"The Gnostics of the 12th century believed that the transcendental Logos (the Word, the creative expression of the transcendental Mythos) could be cognized as "an intelligible sphere whose center is everywhere and circumference nowhere."
-- Joseph Campbell,
The term pseudosphere is used in science and geometry to refer to a spherical array of an infinite number of vectors, each equal in length, thus defining the surface of a spherical plane by their tips. A hyperspherical polarity is simply a pseudosphere in a higher dimension (like a four-dimensional pseudosphere) whose vectors are infinite, representing a pure polarity -- a polarity between a singularity and a higher-dimensional infinity. So, the Hyperspherical Polarity refers to the polarity between the Universal Singularity and the Infinite Void. The "space" around the Universal Singularity represents pure nothingness, in which the concept of spacetime as we know it is meaningless. (Keep in mind that even the empty space in our physical Universe is not nothingness.)
Imagine, as far as it is imaginable, the state of the Universe at the Planck time of the Big Bang (a mere 10
The Singularity of the Big Bang is not localized in spacetime -- space and time did not exist at t=0, and we cannot point to the location of the Big Bang in space (except in a strangely reversed sort of way by pointing in every direction at once into the blackness of infinity all around us). In the same way, the Hyperspherical Polarity is a purely ethereal dynamic condition that cannot be localized in spacetime, and yet is the source of our entire physical Universe of experience, from the Big Bang to the present, no matter where or when we happen to be.
It is important to realize that what we are visualizing is a single state of polarity which is spherical in nature. Rather than being composed of an infinite number of rays as such, it is more like a single polarized spherical condition with an infinite radius -- a pure homogeneousness, a single reality representing all possible realities superimposed. The Hyperspherical Polarity is an undifferentiated and timeless state of spherically projected force -- the Force, if you like. (Polarity is the manifestation of force, whatever force is ultimately -- force is understood in metaphysics as the same thing as consciousness, and to the mystics as the same thing as spirit.)
An equatorial ring-aspect sliced from a four-dimensional sphere would actually be a spherical plane in three-dimensional space. In the same way, a ring sliced from your light-cone represents a spherical plane surrounding you in space. Your consciousness point of reference is the singularity in the center of that spherical plane, which is the singularity at the tip of the light-cone in the higher-dimensional Minkowski model. It is also a singularity on the universal hypersurface of the present (which, remember, is never seen as such).
Although a four-dimensional hypersphere is transcendental to our experience and is beyond a precise representation in three-dimensional space, we can still expand the geometric concept even beyond that. Just as a three-dimensional sphere is a ring on a four-dimensional sphere, that four-dimensional sphere could just as easily be a ring on a five-dimensional sphere, which could be a ring on a six-dimensional sphere, as so on, ad infinitum -- the geometry provides for an infinite number of higher dimensions, though all are centered around and defined by the same Universal Singularity. Also, note that a ring-aspect is actually a circular array of an infinite number of vectors itself, and that there are an infinite number of possible ring-aspects on a single hypersphere.