If It Walks Like A Duck And It Talks Like A Duck, It's A Papaya

"It's not the one thing nor the other that leads to madness, but the space in between."
--Jeanette Winterson, Oranges Are Not the Only Fruit

In a nutshell, the cunundrum scrutinized by many-a-sleuth since the Dawn of Civilization has been;
One Duck--But, Which Duck?

Purportedly, the first to pop the question were the Greeks--and it would have sounded something like:
Mia papya amma pya papya?
(Μια πάπια, αλλά ποια πάπια?)...

But, what's any of this to do with Gravity? One might wonder.

In Greek the query whether it is 'One Duck' or 'A Duck' is one and the same. Only interesting as far as 'One' indicates an exact Duck, yet any generic/indefinite duck will do for 'A Duck'.The word for duck in Arabic, by the way is, ' بط' , pronounced 'but' , or more precisely, 'butt'. A closer look at the image of the yellow ducks swimming past the view finder reveals a series of less obvious white ducks gliding by in the opposite direction. They are embedded in the 'negative space' defined by the more apparent yellow ducks.

The white ducks are what we would call a 'ghost' since they have no definite outline. And just like a ghost they trace a faint, weak or greatly reduced appearance, or possibility of something. But, butt, ' بط'...!
...where at first glance it seemed the marksman is about to miss their target, we now see that they are sighting their quarry and are right on target--an interstitial white 'sitting' duck.

In this example, the interstitial space (white ducks) is the gap or 'empty space' between spaces full of structure or matter (yellow ducks). It is what the Japanese call Ma (間) ...roughly translated as "gap", "space", "pause" or as "the space between two structural parts". Now, before you 'O, Ma!' me, or toss another 'grammar nazi' my way (an apparent euphemism for 'childbirth makes women scatterbrained'), You should know the reason I am bringing up the Japanese is not because you have a Japanese girlfriend at present. But, for the more important reason that while this 'interval' or 'negative space' is understood to be experientially 'concrete' in Eastern philosophy, in that its very absence or voidness is considered to lend value to the overall tangible structure or give shape to the whole, there is no equivalent single word for Ma in the English language--a somewhat serious omission.

The buzz words being interstitial and negative space, right? Somehow, that should bring us back down to Earth, so to speak, and the 'matter' under consideration--persnickity Gravity.

For denizens of the Planet Earth, 'Knowing' about Gravity is a little like 'Seeing' a Ghost.

Just like in the Duck example, where the ghostly Ma white ducks were plucked from the negative space or gap between the substantive yellow ducks, Gravity's ghost is traced by the Tide. Oceanic Tides can serve to 'map' Gravity since it has a direct affect on them.

For the example here, a regular ocean tide at the shore gives some insight as to the basic Geometry of Gravitational (wave dual) affects on a particular day at a particular time and particle location.

And just like Flatlanders cannot fully realize what Higher Dimensional objects and forces Really are in Lower Dimensions (Consider Flatlanders, Emily and Bob, projecting a hyper-cube...etc.); the best we can hope for is a reasonable inference of what these 5th, 6th, or higher dimensional things' true nature is based on our limited perspective of what we are afforded by our 3rd or 4th dimensional existence (remember Chicken Little?).

That is to say, we need to look for their traces in the 'empty' spaces.

The sine function, osculating circles, parabolic sand washes and the triangles indicating the carved sand bars were overlayed onto the image by drawing them in, not actually modeling them mathematically. These shapes, while selected intuitively, have geometries that can be explicitly related by arithmetic relationships.

The simplest case is imagining that a cylinder is cut at an oblique angle such that the cross cut is an ellipse. Unwrapping the cylinder reveals a sinusoudal curve. In a similar fashion, the circles, parabolas, and triangles are derived from conic sections and projections (unwrappings/unfoldings). (Implicit formulas for such profiles/projections available at "Unwrapping Curves from Cyliders and Cones," The Mathematical Society of America, 2007 )

Simple inspection of the underlying geometry for this moment in time by the sea shore--the osculating (kissing) circles, parabolic sand washes, the tide's sinuous wave front, and the triangles depicting the carved sand bars-- gives a good lower dimensional representation (affine) of Gravity's extra-dimensionality (interstitial nature) and just what it might Really be all about.

Having spent time mulling over the implications of a mathematically perfect Solar System, as solargeometry eloquently puts it, it would be reasonable to expect infinitely recurring self-similarity throughout the Universe. And, by extension since Gravitation is a part of the Universe, wherever we measure or observe its effects can serve as reasonable isomorphs in our 4-dimensional space-time domain to understanding Gravity's exact nature.

