### On the Irregular, Page 2: Full Circle

Take this extension of the sides further, and extend each of the sides of the pentagramma mirificum until they become full great circles.  This will create a very interesting tiling of the sphere, but, to make this process clearer, first investigate how this will work with a simple arbitrary triangle; simply take a spherical triangle and extending its three sides to complete great circles. The animation below shows this, and allows the user to create any spherical triangle he or she wishes (remember most all animations can be spun around by using the arrow keys on your keyboard).

A second triangle (green) is created on the opposite side of the sphere, which is the same as the initial triangle (blue), but, different.  It is seen with a little thought, this second triangle has the same angles as the first, and the same sides as the first, but it is still different.  The difference can be noticed if we imagined sliding the second triangle across the surface of the sphere, and tried to place it on top of the first; play with the animation above.  It can't be done. The second triangle won't ever line up on the first, because the two triangles have a “mirrored” relationship.

Now, taking this idea to the pentagramma mirificum, extending the sides into great circles, how does this divide up the sphere?s  The first animation is simply the five great circles (spin the sphere with the arrow keys to see how the sphere gets divided up); the second animation shows in red the initial pentagramma mirificum, and a second, mirrored, pentagramma mirificum that is created on the other side! The third animation has just the two pentagramma. (In each of these, the sliders can be adjusted to see how this will look with any irregular pentagramma.)

Interesting!  Now, think back to the question raised earlier, respecting the locations of the poles of the sides of the pentagramma mirificum. With the fives lines of a pentagramma, one of the poles for each line is, as a corner, part of the pentagramma itself, but what about the second pole from each side?  Give what we have demonstrated so far, where do these second poles for the red pentagramma mirificum lie?

Here lies something very interesting and singular about the pentagramma. But, now move on to the second section of this part of our investigation.