JEHOVAH’S CONIC SECTION TEMPLATES

Contents

I. Introduction

The created order (nature oftentimes henceforth) is largely curvilinear in its design and outworking. It is largely complex curvilinear. In contrast to curvilinear stands rectilinear, i.e., “straight”—square or rectangular, or planar and linear angular symmetry. For example, atomic arrangement in a crystal lattice (variously triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, cubic arrangement). Imagine the blades of grass in a pasture each in its place laid out on a rectilinear grid (an idealization). Or consider a line that connects the center of one’s eyes, and that is perpendicular to an imaginary line that intersects said eye-line and passes through the nose to the tip of the chin, each with precise lengths about some mean and standard deviation. One could go on and on here, so clearly symmetries that are decidedly rectilinear are plenteous in nature to be sure, but curvilinear, complex curvilinear, is more so. And why?

(1) Form follows function. Jehovah designed our universe with complex local spatial curvature, so that balls can roll downhill, and so that humans do not float off the surface of the earth whilst rebounding a basketball, etc.

2) Curvilinear symmetry morphs from rectilinear symmetry. As concerns our study, imagine rotating one end of a line about a second, fixed line, while keeping the other end of the rotating line fixed—this will produce a cone (Fig. 1). And once we have a cone, we can intersect it with a plane (which is rectilinear) at various angles to produce the “curvy” so-called conic sections: a circle, an ellipse, a parabola, and a hyperbola (Fig. 2).

The conic sections are design templates (design “kernels”) which the Creator Jehovah stamped into His creation (thus we hold), and with which curvilinear nature all around us identifies to some degree, oftentimes in combinations of said conics (by conics we mean conic sections). Our world is 3D, so the conics rightly extend so as surfaces of revolution: circle-> sphere, ellipse->ellipsoid, parabola-> paraboloid, hyperbola->hyperboloid. This study is interested in these design templates, from their aesthetics, to the mathematics which perfectly stands in place of them, to their exceeding practicality in those functions that call for a particular conics form.

The conics’ shapes, whether 2D or 3D, should be thought of as the physical manifestation of a computation. (Please note: the computation came first!) We mean to say, when one is looking at say a hyperbola, one is looking at, besides the beautiful form, an equation.

Each computation space, the rectilinear one, and the curvilinear one, consists of bases which comprise a coordinate system particular to that space. A coordinate system is a convenient and orderly and repeatable means of measurement based on coordinate directions. And a basis, very crudely put, is like a bag full of variously sized rulers that can be concatenated in unique ways to be able to reach any point in its respective computation space. For example, a possible basis for a checkerboard computation space might consist of one-square length horizontal and vertical rulers. If one can uniquely concatenate these rulers in such a way as to reach any square on the board, one has a decent checkerboard basis to work with for measurements and computations based thereon. (Two rulers here, with a length of one board-square, one horizontal, one vertical, comprise the checkerboard basis.) But maybe you say, computation on a checkerboard?! Indeed, computation and measurement are intimately related: what is the area of the checkerboard? It is its length times its width. The horizontal/vertical, one-square length basis allows one to conveniently determine checkerboard length and width. But suppose one of the basis rulers were set at a weird angle—that would not be a convenient way to determine this length and width because now one would have to do some fancy trigonometry or slick linear algebra to get at the length and width. One could still reach all the squares, but painfully so without the right skills. Intelligent choice of bases is the stuff of intelligent mathematics and physics. Any engineer or scientist worth their salt knows that deciding upon an appropriate basis is always step one in any problem-solving scenario. This is true in any field of endeavor, not just mathematics and science. For example, an eight-ounce measuring cup is a good basis for baking a small cake (the computation if you will). With such a basis, of “length” eight ounces, and “direction” full or empty, one can collect pertinent ingredients and accurately reach all pertinent quantities in the cake-bake computation space quite conveniently. As an exaggerated contrast, a wheelbarrow would not do so good here—not least accuracy would suffer. A nutcracker is a better basis for cracking walnuts than a sledgehammer, and so forth. (We are quite stretching the idea of a mathematical basis for illustration purposes. Just to be clear, by definition, a mathematical basis spans a vector space, that is, linear combinations of the basis vectors allow one to reach any point in the vector space. Moreover, and importantly, bases are comprised of linearly independent vectors—this precludes degeneracy and redundancy in computations via the basis.) When there is a need for measurement and computations based thereon, proper bases choice is a boon and is oftentimes the difference between success and failure in problem solving. All that to say this: we hold that bases were fundamental to our Creator’s creative modus operandi—He used them in exceedingly elegant, ingenious ways. The conic section templates of interest in this study are part and parcel of that exceeding elegance and ingenuity. Praised be our Creator God Jehovah.

