Trix Question

Q:  How many non-Cooper Union Computer Engineers does it take to tune this system?
A:  All of them!
Ah.All, Again...
Q: And how many Cooper Union Computer Engineers do I need?
A: Just one, Hhheenryyyyy!

~~~~~~~~~~~~~~~~~

Ahem, St%*#; it's not gonna work if You throw your voice over other people--this morning it was Frank!
So, excuse me for being non-responsive by giving You the Silent Treatment.

Just like yesterday's, 'Are You Blind?'

I heard You project from one corner of the coffee shop on my way out of there.

Here's the answer back in case You are Deaf!

"No, I'm not blind! But I'm not Stupid, either!"

Oh! So, I now hear You callin' me a Genus!

I take it we can both finally move on, now that our short-lived mutual infatuation has worn off, unless You are envisioning some sort of inter-species interaction. Genus,  eh?

The modern human species comprises the only extant species of the genus - Homo sapiens; and now that I Am a Genus, (Genus, He Says!), by its very definition, that makes  Me the Alien!

An apt description, ar.  وصف  w.ss.f,  considering what this post is all about.

But all inanity aside, I will now attempt to describe  ( وصف) how information is never lost, only masked (it is preserved in the Zero Point Field).

And it only takes for someone to be well versed in a chosen area of interest to See It for What It Is--should one subscribe to Mr. Bremby's pseudo-randomness philosophy underlying motor vehicle license plates.

An attempt or demonstration on my part to illustrate as much since "He Indicates (Instructs/Demonstrates/Pictures) Using 119."

(BTW, thank You for the g'mmee ("For you," He said), it was so nice to see You the other evening  KIDR1--had we met prior to that occasion?)

Let's Pretend I AM A Quran Maven for the Purposes of This Example!

The possible combinations using the string triad 119, reading left-to-right, without transposing the numbers:

1; 1:1; 1:19.  11; 11:9.  119;

Surah 1 has only 7 Verses and there are only 114 Surahs in the Quran, leaving Chapter 1, Verse 1:1 and Chapter 11,  and Verse 11:9 as viable Surah and Sign substrings.

In the interest of saving Time and Space, I will only parse out the individual Signs and leave the whole Surahs for when I can devote more attention to this speculative realm than that allotted between dinner and bedtime, wield a staff of umpteen many, and have a wee bit more computing power than my MacBookPro can handle.

Surah 1 Al.Fatiha (The Opening Chapter)

بِسْمِ اللّهِ الرَّحْمـَنِ الرَّحِيمِ (1:1   
1:1 In the Name of God, The Gracious, The Merciful.

Surah 11 Hud (The Prophet Hud)
وَلَئِنْ أَذَقْنَا الإِنْسَانَ مِنَّا رَحْمَةً
ثُمَّ نَزَعْنَاهَا مِنْهُ
إِنَّهُ لَيَئُوسٌ كَفُورٌ (11:9  
11:9  And if We Impart to Humanity (Man) a taste (a modicum) of Mercy ( رَحْمَةً  Rahmatan) from Us
and then take it away from them (him);
they (he) abandon(s) all hope (despair)...blaspheme (forget Our Past Favor, revert to infidelity)

Then the possible combinations reading right-to-left, without transposing the numbers:

9;  9:1;  9:11.  91; 91:1.   911.

And, again, since there are only 114 Surahs in the Quran, 911 is not a viable Chapter number, leaving only Surah 9, Verses 9:1 and 9:11; and Chapter 91,  Sign 91:1 as usable substrings in this particular example.

Surah 9 At.Tauba (Repentance)
بَرَاءةٌ مِّنَ اللّهِ وَرَسُولِهِ
 إِلَى الَّذِينَ عَاهَدتُّم مِّنَ الْمُشْرِكِينَ (9:1
 9:1 God and His Messenger Absolve (hold guiltless) those you have made pacts with from among the pagans (idolators, infidels).

فَإِن تَابُواْ وَأَقَامُواْ الصَّلاَةَ وَآتَوُاْ الزَّكَاةَ
 فَإِخْوَانُكُمْ فِي الدِّينِ
 وَنُفَصِّلُ الآيَاتِ لِقَوْمٍ يَعْلَمُونَ (9:11
9:11 Should they repent, and gather to pray, and give to charity;
they are your bretheren in Faith.
