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  The Unicursal Hexagram as Hyperbola
General Thelema Posted by Xnoubis on July 16, 2000 @ 09:45 AM
from the conic-relief dept.

And now I will expound the greatest interpretation of the Unicursal Hexagram that the world has ever seen!

-- Wait, that's hyperbole, not hyperbola. Sorry.


I have to say that I've never felt the attraction that some magicians have for using the Unicursal Hexagram as a planetary glyph. (If you don't know, the Unicursal Hexagram is illustrated in the red on black icon in the upper right hand corner of this article.) The conventional Hexagram, yes. Just as the Pentagram symbolizes the realm of the five elements (and the five senses) with its five point, so the Hexagram symbolizes the planetary realm with its six points, plus the central sun.

It's that central sun that's the problem. Since it's not attributed to a point, the traditional Golden Dawn method for invoking or banishing the sun is to trace all of the six other Hexagrams in one motion. This is a marvellous isometric exercise for the upper arm, but is time consuming, and thus many magicians, Isreal Regardie among them, switched to the Unicursal Hexagram, so that they can trace one solar Hexagram from the central point. (This creates a new problem: what do the four directions coming out of the central point mean? Which has lead to further ingenuity, etc.)

This doesn't seem satisfactory to me. For one thing, an Invoking Hexagram traditionally moves deosil; a Banishing Hexagram, widdershins. There's no way you can trace a Unicursal Hexagram without going both ways, back and forth.

But mostly, it's because I think I get what Crowley was saying when he introduced his use of the figure in The Book of Thoth, p. 11, "The lines, however, are strictly Euclidean; they have no depth." At first, I thought he was just babbling about the fact that the design uses lines of varying width, but that this is only for decoration. Then it occurred to me that he might mean just the opposite.

A digression might be in order. In geometry, certain curves can be defined and classified as "conic sections":


Weisstein, Eric W. "Conic Section." Eric Weisstein's World of Mathematics.
http://mathworld.wolfram.com/ConicSection.html

Instead of a technical explanation of how these curves differ, let's use the time honored "flashlight on the ground" analogy. If you shine a flashlight straight down on the ground, it will light up an area in the shape of a circle. This circle has one focus, and the circumference of the circle consists of those points which are an equal distance away from the focus.

If you tilt the flashlight just a bit, the circle will become an oval, otherwise known as an ellipse. What happens is that the one focus becomes two foci, and that the edge of the oval consists of those points at which the distance from focus 1 plus the distance from focus 2 are equal. If that sounds unfamiliar, it's enough just to know that, while a circle has one focus, an ellipse has two.

Now, what happens when the body of the flashlight is held parallel to the ground? The resultant curve is called a parabola. In this case, the ellipse has lengthened to the extent that one of the foci is nearby, and the other focus is an infinite distance away. Not only does the "oval" never close, the arms of the curve never come any closer together, becoming instead more and more parallel without ever entirely getting there.

The hyperbola is the strangest curve of all (no exaggeration!). We can start to picture it by imagining that we tilt the flashlight just a little further so that its body is angled slightly away from the ground, but enough so that some of the ground is still lit. This forms half of a hyperbola. In fact, the flashlight analogy breaks down at this point. Return instead to the conic diagram illustrated above.

In a hyperbola, the second focus, which was located at infinity in the parabola, has come all the way around the universe to land near the first focus from the opposite direction! This is a surprising turn of events, but it's a natural outcome of the formulae used to determine these curves.

Is it apparent what I think this has to do with the Unicursal Hexagram yet? The Unicursal Hexagram is a flashy symbol of the hyperbola, or at least of the same space-bending property that the hyperbola also possesses.

When Crowley says that the lines have no breadth, I think it's a clue that the widths are intended to imply perspective (they have the same breadth). The thick lines are those closest to us, and then the lines which cross in the center are moving sharply away from us, going through the "point of infinity" and back. And the "strictly Euclidean" business? Strictly a lie.