Please allow a bit more background for the sake of practice. Consider the ancient Israelites, they had a basis by which they made their measurements—the omnidirectional cubit (a rectilinear computation space is assumed). Everybody understood that a cubit ran from the elbow, which is a vector tail and pivot in that system, to the tip of the middle finger, i.e., to the head of the vector, a length of 18 inches or 0.5 meters for an adult, on average. Compare the checkerboard basis of length one board-square, strictly horizontal and vertical—which is the better basis? Answer: they are both good bases in their respective computation space, i.e., for what they are intended to measure, ultimately to be folded into a relevant computation; again, computation leans heavily upon a priori measurement of some sort. The Ark of God, to three significant figures, was 2.50x1.50x1.50 cubits long, high, and wide, respectively (Exodus 25:10). Please note that 2.50/1.50=1.67—sitting right nearby the “golden,” i.e., pleasing,” ratio of 1.62 found throughout nature; the .05 delta is surely for practical reasons of measurement, i.e., God did not say, ‘…Make the Ark 2.43 cubits long and 1.50 cubits high and wide...’! (2.43/1.50=1.62.) How to measure a 0.43 partial cubit in antiquity?! No, no, let us appreciate that one can eyeball half a cubit fairly well, good enough and no big deal, nevertheless, God had them sitting right on His Golden Ratio here taking into consideration and yea despite their limitations! (Not an accident by the way.) So, cubits comprise a (rectilinear) basis; one can concatenate enough of them or parts thereof to measure any distance anywhere. (Given its rectilinear flavor, not a good basis for curvilinear computational space—it could be done, but with difficulty. A cubit basis befits computing in antiquity just fine.) Moreover, it was a convenient form of measure for them (notwithstanding, arm lengths varied!), and again, that’s the thing about bases—they are supposed to be convenient, hence some bases do not lend themselves well to computing as pointed out above, while others do. Thus, circular-type problems are best computed via a circular-type basis called a polar basis, wherein the radius r of the circle and the counterclockwise angle theta from the horizontal determine the final makeup of the basis. But then there are symmetries that are circular, yet not flat, having some height as well, like a piston for example, so working out a problem that has that sort of symmetry would be best approached using a cylindrical basis, which includes a height h as well as a radius r and some angle theta. And then there are problems that have spherical symmetry, like electrical and gravitational potential problems, where it is best to use a spherical basis by which to compute this and that. Spherical bases should have a makeup based on a radial component r, and two angular components, one angle being polar, from the vertical to the horizontal, the other angle azimuthal. By God’s grace, spherical-symmetry computing allowed humankind to derive the electron configuration and dynamics of His elements upon which all of chemistry is based and it goes without saying how important chemistry is to vital human living day by day. This grace came by way of spherical harmonics deriving from multiple separation of variables operations in the way of solving the [spatial] time independent Schrodinger equation consisting of a Laplacian (second-order spatial derivatives summation) based on a spherical basis. To here, then, a brief introduction; let us now look a little closer at Jehovah’s conic section templates.

II. The Circle

Plato said that “the idea of a perfect circle is the form of the circle.” He intimates that a perfect circle is a mental construct, no more. We know the idea, we even know the form mathematically (and are thus apt to consider it perfect), but we cannot take the perfection, the idea in the dress of its form, from mind to matter, from the abstract to the real.

Suppose a real circle is, by definition, a perfect circle (premise), we would not expect to find any real circles manifest in nature if Plato is right (tentative conclusion). Is Plato right?

