The Octahedron is the Complexification of the Tetrahedron (Part 2)

This is a continuation of part 1.

In this post, we will show that we consider the 4 points of the tetrahedron in , the group of complex projective transformations that preserve this set of 4 points is isomorphic (the full symmetry group of the tetrahedron).

The natural homomorphism from the transformation group to the permutation group of the 4 points is the action itself. We will now show that this homomorphism is an isomorphism.

To do this, it is easier to consider extending a permutation of points to a transformation. Through this construction, we will find a projective transformation permuting the 4 points in an arbitrary way.

Given the 4 points of the tetrahedron in , there correspond 6 different lines passing through the points. Since complex projective transformation maps complex lines to complex lines, we can choose such a transformation that sends one of the 6 lines to the line at infinity. Given homogeneous coordinates , the affine chart we consider is and so, the line at infinity is given . Denote and to be the two points that lie on this line at infinity. Choose an affine coordinate on the affine chart such that the x-direction is given by and the y-direction is given by . Then, and are expressed as and , respectively.

By a suitable translation, we can set one of the remaining 2 points, say , to be at the origin on the affine chart. Thus, becomes .

There remains one non-zero finite point, . By a suitable homothety, we can assume to be .

The order of the 4 points in this construction is arbitrary. Thus, there always exists a projective transformation that maps the 4 points, in any order, to the 4 points .

It follows that the image of the above homomorphism from the group of projective transformations preserving the set of 4 points to the permutation group of the 4 points is surjective.

We claim this homomorphism is also injective. It suffices to show that if the order of the 4 points is fixed, then the projective transformation is also fixed.

Claim: the projective transformation fixing the 4 points is trivial.

Since this projective transformation maps and to themselves, it preserves the line at infinity. Hence, it is an affine transformation. In affine coordinates, this transformation can be expressed as for some . Since is preserved, it follows that the transformation is linear, i.e. . Since the direction of the coordinate axes x and y are preserved, . Since is preserved, . This proves the claim.

Hence, the homomorphism is an isomorphism. The group of projective transformations the preserve the set of 4 points of the tetrahedron in is indeed isomorphic to the permutation group of the 4 points, .