Winding Down? Knotting Doing!

Guys! Guys! 
You have to stop with the 'beautiful wife' routine. 
You do realize that had the front men not all offed themselves or been  killed, you would all be married to them, right?--I'm just the interpreter, not  the Messenger. 
And, Michael, must You always act like You have all the time in the world(s)? 
{And to my morning companion, here today 10.29.2012--no, Dear, I do not know who you are, but I do see you and I can hear you just fine; at this point You tell me what I can do for You and what exactly is going on around here...as for what I want, I think I made myself very clear.}

California Red Tail Hawk
One Helpful Community
Among Many

Surah 3 Al'3mran (The Family 'Imran--Zacharia, Mary, John, Jesus...)
ضُرِبَتْ عَلَيْهِمُ الذِّلَّةُ أَيْنَ مَا ثُقِفُواْ إِلاَّ بِحَبْلٍ مِّنْ اللّهِ وَحَبْلٍ مِّنَ النَّاسِ وَبَآؤُوا بِغَضَبٍ مِّنَ اللّهِ وَضُرِبَتْ عَلَيْهِمُ الْمَسْكَنَةُ ذَلِكَ بِأَنَّهُمْ كَانُواْ يَكْفُرُونَ بِآيَاتِ اللّهِ وَيَقْتُلُونَ الأَنبِيَاء بِغَيْرِ حَقٍّ ذَلِكَ بِمَا عَصَوا وَّكَانُواْ يَعْتَدُونَ (3:112  
3:112 They were pummeled/repeatedly struck with humiliation (abased/humiliated time and again) for what they did, but for those who found (were edified) by a rope (lifeline/help) from God and a rope (lifeline/assistance) from people (Nas==Communities); and had earned Gods' Condemnation and their dwellings were repeatedly battered (var. literally, they were repeatedly sedated/subdued); in this manner (they were requited) because they belied God's Messages (Signs) and killed the Prophets (Messengers) without justice (just cause). And because they disobeyed and were defiant (obverse).

Raven
genus: Corvus

{I do wish to thank both God and the communities that reached out to help me...this one (shown above right) actually knows where to find me even when I'm on the bus...

And this one here (shown left) can't stop talking about it.

Surah 25 Al.Furqan (The Criteria, The Discriminant)
أَلَمْ تَرَ إِلَى رَبِّكَ كَيْفَ مَدَّ الظِّلَّ
 وَلَوْ شَاء لَجَعَلَهُ سَاكِنًا
 ثُمَّ جَعَلْنَا الشَّمْسَ عَلَيْهِ دَلِيلًا (25:45
25:45 Can you see how your Sustainer extends the shadow/shade and if He Wills, He Can Make it stand still (stationary, like in a housing, literally==living/abiding/dwelling, tented, featureless, constant/tranquil~subdued), but We Made the Sun its Guide (leads it, betokens it, informs or indicates it, makes it known/publicizes it).
}
  ...I heard, it means Everybody gets to go Home--everybody but Me (I saw this coming), because apparently I don't want to remember who I R--they made it unbearable for me to even hear my name called  (they made my past home unlivable, no chance of any reconciliation no matter how much I love my children--that, and the fact that He now 'espousing' my cause is some mocked up version of Shallow Hal and keeps asking 'how does she look?' Is that why I am still spinning my wheels here?)


BTW I never represented myself as being anything other than Who I Am--My Father's Daughter, and He is still of the opinion that This is no way to treat a Friend! (How are my kids doing?--I only get to see the dark aftermath).

Excerpt From: Mathematical Approaches To Biomolecular Structure and Dynamics, by Jill P. Mesirov, Klaus Schulten, DeWitt L. Sumners

The complexity of a knot is measured in a natural way by the lattice invariants. In an intuitive manner, M(K), A(K), N(K) as well as m(k), n(K) and r(k), measure the relative lengths, or areas necessary to realize a given knot. 

These measures are all of a sub-additive nature, and the authors conjecture that the last three are all additive. 

Table I (excerpt below) indicates the relative complexities of knots, as measured by M(K). It is instructive to compare this table with rankings obtained by other measures, such as the Mobius energies of knots (reference). That spectrum generally reflects the ranking in Table I, but since energies are real numbers (Re), the ranking contains no ties.

