One way to investigate the most general characteristics of a geometrical surface, is through investigating closed “cuts.” With a spherical-type object, if any closed cut (example shown in red) is made on the surface, there will be a region that is isolated from the rest of the surface (shown in green):
Even with surfaces which may be very deformed, or irregular, this property can still hold true (that any cut that closes on itself will isolate a region):
All of the "platonic solids" and their truncations we investigated above, fall into this same category, because, for them, any closed curve will isolate a region; this is a category of shapes which clearly share a characteristic; they have the same "topology" as a sphere. However, some geometrical surfaces can have a closed cut that does not isolate any region (they could still have a closed cut that would isolate a region, but the difference is that they can have a closed cut that would not isolate a region, something which was impossible in the surfaces of a spherical topology):
Again, the irregularity of this shape is irrelevant, it could look like a coffee mug or a perfect torus, the difference is in the hole in the surface, which gives it a higher potential. If we proceed in testing the limits of this surface, by connecting a second cut to the first, we find that it is possible to do this without isolating a region:
It is still possible to get from point A to point B with out crossing any cuts:
But any third cut we make, connected to one of the previous two, will always isolate a section (that is, if a section hasn't been isolated already); this is a limit for surfaces of this topology:
This is a characteristic of surfaces topologically equivalent to a torus. The investigation can continue if we add a second hole. What is the maximum number of cuts we can make before isolating a section of this surface:
Working from this topological characteristic, lets re-investigate the compound mirificum. With the aid of the map developed above, we can look at cuts on the compound mirificum; when working on a compound of octahedra (or any polyhedra in general) the cuts we make must travel along edges, and they, like we did above, must be closed cuts, with the second, third, etc. cuts will close on a previous cut. Here are a few attempts to make as many cuts as possible, before isolating a region of the compound mirificum (with the isolated sections penciled-in).
The reader is encouraged to play with this for themselves; but be cautious, the map can be a bit tricky at first, so, at the same time, have an octahedron in hand, or multiple octahedra, connected in the appropriate fashion, it helps. The author has not been able to make more then 2 cuts without isolating a section (when making the cuts in the fashion described above), meaning that, if that three cut limit holds true, the compound mirificum is of torroidal topology! Interesting. Perhaps this was eluded to in the map, where the five octahedra interconnect in a specific way, such that they form a ring, having a hole in the middle.
The investigation thus far has demonstrated some fundamentally significant properties of the pentagramma mirificum, and has posed some provoking questions. Repeatedly, and through different paths of investigation, the properties of the pentagramma mirificum where shown to be properties of the sphere itself! Things which would never have been seen from a simple empirical investigation of the sphere. The questions raised about the pentagramma (like the nature of it's projection, and it's compound's topological characteristics), are questions about the nature of spherical geometry itself.
There is more to come, as the LaRouche Youth Movement uncovers more of what Carl F. Gauss discovered in this beautiful construction.