Did you know if you make a 4x4 matrix where the columns are the (x,y,z,1) vertices of a tetrahedron, the rows of its inverse are the coefficients of the plane equations of the faces of that tetrahedron? You can also use the adjoins. Geometry is fun.
But I didn’t know that rational tetrahedra (dihedral angles all rational multiples of π) were classified in 2020 https://arxiv.org/abs/2011.14232
Nice! And it's pretty easy to see why:
- in homogenous coordinates the plane equation is just dot(point, plane) = 0
- for every column (i.e. point) of a 4D matrix, exactly 3 rows (i.e. planes) of its inverse will have a dot product of 0
Accidentally discovered that when I was bored during calculus class :]
Start with the cube, and notice that when you deform it into the general parallelepiped, the volumes of all the tetrahedra stay equal to eachother.
However these 6 tets are not necessarily congruent, when I commented I was visualizing the case when the parallelepiped has equal side lengths, but in the general case the 6 tetrahedra are not congruent. However each of the 6 is congruent to at least 1 other out of the 6, since the parallepiped has reflection symmetry.