Here, in Part I, we will just begin to touch upon some initial, fundamental properties of the "pentagramma mirificum." Originally discovered by a 15th century scientist, John Napier, the investigation of this construction was greatly furthered by the work of Carl F. Gauss in the 19th century, though Gauss never published anything on the subject. All that is know to this author, are some of Gauss' personal notes respecting his breakthroughs investigating the Pentagramma Mirificum, published in his collected works. The individual who compiled Gauss' collected works mentions that there are more pages of notes from Gauss on this subject, but they were not included in the collected works. More of the background is included in two other articles on this webpage, "On the Spherical" and "On the Not Spherical," by Meghan Rouillard. Here we will get into a geometrical investigation, through which sense-certainty will be shown to be incompetent, and the assumptions of Euclidean geometry irrelevant; for who, when simply looking upon a sphere, could come to the conclusion that the sphere inherently has a 5-fold periodicity, and that it generates certain specific relations which we see in the "physical" world in living processes?
The Geometry of the Sphere
To fully grasp the significance of the pentagramma it is necessary to discover how to work in a spherical geometry. With the correct tools this is accomplished quicker, so lets discover the tools. Another way to put this question is, “what are knowable actions on the sphere?” Most readers will respond with a confident affirmation if asked whether they have a conception of what a “straight line” is. Take a second to reflect on this question if you have not, and do so also if you have. Now what happens if you try to apply your conception of a straight line to the sphere?
First, briefly, what does it mean to be working on the sphere? We are dealing with the sphere as a surface, not a volume, investigating the nature of this curved surface, the sphere. To get at this, one could ask what are the characteristics of the least arbitrary action in (or on) the spherical surface? Most have, as in a geometry class or independently for fun, played with constructing figures, squares, triangles, rectangles, etc. on a plane. But what happens if you try to create the same constructions on a sphere? (For example: a square on a plane is recognized by its having four equal sides and four right angles, what happens in attempting to construct this figure on a sphere?) Take the simple case of a line: a line on a plane can close by connecting to itself or can extend indefinitely with out ever closing, eg a straight line on the plane does not close, but, as will be shown just below, a straight line on a sphere does close.
A straight line on a plane will continue on to infinity, while a straight line on a sphere closes on itself; it is a simple concept, but it already shows a significant difference in the geometries.
If we think back to an arbitrary plane and imagine two locations on that plane, most will have an immediate conception of what the straight line between those two locations would be; but what happens if we have two locations on a sphere? What is the straight line between them? How can we determine and know that? In performing experiments, and playing around, the reader will see there is more then might be expected. What if, from here, we try to develop a general concept of “straightness?” If we look to the straight line in a plane, how could we characterize this action? One property we can identify is that this is the shortest distance between the two locations, so lets apply this conception of straight action in a spherical geometry. This can be experimented with, by simply taking a string and creating the shortest path between two locations you choose on the surface of your sphere (remember, we are working on the spherical surface, so no cheating by cutting through the sphere). Here we now have a more general concept of the characteristic of straight action outside of, and applicable to both the plane and spherical geometries; keep in mind though, this is a working conception for us thus far, further application of this concept to other geometries, including physical (e.g. gravitational), would require further experimentation to determine if this concept is or is not applicable (the experimenter may be surprised).
Lets look at this spherical straight line a little further. In the animation we see the green line moving on the sphere until it creates the shortest distance between the two locations (dark green); the dark red line cuts through the sphere, as the shortest line through empty space between the two locations. The rotating light blue line shows how the red and green lines lie on a plane which goes through the center of the sphere.
This plane defines a unique circle across the sphere (it also being the largest circle possible on the sphere), which corresponds to a straight line on the sphere. This is called a “great circle.” So any spherical straight line can be extended until it closes, forming a great circle. Note the obvious change mentioned earlier, the same type of action, applied in the plane and in the sphere, have completely different results, with a straight action in the sphere defining a connectedness we would only get from a curved action in the world of the plane.
We now have a first tool with which we can traverse spherical geometry.