A perfect circle has every point around its circumference perfectly equidistant from its center, a distance precisely known. We have a difficulty here. Imperfections associated with the Fall of humankind gave rise to a so-called uncertainty principle (perfection knows no uncertainty on the other hand). We may get close to that via today’s technology at the macroscopic level, but the uncertainty principle (in quantum mechanics) tells us that this will never be fully realized. Why? Each point around the circumference would have to be placed at some precisely known position X that is exactly equidistant from the circle’s center (=no uncertainty in the position), but the uncertainty principle effect, which is pronounced at atomic and subatomic scales, tells us that the uncertainty in the position of something, dubbed DeltaX, must be greater than or equal to a constant divided by the uncertainty in the thing’s momentum P, dubbed DeltaP; thus we have for the position | momentum uncertainty principle the following:

DeltaX >= constant/DeltaP.

A perfect circle would require DeltaX to be 0, so to build a perfect circle we would like to have:

0 >= constant/DeltaP,

and the only way we can get that to happen is if the uncertainty in the momentum, DeltaP, were infinite (constant/infinity=zero). But what exactly does an infinite uncertainty in something’s momentum mean? It means that its motion is spread out over all possible positions X! In short, the thing is not localized in any way whatsoever if DeltaX were 0. A real circle is a physical thing comprised of atoms and electrons—we must put some graphite down on paper or light up a pixel or whatever to physically construct it, and there’s the rub, whatever stuff we use, and try to squeeze the uncertainty in its position down to zero, it will turn on us at atomic scales and become less and less localized the harder we squeeze. So, (Mr.) Plato was right, we can put a piece of graphite exactly where we want to in our mind following a mental mathematical form understood to be a circle, but not on an actual piece of paper, nor even can we get around this electronically via a computer and a computer screen. Of course, we can sure get close, close enough macroscopically, but we cannot be perfect here, physically speaking. Thus, in keeping with our premise, we do not find real circles stamped into nature, but we do find the rest of the conics no problem, because they are betrayed by degeneracy, which is the norm post-Fall. It is ever a wonder how Jehovah out of a disaster brought about lovely things like the conics. Compare what He did post-Flood, same thing, lovely things all about in the face of another massive disaster. Sin is the disaster out of which Jehovah makes a way to bring about lovely things, like the Redeemed, the post-Flood earth, the conic sections…

So, we must work with the form of circles, in our mind, going forward. Thus, we shall (must) proceed. In keeping with our constraint, please consider this mental construct (Fig. 3). Circles have a ring of eternality about them, do they not? Jehovah must have had that in mind when He formed the earth (Isaiah 40:22, 66:22, 2Peter 3:13).

When one divides the circumference of any circle by its diameter a peculiar constant always emerges, namely pi, or 3.141592… (Fig. 4). Why should that be? It is because change in the circumference of a circle with respect to change in its diameter shakes out to be precisely the peculiar constant pi, always. But why should that be? It bespeaks (1) immutability, and (2) eternality. As to (1), imposed change that always lands on the same constant is presumed change, it is a summary-statement of immutability. As to (2), it follows from (1), in that immutability demands eternality, or non-existence, but circles do exist, though we cannot see them for what they truly are, as for example this perfect form (great idea by the way):

r*e^(i*theta), (as theta swings from 0 to 2 pi),

or this perfect form (another great idea):

{{x,y}}.{{1,0},{0,1}}.{x,y}

A circle has no beginning and no ending. God helps us here to realize that if something as simple as a circle can thus be, how much more so the self-proclaimed Eternal Mind behind a circle (Revelation 1:8, 22:13)?

III. The Ellipse

Figure 5 for the basic shape and characteristics (cf. Fig. 2). Jehovah God built an elegant “switch” into His conic section templates so that He could easily transform the shapes from one to the other. The switch is the mixed term b in the equations 1-4 (Fig. 2). The ellipse has ac > (b/2)^2 as in the case of the circle. This is because a circle is just a special case of an ellipse wherein the two foci coincide; a circle has no mixed term b whatsoever, it is “spectrally pure” in that regard. With the ellipse comes some degeneracy—we are not spectrally pure here anymore, but still positive definite, which means there exists a well-defined minimum, which is to say that the ellipse is “upward inclined/sloping,” like a bowl. Knowledge of problem maxima and minima are fundamental to practical computing. (When a pattern stops changing at some point in a well-defined interval, it signals either a local maximum, minimum, or inflection/turning at that point. If one discovers that the rate of change (a second derivative) at that point is zero, one has discovered an inflection point in the pattern, if the rate of change at that point turned negative, one has discovered a local maximum, if the rate of change turned positive, one has discovered a local minimum. In this way “curvy” things like conic sections have precisely placed extrema and inflection points that betray a computation Fig. 5d.)

