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

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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

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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 1 Dimacs Algorithmic Mathematical Art

Jean-Marie Dendoncker

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math
  • 0. A picture of the context

2 Dimacs Algorithmic Mathematical Art

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?”

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.1 Arithmetic algorithm: tables of multiplication

3 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

4 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

5 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

6 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

7 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

8 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

9 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

10 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

11 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

12 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

13 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

14 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

15 Dimacs Algorithmic Mathematical Art

What if the basic curve is an ellipse instead of a circle?

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

16 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

17 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

18 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

19 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

20 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

21 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.2 Geometric algortihm: wavefronts in a two dimensional representation

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Solution:

Evolute ellipse Tetracuspid curve

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3 Geometric algortihm in a three dimensional representation

23 Dimacs Algorithmic Mathematical Art

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

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.1 Wavefronts in a three dimensional representation

24 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.1 Wavefronts in a three dimensional representation

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Some properties of the wavefront surface

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elliptic ridge focal point of the basic ellipse Point of curvature

  • f the

basic ellipse

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.2 Cardioid and nephroid in a three dimensional representation

27 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.2 Cardioid and nephroid in a three dimensional representation

28 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.3 Hyperbolic paraboloid

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Using the same way of curve stitching to visualise a parabola it’s possible to do the same in space.

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.3 Hyperbolic paraboloid

30 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.3 Hyperbolic paraboloid ??

31 Dimacs Algorithmic Mathematical Art

I have a little problem: ‘ There’s a hole in my bucket ‘

(and my mother) by Harry Belafonte

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.3 Hyperbolic paraboloid , NO

32 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.4 Conoid

33 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.5 Surface of Scherk

34 Dimacs Algorithmic Mathematical Art

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 . It is formed by shifting in perpendicular planes without losing contact with each other the two curves ,

x cos y cos ln  z

with integration constants

1 0)

cos( ln 1 ) ( c c cx c x g    

1 0)

cos( ln 1 ) ( d d cx c x h   

1 1

, , , d d c c

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.5 Surface of Scherk

35 Dimacs Algorithmic Mathematical Art

as a translation surface as a minimal surface

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.6 Elliptic surface

36 Dimacs Algorithmic Mathematical Art

2 2 2 1 1 1 2 2 2 2 1 1 1 1

ˆ ˆ 3 1 , 3 1 Q O A Q O B A O B O B O A O   

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.6 Elliptic surface

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.6 Elliptic surface

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Some properties of the elliptic surface

1. Contour lines

  • n

¼- ½- ¾

  • f the

distence between the two ellipses . 2. The angle between the rulings and the plane of the ellipses is constant. 3. The length between two connected points P1 and P2 is constant.

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.7 Two experiments : the Euler characteristic

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Find the mystery of Euler Exercise 1 Exercise 2 Exercise 3 Only after making the real models, more children understood the general calculating method.

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.7 Two experiments : the mobile hyperboloid

40 Dimacs Algorithmic Mathematical Art

Let ‘s try

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 1. Projects : IT’S MATHEMAGIC
  • Van Maat tot Math

1.3.7 Two experiments : conclusion

41 Dimacs Algorithmic Mathematical Art

I can conclude as Prof. Eisenberg of the University of Colorado describes in his paper ‘Mathematical Crafts for children: Beyond Scissors and Glue’, that mathematical crafts have to be seen as a strong element of mathematical education. It’s clear that the use of algorithms gives to young underprivileged children a better structure not only in the use of mathematics but also in their lives.

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.1 Finite geometry

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A second type of examples of a practice of algorithms in mathematics is the use of graph theory. The goal of this project was to make some representations

  • f models that occur in a finite space.
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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.1 Fano configuration (7,7,3)

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This problem has no concrete solutions. Instead of using lines as real lines, we represented them as

  • points. Lines are then nothing else then a sequence of

three points. After numbering the points respectively 1, 2, 3, … 7, it is possible to find the lines as

{1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 6}, {2, 5, 7}, {3, 4, 7}, {3, 5, 6}

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.1 Fano configuration

44 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.2 Desargues configuration (10,10,3)

45 Dimacs Algorithmic Mathematical Art

To make a realizable spatial configuration, you number the points as 12, 13, 14, 15 23, 24, 25 34, 35 and 45 The lines are formed by three points with only three different digits. Doing this you obtain: {12, 13, 23}, {12, 14, 24}, {12, 15, 25}, {13, 14, 34}, {13, 15, 35}, {14, 15, 45}, {23, 24, 34}, {23, 25, 35}, {24, 25, 45} and {34, 35, 45} Solution:

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.2 Desargues configuration

46 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.3 (15,15,3) Tutte configuration (15,15,3)

