POLAR    MATH    ART

                                                                   by Leen Gillis and  Caroline Verbist           

Polar coordinates and complex numbers

At the end of the 18th century three men - Caspar Wessel (1745-1818), a self-taught Norwegian surveyor, Jean Robert Argaud (1768-1822), an equally self-taught Swiss book-keeper and Carl Friedrich Gauss (1777-1855), the Prince of mathematicians, worked independently on geometrical interpretation of complex numbers. 

A straight line being filled with the real number set, the plane is filled with the complex number set. Besides the cartesian coordinates we can reach each point in the plane with a vector, starting from the origin of the plane  having  length R and  moving counterclockwise from the positive x-axis if the angle t increases . As these polar coordinates (R,t) or (r, q) determine completely a point in the plane, a complex number is completely determined by its modulus R and its argument t.

        for positive r

                for negative r

 

Art and polar graphs

We were fascinated by the beauty of the graphs using polar coordinates. We tried them out using the programme 'Graphmatica' and the possibilities of  the graphical calculator TI-83.  

     R = (1.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1))) )
     R = (2.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1))) )
     R = (3.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1))) )
     R = (4.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1))) )
 
       

       R= (1.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1))))
       R= (1.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1)))) - 0.5
       R= (1.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1)))) - 1
       R= (1.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1)))) - 1.5
       R= (1.5) ( 1-abs (-sin (6(t-1)))+ 2(cos(2(t-1)))) -2
 
 

 

     R= (2.5) ( 1-abs (5sin (6(t-1)))+ 2(cos(2(t-1))))
     R= (3.5) ( 1-abs (5sin (6(t-1)))+ 2(cos(2(t-1))))
     R= (4.5) ( 1-abs (5sin (6(t-1)))+ 2(cos(2(t-1))))
     R= (5.5) ( 1-abs (5sin (6(t-1)))+ 2(cos(2(t-1))))

      R= (4.5)(1-abs(5sin(6(2t-1)))+ 0.5(cos(4(3t-5))))
      R= (5.5)(1-abs(5sin (6(2t-1)))+ 0.5(cos(4(3t-5))))
     R=(4.5)(1-abs(2sin(4(2t-3)))+ 1.5(cos(4(3t-3))))
 

      R=(3.5)(3-abs(sin (4(3t-1)))+ 4(cos(2(4t-5))))
      R=(4.5)(3-abs(sin (4(3t-1)))+ 4(cos(2(4t-5))))
      R=(5.5)(3-abs(sin (4(3t-1)))+ 4(cos(2(4t-5))))
 

    R=(4a sin(t+1) (cos(t)) / ((cos(t))^3 + (sin(t))^3) {a: 1,8, 1}  

   R=(4a sin(4t+2) (cos(3t-2)) / ((cos(2t+1))^2 + (sin(5t+2))^3)
                                         {a: 1,8,1}

 R=(4asin(4t-3) (cos(3t+2)) / (6(cos(2t-4))^2 + 2(sin(5t+2))^6)
                                       {a: 0, 5, 0.5}


5R= (a sin(t+3)^2 + cos(t+6)^3)/((cos(3t-8 ) )^2+ +(sin(4t+5))^4)    {a: 4,7,11,16,22,29.37}

            R= a 3 sin(t-2)^2 +4cos(2t-5) {a: 0, 4, 0.5}
         R= a   sin(4t-2)^2 +4cos(2t-5)^2 {a: 0, 4, 0.5}
         R= a 2sin(4t-2)^2 +4cos(2t-5)^2 {a: 0, 4, 0.5}
         R= a 3sin(4t-2)^2 +4cos(2t-5)^2 {a: 0, 4, 0.5}

 

 

Sources 

         Makers of Mathematics, Stuart Hollingdale, Penguin Books
       Graphmatica, P. Bogaert 
         http://archives.math.uk.edu/visualcalculus/

 

BACK to OLVP page