A very nice pattern based on Euler's identity

Euler's identity is one of the jewels of Mathematics, it blends in one sight some of the most significant constants of the mathematical universe:

I tried to visualize how would look the left part of the identity in the complex plane by using the following function:

It is a tryout of a generic version of the left side of Euler's identity. Applying the change e^{ib}=cos(b)+sin(b)i, the following function is obtained:

For instance, to be able to see the patterns, taking k1,k2,k3 in the interval [−10,10] stepping in by 0.1, and calculating all the possible combinations of triplets (k1,k2,k3) and their values f(k1,k2,k3), this is the pattern that it generates when the complex values a+bi are represented in Cartesian coordinates (a=x,b=y):

A closer approach to 0+0i:

Euler's identity f(1,1,1)=e^{pi*i}+1=0+0i would be located at the center and the rest of points are the closer values of the more generic function f(k1,k2,k3) as it was defined above for the example. As expected is periodic and symmetrical. This is a little video:

Isn't it hypnotic? I have asked at MSE about this pattern, as far as I know it might be the fist time that has been plotted. I hope somebody will be able to propose some insights about it!  

Update 2016/01/15: Professor Brent Yorgey kindly gave a very nice explanation of the pattern at MSE!