square root gauss 001
Every complex number except 0 has two square roots (theese have opposite signs). Therefore
one cannot define the square root as a function on the whole complex plane. The standard
choice is to define the square root in the Gaussian plane except on the negative real axis.
It is defined as a differentiable inverse function of the squaring function. On the negative
real axis one defines square_root(-r^2) = + i*r.
square root riemann 001
By applying the stereographic projection we can als view the previous image sequence on
the Riemann sphere.
square root cartesian gauss
The image, under the complex square root, of the previous cartesian grid: viewed in anaglyph
on the Riemann sphere.
The complex square root can be applied to any grid in the left or the upper half plane.
The left grid was made with the Moebius transformation z ⟶ (z-1)/(z+1), it moves the
polar centers from zero and infinity to +1, -1. The right grid, the image under the complex
square root can be reflected in its straight boundary lines to obtain a grid for the whole
plane with four polar centers. Such grids are used for the parametrization of minimal surfaces
like the 4-noids or the saddle towers.
SquareRootMoebius2
The complex square root can be applied to any grid in the left or the upper half plane.
The left grid was made with the Moebius transformation z ⟶ (z-1)/(z+1), it moves the
polar centers from zero and infinity to +1, -1. The right grid, the image under the complex
square root can be reflected in its straight boundary lines to obtain a grid for the whole
plane with four polar centers. Such grids are used for the parametrization of minimal surfaces
like the 4-noids or the saddle towers.