Self-similarity is the flip side of the Duality coin. And just like opposites, or duals, serve to balance each other and teach us something about the Nature of the World by comparing and contrasting these opposites, self-similarity serves as an identity scale from the very big to the very small (You Do It! I simply don't get paid well enough to justify working that hard, http://en.wikipedia.org/wiki/Discrete_Gaussian_kernel#The_discrete_Gaussian_kernel) to teach us about those things that we cannot access readily due to some limitation. In the case of Gravity, that limitation seems to be that it is extra-dimensional.

{It may very well be the case that self-similars do not only exist for inanimate objects, but People as well, considering the Quran has this much to say about them:

نَحْنُ أَعْلَمُ بِمَا يَقُولُونَ إِذْ يَقُولُ أَمْثَلُهُمْ طَرِيقَةً إِن لَّبِثْتُمْ إِلَّا يَوْمًا 20:104

20:104 We know what is involved when the 'likes of them' (those mentioned before, those like them, their representations) say, ‘You went about (were broadcast/remained/stayed/tarried) in this way (method/system/fashion/channel/styling/mode/style/attitude) but a Day'.}

A good place to start looking for self-similars in Gravity's case is by comparing the Curvature of a cylinder to the curl of a Tidal wave.

A cylinder can be given by a flat metric, a 2-D rectangle (planar shape) that is curled (rotated by an angle 'alpha' about its axis) so that the short ends meet. When rotated, its various profiles present as lines, ellipses or circles depending on the viewing angle and axis of rotation.


Curvature characterizes how well Pythagorean’s theorem holds for spatial distances.

A space or space-time with zero curvature is called flat.

If an object/line/field/surface/n-space is ‘flat’ its curvature is zero, then Ω = 1, and Pythagorean’s theorem is valid.

If Ω >1, there is positive curvature (or locally convex), and if Ω <1 then curvature is negative (locally saddle, or hyperbolic--i read it, is it supposed to make me feel any better, or make me launch into a retelling of the story about ذو القرنين ḏū al-qarnayn 'the 2-horned one', as He was already mentioned in earlier posts 'where the 2 (horns?) meet').

In either of these two cases, the Pythagorean Theorem is invalid. The point here being that this type of thing is only detectable on a Cosmic scale (1E26 m); and sometimes that's how big the Truth gets, and they still callin' me a liar! (As an aside, do i get an invitation to the wedding the crew from 'paranoia' were planning at Starbux this morning 02-26-2011 @8:00am? That is to say, the guy on the phone was so loud about it, he should've invited everyone in the joint...I haven't been to one in about 5 years.)

In a plane, curvature is a scalar and in higher dimensions it is described by curvature vectors/tensors.

For a circle, the curvature is equal everywhere, and is defined as the reciprocal of its radius.

For smooth curves, curvature is defined as the curvature of the osculating circle at each point. (Osculating circle based algorithms can also be used to avoid a catastrophe.)


...back to the Sea...

A yellow dot is wrapped onto a cylinder of decreasing radii in the Figure (a)-(f) above.

The edge effect in (f) is reminiscent of waves breaking on the shore.

Pictured here is a yellow dot resting on slack water in an ocean (A), a momentary lull before the tide turns, and something that models a point in a plane (or a Flat field).

In (B) the red arrow indicates an Acceleration field which causes the yellow dot to rise in a trajectory modeled by a geodesic along the surface of a cylinder
(C)--an example of Curvature as the Obstruction to a Flat Field.


 {If You notice how long this post is taking for me to get around to doing it's because I've been sidetracked, thinking I almost made a friend, but that went South quickly when he reminded me that 'human contact is a good thing'. R U kidding?Did you see what happened the last time I came anywhere near them?! So, good try, but no thanks!}

An Observer looking directly down from the Sky  (i.e, a Fixed Point of View, far away) sees the yellow dot translating along a straight line (not following a curved path, since they cannot observe the rotation, α=0, ie. it would appear 'still' in that respect, no longer arcing), its straight line profile traces a path:

p(t)= t


Which is esentially a line given by:

z = ct


The yellow dot's ordinate as it rolls along the curl of a cylinder that's rotating around its central axis in this manner as a function of the angle of rotation about the cylinder's central axis, α, is given by:

zα = p(t cos α +√(r^2- t^2 )sin α

Which is a curve of the form:

zα = t cos α +√(r^2- t^2 )sin α


On rearrangement, is essentially an ellipse (possibly degenerate) of the form

Ax^2 + Bxy + Cy^2 = 1 given by:

zα^2 − 2zαt cos α + t^2 = r ^2 sin^2 α


where:

p = the profile function
t = projection of the point along the x-axis
r = radius of the cylinder
α = angle of rotation about the cylinder central axis
z = the distance the projected point travels parallel to the y- axis as it follows the curl of the cylinder through angle α
(http://www.mamikon.com/USArticles/RollingConesCylinders.pdf)
The expression for zα resembles x'= x cos α + y sin α for changing coordinates from the xy- system to the x'y'-system by rotating the axes through an angle α.

