That thing has bothered me for a while and I think I finally found why this is impossible. In order for the shape to be impossible ADEB, BEFC and ACFD have to

That thing has bothered me for a while and I think I finally found why this is impossible. In order for the shape to be impossible ADEB, BEFC and ACFD have to be coplanar. If you add a hidden AF or CD edge, making ACFD not coplanar, it becomes possible. To make things simple, let's assume a simple, orthogonal projection (discard the z). We will use degrees of freedom for that. All vertices have their x, y position fixed, that's their 2D coordinates in the drawing, we don't know z yet because it is discarded by the projection. The problem becomes: for each vertex, find the z coordinate. The constraints are: ADEB, BEFC are coplanar, DBFE are not (making the entire thing a flat shape would be cheating). So, make the z coordinates for D, B, F, E anything we want, it will make a small tetrahedron, nothing wrong with that. Now, because A has to be in the DEB plane, its z coordinate is fixed, but it can be calculated, no problem. Same thing with C, it is in the BEF plane and its z coordinate is fixed and can be calculated. You can do that in every case. So as long as you are not looking behind the "pyramid", you can always find a 3D shape that matches the projection. Now if we add the constraint that ACFD are coplanar, that's when you have a problem. All points are fixed, and you have to play with your 4 degrees of freedom that are the z coordinates of D, B, F, E to make ACFD coplanar. Playing with the z, you can do translation, scaling on the z axis and shearing on xz and yz, that's 4 independent transforms, all your degrees of freedom are used up. None of them help making your vertices coplanar, except if you flatten everything, which, as said before, is cheating. Now, maybe we can also make the impossible possible using fancier projections. Perspective projection has a FOV parameter, it may be an extra degree of freedom we can play with, but that enough maths for today. EDIT: missed a transform, now, it works