The Hexagrammum Mysticum Using Parametric Equations

Do what thou wilt shall be the whole of the Law.

Any ceremonial magician knows that the hexagram plays an essential role in any and all solar workings, having appeared for time immemorial in Hindu symbolism, Dutch hex signs, etc., etc.  As one grows as a magician and the sweetness of Thelema brushes over the soul, the typical Qabalistic imagery of the old is reinforced, if not outright replaced, by Thelemic symbols.  Of those symbols, the strongest, perhaps, most Thelemic, is the unicursal hexagram.   The unicursal nature of this symbol simply means that the hand does not need to be lifted to construct it on paper; it is continuous.  Blaise Pascal, at the age of 16, had first used the figure in 1640 to describe his hexagrammum mysticum, to prove that six points placed on the perimeter of a conic object will have their opposite edges intersect collinearly.

f(x,y)=(sin(12t), cos(18t))

One can generate the Thelemic hexagram with a relatively simple Lissajous parametric equation: f(x, y) = (sin(12t), cos(18t)).  Starting in radians at t=0, the hexagram will begin at the topmost point, and, moving clockwise, will hit the corners of hexagram and the center at t= c(π/24), where c is a whole number.  An interesting variation, which begins the drawing at the origin, is f(x, y) = (sin(12t), sin(18t)), which, by the way, also moves clockwise.  It is interesting to note that the ratio of the two periods, (12/18), is .666 repeating, a number entirely appropriate for the symbolic nature of the glyph.  In fact, as one begins with 2/3 as the initial ratio of periods, and moves through integer multiples, like 4/6, 6/9, etc., one can see the evolution of the figure as it approaches more linearity at 12/18.   However, an exception to the pattern is  f(x, y) = (sin(10t), cos(15t)), where the pattern creates an amazing “Eye” reminiscent of Atu 16, “The Tower.”

f(x,y)=(sin(10t), cos(15t))

Going back to the original equation    f(x, y) = (sin(12t), cos(18t), there is another interesting property if you look at the figure as two “arrowheads,” one pointed up, the other pointed down.  The area of each “arrowhead,” or the two obtuse triangles combined, is exactly 1 square unit.

f(x,y)=(cos(2t), sin(3t))

Another interesting variation occurs when we take the original f(x, y) = (sin(12t), cos(18t)) equation and simply switch the coordinates and preserve the 2:3 ratio as f(y, x), or f(x, y) = (cos(2t), sin(3t)).  In other words, we kept the same ratio between the period for the x and y coordinates, and then graphed the of the original equation.  Amazingly, this equation produces the ichthys glyph from early Gnostic Christianity.  It certainly appears that the Unicursal Hexagram, and one of the earliest symbols for Christianity are mathematically related. Though this coordinate switch does not mean that they are inverses of each other, it is the first step in the typical procedure one would use to determine whether two functions were inverses of each other.

Love is the law, love under will.

Frater Persevs