The tidal wave develops along the arcing surface of the imagined cylinder whose shape, depending on the symmetry of the acceleration, may be a circle, an ellipse or degenerate ellipse --an example of Tidal Forces as the Obstruction to an Acceleration Field.

This is readily imagined if one were to change the Observer's point of view, such that they are now sighting directly down the cylinder's central axis. One way to do this is to move the Observer's veiwing plane by 90 degrees by actually repositioning the Observer, the other way to do this is to 'tilt' the cylinder itself by α = π/2 to the original viewing plane.

In either case, the profile function is then given by:

zπ/2 = p(√(r^2- t^2 )

Which is essentially a circle given by:

z^2 = r^2 - t^2

Time and Tide are, respectively, bound to Space and Acceleration akin to the manner in which Polar coordinates are related to Cartesian coordinate systems. The relationship is difficult to fathom directly since information is lost (more accurately, 'masked') when moving from one system to the other. And looking at something (information) in one system makes it seem like something entirely different in the other, when in fact it (the information, representation) is essentially the same exact thing.

Words can only take us so far and that's why pictures were invented. And by the fortunate happenstance of a searchable world wide web, wikipedist Kieff (who for whatever reason does not enjoy being Brazillian, or is that getting a Brazillian?) has actually expressed this concept of converting from Cartesian to Polar coordinates pictorially--a similitude for how Curvature of Space-Time is the same as Tidal Forces."

As for seeing me cycle through every mood but Contrition, when have You ever known me not to oblige. If it's contrition You want, here it is: I am genuinely sorry I didn't kill'em all!}

Good things come to those who wait, or as in this case, to those who procrastinate. The delay in getting on with wrapping and unwrapping (unfolding) curves has allowed enough time for this treatise to surface on the web "Unwrapping Curves from Cylinders and Cones," by Tom M. Apostol and Mamikon A. Mnatsakanian, May 1997, The Mathematics Association of America. (reference added March 28, 2011)

Since it seems Gravity can only be encompassed from a Higher Dimension we can only Infer something about its inherent characteristics by its Lower Dimension isomorphic projection on Tides. In this case, Ocean Tides serve as the 4th dimension space-time model of Gravity’s true Higher Dimensional 'Wave' nature.

The Ocean tides give a direct window into the Wave nature of Gravity since modeling tidal behavior can serve as a reasonable affine in Euclidean Space to Gravity’s n-dimensional space affects on them.As we know Tides vary, Gravity, too, may not be as constant as we think, but also instantaneously variable and fluctuating (just like Energy, Light, and other Electro-magnetic radiation, and we have a basic understanding of how Matter interacts/correlates with such Fields).


Gravity's Wave Duality Can Be Inferred From Geometry of Tidal Affects



The pictures can only do so much because they are still rendered in 2-dimensions but a good modeling program should be able to tease out the Unifying Physics of this phenomenon we call 'Gravity' and the finer points pertaining to fractals, duality, and that niggling issue of 'missing mass' (I mean, if we Know it's missing, doesn't that mean it Ain't?--O, Gee :-() that brings us back to 'There ain't no God'!--i kid, i kid, i jest i jest, b( i am soooo depressed)-er).
Some days, when the tide washes up in even more circular fronts, the effect is even more pronounced.


I'm almost certain some bright talent can put all this "As Above, So Below..." to Music just as some time ago William Blake wrote a Poem about it.


'To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour...