{In knot theory, the crossing number is the minimal number of crossings of any diagram of the knot. It is a knot invariant.

By way of example, the unknot has crossing number 0, the trefoil knot 3 and the figure-eight knot 4.

There are no other knots with a crossing number this low, and just two knots have crossing number 5, but the number of knots with a particular crossing number increases rapidly as the crossing number increases.

Tables of prime knots are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular, except that torus knots are listed first). The listing goes 3₁ (the trefoil knot), 4₁ (the figure-eight knot), 5₁, 5₂, 6₁, etc. This order has not changed significantly since P. G. Tait published a tabulation of knots in 1877.}

Excerpt from Table I: In order of increasing energies, the knot crossing numbers are: 

0₁, 3₁, 4₁, 5₁, 5₂, 6₁, 6₂, 6₃, 7₁, 7₂, 7₃, 8₁₉, 7₄… 

The knot 8₁₉ , which one would reasonably expect to have a higher energy than any knot with crossing number 7, appears nevertheless before some knots with crossing number 7. The ranking by M(K) compares well with this progression, except the 8₁₉ appears even earlier in the sequence. [Refer to fractional linear groups LF(4,2)] 

An indication that something 'shady' is going on...

Alif: If you're Jinn, how come she can see you?

Vicram: Easy, I just turn sideways!

--[paraphrasing from the book Alif--The Unseen, by Willow Wilson]

Surah 21 Al.Anbiya (The Prophets, Messengers, Warners, Waker-Uppers)
وَذَا النُّونِ إِذ ذَّهَبَ مُغَاضِبًا فَظَنَّ أَن لَّن نَّقْدِرَ عَلَيْهِ
 فَنَادَى فِي الظُّلُمَاتِ أَن لَّا إِلَهَ إِلَّا أَنتَ سُبْحَانَكَ
 إِنِّي كُنتُ مِنَ الظَّالِمِينَ (21:87

21:87 And He of the Great Fish (Jonah) when He went off in a wrath thinking We Had no power over Him. And He called out when He was in Darkness/Shadow (probably abysmally depressed like me), "There Is No God other  than You! I Extol You! Truly, I have been among the Shadow People (Jinn)."
 (primarily taken to mean Oppressors, but I learned otherwise, first hand, since I didn't oppress anybody and they try to make me believe this is a great Witness Protection Program).
'shadows' ('tthil.m' ظِلٍّ , ظِلٍّم  )  shadow, dusk, outline, dark, ghost, silhouette, gloominess, constant companion==to follow someone closely to learn from them or stalk/spy on them, (Jungian, the primitive self), (phys. Darkened shape of a surface that falls behind somebody or something blocking the light), a lesser remnant of the original /inferior copy (shadow of oneself)

v. ظِلٍّم  shadow, n. injustice, oppression, abuse, inequity,  wrong

Surah 2 The Cow (Al.Baqara)

هَلْ يَنظُرُونَ إِلاَّ أَن يَأْتِيَهُمُ اللّهُ فِي ظُلَلٍ مِّنَ الْغَمَامِ وَالْمَلآئِكَةُ وَقُضِيَ الأَمْرُ وَإِلَى اللّهِ تُرْجَعُ الأمُورُ (2:210
2:210 Will they not see until God Appears to them in a Shadow of Clouds (overcast clouds) with the Angels and the matter is decided (the Order is Executed); indeed, all things revert/return/go back to God (for Judgment).

Surah 13 Thunder (Ar.r'3d)
وَلِلّهِ يَسْجُدُ مَن فِي السَّمَاوَاتِ وَالأَرْضِ طَوْعًا وَكَرْهًا وَظِلالُهُم بِالْغُدُوِّ وَالآصَالِ (13:15
13:15 And before God bows down all that is in the Heavens and the Earth, willingly or begrudgingly, sooner or later (lit. on the morrow(ظِلالُهُم, or var. their shadows)  or in the original  (الآصَالِ var. original creation/form)).

Surah 26 Shu'3ara (Poets)
فَكَذَّبُوهُ فَأَخَذَهُمْ عَذَابُ يَوْمِ الظُّلَّةِ إِنَّهُ كَانَ عَذَابَ يَوْمٍ عَظِيمٍ (26:189
26:189 So they belied him and they were beset with a Day of Shadows (Darkness/Gloom); indeed, it was a Day of Supreme Punishment (torment).

