Intro

We shall begin with one of a number of methods of construction of the pentagramma mirificum. Begin with an arbitrary spherical right triangle. If nothing else is available spherical fruit, such as a grapefruit, and rubber bands or a pen will suffice for a first approximation; however to fully grasp the significance of the pentagramma it is necessary to discover how to navigate through 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 general 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 we must ask what happens when we try to apply our 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 fundamental relations of this curved surface, the sphere; one could ask what are the characteristics of action in the spherical surface? Most have, as in a geometry class, played with figures, squares, triangles, lines, etc., on a plane. What happens when doing similar things on a sphere? Take the simple case of a line, a line on a plane can close, or can continue on to infinity, eg a straight line, but, as will be shown just below, a straight line on a sphere closes.

To some it may seem like a simple concept, but there is great significance in the fact that a straight line on a plane will continue on to infinity, while a straight line on a sphere closes on itself. At first one may simply note that they are different, very different.

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 preforming experiments and playing around the reader will see there is more then might be expected. What if we try to develop a concept of the general characteristic of action 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 crating the shortest path between two locations you choose on the surface of your sphere (remember, we are working in 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), will require experiments to see 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 circle across the sphere; this circle is the largest circle on the sphere, 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 on the plane and on the sphere have completely different results, with a straight action in the sphere we get 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 knowably 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 a very specific location. 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, it 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 two spherical straight lines, great circles. 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) all measured in angles.

For more on spherical geometry see Kaestner [Link XXX] 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 in spherical geometry. They are not solely products of a type a action, as one might consider geometric figures to be, because they are products of a type of action (straight line) on the sphere, and are not products of the same type of action in a geometry such as the plane. We can then see how they represent the properties of the geometry we are working with; the great circle and pole relations are 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. If nothing else is available spherical fruit, such as a grapefruit, and rubber bands or a pen may suffice for a first approximation; however the proper tools make this more understandable and easier. One can purchase erasable sphere's 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 a firm (plastic or other material) arc of 1/4 a circle only slightly larger then the size of a great circle on the sphere one is working with. On one side of the arc is a point to pivot around, and on the other is a pen to mark the lines with.

-pic of lenarte sphere and compass (label lenarte sphere)

Construct an arbitrary 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 AB and AD are perpendicular, as can be see in the great circle pole relations.

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 draw an arc of a great circle, using D as a pole, in the area we approximate this 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. 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 EB 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). These intersect at P, for reasons which will become clearer below. Thus we've constructed a second spherical right triangle, DEP.

If we apply the same actions to the second spherical triangle we will generate a third spherical right triangle. And again, as we did to the first to 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 get a fifth spherical right triangle, and we end up touching the other side of our first spherical right triangle. If we try to do this again to the fifth triangle we regenerate the first triangle.

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 still form a pentagramma mirificum. There is one case which is the most regular, where all the 5 triangles are the same isosceles right triangle. The regular case will be investigated below.

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 the pentagramma mirificum the chain of triangles will always intersect one pole of each side, 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.”

Uniqueness

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 as the property of similar triangles? There is no such property in the sphere (but there are other relations to discover, see Kaestner [Link XXX] 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 these are still one type of a larger class of 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 (CLICK HERE FOR A PEDAGOGY ON DETERMINING THE REGULAR PENTAGRAMMA). 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 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.

As R grows how do each of these values change? Here is a chart with the left/right axis being the value for R and the up/down 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 values 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 of the 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 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 as 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 two intersections again. What about other trigonometric functions (LINK TO PIC OF TRIG FUNCTIONS) of these elements? Play with the following animation.

*(Note: when the curves leave the view they are going to infinity, which is not shown so that one can see the rest of the curves. Also, when there is a straight vertical line 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 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.

-Animation / PROOF?... Show its algebraic (1+Sqrt[5])/2...

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 by a higher principle or power generating what we perceive. A problem encountered today is the mysticism thats has shrowded 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 an unexplained, mystical formula, such as the inverse square law, 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 discover, 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 synthetic 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 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 seen as curves 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; but these ratios change as the pentagram changes size.)

Again we see the symmetry across the 90 degrees pentagram. There are many intersections throughout indicating ratios having the same value at different size spherical pentagrams. Is there anything else interesting?

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 some ratios are the same so there are only 27 different curves. All 27 curves begin as only 9 different values, but they immediately diverge, only to converge again, thins time with only 5 different values, when R=~48.03, the pentagramma mirificum. These values of the pentagramma 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, a map. 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 unique 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 great circle, and thus lies, completely, in one plane. 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 occures.)

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?

The pentagramma mirificum stands as the most unique case, having the most convergence while still embodying the curvature of the sphere. Take note!

At this point in our investigation an interesting question can be posed: if we where 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.

So through all this charting and plotting we see the pentagramma mirificum emerging as the unique case amongst its similar constructions. But, that was also knowable from the previous investigation of it's construction from a chain of spherical right triangles! What was the most fundamental synthetic constructive 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. In the pentagramma mirificum this pole-great circle relation is dominate 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. Also, in the most regular case of this cardinal construction of spherical geometry there exists, uniquely, a convergence on the golden ratio. Mirificum indeed.