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Albrecht durer ppt

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Prof Suzuki
7 June 2011

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Albrecht durer ppt

  1. 1. Albrecht D ürer: Art, Geometry, and Printing <ul><li>&quot;Without [the art of measurement] no one can become a true artisan.&quot; Albrecht D ürer The Painter's Manual (1525) </li></ul>
  2. 2. Welcome to the Renaissance! <ul><li>Vasari's Lives of the Artists: &quot;Rebirth&quot; of classical forms. </li></ul><ul><li>But: What are classical forms? </li></ul>
  3. 3. Medieval or Classical? <ul><li>Notre Dame, Paris </li></ul><ul><li>Crude, barbaric. (&quot;Gothic&quot;, but actually, medieval) </li></ul><ul><li>Pantheon, Rome </li></ul><ul><li>Domes, columns </li></ul>
  4. 4. Medieval or Classical? <ul><li>Book of Hours (Duke of Berry) </li></ul><ul><li>Non-realistic </li></ul><ul><li>Boscoreale (Naples) </li></ul><ul><li>Realistic depiction of space </li></ul>
  5. 5. Medieval or Classical? <ul><li>Merovingian minuscule. </li></ul><ul><li>&quot;Form of torture&quot; </li></ul><ul><li>Latin capitals. </li></ul><ul><li>Easy to read. </li></ul>
  6. 6. A Quest for Lost Arts <ul><li>How did the Romans and Greeks make their art? </li></ul><ul><li>Many artifacts, but few documents. </li></ul><ul><li>Reverse engineering! </li></ul>Raphael's School of Athens (1511)
  7. 7. A Clasical Education <ul><li>&quot;Libera&quot; (free men) vs. &quot;Manus&quot; (hand work) </li></ul><ul><li>Plato's educational system ( The Republic ) </li></ul><ul><ul><li>Arithmetic (number properties) </li></ul></ul><ul><ul><li>Geometry (Euclidean) </li></ul></ul><ul><ul><li>Music (as applied arithmetic) </li></ul></ul><ul><ul><li>Astronomy (as applied geometry) </li></ul></ul><ul><li>Math is all the liberal arts </li></ul>
  8. 8. The Degeneration of Philosophy <ul><li>Martianus Capella (fl. 420) </li></ul><ul><ul><li>The Marriage of Philo and Mercury </li></ul></ul><ul><ul><li>Geometry reduced to areas, lengths, geography </li></ul></ul><ul><li>Boethius (fl. 500) </li></ul><ul><ul><li>Quadrivium </li></ul></ul><ul><ul><li>Basis for medieval education </li></ul></ul>
  9. 9. The Rebirth of Western Mathematics <ul><li>Islamic Civilization </li></ul><ul><ul><li>Preserved and extended Greek geometry </li></ul></ul><ul><ul><li>Abu'l-Wafa On the Parts of Geometry Necessary for Artisans </li></ul></ul><ul><ul><li>Geometric designs </li></ul></ul><ul><li>The Spanish Connection </li></ul><ul><li>Campanus of Novara (fl. 1300) </li></ul><ul><ul><li>New edition of Elements </li></ul></ul><ul><ul><li>Geometry more than lengths and areas </li></ul></ul>
  10. 10. The Mathematization of Art <ul><li>Cathedral Building </li></ul><ul><li>Jordanus Nemorarius (fl. 1225): On Weights </li></ul><ul><li>Filippo Brunelleschi (1377-1446) </li></ul><ul><ul><li>Architect of duomo </li></ul></ul><ul><ul><li>Perspective </li></ul></ul><ul><ul><li>Art is applied math! </li></ul></ul>
  11. 11. Leon Battista Alberti (1404-1472) <ul><li>On Painting (1435) </li></ul><ul><ul><li>Painting as a liberal art </li></ul></ul><ul><ul><li>Mathematical basis </li></ul></ul><ul><li>On Architecture (1452) </li></ul><ul><ul><li>Study of Roman ruins, Vitruvius </li></ul></ul><ul><ul><li>Importance of ratios and proportion </li></ul></ul>Santa Maria Novella (Florence)
  12. 12. Vitruvius Ten Books on Architecture <ul><li>Inspired Alberti, Pacioli, da Vinci </li></ul><ul><li>Importance of ratios </li></ul><ul><ul><li>Face: Body = 1:10 </li></ul></ul><ul><ul><li>Arms: Legs = 1:1 </li></ul></ul><ul><ul><li>Foot: Body = 1:6 </li></ul></ul><ul><ul><li>Forearm: Body = 1:4 </li></ul></ul><ul><li>Aesthetics quantifiable! </li></ul>da Vinci's &quot;Vitruvian Man&quot;
