Drawing S(3)

Although nothing to do with likelinesses per se, it can be useful to exhibit a sample of points from S(3) as points inside a triangle.

There are six ways to do this: three are rotations of one-another, and the other three are their reflections. These are essentially all the same mapping, just presented in different ways. The one I use is

x = f(2) + f(3)/2

y = (√3/2).f(3)

where ( f(1),f(2),f(3) ) is an element of S(3) and (x,y) is the point in the diagram to which it maps. This is the one I used to produce the above diagram, showing the effects of a contraction.

Using this mapping, (1,0,0)  maps to (0,0); (0,1,0) to (1,0); (0,0,1) to (1/2,√3/2). This is an equilateral triangle of side 1 but, due to scaling, the length of the side is irrelevant and the triangle might not plot out on-screen as equilateral.