On the Irregular, Page 1: Projecting Off, and On, the Sphere

In part 1, the case of the regular, or golden, pentagramma mirificum stood singular; a peculiarly unique instance in what, without our maps (which showed the changes in the different elements of the spherical pentagrams, and the changes in the element's functions and ratios of the functions), or without a knowledge of the process of construction, may otherwise appear to be a smooth, continuous series of regular spherical pentagrams.  It is when this fundamental characteristic of spherical geometry, the great circle-pole relation, is contained in the pentagram that this harmony emerges. Now recall that this fundamental great circle-pole relation, as we saw in the step by step construction of the pentagramma above, is not only in the regular case, but in all irregular pentagramma mirifica as well.  Then, if we dig, what other relations can be found to exist in all cases of the pentagramma mirifica? What will these tell us, and what will they telling us about?

In his investigation of the pentagramma mirificum, Carl F. Gauss investigated the gnomonic projection of the pentagramma (also known as the central projection, for points on the sphere are being projected from the center of the sphere to the plane; this is seen by the green lines in the animation below). This is shown in the animation below, where there is a pentagramma mirificum in orange on the sphere. To show how the projection works: the green lines demonstrate the projection of the pentagramma's five outer corners, from the center of the sphere, to a plane which is tangent to the sphere (tangent at the point shown in red). The entire pentagramma mirificum is projected to this tangent plane, forming the black planar pentagram. The reader can use the first two sliders to adjust the pentagramma mirificum, and cycle through all the possible pentagramma mirifica; the second two sliders adjust the point of tangency of the plane on which the pentagramma is being projected (the tangent point is shown in red).


*NOTE: most all animations can be spun around by using the arrow keys on your keyboard. Also note the animation breaks down if the user moves a point (or the points) being projected to 90 degrees or more from the point of tangency, because of the nature of a central projection.

Gauss investigated the very interesting characteristics of this projected pentagram. At one point he looks into the provoking fact that the five points of any pentagramma projected onto a tangent plane will define an ellipse. This then leads into some interesting questions. For why would this construction, which is unique to the sphere, define something which, it would seem from an empirical analysis, the sphere would have no reason or ability to generate? This paradox, and more of Gauss' investigation is taken up in two other articles (CLICK HERE), but, for now, let us return to the sphere and see what else can be uncovered.

Construct any pentagramma mirificum from an arbitrary spherical right triangle, as above.  What does it minimally take to define a specific pentagramma mirificum?  And, in different pentagramma mirifica what characteristics remain?  Play with the following animation.

In the animation above, the two sliders control the size of two sides of the top, multi-colored triangle. The triangle, as was discovered in Part 1, will always have its outer angle equal to 90 degrees, and the two sides controlled by the slider are the sides forming this 90 degree angle. With a little reflection, it can be seen that the length of these two sides (displayed as an angle, shown above the slider), and that they meet at a 90 degree angle, define one specific triangle. Remember also, a specific spherical right triangle defines a specific pentagramma mirificum, so these two sliders allows the user to move through all possible pentagramma mirifica.

One characteristic that remains unique, and of fundamental importance, is the self polar property.  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 extended on each end by equal lengths, which are sides from its two neighboring triangles, and the sum of a pentagon side and one of its two neighboring sides is always 90 degrees.  This characteristic was used in the process of construction the pentagramma in Part 1.

Quickly we take note of another property in all pentagramma mirifica, that a side of the pentagon is always equal to the triangle angles opposite it; i.e. in the image below, Y = b.

If the reader is not sure why X = a + b + a , see the image below:



Our geometrical investigation now quickly takes us into the characteristics of spherical triangles. One can take any arbitrary spherical triangle, but we will use one with a right angle because it is the case with the triangles of the pentagramma mirificum.  Any spherical triangle is made of three sides, straight lines, and, remember, on the sphere any straight line always defines two unique “poles.”  Taking one pole from each side of the triangle we have:

As in the following diagram, if we connect these three poles, a second triangle is formed, it being a function of the first:

This second triangle is clearly related to the first, for it is made of the poles of the first's sides, but how exactly is it related?  Start with a side of the second triangle, e.g. the side connecting the "pole of A" with the "pole of C."  The measure of this side is the distance between theses two poles; could this distance between the two poles be related to the two sides belonging to the two poles?  Those sides are A and C, of the first triangle, and they are separated by the angle β.  As we investigated in Part 1, the two great circles formed by extending those two sides (A and C) would be also be separated by angle β.  Then what would be the angular separation of the poles of these two great circles?  Investigate the following diagram, or refer back to Part 1 if this is not clear.

Because this side of the second triangle is equal to a corresponding angle of the first triangle, then, necessarily, this will be the case for the two other sides of the second triangle as well.  And inverting, in looking for the value of the spherical angles 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 greatly aids the discovery process.

The user can play with this interactive animation to get a sense of how this interesting second spherical triangle (green) changes when the initial spherical triangle (blue) changes:

Something interesing has been stumbled across, the poles of any spherical triangle form another triangle, which is an interesting kind of inversion of the first. Perhaps this could be called it's "polar compliment" (the reader is encouraged to investigate what happens if the other poles from each side is taken; also if similar "polar inversions" occur with other geometric shapes on the sphere, and if so what are their characteristics).  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, a corner of another triangle to be exact (the curious minds can also ask "What about the other, second pole of each side?").

So then, let us fill in these second, polar triangles in our construction of the pentagramma (one triangle of the pentagramma and it's polar compliment are highlighted in blue).


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 (we will call these diagonals); for reasons noted above, each of these five diagonals will always have a length of 90 degrees.


In looking at the just discovered second spherical pentagram inside the pentagramma mirificum, taking one of the former's sides (each of which are always 90 degrees in length, and are diagonals of the pentagramma mirificum), it is connected with two other lines of 90 degrees in length which connect to eachother, the three form one triangle which has three 90 degree sides:


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 (each triangle is shown in a different color, and parts are dashed when two lines overlap; the user can see how these triangels move as he or she uses the sliders to make different pentagramma mirifica, because, remember, these five self-polar triangles will be formed by all irrregular pentagramma mirifica):

(*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, where it will get even more interesting, but first let us extend our knowledge of the pentagramma's relations on the sphere.  The pentagramma mirificum, without adding the five diagonals, is simply made of five different lines:

Now if we extend each of these lines, in both directions, until they intersect another line, a new, larger pentagram is constructed.  Investigate this “extended pentagramma” through the following animation.

As the pentagramma is made of five 90-90-90 triangles, the "pentagramma extended" is made of five 90 degree "lunes" (a lune is a two sided, closed figure).  Here, one lune is shown in blue, inside the pentagramma extended, find the other five.

This gives to recall the old 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,” the astronomer said to a gathered group.  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 them all systematically contained in his books, Euclid's elements.  “This figure your describing, what is it called?” the “master geometer” had asked.  “Why the 'lune,' ” replied the astronomer.  So, as not to 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 they wondered aloud amongst themselves, “what happened our strange neighbor,” they began to refer to him by the only word they heard still repeatedly emanating from him home, asking, "where is that lune?"