“All planetary objects have elliptical orbits.” This is Johannes Kepler’s (1571-1630) first law of planetary motion, a law derived from observations made largely by Tycho Brahe (1546-1601). The eccentricity of the earth’s orbit about the sun, which is situated at one of the foci of the ellipse, is low (=nearly circular), at .0167, thus the sun sits nearly atop the other of the two foci (Fig. 5b , 5c.). Venus and Neptune’s orbits have the least eccentricity at 0.007 and 0.009 respectively, and Mercury has the highest eccentricity at 0.206. While certainly elliptical, the orbits of the planets are nearly circular, thus it took much observation to finally realize that the planets orbited elliptically (thanks to Tycho Brahe’s labors, and computations made by Johannes Kepler based on his and largely Brahe’s observations). Compare the eccentricity of the sample ellipse to get a feel for the planets’ eccentricity. The elliptical orbits of the planets are a major in-stamping into nature of the conics by Jehovah as posited in the introduction.

IV. The Parabola

The modern Latin, “parabola,” derives from the Greek “PARABOLH,” i.e., the preposition “PARA,” (near/genitive, beside/dative) + the verb “BALLW,” (cast/throw), literally, to cast beside; conceptually the idea is that of an analogy, or comparison (e.g. Jesus’ parables, Matthew 5:14-16, 7:1-5, 9, 16-17, 12:24-30, 13:1-23, 13:24-30, 13:31-32, 13:33-34, 13:44, 13:45-46, 13:47-50, 15:10-20, 18:10-14, 18:23-35, 20:1-16, 21:28-32, 21:33-45, 22:1-14, 24:32-35, 24:45-51, 25:1-13, 25:14-30). The mathematical sense follows from the ancient Greek astronomer and geometer Apollonius of Perga (c. 240-190 BC) and his conic section studies; Apollonius built on the work of Euclid (fl. around 300 BC) and Archimedes (c. 287-212 BC). You will notice that the parabola is an even function, there is symmetry across the y-axis, hence the arms are “cast beside” each other across the y-axis (Fig. 6, cf. Fig. 2). It is easy to confuse a catenary shape with a parabolic one—the catenary, for example a chain hanging between two posts (Figures 7, 7a) is a hyperbolic function (a hyperbolic cosine), while the parabola is a quadratic function per figure 6 (the independent variable X is squared).

V. The Hyperbola

“Hyperbola” derives from the Greek “hUPERBOLH,” i.e., the preposition/adverb “hUPER” (beyond”) + the verb “BALLW,” (cast/throw), literally, to cast beyond; conceptionally the idea is that of overthrown/excessive, an exaggeration or overstatement. Mathematically we have two halves “cast beyond” and on either side of the y-axis (Fig. 8, cf. Fig. 2).

Jehovah’s hyperbolas have a great load-bearing shape that maximizes surface area in a minimal amount of space, ideal for things like tall nuclear power plant water towers that require strength and heat dissipating capacity in the extreme.

When two stones are thrown into a pond, concentric circles moving outward form around each impact point, and their intersections tend to form hyperbolas (Fig. 8f). Long-range navigation (LORAN, developed in the USA during WWII) borrows Jehovah’s idea. Here a receiver measures time delays between pairs of radio sources, in turn, lines of position in the form of intersecting hyperbolic arcs (like the intersecting ripples mentioned) are mapped to give the precise locations of the sources (the “impact points” in the pond example).

VI. Concluding Comments

We have tried to present a simple, yet mathematically sound discussion of the Creator Jehovah’s conic section design templates in this study. Our approach was minimal verbiage supported by maximal visuals. Most of the technicals appear in the captions to the pictures and plots we presented, as well as the plots themselves, though without exception, we have barely scratched the surface as concerns the technicals. We have presented the “bare bone” basics, but it is enough to gain an appreciation and working feel for Jehovah’s conic sections.

