(* If the reader is not sure why X = α + β + α, CLICK HERE (simple animation or image showing how to measure spherical angles, using the idea of planes rotated))
On the Irregular
Above the case of the regular, or golden, pentagramma mirificum stood singular; a peculiarly unique instance in what, without our maps (of the elements, their functions, and the ratios), or without a knowledge of the process of construction, may otherwise seem to be a smooth, continuous series of regular spherical pentagrams. It is when this fundamental characteristic of spherical geometry itself is built into the pentagram that this harmony emerges, and this relation, as we saw in the step by step construction of the pentagramma above, is not only in the regular case, but in all. Then, if we dig, what other relations can be found to exist in all cases of the pentagramma, what will these tell us, and what are they telling us about?
[INSERT WORK ON PROJECTION DEFINING AN ELLIPSE HERE, CITE GAUSS' HAVING DONE THIS IN FRAGMENTS (so running into the artifacts of elliptical functions comes up in the beginning and end of part 2]
Lets return to the sphere...
Construct any pentagramma mirificum from an arbitrary spherical right triangle, as above. What does it minimally take to define a specific pentagramma? And in building different pentagramma mirificums what characteristics remain? Play with the following animation.
One characteristic that remains unique, and of fundamental importance, is the self polar properties. What else? Investigating the relations amongst the sides of the triangles and the sides of the pentagon yields another interesting characteristic: any side of the pentagon is capped on each end by equal sides from usually-different triangles, and the sum of a pentagon side and one of its neighboring triangle sides is always 90 degrees. This was defined as part of the construction process above, but it can be noted here why it is generally true, for the pentagramma could be constructed in different manner.
-ANIMATION: PROVING 90 DEGREE SUM OF SIDES (X IS POLE OF α) ----------- XXXX
Also note that there is an interesting relation where a side of the pentagon is always equal to the triangle angles opposite it.
(* If the reader is not sure why X = α + β + α, CLICK HERE (simple animation or image showing how to measure spherical angles, using the idea of planes rotated))
Next take any spherical triangle; we will use one with a right angle because it is the case with the triangles of the pentagramma, but this proof is more general. The triangle is made of three sides, straight lines, and on the sphere a straight line always defines two unique “poles.”
As in the following diagram, if we take one of the poles of each of the three sides of the triangle (always taking three poles on the same side of the triangle), and connect these three poles, a second triangle is formed.
This second triangle is clearly related to the first, for it is make of the poles of the first's sides, but what exactly are the values of this second triangle? Start with a side of the second triangle, for example, the side connecting the pole of A with the pole of C. The measure of this side, is the distance between theses two poles; is this distance connected to the relation of the two sides with which these poles are connected? Those sides are A and C on the first triangle, and they are separated by the angle β. The two great circles formed by extending those sides would be also be separated by β, right? Then what would be the angular separation of the poles of these two great circles? Investigate the following diagram.
So this side of the second triangle is equal to a corresponding angle of the first triangle. Would not this be the case for the two other sides of the second triangle as well? And inverting, in looking for the value of a spherical angle of the second triangle, wouldn't the same relation apply, this time with a side of the first triangle being equal to a corresponding angle of the second triangle; one just has to note where the poles of the sides of the second triangle would be. Developing constructions and drawings of ones own greatly aids the discovery process.
Here we've stumbled across something interesting, that the poles of any spherical triangle form another triangle, which is an interesting kind of inversion of the first (the reader is encouraged to investigate what happens if the other set of poles is taken; also if similar inversions occur with other shapes). So where does this come up in the pentagramma mirificum? Why, in the pentagramma mirificum of course. For the pentagramma is self polar, meaning that a pole from each side of the pentagramma is a part of the pentagramma; in fact a corner of another triangle.
[?WHAT ABOUT THE SECOND POLE OF EACH SIDE OF THE PENTAGRAMMA; JUST POSE THIS AS A Q HERE, AND ADDRESS THE ANSWER BELOW WITH THE SECOND PM?]