And We Separate/Segregate/Partition/Adjudicate/Disentangle/Detail/Itemize/Partition/Spell out...نُفَصِّلُ the Signs (Quranic Revelation) to those who understand/can teach/are knowledgeable/for everyone to learn.

Surah 91 Ash.Shams (The Sun)
وَالشَّمْسِ وَضُحَاهَا (91:1
91:1 And the Sun and its Brilliance/sunshine/luster (var. waking, attention (shedding light on something), clarifies, explicates//its victims/sacrifices)

(If the triune was less than or equal to 114, it would be indicative of using that whole Surah to 'inform the field'; and the numbers do not necessarily have to be a triad, they could be quaternary or longer, as long as the string holds up to the numbering conventions in the Quran. For instance, Surah 17 has 111 verses, so a sequence of xxxxx digits could still make sense in this context, since Sign 17:111 exists, there is a Surah 111,  a Verse 11:17, 11:71, and other sensible substrings).

Taking all the above listed substrings for 119 and transposing Chapter:Verse --> Verse:Chapter yields:

1:1;  19:1;  9:11; 1:9; 11:9; and 1:91

The only viable additional substring to the set established so far is 19:1;

Surah 19 Maryam (The Virgin Mary)
كهيعص (19:1
19:1 kaf. ha. ya. 3'ayn. saad.

The series of 5 Muqati3aat,...  ص ع ي ه ك ...(K. H. Y. 3'ayn. Ssaadd)...

{Aside: added Nov 29, 2012--Good Morning,  Mr. אחד ויחיד  (or א,שהד was it? I can't read it, You Know, I go by pattern recognition).

Frazz Comic Strip
Pattern Recognition 'Really Really False'
Appeared in the January 13, 2013 Los Angeles Times

--isn't it a relief to finally do away with that thing behind Your ear, anyway? Isn't that much better now--so who cares why?  And I am happy to hear You don't care who she is, because You.Know.Who is still asking 'How does she look?'  I think to move things along quicker we should all start agreeing she looks 'good enough,' because just saying, 'beautiful' and 'gorgeous,' doesn't seem to cut it for Him. But, no matter what You all settle on for her looks, I want to be Independently Wealthy!}

Thanks! You heard right--I can be the gorilla in the window and You get to play at Ogee & Take me home--I could use some west and wee wax a shin, at last!

Moving right along...in Arabithmatic terms:

ص  = y

in complex analysis: ع = z = rÐj =   ل Ð ي

where:

ي  = φ (an angle)

ه = e (natural log base~2.718...)

ك = r dφ or κ (both the Greek kappa and the Arabic kaf have the same abjid/gematria value==20; so they have equivalence in scientific/arithmetic notation). Another take on what Kaf may mean, in addition to the modern accepted arithmetic usage, is Lam with a Hamza embedded in it: 
laam==ل

kaf==ك

hamza ==ء  which represents the variable d, or just as readily the operation d (derivative of  ل dφ );  r dφ

{I hear You. I am sure that You don't intend for me to feel the way I do about hearing you, but I can't help it...BTW, so far the sound checks out great! Can they find better applications for this than making it sound like they are following me around all over town? So, when is the persecution over?}

Back to the quirky math lesson...Just for You! (I know, I know; it's not what you asked for, need or want, but I will give it to You, anyway...How do You like that? So, by the same token, please explain why I keep hearing You say, "Just for you.")

First attempt when taking the Bludgeon approach is to make a direct substitution: (I make no representation other than I am guessing at this point & I realized once I got through the problem , I had left out the r in r (cos φ + i sin φ) term, so let's just assume r=1, as in a unit circle).

20~19:1   (e)(φ)  r( cos φ + i sin φ) y

Kaf= κ   is something like     Ha = (e) Ya =(φ) 3ay'n = z SSAAD= y

The Bludgeon Approach: direct substitution:  κ =(e)(φ) r(cos φ + i sin φ) y
assume φ is an angle (in radians) or variable t, the general form; (oops just realized the omission of r in the rÐj term, but rather than rework all this just pretend it's a Unit Circle, r=1)

and alternate form

has roots y= 0;  y =0  or t =0

and the 3D contour graph (mesh) looks like (thanks to wolframalpha.com):

That's right, a Flying Carpet!