Surah 27 Naml (Ants)
قِيلَ لَهَا ادْخُلِي الصَّرْحَ فَلَمَّا رَأَتْهُ حَسِبَتْهُ لُجَّةً وَكَشَفَتْ عَن سَاقَيْهَا قَالَ إِنَّهُ صَرْحٌ مُّمَرَّدٌ مِّن قَوَارِيرَ قَالَتْ رَبِّ إِنِّي ظَلَمْتُ نَفْسِي وَأَسْلَمْتُ مَعَ سُلَيْمَانَ لِلَّهِ رَبِّ الْعَالَمِينَ (27:44
 27:44) He (Solomon) said to Her (Queen of Sheba), "Enter the path (walkway into the palace), and when She saw it, She deemed it a vast expanse of water (var. an abyss), and She uncovered Her calves (lower legs/shins so as to wade into it). He said, "It is a path paved (tiled) with glass (quartz). She said, "O, my God. Truly, I have been oppressing (sinning/shadowing) myself, and I surrender (believe) as Solomon (does that)  God is the Sustainer of the Worlds."

Surah 35 Al.Fatir (The Originator, The Only One Who Creates from Nothing)
وَمَا يَسْتَوِي الْأَعْمَى وَالْبَصِيرُ (35:19
وَلَا الظُّلُمَاتُ وَلَا النُّورُ (35:20
وَلَا الظِّلُّ وَلَا الْحَرُورُ (35:21
35:19 And they are not the same, the blind (spiritually obtuse) and the seeing (enlightened)
35:20 and neither is the Shadow/Darkness and the Light
35:21 and neither the Shade/Coolness and the Heat/Turmoil

Surah 56 Al.Waqi'3a (The Event, The Inevitable)
وَأَصْحَابُ الشِّمَالِ مَا أَصْحَابُ الشِّمَالِ (56:41
فِي سَمُومٍ وَحَمِيمٍ (56:42
وَظِلٍّ مِّن يَحْمُومٍ (56:43
لَّا بَارِدٍ وَلَا كَرِيمٍ (56:44
إِنَّهُمْ كَانُوا قَبْلَ ذَلِكَ مُتْرَفِينَ (56:45
وَكَانُوا يُصِرُّونَ عَلَى الْحِنثِ الْعَظِيمِ (56:46
56:41 And the denizens of the Left (North), what of the denizens of the Left (North)
56:42 In Poisons (Toxins) and Intense Heat (Bathed in a Severe Heat)
56:43 And a Shade of Severe Heat (Intense Heat, s.t. even the shade offers no comfort or coolness)
56:44 Neither cooling nor comforting (kind)
56:45 Prior to it they were opulently (wasteful) (طرف  homonym==marginally on the wrong side (erring))
56:46 And they were on their way to a terrible sin (supreme reneging, perjury)

Surah 77 Mursalat (Transmitters, The Winds, Carriers)
انطَلِقُوا إِلَى ظِلٍّ ذِي ثَلَاثِ شُعَبٍ (77:30
77:30  Experience (split, separate, disassociate) to a Shadow (of) 3-Branches (people, states, listeners, etc.)

Sign 77:30 is Nothing, if not Fascinating. 

Literally, and in context, it directs the reader to experience that very thing they are wont to deny; that being the Resurrection or the capacity for that Final Justice we hear so much about, but seems rarely ever experienced during our stay on Earth. 

And in order to be able to do so, the Person is made to 'split' or 'separate', 'shoot' (as in be discharged rapidly, like firing off a gun, and that's where some classically interpret it as the noun 'fire' or 'flame'), 'divorce'  (oneself of a situation, the antonym for which is 'marriage') in order to experience a  'shadow' or ظِلٍّ , tthil.n of a 3-fold nature, a triad, of sh3b شُعَبٍ.  

The primary meaning of شُعَبٍ  includes 'people,'  'state,' 'folk,' 'rank-and-file (soldiers, workers, columns), 'audience,' 'branch,' 'tree branch'; and the verb form implies 'to pay money,' 'to branch off,' or 'ramify'. 