  13. 13. Every Artist a Geometer... <ul><li>Piero della Francesca (1415-1492) </li></ul><ul><ul><li>&quot;Flagellation&quot; one of oldest to use true perspective. </li></ul></ul><ul><ul><li>Treatise on the Abacus </li></ul></ul><ul><ul><li>Five Books on the Regular Solids . </li></ul></ul>
  14. 14. ...and every Geometer an Artist <ul><li>Luca Pacioli (1446-1517) </li></ul><ul><ul><li>Summary of Arithmetic, Geometry, Proportion, and Proportionality (1494) </li></ul></ul><ul><ul><li>The Divine Proportion (1509), includes Francesca's Five Books (illustrated by da Vinci) </li></ul></ul>
  15. 15. N ürnberg in the Fifteenth Century <ul><li>Political center (Diet meetingplace) </li></ul><ul><li>Oldest paper mills in Germany </li></ul><ul><li>Printing center (Regiomonantus's press) </li></ul><ul><li>Weltchronik of Wolgemut </li></ul>
  16. 16. Albrecht D ürer <ul><li>Born N ürnberg, 1471 </li></ul><ul><li>Education of an Artist </li></ul><ul><ul><li>Apprenticed to Wolgemut </li></ul></ul><ul><ul><li>Carved wood blocks for print illustrations </li></ul></ul><ul><ul><li>Two Italian trips (1494-5, 1505-7) </li></ul></ul><ul><ul><li>Met Pacioli, studied Alberti, Euclid, Vitruvius </li></ul></ul>
  17. 17. The Painter's Manual (1525) <ul><li>&quot;Nothing is more annoying to men of understand in than a blunder in a painting...&quot; </li></ul><ul><li>Practical geometric construction for artists. </li></ul><ul><li>Theoretical geometry </li></ul>
  18. 18. Artist's Problem #1 <ul><li>How do you draw an object? </li></ul><ul><li>Composition of basic shapes. </li></ul><ul><li>Straight lines and circles... </li></ul><ul><li>...but other curves (e.g., helix) required for &quot;realism.&quot; </li></ul>Pope Benedict in Munich
  19. 19. Basic Drawing <ul><li>Straight lines and circles easy to construct. </li></ul><ul><li>Paraphernalia of daily life often includes other shapes. </li></ul><ul><li>Crozier: Spiral and circular arcs. </li></ul>Croziers from Metropolitan Mustum of Art
  20. 20. D ürer's Spirals <ul><li>Divide AB into fourths at C, D, E. </li></ul><ul><li>Bisect D, F. </li></ul><ul><li>Draw semicircles AB, BC, CE, ED with centers alternating D, F (final semicircle DF with center at midpoint). </li></ul><ul><li>Non-Archimedean spiral. </li></ul>
  21. 21. Plan and Elevation <ul><li>Representing space curves. </li></ul><ul><li>Dual viewpoint. </li></ul><ul><li>Detailed instructions on how to construct. </li></ul><ul><li>Adapted from architectural plans. </li></ul>
  22. 22. Conic Sections for Artists <ul><li>The ellipse as a conic section. </li></ul><ul><li>D ürer invents German name ( &quot;egg-shaped line&quot;) </li></ul><ul><li>Similar constructions for parabola (&quot;burning line&quot;) and hyperbola (&quot;fork shaped line&quot;). </li></ul><ul><li>First contour maps? </li></ul>
  23. 23. D ürer's Polygons <ul><li>Simple construction for regular hexagon: no need to change compass opening. </li></ul><ul><li>Join alternate vertices for equilateral triangle. </li></ul>
  24. 24. The Heptagon <ul><li>Side of heptagon is half side of equilateral triangle. </li></ul><ul><li>Method given by Abu'l-Wafa ( On the Parts of Geometry Necessary for Artisans ) </li></ul><ul><li>D ürer's method new to Europe? </li></ul><ul><li>Kepler: &quot;Obviously&quot; too short. </li></ul>