Let us take up another geometrical property many are familiar with. In a plane we begin with a straight line, and look at another line rotating around one point on the first:
What are the least arbitrary, or most singular points in this process? We see at two points in the action the two lines converge, and also there are two points in this process where the second line is exactly in-between these two convergences. At these in-between points the lines are “perpendicular.” If we have two lines perpendicular to a first it can be seen how, in the plane, they will never intersect each other.
What happens when applying the same actions in the sphere?
The dark blue and green lines are both perpendicular to the dark red line, but now on the sphere we see they intersect at the red points at the top and bottom. Notice that they intersect at very specific locations. If we take the dark blue line (great circle) and rotate it, always keeping it perpendicular to the dark red line, it intersects the green line always at the same two red points.
These two red points are the “poles” of the dark red great circle; notice that any line perpendicular to the dark red line, if extended, will pass through these poles. Any spherical straight line will have two poles.
Lets take one second to examine the spherical triangle we've created here. It has two right angles, something paradoxical to one accustomed to thinking in planar geometry. When we add the third angle their sum is over 180 degrees, and not only that, the sum varies, depending on how large we make the third angle. If we wish to measure this third spherical angle we simply measure the amount of rotation separating the two spherical straight lines (great circles) that form the angle. Just like how angles are formed in the plane, we are measuring the rotational separation between two straight lines using a metric of 360 degrees as a full rotation.
Also notice that because the sides are sections of great circles they are measured as arcs, or angles; i.e. distances on the sphere are measured as angles. So with this spherical triangle we have 6 parts (3 sides and 3 angles), which are all measured in angles.
For more on spherical geometry see the work of Carl Gauss' teacher, A. Kaestner on spherical Trigonometry and the LaRouche Youth Movement pedagogical website for Kepler's Harmony Of the World (Book 1 - Golden Section - Regiomontanus and Spherical Triangles).
The great circle and pole relations emerge quickly and often, as fundamental, in spherical geometry. They are not solely products of a type a action, as one might initially consider geometric figures to be, but they are products of a type of action (straight line) in a specific geometry (the sphere), while they are not products of the same type of action in another geometry, such as the plane, or even any other a geometry but the sphere. We can then see how the great circle-pole relation uniquely represent the properties of the geometry we are working with: they are the fundamental relations intrinsic to a spherical geometry (understand and remember this, as it will be of importance in the pentagramma mirificum).
Construction of Pentagramma
We shall begin with one of a number of methods of construction of the pentagramma mirificum. Begin with an arbitrary spherical right triangle. Trying to construct this by drawing on your paper will be difficult; if necessary, balloons, balls, or spherical fruit, and rubber bands, string, or pens, can suffice for a better first approximation; however the proper tools aid in the discovery process. One can purchase erasable spheres complete with "spherical compasses," or one could construct similar tools. The key to constructions on the sphere is the "spherical compass" used to make straight lines, great circles. The compass used to make spherical straight lines is simply an arc of 1/4 (or 90 degrees) of the great circle of the sphere one is working with. Fix one end of the 90 degree arc as a point to pivot around, and the motion of the other end will trace out a great circle (If this is not clear investigate the 6th animation of this pedagogy, located above, and look at the photos below of the October 2007 LaRouche Youth Movement California Cadre School).
To reach the pentagramma mirificum, begin by constructing an arbitrary spherical right triangle, whose only constraint is that each side is less then 90 degrees. If one is not sure how to do this, start by choosing a pole (P) for the first side, then draw the first arbitrary side, AB, (just make sure it is less then 90 degrees). Also mark point C by extending AB to a total length of 90 degrees, so that AC=90 degrees. Next we use C as our pole and draw our next arbitrary side, AD (this side also has to be less then 90 degrees); notice that, by construction, AB and AD will be perpendicular.
Now if we can find the pole for the line that passes through D and B we can draw our last side; we know this pole will be 90 degrees from D, so we can use our 90 degree spherical compass to draw an arc of a great circle, using D as a pole, in the area we approximate the sought pole to be in. We know B will also be 90 degrees from this sought pole, so we can draw a similar great circle arc using B as a pole. Where these two arcs intersect will be the pole (Q) for our last side (BD).
