Suppose we had a $16$-cell whose eight vertices have the coordinates given in the $16$-cell Wikipedia article:
(±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1)
If we viewed this $16$-cell as one of the two demitesseracts of a certain tesseract (meaning those eight vertices are also vertices of that tesseract), what are the possible coordinates of the other eight vertices of the tesseract?
I understand from this answer that there can be two solutions to this, and I'd like to know both solutions.
It seems the solutions were practically already there in the answer post I linked to. I just had to divide everything by two in order to get the following:
First solution: $(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})\text{ with odd number of }-\frac{1}{2}$
Second solution: $(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2})\text{ with even number of }-\frac{1}{2}$