Substituting 19:1 in the general form, for the variables in order (phi, or psi or t =19 and y=1):   yields     19 e^(1+i 19)(1)

alternate representation for which:

Note that z= 1, see ع above

...a transcendental number,  with the polar coordinates: 

r = 51.6474, φ = 8.61981 degrees--
what the angle value y = 8.61981 means:

it's an arc length along the Earth's equator of 596 miles (518 nmi, 960 km), 
an arc  length at 45 deg latitude of 422 miles (366 nmi, 679 km)

and correlates to:
~ 0.37 x the tilt of the Earth's axis relative to its orbital plane (~23 deg)
~2.2 x the tilt of the angle of the Leaning Tower of Piza (~ 4 deg)

interesting to note that the numeric value of r = 51.6474 correlates to:

~ 2.2 x  the tilt of the Earth's axis relative to its orbital plane (~23 deg)

____________________________________

By extension this same relationship κ =(e)(φ) r(cos φ + i sin φ) y and

in context with Surahs 13 Ar.Ra3d (Thunder) and 21 Al.Anbiya (Prophets), Verses 13:41 

and 21:44, respectively (the Quran as a Knowledge Object):

أَوَلَمْ يَرَوْاْ أَنَّا نَأْتِي الأَرْضَ نَنقُصُهَا مِنْ أَطْرَافِهَا 

وَاللّهُ يَحْكُمُ لاَ مُعَقِّبَ لِحُكْمِهِ وَهُوَ سَرِيعُ الْحِسَابِ (13:41 

بَلْ مَتَّعْنَا هَؤُلَاء وَآبَاءهُمْ حَتَّى طَالَ عَلَيْهِمُ الْعُمُرُ 

أَفَلَا يَرَوْنَ أَنَّا نَأْتِي الْأَرْضَ نَنقُصُهَا مِنْ أَطْرَافِهَا 

أَفَهُمُ الْغَالِبُونَ (21:44 

Both verses refer to shorting/reducing/making a gap or shortage نقُصُ in the 'Earth's' extremities, sides, tips, ends, parts or even parties (referring to folks/communities/people) طرف. 

In this case, the function (e)(φ) r(cos φ + i sin φ) y , shows a conspicuous 'clipping' or reduction of the corners in the imaginary contour plot for this function; a definite pattern that emerged when plugging the values 13, 21 or 19 in for the y term, and on cursory examination, the results seem to point to an apparent threshold at values >=9: 

Such that: 13 (e)(φ) r(cos φ +/- i sin φ), or 19 (e)(φ) r(cos φ +/- i sin φ) or 21(e)(φ) r(cos φ +/- i sin φ) produced these type of 3D contour plots:

Imaginary Contour Mesh f(t) = 21 e t r (cos t + i sin t) from 19:1 Muqati3'at
Corners Reduced Per 13:41 and 21:44
Compare to 'Flying Carpet' Above

Imaginary Contour Mesh f(t) = 13 e t r (cos t - i sin t) from 19:1 Muqati3'at
Reduced or Showing Gap at Extremities (parts, corners) Per 13:41 and 21:44
Compare to 'Flying Carpet' Above

________________________________________

A second approach: 
Assuming the derivative form of κ  as  κ' =(e)(φ)r(cos φ + i sin φ) y

For a Unit Circle, the general equation and solution are:

i e.,  k'(φ)) =e(φ) z y == e (φ)(cos φ + i sin φ) y

An ordinary first-order differential equation

the solution for which:

If the radius is not equal to 1 the differential equation  k'(φ) =eφr(e^iφ) *y yields the following general solution:

for simplicity, stick to the unit circle, and keep r =1 st: κ'(φ) = e φ (e ^(iφ))y

then

k(φ) = c1 +[e^(iφ)](e- i*e*φ)y

this function returns a complex solution:

 c1 = k(φ) ,   φ  = -i


and a real solution:


c1 = k(φ) - ye, φ = 0



Interesting, more so, since setting k(φ) = 0, produces a trace c1 reminiscent of the Time Equation, by choosing y carefully:

since c1 = - e^i(φ) [ e - e i φ] y

See Time Equation
Superposition of Sinus Curves
Sine and Cosine

----

Sound Check: I hear 'awesome' 'awesome' 'ahhh, sum?"--STOP. Who are you foolin'? I have clearly demonstrated I can't even do high school math, unless you mean Genius, like Daffy Duck, who happens to be a Super Genius!--Just make it easy for me and simply tell me what everyone else does at this level, I will follow suit.

