## Complex Function Graphs

*Graphs of the real part of the exponential function and the imaginary part of the logarithm, together with a "tetraview" for the exponential function.*

This film treats graphs of complex functions

Color certainly is a considerable help in analyzing these objects. Coloring the axes and displaying a coordinate frame during rotations is a very effective way of keeping track of positions in real four-dimensional space. The authors have devised other methods for displaying complex function graphs, both of which have played central roles in expositions of art and mathematics, most recently in Lisbon [16].

A method with promise is the "tetraview", a way of displaying the real and imaginary parts of a function and its inverse relation in such a way that we can see the deformation from any position to any other by selecting edges in the one-skeleton of a tetrahedron whose vertices represent the four main projections from 4-space. We can also consider the intermediate position at the barycenter of the tetrahedron, the average of the four extreme views.

*w*=*f*(*z*) considered as parametric surfaces (*x*,*y*,*u*,*v*) in 4-space, where*z*=*x*+*iy*and*w*=*u*+*iv*. In each case orthographic projection into (*x*,*y*,*u*) is used to get the graph of the real part of*w*, then rotation in the*uv*plane gives (*x*,*y*,*v*), the graph of the imaginary part of*w*. Rotating the original graph in the*xv*plane leads to (*y*,*u*,*v*), the graph of the imaginary part of the inverse function of*f*. Finally, projection to (*x*,*u*,*v*) gives the graph of the real part of the inverse function.A particularly interesting example is the exponential function*w*=*e**z*with the inverse relation*z*= log(*w*). The graph is given by (*x*,*y*,*e**x*cos(*y*),*e**x*sin(*y*)) in 4-space. Projection to (*x*,*y*,*u*) gives the real part of the exponential. The projection (*y*,*u*,*v*) gives a right helicoid which represents the imaginary part of the Riemann surface for the logarithm. The projection (*x*,*u*,*v*) gives a surface of revolution of a real exponential function as the real part of the logarithm [3].Color certainly is a considerable help in analyzing these objects. Coloring the axes and displaying a coordinate frame during rotations is a very effective way of keeping track of positions in real four-dimensional space. The authors have devised other methods for displaying complex function graphs, both of which have played central roles in expositions of art and mathematics, most recently in Lisbon [16].

A method with promise is the "tetraview", a way of displaying the real and imaginary parts of a function and its inverse relation in such a way that we can see the deformation from any position to any other by selecting edges in the one-skeleton of a tetrahedron whose vertices represent the four main projections from 4-space. We can also consider the intermediate position at the barycenter of the tetrahedron, the average of the four extreme views.

**The Gauss Map, A Dynamical Approach**