To ramify means to divide into branches but with the unintended consequence of making a situation more complicated, and making the end result more difficult to achieve (becomes asymptotic).

While I know this isn't about me, I can't help but wonder why this unGodly, reprehensible thing happened to me anyway since I never contested the validity of the Resurrection, or God, or anything that would make me a likely candidate to Experience 77:30 first hand, but I suppose it always goes back to 'them' and so it must be for 'their' benefit. 

{I got my answer; 
Surah 8 Cattle (Chattle, Property)
  وَاتَّقُواْ فِتْنَةً لاَّ تُصِيبَنَّ الَّذِينَ ظَلَمُواْ مِنكُمْ خَآصَّةً
8:25 وَاعْلَمُواْ أَنَّ اللّهَ شَدِيدُ الْعِقَابِ

8:25 And beware (have a care, fear that) a trial (var. temptation/test) does not only touch those (affect those)  who are lost (in the dark//shadows//oppressive) among you exclusively; and know that God Is Severe (Stringent) in Punishment. 

Mani:
Don't Ever Let Me
 Catch You Impersonating the Queen of England, Again!

Due to Signs 56:41-56:46, 
this is where I console myself since Raphael only goes out to other people; Thanks for looking in on Mani.

(I had no idea it was that close!
--&Tommy, quit sneaking up on your father...remember to put rotational matrix here)

(I take it I paid it so far forward that I can come back as Hitler, Pol Pot, Attila the Hun, and Christopher Columbus and Pizarro, and de Gamma and Alexander the Great and all the Caesars and some future Darth Vader or 2 and then some, oh, and Mom!)

}

Surah 16 Bees (An.nahl)
أَوَ لَمْ يَرَوْاْ إِلَى مَا خَلَقَ اللّهُ مِن شَيْءٍ يَتَفَيَّأُ ظِلاَلُهُ عَنِ الْيَمِينِ وَالْشَّمَآئِلِ سُجَّدًا لِلّهِ وَهُمْ دَاخِرُونَ 16:48
16:48 Or can they not see (are they so unaware) that whatever God Creates of anything its shades (shadows) meet from the Right and the Left (var. the North), prostrating humbly to God (دَاخِرُونَ  var. while they are unassuming, or when they are separated, partitioned, set apart (the 3 states are not united, so that they are not fully aware? See Sign 77:30 above about partitioning shadows/columns).

My only consolation is that similar things have happened to better people. Some of whom may have slipped under the Radar. Like Enoch, for example, while in Genesis we get to learn the names of many progenitors, how long they lived, the children they had and that they 'died', Enoch didn't get to 'die' but got to take a walk with God and became 'Not'. (Like Mariam, Elijah, others throughout time who 'died' without a name, and perhaps even the Queen of Sheba).

Genesis 5:24
Enoch walked with God; and he was not, for God took him.


And while Enoch went for a 'walk', Elijah got taken for a 'ride.'

2 Kings 2:11

As they were going along and talking, behold, there appeared a chariot of fire and horses of fire which separated the two of them. And Elijah went up by a whirlwind to heaven.

And while in 'broad strokes' there is a reference that it was not much fun for Mother Mary, either, at least she had a head's up warning, which is more consideration than what I got, and She got a Fair Baby out of the Deal, and we all know not to even say word one about my baby, okay.

{Aside: That's why I tried my best to keep this from happening to any other unsuspecting mark, and since they didn't even take the trouble to make sure I was totally unconscious when they did this (and other unmentionable things to me), I must insist that You Do Not Resuscitate, (if they haven't got around to remanding the body to its family, which they should have done back in 2000).

For one thing, just saying, 'It's not happening to you', while I surfaced to see what they were doing, doesn't mean I could ever forget it or that by magic it didn't happen--so thanks for making me too dirty to ever hope for anything but kissing the dirt). That's Why, if it turns out that I am wrong, and there really is no God, they will all find themselves in a position where they Pray There Is One in earnest...Dad?What are You so proud of bringing her down? Was I ever such a high flyer?!!! Just take a look around.}

In  A Fourier Theory for Cast Shadows, (Ravi Ramamoorthi, Melissa Koudelka, and
Peter Belhumeur, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 27, NO. 2, FEBRUARY 2005 Feb. 2005), the authors uncover the mathematical convolution structure of cast shadows and analyze them using mathematical convolutions and Fourier basis functions.

