4DLab - Critical Thinking  

Function domain for this example: A complex rectangle of 4 positive radians on the X-axis and 2 positive radians on the iZ axis with 4 subdivisions per unit, the Y^~-axis going through the center of  the rectangle as show in the figure at right. 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 the rectangular domain 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 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. Note how the surface generated by this strip wraps around the function subdomain.

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 a strip of the rectangular domain defined in Fig. 1. The strip goes from X = 1 to X = 2 all the way along parallel to to the iZ axis.

Figure 1-d shows the surface map produced by the positive side of the complex domain of the plane X-iZ.

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 sin function. Now we see that this figure is just the planar projection of a more general function 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 View: Top. The observer is looking from the positive Y^~-axis toward the origin of coordinates. This is the same Fig. 1 seen from above.

Picture View: Frontal. 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.

Picture View: Axonometric. 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.

Figure 6 is the same Fig. 5 seen from the iZ axis toward the origin. Now it can clearly be seen that the real function y = sin(x) is a very special mapping of the transcomplex sin 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.

Below is shown the resulting transformation of the transformation of Da Vinci's Mona Lisa under the transcomplex trigonometric sin map.