Some Solid Connections, Page 1: The Pentagramma Solid

Now we will take a step further into this very unique construction, the pentagramma mirificum. When the five sides are extended to complete great circles, which process itself is characteristic of spherical geometry, there is a very interesting division of the sphere that results.  To investigate this division of the sphere, take the regular, or golden pentagramma, and look at the “solid” it generates.  The pentagramma mirificum solid can be defined in the same way that three great circles, dividing the sphere evenly, define a solid called the octahedron (the octahedron is constructed from these three great circles simply by connecting, through the sphere, each point of intersection of two great circles on the sphere; the octahedron is in black, and the animation can be spun to get different points of view).

   

Another animation with the octahedron in red, without the sphere, but keeping the great circles:

    

Notice the specific, very non-arbitrary, i.e. regular, way that these three great circles are set up to generate the octahedron: they divide each other evenly, with only two intersecting at any one point (i.e. each great circle is divided exactly into four equal parts by the other two great circles).

Now, using the pentagramma mirificum, with its five great circles, we apply the same idea of connecting intersection points, thus creating a pentagramma solid. In the four animations below the pentagramma can be adjusted to look at the irregular solids that the irregular pentagramma mirifica create; the first animation shows the pentagramma solid in red, and the great circles that define it are black.

Here the solid is in black, but each great circle is a different color:

Simply the pentagramma solid, without the sphere or the great circles, looks like this:

Taking the pentagramma solid, and adding solely the pentagramma mirificum that would lie on the sphere (red), and its compliment pentagramma on the reverse side (blue), we have the following:

These animations begin, like most shown above and below, at the regular, or golden, pentagramma, but can then be changed into irregular pentagramma mirifica using the sliders.  But, as a function of the single golden (regular) pentagramma, what is the nature of the resulting, specific division of the sphere?  To aid such an investigation, each of the five different great circles in the following animation is a different color.

And shown here without the sphere:

Again, like with the Octahedron above, with the pentagramma mirificum solid, any great circle will only intersect one other great circle at any point (i.e. the points of intersection of great circles have two, and only two great circles intersecting).  Each each great circle is divided in a nearly, but not, regular fashion: alternating arcs of ~38.17 degrees and ~51.47 degrees, with each pair adding up to 90 degrees, resulting in the great circle's being divided into four such pairs (this should be clear from the work above).

This interesting division of each of the great circles of the golden pentagramma mirificum define an irregular octagon; look back to the above animations of the pentagramma solid, seek out these octagons.  The octagon has two different sized sides, one with subtends ~38.17 degrees, and the other which subtends ~51.47 degrees.

These two different sized sides alternate, resulting in the property, that any two neighboring sides combined subtend a 90 degree angle.  So, with a bit of thought, the following image can be seen: two squares, thus defined, one rotated from the other by either ~38.17 or ~51.47 degrees (depending on which direction of rotation).

The division of the sphere that defines the regular pentagramma mirificum is similar in class (though not quite the same, as we will see) to the type division of the sphere that defines the octahedron (where the sphere is divided by great circles which divide each other evenly, and only intersect two at a point).  So then what is this class, what other polyhedra would be included? The octahedron was generated by three great circles dividing the sphere in this most regular way, and the pentagramma is defined attempting this most regular division with five great circles, so what about other numbers of great circles, such as four or six?