Tesla Girls, Tesla Coils, She's My Tesla Goy-Um!

But Siriusly, an idea that is ahead of its Time is about as popular as the one that goes against the accepted Tide, relegating its proponent to Loopy (Round-the-Bend) status--without further introduction, I give You, Tesla!

"I hold that space cannot be curved, for the simple reason that it can have no
properties. It might as well be said that God has properties. He has not, but
only attributes and these are of our own making. Of properties we can only speak
when dealing with matter filling the space. To say that in the presence of large
bodies space becomes curved is equivalent to stating that something can act upon
nothing. I, for one, refuse to subscribe to such a view."
--Nikola Tesla
(1856-1943)

Tesla's adamant refusal to adopt the little understood yet highly regarded (then as now) views of the vanguard of his day, put him instead in league with the 'luna-tics', aka Renaissance Thinkers and Prophets, or just your run-of-the-neighborhood crazy mother.

While there is no denying that both the Theory of General & Special Relativity do a serviceable job in explaining and even predicting certain phenomena, the very fact that the one necessitates the other hints that they each are incomplete which begs for careful vetting.

And while my undistinguished career is a dim bulb compared to either Tesla's or Einstein's resplendent contributions to Science, it hasn't escaped me that Tesla may have had a real concern with Einstien's Relativity and may not have been simply casting aspersions. It may very well be the case that space itself is not 'curved' due to the presence of matter (heavy objects) since what we are observing/experiencing is a 'projection' of the True Nature of the Matter under discussion.

By 'projection' I am not refering to our simplistic view of holograms, which may serve as a good analogy, but something much deeper than that. Some 'other' physical entity that due to its extra-dimensionality (hyper-space, or whatever the common catch-phrase-of-the-day is), we cannot gauge directly but should be able to deduce its presence much the same way similes, metaphors, and music can convey something as indeterminate as what someone else is feeling without our having to sense it ourselves.

The limited 4D space-time only allows us rare glimses into the larger picture. It's a little like looking out of the porthole (A Gardner Like That One above) of an ocean-going vessel and thinking that's all there is to the ocean. Enough wrangling with the words, time for the GeoMetry (gee, i'm a tree!) using a concrete example.

{And to our left, allow me to call Your attention to the Bliss comic panel of Tuesday May 17, 2011, insinuating that a b**tch explains 'it'; an accolade meaning "Congratulations! __ (Insert Name) has now graduated UFO Academy (summa cum laude)"}
The orbit of the Moon around the Sun is a convex curve approximated by:
(400cos t + cos 13t, 400 sin t + sin 13t)

When expanded this yields:
160000 cos^2(t)+cos^2(13 t)+800 cos(13 t) cos(t)
+ 160000 sin^2(t)+sin^2(13 t)+800 sin(13 t) sin(t)

Which can then be simplified to: 160000 + 800 cos (12t)

And since we're not trying to land anyone on the Sun or the Moon, but simply doing this as an excercise to bridge the esoteric to the familiar, we can say the relationship approximates:

160000 + 800 cos (13t)
or about 40 + cos (13 t)

And, thanks to wolframalpha, the (exagerrated) curve for which appears on the right.

A few things worth mentioning about the Moon's orbit around the Sun is that it is essentially elliptical, it is locally convex, and when we consider the sidereal month of 27.32 days instead of the synodic month of 29.54 days, the ellipsoid is more like a tridecagon.

From studying the concept of 'unwrapping' curves from developmental surfaces, we learned a curve that lies on a conic surface and its ceiling projection are the same type, such that an ellipse projects as an ellipse, a parabola projects as a parabola, and a hyperbola projects as a hyperbola; although, their eccentricity may differ. And we know from observation that the radius vector (r) of objects with elliptical orbits 'sweeps equal areas in equal times'.

That last statement about elliptical orbits is how we were taught to think about orbits in school. Thinking in terms of 'uwrapping' and/or 'projecting' curves from developmental surfaces (such as cylinders and cones) simplifies this relationship since the areas are preserved in much the same way arclength is invariant, eliminating the need to consider the Time, and allowing us to focus on the spacial attributes alone.

{Proofs and more illustrations can be found at the source used here, "Unwrapping Curves from Cylinders and Cones," by Apostol & Mnatsakanian (The Mathematical Association of America, Monthly 114).}

The illustration on the left depicts a conic curve C with its 'ceiling projection' curve C0 (A ceiling projection of a conic curve is like having it lie on the surface of the cone and then the cone is 'popped' open like one would an umbrella). Conic curves projected in this manner have the following attributes:

1. a focus at the Vertex of the cone
2. a directrix L at a line of intersection of the cutting plane and the ceiling plane
3. eccentricity denoted by lambda = tan (alpha) tan (beta)
4. is a polar equation:


r (phi) = r(0)/(1+ lambda*sin(phi))


Having one focus at the vertex of the cone is significant in that we know for elliptical orbits the center of mass of "...the orbiting-orbited system is at one focus of both orbits" and the general assumption is that there is "... nothing present at the other focus."