Abstract- [paraphrase] Cast shadows are important in computer vision applications, such as lighting-insensitive recognition and surface reconstruction.

Due to nonlocal interactions in non-convex regions, formal analysis is difficult.

Real instances that map closely to canonical configurations like a wall, a V-groove structure, or a pitted surface-- in particular, 3D textures whose surfaces include canonical configurations like V-grooves can serve as a basis to develop formal analysis of cast shadows. 

Theoretically, many configurations can be mathematically analyzed using convolutions and Fourier basis functions.

The authors’ analysis uncovers the mathematical convolution structure of cast shadows and shows strong connections to recent signal-processing frameworks for reflection and illumination.

Considering lighting variability over nearly flat textured surfaces such as a wall or moss, assuming a fixed Point Of View, the expectation is that variation in illumination (the same purpose the Sun serves as it courses the Sky throughout the day) should correspond to a simple Lambertian cosine-dependent relationship. Cast shadows play a major role, however, that leads to effects that are quantitatively described and mathematically explained here.

Surah 25 Al. Furqan (The Criterion, 'The Norm, Standard, Gauge/Gage, Principle, Measure...by which to decide/discern/judge/establish... wrong from right (True from False; Light from Shadow)')
أَلَمْ تَرَ إِلَى رَبِّكَ كَيْفَ مَدَّ الظِّلَّ 
وَلَوْ شَاء لَجَعَلَهُ سَاكِنًا
 ثُمَّ جَعَلْنَا الشَّمْسَ عَلَيْهِ دَلِيلًا (25:45   
25:45 Do You See How Your God (Lord, Sustainer) Extends (Lengthens) the Shadow (Shade), and if He Wills He Can Make it Stand Still; and then We Made the Sun to Have Influence Over it (its guide, pilot, director).

Cast shadows have a simple convolution structure amenable to Fourier analysis, meaning a strong link between the mathematical properties of visibility and those of reflection and illumination (but ignoring cast shadows) for which Basri and Jacobs and Ramamoorthi and Hanrahan have recently derived signal-processing frameworks.  

The incident flux of radiant energy per unit area, or irradiance, is a convolution of the lighting and the clamped cosine Lambertian reflection function, an analogous result for cast shadows as convolution of the lighting with a Heaviside step function canalso be realized, which also generalize Soler and Sillion’s convolution results for shadows when source, blocker, and receiver are all in parallel planes—for instance, V-grooves that do not contain any parallel planes.  

The paper builds on a rich history of previous work on reflection models, and the properties of 3D textures. While the analytic formulae are derived considering the standard V-grooves used in many of these previous reflection models, it differs in considering cast shadows because of complex illumination, deriving a convolution framework, and analyzing the eigenstructure of visibility, aiming to derive a particular convolution formula for shadows directly, rather than considering a general integral involving the illumination, reflective properties of the surface, and visibility.  

To make the math simple, the authors begin in Flatland, a 2D slice through the viewpoint. A V-groove model, where the visibility is locally monotonically changing, serves as the model to develop the Convolution Formula for Shadows to find the irradiance E(x,β) as a function of groove angle β= [-β₁, +β₂] and the distance along the groove, x. 

Four Common Shadow Scenarios

Without loss of generality, only the right side of the groove is taken into consideration and the incident cosine term is neglected. {This cosine term may be folded into the illumination function if desired, as the surface normal over a particular face (side) of the V-groove is constant.} 

In a nutshell formalizing the idea of convolution for cast shadows suggests using the frequency space basis but that many more basis functions are required above the order 2 harmonics that are adequate for irradiance. And the results indicate a theoretical link between cast shadows and convolution formulae for their dual--irradiance, and more general non-Lambertian materials.

The authors derived the following orthonormal basis functions as being useful for analyzing cast shadows in 3D textures:

Orthonormal basis functions for 3D cast shadow analysis.
Boundary Conditions (Limits) for Odd Legendre Polynomials are automatically satisfied since they vanish at θ= π/2 or cos θ=0.
 Using only odd terms is similar in some ways to spherical harmonic constructions over the hemisphere.