  25. 25. How Close to Exact? <ul><li>H is vertex of regular heptagon. </li></ul><ul><li>For circle with diameter 20 feet, approximation gives side of heptagon as 8 feet, 7.92 inches vs. actual side of 8 feet, 8.13 inches. </li></ul>
  26. 26. Pentagons <ul><li>Bisect radius OA at C. Join CB, and mark CD equal to CB. BD is side of pentagon. </li></ul><ul><li>Method given by Ptolemy in Almgest </li></ul><ul><li>Exact construction. </li></ul>
  27. 27. Rusty Compass Pentagon <ul><li>AB as side, draw circles with centers A, B and radius AB. </li></ul><ul><li>Join CD. Draw circle with center D and radius AB. </li></ul><ul><li>Draw EG to H and FG to I. </li></ul><ul><li>Make IJ, HJ equal to AB. ABHJI is equilateral pentagon. </li></ul>
  28. 28. Also Approximate <ul><li>Pentagon is equilateral but not equiangular </li></ul><ul><li>Method from Geometria Deutsch of Matthias Roritzer </li></ul><ul><li>&quot;Rusty compass&quot; problems appear in Islamic geometry (Abu'l-Wafa gave exact construction for pentagon). </li></ul>
  29. 29. Tiling Patterns <ul><li>Floor covering using polygons </li></ul><ul><li>Regular tiling using squares, triangles, octagons. </li></ul><ul><li>Heptagonal Tilings </li></ul><ul><li>Pentagonal Tilings </li></ul>
  30. 30. A Near Miss... <ul><li>Pentagons </li></ul><ul><ul><li>Joined rosettes </li></ul></ul><ul><ul><li>Fill space with other figures </li></ul></ul><ul><li>Kepler (1619) </li></ul><ul><ul><li>Critiqued D ürer's heptagon </li></ul></ul><ul><ul><li>No mention of Dürer tilings </li></ul></ul><ul><li>Penrose (1974) </li></ul>
  31. 31. ...Or Not <ul><li>Problem: How do you represent three-dimensional objects schematically ? </li></ul><ul><li>&quot;Framework&quot; to show structure (Leonardo's drawing of rhombi-cuboctahedron from Pacioli's Divine Proportion ) </li></ul>
  32. 32. D ürer's Polyhedral Nets Dodecahedron Rhombicuboctahedron
  33. 33. Mapping the World <ul><li>Can &quot;unwrap&quot; sphere in same way (D ürer). </li></ul><ul><li>Spherical projection problem. </li></ul><ul><li>Waldseem üller map (1506) first to use designation &quot;America.&quot; </li></ul>
  34. 34. Artist's Problem #2 <ul><li>How do you make a painting look like a real scene? </li></ul><ul><li>More distant objects look smaller (foreshortening). </li></ul><ul><li>Convergence of parallel lines (perspective). </li></ul>A Building in N ürnberg
  35. 35. Foreshortening <ul><li>Used to convey sense of distance. </li></ul><ul><li>D ürer used (St. Jerome woodcut) </li></ul><ul><li>Mathematical basis: Objects of equal size subtend smaller angles at greater distances. </li></ul>
  36. 36. Practical Design <ul><li>Inscriptions on buildings and columns. </li></ul><ul><li>Equal arcs make for equal apparent size (from vantage point). </li></ul><ul><li>Modern street writings. </li></ul>
  37. 37. Perspective <ul><li>Radial symmetry about vanishing point. </li></ul><ul><li>Not in Classical Art! </li></ul><ul><li>Renaissance invention. </li></ul>
  38. 38. Woodcuts and Paintings <ul><li>D ürer learns &quot;secret&quot; during 1506 visit to Italy. </li></ul><ul><li>Explains technique in Painter's Manual. </li></ul>
  39. 39. Ratios of Gold... <ul><li>Pacioli's Divine Proportion </li></ul><ul><ul><li>Ancients constructed temples and art using key ratios. </li></ul></ul><ul><ul><li>Ratio between diagonal of pentagon and side is &quot;Golden Ratio&quot; </li></ul></ul>
  40. 40. ...and Dross <ul><li>More natural to use Fibonacci sequence </li></ul><ul><li>ALL Vitruvian ratios rational. </li></ul><ul><li>NO evidence for deliberate use of side/diagonal of pentagon. </li></ul><ul><li>Pacioli doesn't use golden ratio as basis. </li></ul>
