Connecting Things

            With what has been seen so far let us return to a characteristic of the pentagramma brought up above.  It was seen that every pentagramma mirificum, not just the regular, define five 90-90-90 triangles; and, as can been known from the investigation above, 90-90-90 triangles result from the division of the sphere that generates the octahedron.

(*Note: You may need to move one of the sliders to get the triangles to appear)

            Then it is necessary that any pentagramma mirificum will define five octahedra, whose orientation with one another change depending of the irregularity, or regularity, of the pentagramma.

            Here are the five octahedra, and each octahedron's three great circles, though some octahedra will share a great circle (as is evident from above).

            Here is the pentagramma and the five octahedra:

            Here is the same without the sphere:

            And here are just the five octahedra:

            The five octahedra together define what is called a “compound” solid, and, as a compound, exists as an artifact of the pentagramma mirificum.  Play with the animations above, and make one for your self.

            (*Note that this is not the compound of five octahedra, as in the only compound of five octahedra, but it is a very interesting compound of five octahedra, as we will see below; we can call it the compound mirificum)

            Take note of how the different octahedra connect.  Begin by looking (on the animations above, or the one you made) at a corner of the pentagon of the pentagramma: two octagons come together at each of these corners, and, as should be evident from the work above, the same two octahedra will come together at the opposite side of the sphere as well.  So each of the octahedra share two opposite vertices, and thus one of their axis, with another; this actually occurs twice with each octahedron.  Looking at the following picture (the octahedra are red, white, silver, brown, and wood colored, and the rubberbands have been added to show where the pentagramma solid would be): take the red octahedron, it connects to the brown and wood colored octahedra at the indicated places (which are corners of the pentagon of the pentagramma).

            However, the red octahedron does not connect to the white or silver octahedra; the wood and brown do connect to the silver and white though.  So traveling along the edge of the red, one could, through crossing at a shared vertex, pass to the wood, and from the wood travel to the white, across the vertex they share.  So all five are interconnected in this chain-like fashion.  Looks complex, at least as it appears visually; but what if we, as we did in another way in part one, develop a map.  How many faces, edges and vertices does an octahedron have? (If you haven't built one, build one and find out.)  Then the following image will suffice as a planar map of one octahedron (providing that we treat the largest, outer triangle as one of the faces, along with the other 7 triangles, totaling the 8 faces of an octahedron):

            Now, in our compound, one octahedron connects to another through sharing one pair of opposite verticies, so we could take two such maps, and place them vertex to vertex, and we have one of the connections:

            However, a connected pair of octahedra share two verticies; no worry, we just have to find which other two vertices will be connected, which is defined by the first connection, and we can represent this second connection with a dashed line like so:

            Thus, the map of the compound of five octahedra will look like this (the colors are to correspond to the one I made, the picture above; also note the dashed green lines do not represent a connection when two of them cross, only that they are strictly connecting two verticies of two octahedra):

            What may look, at first, to be a jumbled mess, now has some of its relations mapped out, showing a beautiful symmetry; but what can we make of this compound mirificum now that we can see it's connections?  An interesting characteristic of this figure is how it is interconnected, with any one octahedron not being connected to all the others directly, only to two specific others, but still reaching some through others.  An investigation of these interconnections leads to the question of “topology.”

            One way to investigate the fundamental characteristics of geometrical shapes is through “cuts.”  With a spherical object, if any closed cut (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), which could not be entered, or exited, with out crossing the cut (red line here):

            Even if the surface is very deformed, or irregular, this property still holds true, any cut that closes on itself, “closed,” will isolate a region:

            So all of the platonic solids we looked at above would fit into this category of any closed curve isolating a region; this category of shapes we can call topologically equivalent to a sphere.  However, some geometrical surfaces can have a closed cut, but not isolate a region:

            On this surface, one could make a closed cut that will isolate a region, but not every closed cut will, like was the case before; the irregularity of this shape is irrelevant, it could look like a coffee mug, but that there is this hole in the surface clearly gives it some different properties.  If we proceed by connecting a second cut to the first, we find that we still haven't isolated a region:

            It is still possible to get from point A to point B with out crossing the cut:

            Only when a third cut is added, through this method of connecting it to a previous cut, can we actually isolate a section, such that any three closed cuts, each connecting to another, must necessarily isolate a region:

            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 required to first isolate a section of this surface:

            Working from this topological characteristic, lets re-investiagte the compound mirificum.  With the aid of the map developed above, we can look at cuts on the compound mirificum; when working on such a compound of octahedra (or polyhedra in general) the cuts we make must travel along edges, and they, like we did above, be closed curves, where the second, third, etc. 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.

            The reader will find joy in attempting this for him or herself; but be cautious, the map can be a bit trick at first, so, at the same time, have an octahedron in hand, or multiple octahedra, connected in the appropriate fashion, is necessary.  I have not been able to make more then 2 cuts without isolating a section (when making the cuts in the fashion describe above), meaning that the compound mirificum shares in topological characteristics with the torus!  Interesting.  This is 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 torroidal topology corresponds to elliptical functions
                        -this compound, as an artifact of the pentagramma, has the characteristics of an elliptical function!

            -again, we have come back to the pentagramma, this fundamental construction of the sphere,     representing the properties of the sphere itself (through the pole-gc relation, the convergence,      and through the division of the sphere), leaving artifacts, or shadows of an elliptical geometry,    which is strange, for the sphere initially gives no empirical evidence of anything elliptical, it   looks to be a simple surface, with constant curvature, a single rate of change, not a changing       rate of change like the ellipse, yet here, with the 5 octahedra compound, and above with the             projection [A SECTION TO BE ADDED TOWARDS THE BEGINGIN OF PART 2,  ADDRESSING THE GNOMONIC PROJECTION OF THE PENTAGRAMMA DEFINING   AN ELLIPSE] we see shadows of a higher geometry...