a defined position of light, represented by art with lines and colours on a given surface.

11. Piero della Frencesca, Flagellation (1470) It wasn’t till 1450 that Pierro della Francesca wrote out the mathematical arguments for why the techniques worked. Where is the “visual pyramid”? Note the tile floor!

12. Alberti’s Method for drawing tile floors • Now we’ll finally see how to draw a tile floor. • We’ll do this on the handout.

13. Mark your vanishing point on the horizon on the canvas. You are looking “through” the canvas to this point. • We are drawing a 5 x 5 tile floor. From each mark on the bottom of the canvas draw a line to the vanishing point. • Now mark a point to the right of the canvas on the horizon. The distance from the frame on the horizon should equal the distance of your eye from the canvas. • Draw a diagonal from this point to the bottom left corner of the frame. • Wherever the diagonal intersects your six lines, these will be the places where the horizontal lines of the tile floor intersect the five lines. • Finally, color in every other tile.

14. A hint at why this works • We created the red triangle so that it is congruent to the blue triangle. • Then using basic geometry about triangles, you can prove that Alberti’s method works.

15. Albrecht Durer’s grid • Soon, other artists developed even more sophisticated techniques. • Durer (1600s) introduced the notion of a transparent grid by which to sight the figure to be painted. • The artist looked through the grid and sketched what he saw on a canvas with a similar grid.

16. Here’s an engraving by Durer showing how the method works.