The Unicursal Hexagram:
"The cone is cut with an infinite ray; the curve of hyperbolic life springs into being."
Liber VII, V, 35.
The Unicursal Hexagram appears to have been included with one version of the Thoth tarot deck just to fill in the blank spaces in the printing process. Crowley probably did not intend it to be included in the deck; that being said he does include it in the BoT.
And that is the interesting question to me; what did the UH mean to Crowley? And why did he include it in the BoT without any other apparent textual reference to it? He describes it as 'purely euclidean, the lines have no breadth'. The 'lines have no breadth' makes me think of the 'Naples Arrangement'. If they have no breadth then the reference must be to Triangles and the supernal triad.
Although it has been adapted by ritual magicians for use as a planetary hexagram the only ritual in which Crowley used it is Liber V vel Reguli. When I look at the Unicursal Hexagram the varying widths of the lines suggest to me perspective and I see them as lines regressing in parallel to the point of infinity where the Marian Rose is imposed. Yet Crowley states that the 'lines are purely euclidean, they have no breadth". However in his commentary on Liber Reguli, the only ritual in which he uses the UH, he says, "Yet breadth is equally a nothing in the presence of the Cone." Which suggests to me that the nature of the UH is hyperbolic.
Also from his commentary on Reguli I would suggest it represents "the voice of Nature awakening at the dawn of the Aeon .... proclaiming the Word of the Law of Thelema". And that it "reveals the omnipresence of Hadit identical with Himself, yet fullfilling Himself by dividing his interplay with Nuit into episodes, each form of his energy isolated with each aspect of Her receptivity, delight developing continuous from complex to complex."
Also:
"A parabola is bound by one law which fixes its relations with two straight lines at every point; yet it has no end short of infinity....", similarly the initiate, "..may attain to measure fellow parabolas, ellipses that cross his path, hyperbolas that span all space with their twin wings." There is much more in the commentary that connects it with the whole 5/6 11 formula of magick and the 0=2 formula.
You may also note that when an upright and an averse pentagram are imposed within a hexagon you get a Unicursal Hexagram. Although what significance that has, if any, I don't know.
Anyway. A lot of rubbish has been written about the Unicursal Hexagram; so having disposed of my own I shall hereby cease. (And I think I have confused breadth with depth? In which case not the triangle of the supernal triad but the 'line' of Kether - Chokmak? Perhaps our parabolic philosophers can provide some answers?).
Kwaw