Now that we've constructed our initial spherical right triangle we can proceed in constructing the pentagramma mirificum off of this first triangle. Extend the hypotenuse (BD) to a total length of 90 degrees (on either side, for this construction we will extend it to the left to E, so that BE is 90 degrees), and then create a line perpendicular to EB at E, using B as the pole, which extends until it intersects an extended side of our initial spherical triangle (in this case we extend AD). AD is perpendicular to AB (by construction), so an extended AD will reach P, which is the pole of AB (making AP 90 degrees); and to construct EP we used B as the pole (to ensure that EP will be perpendicular to EB at E), and any arc of 90 degrees (E.g. BE) rotating around any point of a line (E.g. point B of line AC) will intersect the pole of that line (P); thus they intersect at P, the pole of AC. Thus we've constructed a second spherical right triangle, DEP, as a function of the first spherical right triangle.
If we apply the same actions to the second triangle we will generate a third spherical right triangle. And again, as we did to the first to get the second, and the second to get the third, we do to the third to get a fourth spherical right triangle. One more time to the fourth, and we've constructed a fifth spherical right triangle. Each of the last four are determined by the first, and the fifth always ends touching the other side of our first spherical right triangle, closing the process. If we try to generate a 6th spherical right triangle off of the fifth in the same way, the first triangle would simply be reconstructed.
Take notice that at each of the star corners of the pentagram (A, E, F, G, H) there is a right angle, this will come up later. What is interesting is that this is a closed process, always generating a chain of exactly 5 spherical right triangles, forming a spherical pentagram, a pentagramma mirificum. Remember we started with an arbitrary spherical right triangle (only making sure the sides were each less then 90 degrees), and beginning with a different spherical right triangle will generate a different chain of 5 spherical right triangles which will form a different pentagramma mirificum. There is one case which is the most regular, where all the 5 triangles are the same isosceles right triangle.
This is one method of construction, from which we can gain an understanding of some of its many interesting properties. It can be seen that our chain of triangles intersect the poles we were using in the beginning of our construction, P, Q, and C. In fact, in the pentagramma mirificum, the chain of triangles will always intersect one pole of each of the sides, or to put it another way, one of the poles of a side is always a corner of another spherical triangle. For this reason the pentagramma mirificum is “self-polar.”
This self polar characteristic leaps out as a particularly interesting property for reasons we have touched upon already, and after a brief investigation into the uniqueness of the pentagramma mirificum from another standpoint we will return to why specific emphasis needs be placed on its self polarity.
As we saw above, the angles of a spherical triangle do not add up to 180 degrees, as they do in a plane, and the sum changes as the spherical triangle changes size.
So the property of angles in a triangle totaling 180 degrees is not a property of triangles. It is a property of triangles in a specific geometry. Then what about such helpful tools we may recall from planar triangles, such as the property of "similar triangles?" Similar triangles on the sphere are very rare, and much less helpful then in the plane; as should be clear by now, any property one may know from one geometry (E.g. similar triangles in the plane) can not be taken as valid a priori, but must be tested in another geometry (E.g. the sphere) before it can be determined to be applicable; the unique properties of any geometry will have to be discovered from that geometry (Again, look to the work of A. Kaestner on spherical Trigonometry and the LaRouche Youth Movement pedagogical website for Kepler's Harmony Of the World, Book 1 - Golden Section - Regiomontanus and Spherical Triangles).
Because we began our construction of the pentagramma mirificum with an arbitrary right triangle, there are an infinity of different pentagramma mirificums possible; yet all these are still one type of a larger class of all spherical pentagrams. To get a sense of where the pentagramma mirificum lies amongst other spherical pentagrams lets look at the regular pentagramma mirificum, where the five triangles making up the pentagramma are the same isosceles right spherical triangle. We will measure the distance from the center of the pentagram to a corner of the pentagon, calling this R. As R grows from 0 to 180 degrees the spherical pentagram will grow (R is the purple line).