Our goal in this study was to show how the conic sections show up in the created order as design templates utilized by the Creator Jehovah; templates used again and again by Him. There was much technical ground to cover and concentrating on that allowed for only minimal address of our goal, so, rather than summarizing all that background here, we shall, by the Creator’s grace and with His help, focus on our goal in these concluding comments and reference back to the technical side as needed.

A good starting point is the circle. There is something markedly special about a circle, that is, a real circle. It is set apart from the other conics. A real circle is spectrally pure, no degeneracy whatsoever, no blemishes, no beginning, no ending, eternal in that sense. We made a case for immutability in the Circle section above based on the unending constant Pi:

3.141592653589793238462643383279502884197169399375105820974944592307…

Eccentricity is a measure of how nearly circular a conic section is; thus, the circle is the standard by which nondegeneracy is measured; eccentricity e equal 0 here. But we find no real circles in nature because nature is fraught with degeneracy; perfection is elusive in nature, and therefore so is a real circle. We rightly assume it exists, there must be a template, it is the king of the conics after all, wearing the crown on the cone, and we can “see” it in our mind, and produce real, but imperfect models per that vision, but that is all. Alas we are constrained to discover degeneracy in nature, that is our lot, because that is all there is that might be discerned. That is why the rest of the conics are manifestly discernible. We discover in them a mixture, a show of spectral purity, starting with the ellipse, which is slightly eccentric—e greater than 0 but less than 1, the parabola—e precisely equal to 1, and the hyperbola, e greater than 1. These other conics then are what we find stamped into nature and so they concern us in this study.

Let us suppose that these other conics are prominent design templates, but as such, as “others,” it begs the question, are they flawed? Did Jehovah utilize flawed templates here? No, they are not flawed, they are degenerate, there is a difference. A flawed template is more than non-practical, it is nonfunctional. These others are practical and functional as pointed out above by way of the various visual examples of in-stamping, and the computations that give rise to the various forms—that is conclusively not flawed; in every instance the form quite follows the intended function. What is more, these others are lovely, aesthetically pleasing, mathematically, and in their manifest form.

Said degeneracy entered by way of the Fall of humankind, because the created order fell in lockstep with humankind (Romans 8:19-23). One can’t help but wonder about the (pre-Fall) Edenic state in a universe where real circles and perfect spheres were possible. (Maybe that should have been the title to this study, i.e., “A universe of real circles and perfect spheres.” Before such a universe returns, Jesus will have to return first amen.) It is too hard to truly grasp that pre-Fall, Edenic state, and quite risky to draw conclusions from any such notions. Better to think forward and work with the facts of biblical and mundane history and mathematics and science and contemplate how Jehovah ameliorated the disaster behind said degeneracy unto practicality and aesthetics, for that is precisely what we think happened. This is not a difficult conclusion, for that is what the facts all about us point to. In this way of thinking, the circle came first, it was perfect, primitive, and certainly real when the Creation dawned. Post-Fall, it morphed into the other conics. That statement is a hypothesis, and we cannot prove it.

An ellipse has degeneracy, but it is a highly practical concept in practice, and there is a loveliness to an ellipse as well—one chuckles at a bobbing bird egg-ellipsoid which, as though seriously confused, wobbles back and forth on the tabletop (bird egg shapes are variously spherical, ellipsoidal, ovoidal, or conical), and yet one must marvel at the practicality of that ellipsoidal shell as concerns its impact-cracking threshold, and generally its hardness and the capacity to protect the life inside. The ellipse serves as the design template here that Jehovah intended all along to stamp into that shell. (Jehovah was certainly not oblivious to the impending Fall.) And we know from our discussion above that the ellipse is decidedly a computation—He Jehovah computed that design template ultimately to be stamped it into His creation here and there again and again (thus we propose). And so it goes for the rest of Jehovah’s conic section templates. Degenerate yes, but lovely and practical, shaking hands with the environs around them and fitting hand-in-glove into those environs, not least because those environs are largely curvilinear by design to begin with.