Every night and every morn
Some to misery are born,
Every morn and every night
Some are born to sweet delight...
We are led to believe a lie
When we see not thro' the eye,
Which was born in a night to perish in a night,
When the soul slept in beams of light.
God appears, and God is light,
To those poor souls who dwell in night;
But does a human form display
To those who dwell in realms of day.' --from Auguries of Innocence by William Blake

O, Gee, again! SomeOne already made it down this far in the post and is wondering where is the Quran tie-in. Here it is:
Surah 20 TaHa (طه )

طه 20:1

20:105 وَيَسْأَلُونَكَ عَنِالْجِبَا

فَقُلْ يَنسِفُهَا رَبِّي نَسْفًا فَيَذَرُهَا قَاعًا صَفْصَفًا 20:106

20:107 لَا تَرَى فِيهَا عِوَجًا وَلَا أَمْتًا

20:108يَوْمَئِذٍ يَتَّبِعُونَ الدَّاعِيَ لَا عِوَجَ لَهُ وَخَشَعَت الْأَصْوَاتُ لِلرَّحْمَنِ فَلَا تَسْمَعُ إِلَّا هَمْسًا


The first sign in this chapter is one set of the Disconnected Letters found in certain Chapters of the Quran. They are pronounced individually and generally do not comprise words that have any meaning, for that reason they have kept their integrity all these years.
For Sign 20:1 they are the letters Tta ط and Ha ه .

In a previous post there was some discussion as to the mathematics in the Quran, so going by conventional Arabic math notation the ط can mean 'Natural Number' and/or 'pi' and/or gematria value for tta being '9'. And the Ha is Euler's number which is the irrational number denoted by 'e' and has the gematria value '5'.

Now, it doesn't take much to combine pi and e to come up with a natural number. The general case is:



The general solution for which is:

For the special case when taking the Chapter value 20 as the variable and the Sign number 1 as the case to test for True, we get: 


  

Which happens to be True.


20:105 And they will ask you about the Mountains/Barrows/Highlands; Say, 'My Lord Will blast them (a) demolition (torpedo them (to) sinking)'

20:105 transliteration

Wa ya s a loo n aka AAani al jib ali

fa qu l ya nsifu ha

rabbee nasfan

20:106 Leaving it a level plane (Leaving it bottoming (simply) wasting)
20:106 transliteration

Fa ya فَ يَ

tharuha qaAAan saf safan
 ذَرُهَا (a word for corn/kernel/seed/scatter/sow 'tthura' and here it is understood to mean 'its kernel' since it's take as 'thuruha' (reflexive); but this is by no means the only way it may be understood--just happens to be the most accessible to me.

safsafan

20:107 You will not see in it any Crookedness (Bend/Curvature) or Ruggedness

20:107 transliteration

La ta ra fee ha
(La=no, then use tau, roh, phi, and Ha==e,
should lead to something meaningful
when 20:107 used for variable/solution)

AAiwajanwala amtan

(and not; d, a? Complex, Imaginary, compile/ until n?)
(more than likely the math is wrong since I have no insight into the physics/math and simply stringing along the Arabithmetic guessing, someone more keen on this can probably recognize it for what it is. the point is the message has to do with learning something new and not necessarily using the Quran to bonk your brother.)


20:108 On a Day they will follow the Caller/Summoner, without Crookedness/Dishonestly/Evasion to Him, and the Voices will be Hushed/Overawed by the Merciful, and you'll not hear anything except murmurs/whispers/soughing sound.

20:108 Transliteration

Ya w ma I thi n
ya ttabiAAoona alddaAAiya
la AAiwaja lahu
wakhashaAAati alaswatu lilrrahmani
fala tasmaAAu illa hamsan

Surah 46 Al-Ahqaf ( الْأَحْقَافِ , Curved Sand Dunes)
وَاذْكُرْ أَخَا عَادٍ إِذْ أَنذَرَ قَوْمَهُ بِالْأَحْقَافِ وَقَدْ خَلَتْالنُّذُرُ مِن بَيْنِ يَدَيْهِ
 وَمِنْ خَلْفِهِ أَلَّا تَعْبُدُوا إِلَّا اللَّهَ
إِنِّي أَخَافُ عَلَيْكُمْ عَذَابَ يَوْمٍ عَظِيمٍ 46:21

46:21 And remember 'Aad's bretheren when He warned his people among the wind-Curved Sand Dunes, and indeed warners came and went before and after him, saying: 'Serve none but God. Truly, I fear for you the suffering of a Momentous/Significant/Great/Extraordinary Day'.

But like I stated earlier, it's taking time to get through this post since I tend to wallow in my own misfortune, knowing this News never makes it Home (with or without outside assistance, given how often the point about hyperbolic curvature has been deleted to date 21/02/2011; the upside is that while so far Gravity is still present I have magically dropped 3 dress sizes while still consuming the same amount of chocolate). {'Grammar Nazi'?! Really? How many more times You gonna say that?}