To get sense of the properties of a spherical pentagram we will investigate four lengths and three angles. The pentagon side, the star side, the pentagram side, the diagonal, the pentagon angle, the pentagram angle adjacent to the pentagon, and the pentagram angle opposite to the pentagon; I called these the seven elements of a regular spherical pentagon.
As R grows how do each of these values change? Here is a chart with the horizontal axis being the value for R and the vertical axis being the value of the side/angle (element); each curve is a map of how a particular element changes
We immediately see a symmetry between the change from 0 to 90 degrees and the change from 90 to 180 degrees. The spherical pentagram defined by R = 90 degrees is a definite singularity point, when the pentagram stops growing and begins shrinking, again an effect of the spherical geometry. We also see various intersections, showing where the values of certain elements are equal. The “diagonal” and the “pentagram angle opposite” curves intersect at a value of 90 degrees, at a point where the R is about 48 degrees; this is the R value of the regular pentagramma mirificum. Here also we see the values for the “pentagram angle adjacent” and the “pentagon side” intersect, and the values of the “pentagram angle” and the “star side,” which are each properties of the pentagramma we will see later.
Now what do the value curves look like if we take the sines of each of the elements (remember distances on the sphere are measured are angles, so we can take the sine of a side).
Again we see the symmetry around R=90 degrees, and at the pentagramma there are again, two intersections. What about the other trigonometric functions of these elements? For reference, here is an image of the different trigonometric functions for a specific angle:
Play with the following animation, where you can change the size of the regular spherical pentagram, and the bar on the right shows the location on the maps corresponding to the size of pentagram; one can also cycle through different maps (for the different trigonometric functions of the elements); also there is a box on top which displays, and can be set to, values of R, and there are boxes below constantly displaying the angle value of each element.
*(Note: when the curves leave the display (above or below) they are going to infinity. Also, when there is a black straight vertical line (e.g. in the tangent map) it is a glitch of the program that plotted the values, it is the computer trying to connect the curve across an infinity; play around with how the trigonometric functions are geometrically generated, and this will become clear.)
When we are dealing with regular pentagons or pentagrams in the plane, there is a significant property where the ratio of the side of the pentagon to the diagonal is the golden ratio. The appearance of the golden ratio often provokes curiosity and attention, especially because of its abundant manifestations in living processes. It often appears as a signature characteristic of a living process, as an invariant mark left behind. A problem encountered today is the mysticism that has shrouded its scientific significance. Looking at it as a magical number or a formula with special force, is the opposite of science, the opposite of the nature and mission of mankind, to discover the lawful, beautiful ordering of the universe and to use the power we gain thus, as knowledge, to further develop. The application of such mysticism is an attack on science; like such unfortunate examples as when modern university classrooms have students believe in unexplained, mystical formulas as if the formula itself were causal, such as the inverse square law posed as the explanation of gravitation. A mathematical formula is not cause. Not to worry for such victims, the actual discovery can be made by retracing the path of the discoverer, and a truthful, lawful ordering the universe can begin to be unraveled, through the discovery of the universal principles governing their own effects with which our senses interact (C.F. the LaRouche Youth Movement's pedagogical pages dedicated to the revival of science: Kepler's New Astronomy, Kepler's Harmony of the World, and soon to be completed Gauss' determination of the orbit of the asteroid Ceres).
Thus the appearance of the golden ratio in geometrical constructions justly draws attention. We see it in the pentagram in the plane, between its side and its diagonal, but what about in a spherical pentagram? We saw before, in the maps above, that the different elements changed in different ways, so the ratios amongst them will not be constant, i.e. they will change as the spherical pentagram changes size, unlike in the plane. Here we have plotted all the possible ratios and their inverses amongst the 7 different elements, totaling 42 ratios, each mapped as a curve here (E.g. at the regular pentagramma mirificum, the length of the diagonal is 90 degrees and the length of the pentagram side is ~38.1727 degrees, so the ratio “diagonal”/”pentagram side” is 90 / 38.1727 = ~2.3577, and the ratio “pentagram side”/”diagonal” is 38.1727 / 90 = ~0.4241; as the pentagram changes size each of these elements change, but in a different way, so the ratio will change).
