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