Let’s talk about a parabola a little bit in this context. Wonderful symmetry here across the y-axis. The eccentricity is high at 1—lots of degeneracy, but a highly practical focusing template for example in Jehovah’s hands, with myriad applications in optics. With such a high eccentricity, the parabola is far removed from being spectrally pure like the circle, but “redeemed” nevertheless, and made serviceable to Jehovah’s Creation. Placing a parabola and a circle side-by-side shows the extent of the “redemption” and attendant serviceability, it is remarkable (Fig. 6, Fig. 3). It is the Potter at His wheel thus redeeming and making it serviceable (Isaiah 64:8, Jeremiah 18:4). It is the same Potter who in a much more meaningful way reshapes broken vessels, even people, and makes them whole and holy—spectrally pure if you will, and makes them serviceable, to Him, to others, to themselves, to their environs (“A Letter of Invitation”).

And the hyperbola…even the name suggests trouble. The eccentricity is greater than 1—this conic is degenerate in every sense of the word in this context (Fig. 8, Fig. 3), yet it is one of the most remarkably serviceable of the “redeemed” conics no question. Thus it pleases the Potter to shape and mold and fashion exceeding uncomeliness unto serviceability and loveliness. Thus it pleased Him to turn a disaster, Sin, around on its ear both at the spiritual level and the mundane, the former by way of His Salvation, the latter by way of His consummate control over all aspects of His Creation into which He intended all along to stamp His lovely conic sections or arcs thereof to make a curved universe so pleasing to experience with its twists and turns and flat runs all mixed together.

Praised be thy elegant Name great Jehovah God, thou eternal, immutable, thou pure, thou holy and lovely. Amen.

VII. Illustrations and Tables

Figure 1. Making a cone.

Figure2 The Conic Sections. Studied intensely by “Apollonius of Perga, following Euclid, and Archimedes; his classifications have persisted. By definition, the conic sections are the red boundaries to the white regions (white for parabola implied). We do not show the other half of the hyperbola which lies in the same place of another cone in a tip-to-tip double cone configuration not shown. Some sources do not include the circle among the conic sections deferring instead to it being a special case of an ellipse. Note that the mixed term b in the equations acts as a “switch” whose setting transforms the shapes from one to the other.

Figure 3. The circle, Pure Squares.

Figure 4. The Circle, Pi.

Figure 5. Ellipse Shape and Basic Characteristics. Compare figures 5a, 5b, 5c just below for a few of Jehovah’s elliptical stamp-ins.

Figure 5a. Conics in-stamping, ellipse. Snowball Hydrangea Shrub Leaf Eccentricity (very high [loveliness out of degeneracy; thus Jehovah transforms the uncomely and makes them serviceable]). Notice the nested pattern of elliptical veins—this comes from negative feedback, something quite fundamental to Jehovah’s design (see our “God’s Amazing Water” video for nature’s fundamental patterns | structure | feedback. We hold that (negative and positive) feedback is the fundamental computation of nature—thus it pleased Jehovah).

Figure 5b. Earth’s Elliptical Orbit (eccentricity exaggerated).

Figure 5c. Earth’s Elliptical Orbit, the Seasons.

Figure 5d. Local Maxima, Minima, Inflections. Curvy things were created with precisely placed extrema and inflections, which betrays a computation.

Figure 6. Parabola Shape and Basic Characteristics. Compare figures 6a, 6b, and 6c just below for a few of Jehovah’s parabolic stamp-ins.

Figure 6a. The “Parabolic Banana.”

Figure 6b. The “Parabolic Rainbow.”

Figure 6c. Parabolic Projectile Motion. A dense projectile spewed from a volcano at an angle of 45 degrees to the horizontal with an initial speed of 100 meters/second follows this parabolic path, with maximum height, range, and times shown.

Figure 7. The Catenary Shape—note the lazy slope at the bottom of the top curve and the top of the bottom curve (top curve is the “hanging chain” catenary, bottom curve is the catenary arch); compare Fig. 7a just below and hanging chain catenaries. Power lines hang similarly. This shape minimizes gravitational and tensile/compressive forces and is naturally assumed (thus it pleased the Creator Jehovah praised be His elegant Name). The problem can readily be solved by way of the calculus of variations and Lagrangian mechanics (it is done in the second or third year at the undergraduate level), but we prefer to solve such problems via negative feedback (faster, simpler, not discontinuity-accident-prone like differential and integral calculus). Feedback (negative, positive) is the fundamental computation of nature as we have posited here and elsewhere.

Figure 7a. A Natural Catenary Shape. Unknown Author, licensed under CC BY-SA-NC.

Figure 8. Hyperbola Shape and Basic Characteristics.

