POLAR MATH ART

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.

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}

by Leen Gillis and Caroline Verbist

Polar coordinates and complex numbers