However, in the search for 'missing mass', based on conic projections of orbits and the relationships between the viewing plane and the observed point P0, and what we know of the observable 'detectable mass', it would be safe to posit that the second focus is actually the barycenter of the 'missing mass' related to P0 ( which in Reality is a projection of the Actual point P; while it cannot be observed directly from the viewing plane, however, its 'gravitational' effects probably are what manifest as the presence of the secondary focus, which makes it appear as if 'there is nothing present at the other focus').

Conic C (figure above left) and its ceiling projection Co share the following relationships:

1. the ratio of the projected curve
Co radial vector (r) to the distance (d) is constant and independent of
P or Po and is the same as eccentricity (lambda) s.t. r/d = lambda = tan
(alpha) tan (beta)

2. the polar equations
of the unwrapped curve (C) and its ceiling projection (Co) are related by a
constant k = 1/sin(alpha) s.t. R(theta) = k r(theta)

3. the area A of the projected curve Co after one complete
revolution (phi =2 pi) of unwrapping the curve C is given by:

A = E cos (beta) = S sin(alpha)= pi*rho*s

(where: E= disk area of conic C, beta = its angle of inclination, S= finite lateral surface of the cone subtended by C and its vertex, alpha = the half-angle of the cone with vertex angle=2(alpha), rho = radius of the cone base, s= unwrapped arclength of C on the surface of the cone.

A generalized conic is a planar curve generated when a conic curve is projected onto a ceiling plane with a focus at the vertex (V) of the cone. In the figure above a generalized ellipse with period interval length 2pi/k, k=5.5, lambda = 0.22 is generated after 1 period only. In (b) and (c) the same generalized ellipse after 5 and 11 periods, respectively.
The figure on the right depicts an approximate annual orbit of the Moon around the Sun as a generalized conic; 1-year orbit is about 13 synodic months, or 13 periods.

According to the figure on the right the period interval length is approximately
55.4 deg =2 pi/k,
giving
k=6.5

Similar shapes found in nature may be describe in the same manner.

{having run my one brain cell to the ground getting this far, time to move on to a refresher exercise otherwise known as "physics from lyrics"} And, yes, while the aim of all the above may seem nebulous, that's more owing to lack of time than understanding on my part, and the fuzziness clears right up given a little attention. A little like when expressing the simple sentiment "all roads lead to Rome" as a refomulated old Irish Blessing, "May all roads lead You Home", which could prove to be a Curse as much as a Blessing, depending on who You find waiting there for You.

The main idea is to derive the orbit as a generalized conic and consider that there is symmetry in the integral of a smooth function f over the closed boundary of a geometric curve S that is equal to the integral of the derivative of the function f ' over the figure S itself. {What she said, expressed succinctly in the equation below, and That's why Math!}


The explicit form of this relationship appears in physics time and again. The linear case as the Riemann integral (calculating the function at the endpoints of a line equals the integral of its derivative on the line); the 2-D case of integrating f over the closed curve bounding a surface equals the integral of f’ on the surface (as in Stokes’ Theorem and Ampere’s Law); the 3-D case as given by Gauss’s Law of integrating a function f over a closed surface surrounding a volume equals the integral of f’ throughout the volume. (from THE MAP OF PHYSICS, EXPLORATIONS OF NEORATIONALISM, Essays in the Nature and Uses of Reason, by Dennis Anthony)

Ceiling projections of a hole drilled (bored) through a cone’s axis.
Compare to Cassini Ovals, below.

Bean Curves, Qurartic Curves have Parametric Given By:

Ceiling projections of a hole drilled (bored) through a cone’s axis. Note: Bean Curves appear in the first column bottom row, (from Unwrapping Curves...)

Cassini Ovals, Quartic Curves Given By The Parametric

{I am no apologist, but You know by now that I pretty much forgot much of my key life experiences up to this point, especially the higher-learning part of it, but it seems I can finagle some understanding of it up to 9th grade (which includes Geometry & Trig). (And, don't be misled into thinking or trying to figure out this weird amnesia; that, while it may seem to be selective, at times I think it is a protection mechanism and at other times it may serve as the only polite way to avoid having to confront the outlandish; especially when random strangers 'say' things about me like 'why can't she remember' in the third person while they stare pointedly at me and know I am present. Since I don't know who 'they' are and none of them seem to ever bother with an introduction, I act like I don't 'hear' a thing--so, go right on talking about me to me like I am not in the room!) }

I am constantly reminded that 'it' is not about Me, and my Ego is not so grandiose nor so fragile as not to permit continuing to operate under such an assumption--that it isn't about Me. Can You imagine how paranoid I would be if I ever thought it otherwise, especially when confronted with song lyrics about "Every Breath You Take" (The Police).