Since the derived basis functions involve Legendre polynomials over q and Fourier basis functions over φ they are closely related to spherical harmonics but specialized to the hemisphere. Using only odd (sum  odd) Legendre Polynomials inhibits representing non-zero functions at  θ= π/2, but this may partially be corrected by adding a zero^th order basis term or a constant.

Their form is a simple product of 1D functions in q and φ and is simpler than for Zernike polynomials, spherical harmonics, or hemispherical harmonics. And for diffuse textures, due to visibility and shading effects, the intensity goes to 0 for lighting at grazing angles. Simply adding extra basis functions for the even Legendre Polynomials (that go to 0 at θ= π/2) precludes orthonormalization, and leads to slow convergence or inaccuracies, however, numerical methods such as Gram-Schmidt orthogonalization procedure are available to enable this practice.The hemispherical basis functions for cast shadows were constructed in three ways:

1. adapted from a domain of cylinder topology;
2. using Zernike polynomials that are adapted from the disk;
3. and hemispherical harmonics adapted directly from the sphere.

In developing a parametric for cast shadows, the main analytic results made it clear that the Net Visibility of Irradiance is a simple Convolution of the Incident Illumination with the Heaviside Step Function that accounts for cast Shadow effects.(Makes it easy to see Your Mom, which means You don't need to look too hard to find your Dad, since He is usually right behind her or just ahead of her, if He's not raiding her panty-drawer).

This naturally led to deriving a product formula in the Fourier or frequency domain:

Product Formula in Frequency Domain for Cast Shadows

Fig. 5 Analysis and Formula Derivation for Cast Shadows in the Frequency Domain
Note: The Non-zero eigenvalues slopes are between -1 and -2, for shadows vs. irradiance, respectively; 
compare to the 'information angle' given at 45 degrees to the axis

The Unfolding

Surah 74 AlMuddathir (الْمُدَّثِّرُ) (The Enveloped/Enfolded/Wrapped Up One)

(74:1) O You Enfolded One (Enveloped/Wrapped Up In Your Mantle)
(74:2) Arise and Warn! (Ascend/Mount Up To Deliver The Message)


You, My Dear, Can give any Message You would like at this point. Since I've been hung out to dry for over 14 years by my reckoning, the only message I feel the need to deliver is in answer to the question, "Why are the planetary orbits elliptical?"

Plane curves (lines, circles, parabolas, sine curves, etc.) that are wrapped onto cylinders of varying radii exhibit a sinusoudal influence, and can be characterised in terms of equations whose solutions are simple 2-Dimensional geometric transformations(rotation, translation, etc.), without resorting to calculus.
A cylinder is a developmental surface it can be 'flattened' or 'unwrapped' onto a plane without distortion--Mathematically expressed as having Zero Guassian Curvature Κ , (intrinsic curvature is Zero). Gaussian curvature is defined as the product of the principal curvatures κ1 and κ2.




Κ = κ1κ2
And since cylinders can be transformed flat without compression or expansion (distortion), unwrapping a curve that lies on the surface of a cylinder preserves the distance between points on that curve. In other words, any arc lengths on their profile unfolds/unwraps/uncurls onto a line segment of the same length. The diagram above shows the simple case where an arc on a right cylinder's circular profile is unwrapped by rolling the cylinder along its side; the end point on the circle projects onto a point t, but unwraps onto a point x. And since the sinusoidal influence is determined by the relationship of the arc length to the linear projection, unwrapping curves that lie along the cylinder surface gives rise to their periodic nature.









By extension, space curves unwrapped from the lateral surface of a right circular cone produce periodic plane curves. (An interesting anolog found in nature of curves unwrapped from conics is the lunar obit).



A line segment (the shortest distance between two points) wrapped onto a right circular cylinder is also a geodesic arc (shortest path) on the cylinder, no matter how tightly it is rolled (decreasing r) and the profile of a geodesic arc on a right circular cylinder is a function of arcsine. * A geodesic on a circular cylinder is always part of a circular Helix.



