Some Solid Connections

            Looking further into this construction of the pentagramma mirificum, when the five sides are extended to complete great circles, there is a very interesting division of the sphere.  Taking the regular, or golden pentagramma, it's division of the sphere can be investigated by looking at the “solid” it generates.  This can be built in the same way that three great circles, dividing the sphere in a specific way, define a relatively simple solid called the octehedron (the octahedron is constructed from these great circles simply by connecting, through the sphere, each point of great circle intersection on the sphere).

            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 into four equal parts by the other two).

            Now, using the pentagramma mirificum, with its five great circles, we apply the same idea and create a pentagramma solid.

            Here is the solid without the sphere or the great circles.

            These animations begin, like most above and below, at the regular, or golden, pentagramma, but can then be changed using the sliders on the side, in the same fashion as above and in part 1, where the sliders determine the value of the two legs of the initial triangle (remember, to get back to the golden pentagramma set each slider to 38.17 degrees).  As a function of the golden (regular) pentagramma, what is the nature of the division of the sphere?  To aid such an investigation, each of the five different great circles in the following animation is a different color.

            Again, like with the Octahedron above, the great circles only intersect two at a point (each great circle is a specific color, but the solid is traced out in black, and defined by the intersections of the great circles).  Also, the divisions of each great circle are pretty regular: alternating sections of ~38.17 and ~51.47 degrees, each pair adding up to 90 degrees, resulting in 4 such pairs (this should be clear from the work above). 

            Here is the irregular octagon defined in each of the five great circles of the regular pentagramma; 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.

            As is seen these two sized sides alternate, resulting in 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 which way you look at it).

            So the division of the sphere that defines the regular pentagramma is similar in class (though not quite the same, as we will see) to the division of the sphere that defines the octahedron.  But what else would be in that class, for we have three great circles in the case of the octahedron, five great circles defining the pentagramma, so what about other numbers of great circles such as four or six? 

[THE “ANIMATION MACHINE TOOL” HERE?]

            In the case of four great circles, there is such a division of the sphere, where each great circle is divided evenly by the others; this division defines a solid called the “cuboctahedron.”

            For six great circles, there is another solid defined by this regular division, the “icosidodecahedron.”

            As always, the reader is encouraged to proved the validity of each of these divisions (3, 4, 5, and 6 great circles) for him or her self.  As for divisions with higher numbers of great circles, I am not sure what is to be found, but here we have something very interesting emerging simply with these four.  The octahedron, cuboctahedron, and icosidodecahedron, are of a single class, being defined by great circles that intersect only two at any point, and which divide each other evenly.  If there are two fully extended straight lines on a sphere, each on will interesct the other exactly twice (as long as they are not on top of one another); where a similar process in a plane will only produce exactly one intersection from two fully extended lines.  So if we have three great circles, any one will be intersected by the other two four times.

                        -ANIMATION:  [SPHERE WITH THREE GCS, ONE BLUE, TWO ORANGE, INTERESCTIONS HIGHLIGHTED; WHERE YOU ARE ABLE TO MOVE THE TWO ORANGE GCS!] ----------- XXXX

            Then, if we can get three great circle to divide each other evenly, each great circle's divisions defines a square, and the three squares form an octahedron.  In similar fashion, four great circles define regular hexagons, and thus the cuboctahedron, and six great circles define regular decagons (10 sides), and thus the icosidodecahedron.

            It becomes very provoking, in that each of these three solids has a very special relation to one, or a specific pair, of a most regular class of solids, the “platonic” solids (where the solid is composed solely of a single, regular shape for all the faces, while all its vertices lie on a single sphere).  Here, for reference, are each of the five platonic solids.

            We are only dipping into a much broader field of different regular and quasi-regular divisions of the sphere, and the thus defined solids; the connection to our investigation thus far comes in when we look a specific type of “truncation” of these platonic solids: taking the midpoint of each edge of a solid, and connecting these will generate a new solid (as a “truncation” of the first, as if we were chopping off sections of the first).  Because of the regular nature of the platonic solids we are truncating, the resulting solids will, like the platonic solids, have all their edges equal, but the faces will not necessarily be composed of a single shape.

            The bisected-edge truncation of the octahedron:

            The bisected-edge truncation of the cube:

            Notice they make the same solid: the cuboctahedron.

            The reasoning for this interesting characteristic is seen in that the octahedron and cube are “duals,” a characteristic of which is that if you connect the centers of the faces of one you generate the other.  Again, the best for the reader it to construct these, both in the rectilinear form (one could use straws, sticks, or something equivalent, or poster board), and on the sphere.

            So here are the spherical arcs that correspond to the edges of the cube, octahedron, and their bisected-edge truncation:

            Here is the bisected-edge truncation of the Icosahedron:

            Here is the bisected-edge truncation of the Dodecahedron:

            Again, as with the cube and octahedron, the icosahedron and dodecahedron bisected-edge truncations produce the same solid: the icosidodecahedron.

            And again, the icosahedron and dodecahedron are of this “dual” relation.

            Also, these are the spherical arcs corresponding to the edges of the icosahedron, dodecahedron, and their bisected-edge truncation:

            For the fifth platonic solid, the tetrahedron, its biescted-edge truncation is actually an octahedron:

            And the dual of the tetrahedron is another tetrahedron:

            So the arcs of the two tetrahedra and their truncation, the octahedron, would look like this:

            So what do we have?  Looking at the five regular solids (the reader will find joy in, independently, discovering for him or her self that these are the only possible regular solids), and taking probably the least arbitrary, i.e. most reasonable, truncation (bisecting each edge), generates three other solids: the cuboctahedron, the icosidodecahedron, and the octahedron.

            These truncations of the platonic solids are the same three solids we came across above, when investigating a specific class of solids from dividing the sphere with only 3, only 4, and then only 6 great circles, where each great circle is divided evenly and only two intersect at once.  Above, when investigating this class, we saw that the closest we could get with 5 great circles generates the golden pentagramma mirificum, where the division of each great circle is not quite equal.

            Reinverting the investigation, if we divide the sphere in the most regular way with 3 great circles an octahedron is defined, which itself defines two tetrahedra, which are duals of eachother, and are each among the regular, platonic, solids.  The most regular division of the sphere using 4 great circles generates a cuboctahedron, which defines the cube and octahedron, two more platonic solid duals.  Using 5 great circles, the most regular division of the sphere generates the golden pentagramma mirificum solid.  And with 6 great circles, dividing the sphere in the most regular way, we have the icosidodecahedrton, which defines last two platonic solids, the icosahedron and the dodecahedron, which are duals.

            The pentagramma mirificum appears when attempting to divide the sphere in the most least arbitrary way; more evidence, in addition to what was seen in part 1 of this pedagogy, that the pentagramma mirificum is an artifact of the fundamental nature of the sphere itself.