So, let's just say, for argument's sake, that it is about Sting (the talent behind the aforementioned lyrics) and what Sting may have been thinking when he wrote the lyrics to "Ghost Story".

Since Sting is an Englishman by birth and speaks English as his first language and was an English teacher, it may be reasonable to assume that he means what he says and says what he means (in English)--making him one of the most literate songsters around--making his lyrics relatively easy to parse for our purposes here.

What is the force that binds the stars
I wore this mask to hide my scars
What is the power that pulls the tide
I never could find a place to hide

What moves the Earth around the Sun
What could I do but run and run and run... -excerpt from " Ghost Story" lyrics by Sting from the album "Brand New Day"

While the lyrics are profound and the experience of hearing Sting's rendition of "Ghost Story" quite moving, the answer to many of his questions is "I don't know." At the end of the song, the listener is satisfied with Sting's jump to conclusion, "I must have loved you."

Really? Is that all? Just an assertion without any proof? Just a series of observations and a lack of post doctorate studies in physics or math?

To demonstrate how well this model runs without Me, I will now show (Computer Command Copy on Write) that Sting unwittingly holds the answers to his own thoughtful questions. (Not by Magic, but by the premise that All Inspiration comes from God--that's why You should just tell me and not have to rouse Mr. Sting up from his sound slumber in the middle of the night to go multi-platinum with it.)


W/hat is= the force F, that binds the stars (Gravitational Binding Energy)...
Ŵ=∫▒〖F dx〗; Ŵ=BE ; F = dŴ/x+c= dBE/x+c



W/hat is= the P(power), that pulls(mechanical work) the tide (tidal)


W/hat moves(equation of motion for) the Earth around (polar/cylindrical/orbit) the Sun

W/hat could I(Current, Impedence, Impulse) do but run + run + run (streaming) (you could take a walk)

W in physics/math can mean a few things directly applicable to the query:

W= watt (unit of Power); W= mechanical Work; W= dimensional (w(width) or ωx (frequency); W=Lambert W function; W = W Boson.


Just a cursory look at the lyrics indicates that they break down the physics in somewhat of a coherent way, in that Work is the amount of energy transferred by a force acting through a distance in the direction (vector) of the force. The units for Work and Energy are the same (joules), scalar quantities, and Watts are units of Power which is the amount of energy expended or work done per unit time (dW/dt)--a differential.

Sting's veiled, yet keen, insight is more evident when W is understood to be the Lambert Function, which provides an exact solution to the quantum-mechanical double-well Dirac delta function model which consists of a time-independent Schrödinger equation for a particle in a potential well defined in one dimension. Applications of the Lambert W function is ubiquitous in physics including, but not limited to, the areas of atomic, molecular, and optical physics.

And while W as width may seem out of place with the other W indicators, it makes perfect sense in this context when considering Feynman's assertion, and it seems only appropriate that Feynman be mentioned along with Tesla and Einstein:

• "These notions of potential and kinetic energy depend on a notion of lengthscale. For example, one can speak of macroscopic potential and kinetic energy,
  which do not include thermal potential and kinetic energy. Also what is called
  chemical potential energy (below) is a macroscopic notion, and closer
  examination shows that it is really the sum of the potential and kinetic energy
  on the atomic and subatomic scale. Similar remarks apply to nuclear "potential"
  energy and most other forms of energy. This dependence on length scale is
  non-problematic if the various length scales are decoupled, as is often the case
  ... but confusion can arise when different length scales are coupled, for
  instance when friction converts macroscopic work into microscopic thermal
  energy."--Feynman

Feynman specifies "length" but the general meaning is a scale of 1-dimension, so, length, width, height, etc. are exchangeable. Of course, what Feynman is really talking about, in physics parlance, is the idea of fractals "As Above, So Below."

The /hat (^ ) operator, usually presented over a value or function means: Fourier transform; the element removed from a set; a unit vector (a dimensionless vector with magnitude 1); or used to denote an estimator or an estimated value x vs theoretical x. Read as x-hat or x-roof, where x represents the character under the hat.

Given the various interpretations of W above, the implications are:

• since Work, Energy, and Power are scalar in our 4D space-time, in order for them to be understood as vectors, they must be moved up a notch (or 2 or more) in order for w/hat to make sense as a unit vector (dimensionally speaking); so, based on the 'lyrics to physics' these relationships hold true in higher order dimensions as non-scalars. {So w/hat if I'm wrong? The Theory Of Everything must include Everything from A Tree Grows in Brooklyn to the Kitchen Sink!}
  
  ..look, a joke can run its course and in this case been taken too far, but I have to say hearing "the wife" reference coming from strangers is offensive to say the least and utterly ignorant & insensitive to put it mildly.{I need to see my Father, stop paying bills & stop getting e-mails at the office from colleagues who verify my e-mail address is not ever in their send log}