I would like to draw a Schlegel diagram of a tesseract to visualize via a Cartesian coordinate system inside the tesseract the symmetry of some four-dimensional points located in a range of integer values no longer than the half of the length of the side of the tesseract. (The questions are at the end of the explanation)
My idea is as follows:
For instance in two dimensions, it is possible to visualize two-dimensional points $(x_1,x_2)$ inside a square whose side length is for instance $s$, where $| x_1 |,|x_2| \le \frac{s}{2}$ if the center of the Cartesian system is located in the center of the rectangle. In the same fashion, is is possible to do the same inside a cube for three-dimensional points $(x_1,x_2,x_3)$, locating the center of the Cartesian system in the exact center of the cube, keeping the same restriction for the values of the coefficients of the points (smaller than half the length of the side of any face of the cube.
I want to apply that idea for a set of four-dimensional points $(x_1,x_2,x_3,x_4)$ in the Schlegel diagram of a tesseract. But in this case I am not sure how to visualize a given point $(x_1,x_2,x_3,x_4)$. My guessing is something like this:
In a static image of the tesseract my intuition is that I should be able to visualize three coefficients of a given point $(x_1,x_2,x_3,x_4)$ for instance let us say for this example that they are $(x_1,x_2,x_3)$ (if the belong to the current position of visualization of the tesseract), but not very sure where should be located the fourth one (always assuming that it should be visible in the current position of visualization of the tesseract).
I would like to ask the following questions:
Is it possible to visualize four-dimensional points in a Schlegel diagram of a tesseract? Is there a known technique to do it correctly?
In the example above I have assumed that in a static view of the Schlegel diagram of the tesseract I can visualize three dimensions of a given point, for instance $x_1,x_2$ and $x_3$. Is that intuition wrong? Where should I plot/locate/visualize the remaining fourth dimension, $x_4$? Is it possible in a static image, or the diagram must be shown in movement so all the dimensions of each point are visualized depending on the "rotation" though the dimensions of the tesseract?
Is this kind of approach in use for visualization of fourth-dimensional problems in some field of Mathematics? Are there online tools (initially I did not find one) to make this kind of visualization?
Any hints are very welcomed, thank you!
UPDATE: I have found a very related question here.
Well, gathering the information regarding the basic theory here and the nice explanation here regarding the projection, I was able to build my own version of the tesseract, and yes it is possible to show a point inside the tesseract, and I was wrong in the assumptions I made regarding the methodology applied to visualize the 4D point. Basically, if we want to show a point inside the tesseract, we need to project the tesseract first, and then project the desired point as well, following the same projection rules.
(The animated gif was generated joining the jpg files with VirtualDub).
UPDATE 2017/02/06: I have included both in the image and the Python code in blue color the reference axes of the point (the projection of the $(x_0,0,0,0),(0,x_1,0,0),(0,0,x_2,0),(0,0,0,x_3)$ points and the reference axes generated with them).