47 Dimacs Algorithmic Mathematical Art

For this purpose the points are numbered as 12, 13, 14, 15, 16 23, 24, 25, 26 34, 35, 36 45, 46 56 . Three points form a line if their numbers contain all the digits from 1 to 6 {12, 34, 56}, {12, 35, 46}, {12, 36, 45}, {13, 24, 56}, {13, 25, 46}, {13, 26, 45}, {14, 23, 56}, {14, 25, 36}, {14, 26, 35}, {15, 23, 46}, {15, 24, 36}, {15, 26, 34}, {16, 23, 45}, {16, 24, 35} {16, 25, 34}. Solution:

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.3 (15,15,3): GQ(2,2)

48 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.4 (45,27,3): GQ(4,2)

49 Dimacs Algorithmic Mathematical Art

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.5 Exercise 1: a colorproblem on a torus

50 Dimacs Algorithmic Mathematical Art

How many colors are at least needed to color an arbitrary map

  • n a torus so that each adjacent ‘country’ has another color?

Solution:

{1, 3, 4} {2, 4, 5} {3, 5, 6} {4, 6, 7} {5, 7, 1} {6, 1, 2} {7, 2, 3} i.e. the Fano configuration

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.5 Exercise 1: model

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.5 Exercise 2: two new olympic disciplines

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Shot put Discus throw

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 2. Higher degree mathematics IT’S MATHEMAGIC

2.6 Other models

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

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  • He formulated his fundamental ideas in

Le nombre plastique Quinze leçons sur l’ordonnance architectonique Brill Leiden 1960 De Architectonische ruimte Vijftien lessen over de dispositie van het menselijk verblijf Brill Leiden 1983

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

57 Dimacs Algorithmic Mathematical Art

STEP 1: first experiment A sheet of paper of 50 cm torn in two equal parts produced in his experiment with 50 people lengths between 24.5 cm or 25.5 cm so that they differ 1/25 to each other.

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

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STEP 2: second experiment

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

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STEP 3: classification of 36 squares

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

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STEP 4: margin, type and order of size

ORDER OF SIZE

40 42 56 75 98 130

M A R G I N

44 58 78 102 135 46 60 81 106 141 48 63 84 110 147 50 66 87 115 153 52 69 90 120 159 54 72 94 125 165

Type I Type II Type III Type IV Type V Type VI

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

61 Dimacs Algorithmic Mathematical Art

STEP 5: the common ratio of the geometric sequence

To establish the basic proportion of the real three dimensional quantity it’s necessary to know the size

  • f the smallest different size.

Dimension 1: I + I = II Dimension 2: I + II = III Dimension 3: I + II =IV Due to this last requirement, there exists a fixed common ratio between the threshold measures. Golden number Plastic number

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

62 Dimacs Algorithmic Mathematical Art

STEP 5: the common ratio of the geometric sequence

So Van der Laan determined exactly the common ratio of the geometric sequence of the different threshold values of the different types of sizes.

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

63 Dimacs Algorithmic Mathematical Art

STEP 6: the extent of the order of size

As I +II= IV than VI-V = (III+IV) - (II+III) = IV – II = I

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

64 Dimacs Algorithmic Mathematical Art

STEP 7: a total system of eight measures

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

65 Dimacs Algorithmic Mathematical Art

CONCLUSION: a system of eight measures

Type Name Ratio Plastic ratio I Small element 1 1 II Great element 4/3 1,3247… = Ψ III Small piece 7/4 1,7548… = Ψ2 IV Great piece 7/3 2,3247… = Ψ3 V Small part 3 3,0795… = Ψ4 VI Great part 4 4,0795… = Ψ5 VII Small ensemble 5 1/3 5,4043… = Ψ6 VIII Great ensemble 7 7,1591… = Ψ7 Δ = 1 H(VIII, VII) = 6,1591 VIII – H(VIII,VII) = 1 I + II = IV

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

66 Dimacs Algorithmic Mathematical Art

STEP 8: morphic numbers

A real number a is a morphic number if there exist two natural numbers k and l so that

l

a a

 1

and

k

a a  1

Only Φ and Ψ are morphic numbers. Kruijtzer, Aarts and Fokkink 2002 Ψ is a morphic number as I + II = IV and VI – I = V

4 3   l k

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

67 Dimacs Algorithmic Mathematical Art

STEP 9: polynomiography

Golden number Plastic number: I + II = IV Plastic number: VI – I = V

1

2

  z z

1

3

  z z

1

4 5

  z z

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

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Nunnery Waasmunster Colors : Floor: tint 5 Wall: 1 Ceiling: 3

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

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  • 3. The architecture of Van der Laan and the use of the plastic number

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DIMACS Algorithmic Mathematical Art May 11 - 13, 2009 J-M Dendoncker

  • 3. The architecture of Van der Laan and the use of the plastic number

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