  41. 41. A Brief History of Typography <ul><li>Roman majuscule (CAPITAL LETTERS). </li></ul><ul><li>Merovingian minuscule (lower case letters, joined with ligatures). </li></ul><ul><ul><li>t, e, i, r: </li></ul></ul><ul><ul><li>er </li></ul></ul><ul><ul><li>et </li></ul></ul><ul><ul><li>te </li></ul></ul><ul><ul><li>ti </li></ul></ul><ul><ul><li>tr </li></ul></ul>Merovingian script, from Luxeuil manuscript. (From http://medievalwriting.50megs.com/scripts/examples/luxeuil.htm)
  42. 42. Carolingian Minuscule <ul><li>Alcuin and Charlemagne </li></ul><ul><li>Reform of Merovingian script </li></ul><ul><ul><li>Basis for modern Roman lower case </li></ul></ul><ul><ul><li>Far easier to read and write! </li></ul></ul><ul><li>Degeneration of Roman type? </li></ul>
  43. 43. Manuscripts and Printers <ul><li>Printing press </li></ul><ul><ul><li>China/Korea </li></ul></ul><ul><ul><li>Print money, playing cards. </li></ul></ul><ul><li>Gutenberg </li></ul><ul><li>What should letters look like? </li></ul><ul><ul><li>Challenge: Make an &quot;A&quot; using the Golden Ratio (or Fibonacci Rectangle) </li></ul></ul>
  44. 44. Classic Fonts <ul><li>Felice Feliciano </li></ul><ul><li>Pacioli </li></ul><ul><ul><li>Letters in appendix to Divine Proportion </li></ul></ul><ul><ul><li>Based on circle and square </li></ul></ul><ul><ul><li>Thick limbs = 1/9 height, thin limbs = 1/2 of thick </li></ul></ul><ul><ul><li>NO golden ratio. </li></ul></ul>
  45. 45. D ürer's Letters <ul><li>Circle and square </li></ul><ul><li>Ratios </li></ul><ul><ul><li>Square into four parts </li></ul></ul><ul><ul><li>Thick limbs = 1/10 height, thin limbs 1/3 of thick. </li></ul></ul><ul><ul><li>Top center of width. </li></ul></ul><ul><ul><li>Base serifs use circle with radius 1/7 of side of square </li></ul></ul>
  46. 46. Gothic Letters, Old Style <ul><li>Squares atop squares </li></ul><ul><li>1/3 division of sides </li></ul><ul><li>Vertical diagonals </li></ul><ul><li>Letters as components of other letters (&quot;i&quot; as basis for u, b, etc.) </li></ul>
  47. 47. Gothic Letters, New Style <ul><li>Old style, with additional serifs, squares. </li></ul><ul><li>Roman type only upper case letters. </li></ul><ul><li>D ürer first (?) to geometrize lower case letters. </li></ul>
  48. 48. Reactions <ul><li>Aesthetics </li></ul><ul><ul><li>Cresci (1560): Curves of actual letters not circles and straight lines. </li></ul></ul><ul><ul><li>Letters &quot;ugly&quot; </li></ul></ul><ul><li>Practicality </li></ul><ul><ul><li>No one learns to write this way! </li></ul></ul><ul><ul><li>Difficult to cut type metal this precisely </li></ul></ul><ul><ul><li>Impression printing has no need for mathematical defined fonts. </li></ul></ul>
  49. 49. The Computer Connection <ul><li>Bit map fonts </li></ul><ul><ul><li>Graininess problem </li></ul></ul><ul><ul><li>One image for each size of each letter </li></ul></ul><ul><li>True-Type fonts </li></ul><ul><ul><li>Letters defined by lines and curves </li></ul></ul><ul><ul><li>Curves beyond circle (splines) </li></ul></ul><ul><li>Printers (and computer screens) </li></ul><ul><ul><li>Ink-jet: True-Type to bitmap to print </li></ul></ul><ul><ul><li>Laser printer: True-Type to laser pointer to print </li></ul></ul>
  50. 50. Some Final Thoughts <ul><li>Thinking componentwise and choosing what is important are essential skills in the creation of art...and mathematics. </li></ul><ul><li>The constructive arts are an underutilized resource in the teaching of mathematics. </li></ul>

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