On the Spherical
In a recent paper by Lyndon LaRouche, “More on Physical Space-Time” and following that in discussions with collaborators, he developed the idea of Sphaerics as actually that which deviates from the purely spherical. Otherwise Sphaerics is akin to a housewife who cleans her house and then says, “Don't walk, talk or breath! I don't want anything to change! It's perfect!” He also emphasized that a mission-oriented human being will engage in scientific investigation from that standpoint, hopefully with an increased capacity to recognize that which deviates from the purely spherical or regular. In an interesting way, this seems to be the subject of Carl Friedrich Gauss’ treatment of the Pentagramma Mirificum, which is a very creative investigation of that which appears to be simply spherical.
During a recent conference on the Eurasian Landbridge in Kiedrich, Germany, LaRouche referenced to Gauss, saying that he “never tells the truth” in the sense that he will give mathematical formulations to prove the fact that his discoveries are valid, but that those mathematical formulations, given as proof, will not reveal the method used by him to make a particular discovery. This author has a sense of the truth of what LaRouche is saying from working on the Pentagramma Mirificum, which is handed down to us in the form of 11 short fragments, complied over a period of decades, probably from the 1820's to the 1840's, and consisting mostly of mathematical formulas. However, what we have been able to discover about the pentagramma mirificum was rarely catalyzed by the calculations in the fragments, but came mostly from harking back to the works of the Greek geometers, Johannes Kepler, John Napier, Kaestner, as well as the French and Italian geometers of the 19century, and also a couple of letters by Gauss, where he discusses as least several things which are on his mind, which are reflected in the pentagramma mirificum implicitly. LaRouche, who at one point was rumored to have said that it would take ten lifetimes to figure out all of the implications of the pentagramma mirificum(!), emphasized to the Ceres group the fact that this work was a good window into the actually constructive thinking of Gauss, his reference to the celestial sphere, and harmonics. LaRouche said that Gauss was looking at “astronomical functions” here. He said that Napier examined this from a 16 century standpoint, Gauss from a 19 century standpoint.
16 century Standpoint
The Pentagramma Mirificum was originally found in the work of Scottish mathematician and astronomer John Napier, who likely gave it that name. He was a contemporary of Kepler; who was very thankful for Napier’s development of logarithms. The pentagramma mirificum comes up in Napier's paper on logarithms, something he developed to facilitate astronomical calculations, like the multiplication of huge sine values, for example. (Just to get an idea, let’s say the actual distance from the sun to the earth’s aphelion is 95 million miles. If we use trigonometry to calculate these distances, as Kepler does in his New Astronomy, and we use a unit of 167,000 to approximate such a vast distance, how great could our error potentially be without using many decimal places in making our calculations using this unit?)
He applies this to solving for logarithms in plane triangles, and then has a second book dealing with spherical trigonometry. The pentagramma construction arises out of Napier's taking one spherical right triangle and extending it sides to 90 degrees in full, and creating a chain, which becomes an irregular pentagramma mirificum. Napier exclaims that here we now have triangles which “in their natural parts are not equal, but such differences are resolved in the context of this construction” . (See Ben’s animations on the construction of the PM, and also The Harmonies of World Page; reference the part on the Golden Section).
Napier does discuss the fact that a gnomonic projection of an arc will project the tangent of that arc, but his discussion is general and he is not specifically discussing the pentagramma mirificum (This would be the treatment of the pentagramma mirificum done by Gauss many years later). Interestingly enough, Napier actually projects the entire construction of the pentagramma mirificum onto the celestial sphere, giving each vertex the letters S, P etc.. which stand for Sun, Pole etc... If the use of logarithms is for calculations of an astronomical magnitude, it would seem to make sense that the triangles Napier is actually thinking of are not little constructions in his notebooks, but are conceptually used for solving astronomical problems. We can think of generating the Pentagramma Mirificum in two ways. One is, what would be the orientation of 5 great circles such that we could create a spherical pentagram that had right angles at all of it’s star points (this is how Napier presents it). We can also think of the difference between a chain of right triangles on a plane, and a chain of right triangles on a sphere—How is each one generated and what are its characteristics? On the plane we get a different periodicity—a chain which closes in on itself after 8.
This is characterized by similar triangles, which don’t exist on the sphere.
