4DLab - Critical Thinking  

Function domain for this example: A complex square centered at the Y^~-axis. The square is shown 2 positive radians on the X-axis and 2 positive radians on the iZ axis with 4 subdivisions per unit.  The observer is looking from above the square domain.

Fig. 1 shows the surface generated by the the square domain above. The picture view is axonometric. The observer is looking from the positive X, iZ, and Y^~ axes toward the origin of coordinates. Note that in this picture the square domain is dwarfed in comparison to the surface generated. This is because each subdivision of the domain is squared, and as we move along the X-axis.

In Fig. 1-a is showing the map of a unit strip of the domain parallel to the X-axis. The strip also starts at x = -2 on the X-axis and ends at x= 2 on the same axis. The width is one radian along the positive side of iZ axis, that is 0 <= z <= 1. We can clearly see that the negative side of the strip maps into the negative side of the iZ-axis, and the positive side of the strip maps into the positive side of the iZ-axis. So, the strip 'moves' smoothly from the back to the front while its mapping also does the same, but also curving from right to left.

Fig. 1-b shows the same Fig. 1-a, but now, with the aid of a grid or a wireframe,  we can see the other part of the domain that was not plotted in Fig. 1-a. 

Figure 1-c shows the mapping of another strip of the rectangular domain defined in Fig. 1-a. This rectangular subdomain goes from x = 1 to x = 2 all the way along parallel to to the iZ axis. The presence of the wireframe illustrates that the behavior of this rectangular region is very different from the behavior of the previous strip. Now everything occurs within their respective quadrants of the complex plane.

Figure 1-d shows the surface map produced by the positive side of the complex plane X-iZ. Note that is subdomain is just the half 'positive half' of the original domain. This surface is at the same time 'half' of the surface depicted in Fig.-1.

Picture view: lateral. The observer is looking from the positive X-axis toward the origin of coordinates. This figure is important because it is one of the classic figures used in textbooks about complex variables when discussing the complex quadratic function. Now we see that this figure is just the planar projection of the more general graph depicted in Fig. 1.

The same Fig. 2 but now showing in color only the map produced by the unit-width strip. The other picture wire frame is the remaining on the original complex domain. This figure is the same Fig. 1-b but looking at it from the positive X-axis toward the origin.

Picture 3 is a view from the top, that, is, the observer is looking from the positive Y^~-axis toward the origin of coordinates. This is the same Fig. 1 seen from above. 

At right is a frontal view of the quadratic surface. The observer is looking the Fig. 1 from the positive iZ-axis toward the origin of coordinates. From Fig. 1-d can be seen that the higher and lower crests are the plots of the domain extremes.

At right, an axonometric view of a thin strip this time centered along the X-axis. This is the mapping of a thin strip 0.25 units to the left and to the right of the X-axis. In this case the function domain is approaching the real X-axis. 

Al right,  Figure 6. This figure is the same Fig. 5 seen from the iZ axis toward the origin. Now it can clearly be seen that the real function y = x^2 is a very special mapping of the transcomplex quadratic function. When the thin strip approaches the X-axis we will have the classical real-valued sin function.

At right is shown the Da Vinci's Mona Lisa which we can taken as the planar domain for our function under study to see how it is transformed in a point by point basis. To approximate the painting's resolution, the  subdivisions of the  complex domain was incremented accordingly. The domain is positioned at the upper left corner of the painting; for this reason, some lower border or right border cropping of the original may occur.

To the right is shown the resulting transformation of the transformation of Da Vinci's Mona Lisa under the transcomplex quadratic map. Note the small outer side of the rectangular domain which is the original Da Vinci's painting.