The profile of such a wrapped line is given by:





p(t)=cr arcsin (t/r) (where; c= constant, r=radius, t= projected point)
In case of a right circular cylinder, eccentricity is 0, where such a line in the xyz plane is given by z=t lies at a 45 degree angle to the generators (a=pi/2), the tightly wrapped line would stack into circular ringlets should the cylinder collapse orthogonally (like a Slinky would when there is no tension). The profile of such a line in this particular case is given by:









p(t)= ct^2





The stacked ringlets (circles) it forms are given by:





z^2 + t^2 = r^2
There are 3 ways to distort these circles into ellipses, by changing the viewing angle either via rotating the cylinder or changing the position of the outside Observer or by deforming the cylinder itself (squeezing it) so that the eccentricity is no longer 0. Changing the viewing angle is simply a matter of shifting perspective (--); physically deforming the right cylinder such that its eccentricity is no longer 0 requires effort (Force).









So, this begs the question, when considering planetary orbits, is it the perspective that makes the orbits appear to be elliptical or something more intrinsic?





From a purely geometric consideration, and realizing that circles are in fact merely ellipses with eccentricity=0, it is strange that the more generic elliptical orbits are the common case and not the perfect circular orbit.





In a 2-body system where a much heavier object is at the center, a circular orbit would be the expected outcome, or the ideal case, in any event. Neptune's largest moon, Triton, comes closest to this ideal case in the Solar system, with eccentricity=0.000016. {That of the Earth's orbit around the Sun is 0.01675 (a far cry from the exaggerated elliptical orbit often depicted in the literature)--based on a cursory look at the limited curricular activity on the web, it's no longer a surprise that the Germans landed first--in my living room--& very good helpers they are, too, even if they do tend to overshop the tomatoes.





The event alluded to happened over a dozen years ago and has nothing to do with my state of mind which was as sound then despite the intractable headache as it is now despite the fact that a casual trip to the local library to pick up a book accompanied by my child has some people conspicuously talking to You about 'stopping her, now!'





Is 'stop her' code for 'shoot her'? (^_~)





And it's a good thing You didn't because it isn't like I don't give You ample opportunities to do just that without the children present; and it would be in exceedingly poor taste if something like that were to happen right in front of their eyes given how much of a predictable, isolated, easy mark I make myself the rest of the time when they are not around.} "<o _<o"





Anyway, back to the present.





That the geodesic is the shortest distance between two points on the surface of a cylinder, regardless how tightly it is rolled up, is interesting from the perspective of an outside observer in that an object or person following this route would appear to be going around in circles (elliptical path), but from that traveller's or object's point of view they think they are travelling in a straight line.





An interesting variation on this theme is if the object or person were to follow a weaving path in a sinusoidal fashion in a plane that is then rolled up into a right cylinder. The sinuous path can represent Time as well as Displacement. One familiar representation of just such a path is the apparent Time Difference due to the Earth's orbit about the Sun (The Time Equation).










A knack for stating the obvious is one of my strong suits. Orbits "unfolded" in such a manner are indistinguishable from displacement due to simple harmonic motion, as found in springs that obey Hooke's Law or the motion of a Pendulum.





In terms of the Geometry, an Orbit is essentially a Straight Line 'Wrapped' around a Developmental Surface (like that of a Cylinder), and its Profile is essentially a Circular (Elliptical) Path.





By extension: Time (t) is a Projection of Orbit onto a Tangential Viewing Plane. Each Point in Time is also a Position (x) in the Orbital path. As that Path (displacement) 'unfolds/unwraps' (theta) onto the Viewing Plane, the Time correlates to a particlular Place (x) on the Orbit's Profile given by:





x= (Time*Theta)/(Sin (Theta))





or x= Time*θ Csc[θ] = ; and





alternatively;











and for the Real Time, and Place and Angular Displacement (Azimuth)











Interstingly has only 1 Real Integer Root at t=0;

Does this mean that In Reality Nothing Really Ever Happens?





(Projecting a Point and/or Unwrapping a Curve from a Developmental Surface is like Stepping Down from a Higher Dimension. When the need to refresh What this is all about arises, we can always go back to How Emily treats Bob, and Where Angels come from--always leaving the residual question, Why?)

"They forgot Y I told U!"
The Argyle Sweater -Confusion at the Alphabet Gang Hideout
Appeared in the March 1, 2014 Los Angeles Times Comics Section

*Unwrapping Curves from Cylinders and Cones, Apostol and Mnatsakanian, 1997, The Mathematical Association of America; pp 392-415

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?}