Very important: The construction is unique to the geometry of the sphere as it arises from the self-polar characteristic of the sphere (pole-to-equator), as every arc in the pentagramma mirificum is one quarter of a great circle, including any diagonal or altitude line produced internal to the construction. This was obviously a quality Napier found very interesting, which we see from his placement of this magnificent construction on the celestial sphere. It is different than simply drawing a regular pentagon on the sphere, which does not reflect this characteristic of the sphere. This is crucial, because any modern day Andy Warhol could draw any construction they want on a sphere, with total disregard for its unique characteristics (only visible to the mind’s eye!). The Pentagramma construction is only possible with aid of the singularity of the great circle and its respective pole, which is not only the distance from any point on the sphere to literally “the greatest circle”, but also the great circle itself, and any portion of it, represents a geodesic (portion of the shortest path) on the sphere.
Interestingly enough this construction embodies one of Kaestner's notable criticisms of Euclid, the idea that two parallel lines will never intersect. In the Pentagramma construction, they always do!
So rules of plane trigonometry are not much of a road map when navigating spherical space (You may be able to solve many problems from Kepler’s New Astronomy, but be lost on the sphere! Such a sense of feeling lost in a new space may be one of the most important things you can realize about geometry!). The reader is encouraged to work through Kaestner’s lessons on spherical trigonometry, especially his table which makes use of proportions to figure out from any two elements of a spherical right triangle, the third, and from that, all 6. (Napier also has a whole section in his book on solving for the third element in a spherical right triangle). It would be useful for the reader to work through this, since to know intellectually, without aid of a flashlight ( which we cannot assume he had!), how a projection of the pentagramma construction works, you will have to figure out all of the elements (sides, angles) of all of the spherical triangles. See two different examples above-- Gauss provides the investigator with only the five sides of the outer triangles. The intersection of two altitude lines will determine where the other three will be (we see below the unique case of three visible altitude lines being sufficient) . But don’t get too attached to this (although its pretty fun navigating the sphere!) With Gauss, we are figuring out the relationships on the sphere, only to depart from it!
Above is the skeleton of the cone representing one possible projection from the regular pentagramma mirificum. The apex represents the center of the sphere, and extending from it are five points which lie on an oblique cut of the cone. The other points represent altitude lines of the projected pentagon.
So there is a bounded, yet unlimited, amount of constructions we can create. Since similar triangles do not exist, these will all consist of different bounding triangles, or one specific chain that will have five of the same triangle.
(To play with this animation: the first two sliders allow you to change the pentagrammma mirificum (such that you can cycle through all possible pentagramma mirificums), the second two sliders control where the plane the pentagramma is being projected onto is tangent to the sphere (shown by the red dot), and the animation can be spun around using the "arrow keys" on the 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.)
Now, what is the likelihood that this would be an arbitrary triangle? What do we know about the regular pentagon on the plane—it is characterized by the golden section, as is the dodecahedron. It would be fair to hypothesize that perhaps the golden section would be present somewhere in a regular pentagramma. Would it appear in the same way as we see in the planar pentagon, where the ratio of the side to the diagonal is the golden section?
(*Above the radius is 1, below the radius is 2; the constuction holds in each.)
Nope. That ratio on the sphere for the side to diagonal of the Pentagramma Mirificum is close, but no cigar-- .57 to 1 on the sphere, for a regular or “Golden” Pentagramma. Try to figure it out!
Here is the side:
The sines of the arcs of the PM is where we find the golden section , although the reader is encouraged to think about whether the golden section implications of the PM are limited to its appearance as amongst the sines of arcs of the construction. ( This author is still investigating that, and doesn’t think that’s the case, based on some clues from a paper by Lyndon LaRouche called “An Economists View of the Pentagramma Mirificum”)
There is a crucial difference between this regular pentagramma mirificum and the others, (one could also ask if this regular pentagramma mirificum is actually “constructible”, since it is expressed as the arcsine of the square route of the golden section, which is then projected onto the sphere as an arc—not easily constructed with a simple spherical compass, without approximating) which we will only get at when expanding our investigation into conic sections and get a whiff of where Sphaerics intersects Gauss’ wok on elliptical functions. This is where we leave the actually Spherical Pentagramma Mirificum!