A former post two years ago described some methods to visualize the roots of quadratic and cubic equations. In the present post I will show a new method that can be applied to the real roots of quadratic equations based in my recent previous post regarding the use of projections of $4$-dimensional points into $3D$.
Basically, we allocate the points inside a $4D$ tesseract whose center will be the origin of coordinates in the $4$-dimensional space and the reference of the position of the points in the internal space of the tesseract, and then the tesseract and its content is projected into $3D$. The example below shows every real root for $a,b,c \in [-10,10] (a,b,c \in \Bbb Z$):
And below there is a zoom showing only the root tuples. As it can be seen, the pattern generated by the set of tuples shows interesting symmetries in the $4D$ space that can be observed in the $3D$ projection of the set of points (kind of mesmerizing!). The movement is required because to be able to visualize a complete $4D$ object through its projection, we need to rotate around one of the $4D$ axes. By doing so, it is possible to see the $3D$ "shadow" of the positions of the points in the original $4$-dimensional space:
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