The conic section {{a,b/2},{b/2,c}}, specifically {{1,2} ,{2,1}} in this example, is a hyperbola of the form ax^2+bxy+cy^2, where a = 1 and b= 4 and c =1 to give, x^2+4xy+y^2 =1, where the trailing equality of 1 chosen is arbitrary for illustration purposes. The red arrows are its principal directions (eigenvectors), and the black arrows point in the unit directions of the standard axes {X,Y}. Among other things eigenvectors reveal how far away from the standard axes the principal directions of this hyperbola are rotated (45°); eigenvectors are direction cosines and the Arccosine thereof gives the angle of rotation for us. This conic section matrix has both a negative and positive eigenvalue {3,-1}, it does have an inverse {{-1/3,2/3},{2/3,-1/3}}, hence this conic section matrix is indefinite (see legend, Fig. 2). Please see figures 8a-f just below for some of Jehovah’s hyperbola-based in-stamp ideas.

Figure 8a. Hyperbola (hyperboloid), the Hourglass Shape.

Figure 8b. Hyperboloid, Guitar Body. The guitar body has the shape of a hyperboloid both for resonance purposes and to conveniently hold and strum it.

Figure 8c. Hyperbolic Paraboloid, the Saddle Shape. Hyperbola into the screen, parabola left to right. Mountain passes have this this shape stamped in. Mathematically, the bottom of the parabola is a local minimum, yet the top of the hyperbola coincident at the same place is a local maximum, hence this point is indefinite. It is represented in matrix terms by way of the intersection of a row minimum and a column maximum as shown in figure 8d just below. (This Phota by Unknown Author is licensed under CC BY-SA-NC).

Figure 8d. A Saddle Point. The 2 in row two column one is a saddle point—a minimum in its row yet a maximum in its column. Though not often helpful in problem solving, saddle points have their place, particularly in game theory and ultimately economic and military strategy, where two-player, zero-sum strategies might have one player looking to minimize payout, whilst the other is looking to maximize gain. Game theory applications are much studied and belong to the field of finite mathematics, itself a very practical branch of mathematics because discoveries made therein readily lend themselves to digital and quantum computing.

Figure 8e. Exponential vs. Hyperbolic Growth. Another sort of curve that Jehovah stamped into nature is the well-represented red exponential curve. When something changes (e.g., grows) in proportion to some constant times itself like so: y’[x]= a y[x], where y’[x] represents the change, and y[x] is the thing itself, and a is some constant, when that happens, we see exponential change (e.g., growth). When something changes in finite time and hits a singularity (an infinity, typically at a discontinuity, break, jump, undefined), we see hyperbolic growth. Hyperbolic growth (or decline) is sudden and immediate, exponential growth is steep and tends toward infinity, but it is not sudden. For example, the reciprocal function 1/x shown in blue, which is an inverse, has a singularity at x=0 (1/x “blows up” at x=0) and the change becomes hyperbolic (“cast beyond”) around the singularity. Inverse scenarios can be like that. Note that the inverse function 1/x is a hyperbola with symmetry through the origin (the {0,0} point), and is thus an odd function by definition (as opposed to an even function like the parabola which has symmetry across the y=axis per Fig. 6). Another example of hyperbolic growth would be overstretching a violin string—at a certain stretch point it suddenly snaps (dubbed catastrophic failure). Emotional loading on human beings can have a hyperbolic response curve—if it is measured (not cast foo far beyond!), it can be a lifesaving “release valve” stamped in by Jehovah praised be His great Name.

Figure 8f. Concentric Wave Intersections. Hyperbolic arcs of intersection from two wave sources employed in long range navigation. (This Photo by Unknown author is licensed under CC By SA-NC.)

VII Works Cited and References

“A Letter of Invitation.”

Jesus, Amen.

< https://jesusamen.org/aletterofinvitation.html >

“Apollonius of Perga.”

Wikipedia.

< Apollonius of Perga - Wikipedia >

“Catenary.”

Wikipedia.

< Catenary - Wikipedia >

“Conic Section.”

Wikipedia.

< Conic section - Wikipedia >

Strang, Gilbert.

Computational Science and Engineering.

Wellesley, MA: Wellesley Cambridge Press, 2007.

Top