So let us fill in these second triangles in our construction of the pentagramma.
Our pentagramma mirificum has generated another spherical pentagram. What are the relations of this second spherical pentagram? It can be seen to be made up of five lines, each connecting two corners of the pentagram of the pentagramma mirificum; for reasons noted above, each of these five lines will always have a length of 90 degrees (if the reader is not sure why: in looking at any one line connecting two corners of the pentagon of the pentagramma, where is the pole of one of these corners).
So now, given what properties of the irregular pentagramma we have uncovered, what further interesting relations can be found? Take one of these sides of the just discovered second spherical pentagram inside the pentagramma (these sides of the second spherical pentagram will be referred to as diagonals of the pentagramma from here on), which is 90 degrees in length; any one of these diagonals is connected with two other connected lines of 90 degrees, forming one triangle.
-IMAGE: 90 90 90 TRIANGLE IN PM----------- XXXX
This is a rather interesting triangle, it is made of three sides of 90 degrees, and each of its angles are also 90 degrees. It is the self polar triangle, and five of them are defined by the self polar pentagramma mirificum.
(*Note: You may need to move one of the sliders to get the triangles to appear)
We will come back to these 90-90-90 triangles below, looking further into the implications, but first let us extend our knowledge of the pentagramma's relation on the sphere. The pentagramma mirificum, without adding the five diagonals to make the 90-90-90 triangles, is simply made of five different lines. Now if we extend each of these lines, in both directions, until they intersect again a second pentagram is constructed. Investigate this “extended pentagramma” through the following animation.
A relation worth noting that emerges here: as the pentagramma is made of five 90-90-90 triangles, the pentagramma extended is made of five 90 degree lunes. Here one is shown in blue, inside the pentagramma extended, find the other five.
This reminds me of the story of an astronomer who discovered a two sided, closed geometrical figure, “i don't know if i am the first to have discovered it, but i have discovered it,” he said to a group gathered. A listener stepped forward, proclaiming himself to have mastered all of mathematics, which, he said, elevated him above the mere imprecise astronomer; for he claimed he knew the laws of the universe! And he had it all systematically contained in his books, Euclid's elements. “This figure your describing, i will go master it, what is it called?” asked the “master geometer.” “Why the 'lune,' ” replied the astronomer. So, to as not forget, the master geometer repeated the name aloud, over and over and over again, on his way home. There, in his study, he sat, Euclid shackled to his side, day in, day out. Week in, week out. Months passes by, and his neighbors saw him no more. When wondering aloud amongst themselves, “what happened our neighbor,” they began to refer to him by the only word they heard still repeatedly emanating from him home...
[CLICK HERE FOR AN ASIDE ON MEASURING SPHERICAL AREA (NOTE THE 90 coming UP IN THE AREA OF THE PM)]
Lets take this extension of the sides further, and extend each of the sides of the pentagramma to full great circles. This creates a very interesting tiling of the sphere, but first investigate what this looks like with a simple arbitrary triangle; simply taking a spherical triangle and extending its three sides to complete great circles.
So there is a second triangle created on the opposite side of the sphere, which is exactly the same as the first, but, it is worth noting, different. This second triangle has the same angles as the first and the same sides as the first, but it is different. The difference can be seen if we tried to slide the second triangle across the sphere, and tried to place it on top of the first; play with the animation above. It doesn't work, the two triangles have a “mirrored” relationship.
Now investigate the following animations, where the pentagramma's five lines are extended to full great circles. The first is simply the five great circles, the second has the pentagramma mirificum and a second, mirrored, pentagramma created by the great circles on the other side with both pentagramma marked off, and the third animation has just the two pentagramma.
Interesting. Now, think back to the question raised earlier, the locations of various side's poles; 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've just looked at, where do these second poles lie? Very interesting!