Extraversion of points defined by half-angles

If point coordinates have half angles in them, there are many extraversions. This pages shows the details.

We know that extraversion of points whose coordinates are of the form (sin A : : ) results in quartile points with lots of desmic linking.

This pages shows the extraversion of a point with coordinates ( sin(A/2) : sin(B/2) : sin(C/2) ). Each circle represents a point and each line an extraversion move. The a-extraversion is, for angles

(A, B, C) -> ( –A , π – B, π – C ) with analogous formulas for b and c versions.

Inside each circle is three letters representing the three coordinates. s represents sin(angle/2) and c represents cos(angle/2).

The extraversion changes will change a point into one of its harmonic associates, so that the 'ex' points are included with the 'extra' points, but s, s, s will never turn to c, c, c because a consequence of the rules is there can never be more than two coodinates using the cosine function.

There are, I believe, 16, different points. There are 4 choices for the first slot, 4 for the second, but only 2 for the third, then we divide by 2 to get rid of equivalent signs.

As in the quartile points we expect lots of lines between these points, but only if we add the extraversions of the harmonic associates of the original point. This produces a super-desmic system, that I find very interesting. Hopefully I will write about it soon.

The rules for a, b, and c are that the product abc = cba = C-harmonic associate. The harmonic associate is inversive, so that (abc)2 = identity.

The dark arrows represent the operation of taking a point to its harmonic associates.