Ross G Caldwell
27042006 23:28 
Unicursal hexagram exercise
After a bit of thinking, I think I can see why Crowley had to insist that the Unicursal Hexagram's lines must be considered "Euclidean".
Here's an exercise 
draw two triangles, one upright ABC and one downward DEF (The letters are the points of the triangles). Make BC the bottom line of the first, and DE the top line of the second. Label the angles, and cut the triangles (they don't have to be perfect triangles for this exercise).
Now, how do you "unite" the two triangles? That is, how do you make two distinct triangles with only one line? Mathematically, they can only be united if *at least* one part of each is identical in both. Then you can create a new shape where both triangles have preserved their identity.
If you overlap the triangles so that you have a normal hexagram, a Star of David, you can do something to unite them.
Slit the lines so that you can make a nice Star of David.
Insert the triangles.
Draw a unicursal hexagram on the Star of David.
Going from top A to bottom left B, you will notice that the next point is on the *second* triangle, point E! And the rest goes from point F on the bottom, to point D, to point C on the first triangle!
Geometrically, you have created a twodimensional single shape out of two distinct objects, where the line BC of the first triangle can be considered geometrically IDENTICAL to the line DE of the second. This transformation happens only at ONE POINT (geometrically) of the design. This is the Rose Point in the Unicursal Hexagram.
It is ONLY if the lines are considered as Euclidean  that is, perfect lines, abstract lines  that this can be done. Because, if the lines have breadth, then they will interfere with one another, and it is always possible that they are not actually meeting  it might be just because our vision is limited, or our measurement is faulty. But if they do not have breadth, and are Euclidean, geometrically and strictly mathematical, at least one point on both can be identical.
It is only because one of the sides belongs BY DEFINTION to both  hence Euclidean (because the lines between points DE and points BC occupy the same position; if they had any extension, occupied any space at all, they would not be Euclidean)  that the Unicursal Hexagram can represent  and even illustrate  the Great Work.
(Note that if you put the top and bottom lines together, so that they are considered mathematicallygeometrically identical (Euclidean), you have created a foursided object where the triangle lines BC and DE are indistinguishable. But in this case you have diminished the creation by making the triangles have an interior line in a tetragram, you have not extended the triangles so that both preserve their unique identities, although paradoxically).