What this illustrates for me, then, is 0 = 2. I feel confirmed in this by the fact that the only place where Crowley actually uses the Unicursal Hexagram is in Liber Reguli, which is first and foremost an exposition of the 0 = 2 formula, as can be seen by the ritual's closing commentary.

And as for the solar Hexagram? Work those triceps!



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  • The Fine Print: The following comments are owned by whoever posted them.


    Re: The Unicursal Hexagram as Hyperbola
    by Mordecai Shapiro on Monday July 17, @10:25PM
    >When Crowley says that the lines have no breadth,
    >I think it's a clue that the widths are intended
    >to imply perspective (they have the same
    >breadth). The thick lines are those closest to
    >us, and then the lines which cross in the center
    >are moving sharply away from us, going through
    >the "point of infinity" and back. And the
    >"strictly Euclidean" business? Strictly a lie.

    Wait a minute. You're not implying that Crowley was not a mathematical genius, or worse yet, that he could make an error?

    • Re: The Unicursal Hexagram as Hyperbola
      by Xnoubis on Tuesday July 18, @04:44PM
      That's what you would have meant if you'd said that. I suspect that Crowley was hinting at his covert meaning by bringing up Euclid in the first place. I think he doth protest too much, sorta thing. (True, there's nothing inherently non-Euclidean about a hyberbola, but the space-bending of the Unicursal Hexagram might recall the hyperbolic cone, which I believe is non-Euclidean.)


      • Re: The Unicursal Hexagram as Hyperbola
        by Aleph on Sunday July 23, @12:18PM
        Hmmm. You may be right here. This calls to mind a verse from Liber VII:

        The cone is cut with an infinite ray; the curve of hyperbolic life springs into being.

        -- Liber VII, V, 35


        • Re: The Unicursal Hexagram as Hyperbola
          by Mordecai Shapiro on Sunday July 23, @07:28PM
          Excellent quotation! Perhaps Crowley's HGA is a better mathematician than he is?


    Re: The Unicursal Hexagram as Hyperbola
    by Patrick Crumhorn on Wednesday July 26, @05:03PM
    A very interesting perspective. Frankly,
    I had never encountered the use of the UH
    as a time-saving device for solar hexagram
    work before, though it makes perfect sense.

    I always saw the UH as the "active" form
    of the "as above so below" symbolism of the
    interlocked triangles. The traditional
    "star of David" Hexagram is the equilibration
    of opposites, whereas the UH is the active
    annihilation of opposites.

    Nema has a wonderful drawing on the cover
    of her chapbook, "The Priesthood" that shows a
    couple seated in yab-yum, viewed from above,
    with their crossed legs and heads forming the
    points of the UH, and the flowering explosion
    in the center pinpointed exactly on the joined
    lingam-yoni. This pretty much sums up the
    formula of the UH for me.

    Re: The Unicursal Hexagram as Hyperbola
    by Matt Yeager on Friday January 25, @07:31AM
    Hey I'm new at this hexagram thing. I'm trying to understand it more through research. I found out about it through Danny Carey from Tool. They're all into it and i'm trying to come to grips with it. Do you listen to Tool at all? Anyway, can you simplify this in anyway or tell me some good background information on it? Thanks for your time. Later,
    Matt

    Re: The Unicursal Hexagram as Hyperbola
    by Sean on Monday March 18, @09:16PM
    Yeah, i also got this from danny carey because he uses it to channel Dameons, i want some shapes to paint on my drumheads so that i can also channel this talent.
    Thanks

    • Re: The Unicursal Hexagram as Hyperbola
      by Sean on Tuesday May 07, @03:45PM
      Yeah, i also got this from danny carey because he uses it to channel Dameons, i want some shapes to paint on my drumheads so that i can also channel this talent.
      Thanks


      • Re: The Unicursal Hexagram as Hyperbola
        by Sean on Tuesday May 07, @10:16PM
        Yeah, i also got this from danny carey because he uses it to channel Dameons, i want some shapes to paint on my drumheads so that i can also channel this talent.
        Thanks


    The Unicursal Hexagram as Hyperbola
    by Xnoubis on Wednesday May 15, @09:23AM

    I have to say that I've never felt the attraction that some magicians have for using the Unicursal Hexagram as a planetary glyph. (If you don't know, the Unicursal Hexagram is illustrated in the red on black icon in the upper right hand corner of this article.) The conventional Hexagram, yes. Just as the Pentagram symbolizes the realm of the five elements (and the five senses) with its five point, so the Hexagram symbolizes the planetary realm with its six points, plus the central sun.