Again we see the symmetry across the 90 degrees pentagram. There are many intersections throughout, indicating when ratios having the same value. Is there anything else interesting?
At this point in our investigation an interesting question can be posed: if we were to take the same ratios for different sized regular pentagrams in the plane, what would the plotted curves look like? These curves, mapping the relations of regular spherical pentagrams, are not simply artifacts of pentagrams per se, but of the properties of pentagrams interacting with the properties of spherical geometry, spherical curvature.
Now what happens when we take the sines of these elements and look at the ratios amongst the sines?
Here we see more of an interesting mapping of the properties of spherical pentagrams. There are 42 ratios (7 elements, so we have all their combinations and the inverse of each combination), but, because some ratios are the same, there are only 27 different curves. All 27 curves begin as only 9 different values, but they immediately diverge, only to converge again, this time with only 5 different values, when R=~48.03, the pentagramma mirificum. These values of the ratios of the sins of the pentagramma mirificum's elements are: the golden ratio, the square root of the golden ratio, 1, the inverse of the square root of the golden ratio, and the inverse of the golden ratio. They diverge again, only to again converge on 5 values at R=90. Then at R=~138.03 we have second, compliment pentagramma mirificum on the other side of the sphere (this second pentagramma mirificum will be investigated below), and at R=180, where the pentagram is again infinitely small (as when R=0).
Here we have the ratios of the sines of the elements of all regular spherical pentagrams. One might notice the curious almost convergence around R=~30, ~150 and R=~60, ~120; its is as if they were tending towards, but could not quite achieve convergence. Only 3 pentagrams stand out as having a unique convergence (R=0 and 180, the regular pentagramma and its compliment, and R=90); even so, in one of these cases, R=0 and R=180, the spherical pentagram is infinitely small, and thus partakes of none of the curvature of the sphere, having similar properties to the flat plane (one can get a sniff of this simply by comparing the the ratios of the elements of the spherical pentagrams when R=0 and R=180, with the corresponding ratios for any regular pentagram in the plane); also, in the R=90 case, looking back to the animation above, it can be seen how the entire spherical pentagram becomes a single great circle, and thus lies, completely, in one plane. So, partaking most in the geometry of the sphere, the pentagramma stands out amongst these three. Now, what are the ratios for the other trigonometric functions of the pentagram elements? Investigate the following animation.
*(Note: some of these plots are the same, like the ratios of Tangents and the ratios of Cotangents; through investigation of the geometric determination of these trigonometric functions and their interrelations, one can see why this occurs.)
The last chart shows all the sets of ratios for each trigonometric function together; when we look at all the ratios of similar trigonometric functions together, which spherical pentagram has the most convergence of values? Having the most convergence while still embodying the curvature of the sphere, and expresses golden ratio relations in the trigonometric functions of its elements, The pentagramma mirificum stands as the most unique case. Take note!
So through all this charting and plotting we see the pentagramma mirificum emerging as the unique case amongst its family of pentagrams. But, that was also knowable from the previous investigation, it's construction from a chain of spherical right triangles! What was the most fundamental relation we found in working in spherical geometry? The great circle-pole relation. This is the most unique property of action in a spherical geometry; it embodies the uniqueness of the geometry of the sphere itself. In the pentagramma mirificum this pole-great circle relation is dominant throughout (e.g. through its self polarity). And it is not only in the regular case of the pentagramma that we have this fundamental relation, but in all constructions of the pentagramma mirificum.
So to recap, we have now discovered that the pentagramma mirificum centers around this most fundamental characteristic of spherical geometry, the “great circle”-”pole” relation, exposing the pentagramma mirificum as a reflection of spherical geometry itself, and, through it's construction from an initial arbitrary spherical right triangle, generating a closed chain of five spherical right triangles, it exposes a five fold periodicity as intrinsic to the sphere. Also, in the most regular case of this cardinal construction of spherical geometry there exists, uniquely, a convergence on the golden ratio. Mirificum indeed.
Here we have only scratched the surface. To investigate a step deeper, move on to Part II; there is still much more rediscovered from the notes of Carl F. Gauss' own investigations as well.