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Algorithmic Mathematical Art: Special Cases and Their Applications - PowerPoint PPT Presentation

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DIMACS Workshop on Algorithmic Mathematical Art: Special Cases and Their Applications May 11 - 13, 2009 DIMACS Center, CoRE Building, Rutgers University Jean-Marie Dendoncker Dimacs Algorithmic Mathematical Art 1 DIMACS Algorithmic

  1. DIMACS Workshop on Algorithmic Mathematical Art: Special Cases and Their Applications May 11 - 13, 2009 DIMACS Center, CoRE Building, Rutgers University Jean-Marie Dendoncker Dimacs Algorithmic Mathematical Art 1

  2. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 0. A picture of the context A primary school with • 100% of the children who don’t speak Dutch at home • 65 % of the children are underprivileged • 45% are refugees “How can we help these children?” Dimacs Algorithmic Mathematical Art 2

  3. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.1 Arithmetic algorithm: tables of multiplication Dimacs Algorithmic Mathematical Art 3

  4. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 4

  5. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 5

  6. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 6

  7. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 7

  8. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 8

  9. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 9

  10. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 10

  11. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 11

  12. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 12

  13. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 13

  14. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 14

  15. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation What if the basic curve is an ellipse instead of a circle? Dimacs Algorithmic Mathematical Art 15

  16. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 16

  17. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 17

  18. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 18

  19. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 19

  20. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 20

  21. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Dimacs Algorithmic Mathematical Art 21

  22. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.2 Geometric algortihm: wavefronts in a two dimensional representation Solution: Evolute ellipse Tetracuspid curve Dimacs Algorithmic Mathematical Art 22

  23. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3 Geometric algortihm in a three dimensional representation A further generalization is to visualize curves in space 1.3.1 Wavefront surface 1.3.2 Cardioid and nephroid 1.3.3 Hyperbolic paraboloid 1.3.4 Conoid 1.3.5 Surface of Scherk 1.3.6 Elliptic surface Dimacs Algorithmic Mathematical Art 23

  24. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.1 Wavefronts in a three dimensional representation Dimacs Algorithmic Mathematical Art 24

  25. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.1 Wavefronts in a three dimensional representation Some properties of the wavefront surface Dimacs Algorithmic Mathematical Art 25

  26. elliptic ridge Point of curvature of the basic ellipse focal point of the basic ellipse Dimacs Algorithmic Mathematical Art 26

  27. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.2 Cardioid and nephroid in a three dimensional representation Dimacs Algorithmic Mathematical Art 27

  28. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.2 Cardioid and nephroid in a three dimensional representation Dimacs Algorithmic Mathematical Art 28

  29. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.3 Hyperbolic paraboloid Using the same way of curve stitching to visualise a parabola it’s possible to do the same in space. Dimacs Algorithmic Mathematical Art 29

  30. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.3 Hyperbolic paraboloid Dimacs Algorithmic Mathematical Art 30

  31. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.3 Hyperbolic paraboloid ?? I have a little problem: ‘ There’s a hole in my bucket ‘ by Harry Belafonte (and my mother) Dimacs Algorithmic Mathematical Art 31

  32. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.3 Hyperbolic paraboloid , NO Dimacs Algorithmic Mathematical Art 32

  33. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.4 Conoid Dimacs Algorithmic Mathematical Art 33

  34. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.5 Surface of Scherk The hyperbolic paraboloid should not be confused with the surface of Scherk (1798-1885). This surface is the only non trivial minimal translation surface. It can be given, with disregard of a translation and homothetic transformation, by the equation . cos y  z ln cos x It is formed by shifting in perpendicular planes without losing contact with each other the two curves , 1 1        g ( x ) ln cos( cx c 0 ) c h ( x ) ln cos( cx d 0 ) d 1 1 c c , , , with integration constants c c d d 0 1 0 1 Dimacs Algorithmic Mathematical Art 34

  35. DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker 1. Projects : IT’S MATHEMA G IC Van Maat tot Math - 1.3.5 Surface of Scherk as a translation surface as a minimal surface Dimacs Algorithmic Mathematical Art 35

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