    It's that central sun that's the problem. Since it's not attributed to a point, the traditional Golden Dawn method for invoking or banishing the sun is to trace all of the six other Hexagrams in one motion. This is a marvellous isometric exercise for the upper arm, but is time consuming, and thus many magicians, Isreal Regardie among them, switched to the Unicursal Hexagram, so that they can trace one solar Hexagram from the central point. (This creates a new problem: what do the four directions coming out of the central point mean? Which has lead to further ingenuity, etc.)



    This doesn't seem satisfactory to me. For one thing, an Invoking Hexagram traditionally moves deosil; a Banishing Hexagram, widdershins. There's no way you can trace a Unicursal Hexagram without going both ways, back and forth.



    But mostly, it's because I think I get what Crowley was saying when he introduced his use of the figure in The Book of Thoth, p. 11, "The lines, however, are strictly Euclidean; they have no depth." At first, I thought he was just babbling about the fact that the design uses lines of varying width, but that this is only for decoration. Then it occurred to me that he might mean just the opposite.



    A digression might be in order. In geometry, certain curves can be defined and classified as "conic sections":





    Weisstein, Eric W. "Conic Section." Eric Weisstein's World of Mathematics.

    http://mathworld.wolfram.com/ConicSection.html


    Instead of a technical explanation of how these curves differ, let's use the time honored "flashlight on the ground" analogy. If you shine a flashlight straight down on the ground, it will light up an area in the shape of a circle. This circle has one focus, and the circumference of the circle consists of those points which are an equal distance away from the focus.



    If you tilt the flashlight just a bit, the circle will become an oval, otherwise known as an ellipse. What happens is that the one focus becomes two foci, and that the edge of the oval consists of those points at which the distance from focus 1 plus the distance from focus 2 are equal. If that sounds unfamiliar, it's enough just to know that, while a circle has one focus, an ellipse has two.



    Now, what happens when the body of the flashlight is held parallel to the ground? The resultant curve is called a parabola. In this case, the ellipse has lengthened to the extent that one of the foci is nearby, and the other focus is an infinite distance away. Not only does the "oval" never close, the arms of the curve never come any closer together, becoming instead more and more parallel without ever entirely getting there.



    The hyperbola is the strangest curve of all (no exaggeration!). We can start to picture it by imagining that we tilt the flashlight just a little further so that its body is angled slightly away from the ground, but enough so that some of the ground is still lit. This forms half of a hyperbola. In fact, the flashlight analogy breaks down at this point. Return instead to the conic diagram illustrated above.



    In a hyperbola, the second focus, which was located at infinity in the parabola, has come all the way around the universe to land near the first focus from the opposite direction! This is a surprising turn of events, but it's a natural outcome of the formulae used to determine these curves.



    Is it apparent what I think this has to do with the Unicursal Hexagram yet? The Unicursal Hexagram is a flashy symbol of the hyperbola, or at least of the same space-bending property that the hyperbola also possesses.



    When Crowley says that the lines have no breadth, I think it's a clue that the widths are intended to imply perspective (they have the same breadth). The thick lines are those closest to us, and then the lines which cross in the center are moving sharply away from us, going through the "point of infinity" and back. And the "strictly Euclidean" business? Strictly a lie.



    What this illustrates for me, then, is 0 = 2. I feel confirmed in this by the fact that the only place where Crowley actually uses the Unicursal Hexagram is in Liber Reguli, which is first and foremost an exposition of the 0 = 2 formula, as can be seen by the ritual's closing commentary.



    And as for the solar Hexagram? Work those triceps!



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