{Honestly, I know You hear me go off on All of You when I get home--evidenced by how what I say in my bathroom at home is in the next morning's syndicated cartoon strip, and You still tittering about 'my wife is gorgeous', 'beautiful', 'genius'---&amp..?}

Non Sequitur Comic Panel
Gorgeous-Like the Grand Canyon~Whilst Gorging On Grapes
Appeared in LA Times Dec 1, 2012

As for You/Him/'We Never been introduced' asking, 'Did you miss me?" 

I don't know, let me hurl the nearest projectile in Your general direction and see by how much! 
Is that where the first flying saucers came from?

And I am an unmarried woman, so who, if it's me You are alluding to, am I married to? 
Am I a polygamist without my knowing? If it turns out I am still married to my ex in some reformulated format then i can only assume I am in Hell for all eternity and will keep up the silent treatment because I choose not to acknowledge such injustice in my life And hereafter.

That is not to say that I will not take You up on your proposed 'cover marriage' just so my daughter can get that Father she always wanted to spoil her rotten like the princess she deserves to be--never saw anyone work harder than she since such a young age to impress her dad just to have his behavior make me sign on to the 'all men are a!@#@*%s' philosophy.}

Why it matters concerning what this Blog goes on and on and on about (decipher in context):

• In graph theory, the connectivity of a graph is given by κ.
• In differential geometry, the curvature of a curve is given by κ.

Right-Hand Side of the Equation
Definite and Indefinite Integrals

How this is applicable to 19:1: 
The most general interpretation of the series of muqati3'at based on the above:

¶ (r e ^{^± i θ}) = e*θ *(rcos θ ±  rsin θ) *y

Left Hand Side of the Equation
Partial Derivatives and Complex Map
Note the Analogy between the Complex Map and Sett Point Paper

Since here we are considering curvature the most general interpretation of the right-hand side of the equation may not be a straight multiplication, but makes sense in the context of curve parameters:
ie. x, y , and z (meaning the pitch (as in the case for a helix), or simply a third axis (as is the case for azimuth)):

x (t) = a cos (t)
y (t) = a sin (t)
z(t) = bt

st;

x (θ)= rcos θ
y (θ)=±  rsin θ
z (θ) = ey θ, where y may be an actual constant value, like 1 or 19, or substitute the y(θ) from the y coordinate, so that  z1(θ) = e*(±  rsin θ)*θ...

Assuming a right-handed circular helix with radius 1, and pitch 2pi*b = 2pi (ey) and using cylindrical parameters [r, θ, h] :
r (θ) = 1
θ (t) = t
h (t) = (et)*t =e(t^2)  which seems appropriate in this case since t^2 is analogous to (sin^2 θ); the square of the sine of an angle, essentially the concept of spread in rational trigonometry, or it may possibly still mean et?)

Several interpretations follow:

Parametric Plots v Unwrapping Curves v Sign 19:1

Ichthus v Solar Geometry and Sign 19:1

Parametric curve of the form x = rcos t,  y = r sin t, and z = e*t* y(t);
Here z(t) = e t sin (t)  the crumpling is mitigated near the Origin, where there is 'no crookedness'  (See Sign 20:108)

Parametric curve, Cylindrical coordinates of the form x = 1,  y = t, and z = e*t;
Here the contour maps show no curvature at all, (See Sign 20:107)

Parametric curve of the form x = 1,  y = t, and z = e*t^2;
Here z(t) = e t^2  the flats and the 'mountains/barrows' depicted int the above alternative parametric curves are now sunken due to the emergent parabolilc function z= et^2, or 'bottoming' at the Origin,  (See Sign 20:105)

Superimposed Sinusoids

Surah 88 Al.Ghashiya (The Overshadowing)
أَفَلَا يَنظُرُونَ إِلَى الْإِبِلِ كَيْفَ خُلِقَتْ88:17
وَإِلَى السَّمَاء كَيْفَ رُفِعَتْ (88:18) 
وَإِلَى الْجِبَالِ كَيْفَ نُصِبَتْ (88:19) 

وَإِلَى الْأَرْضِ كَيْفَ سُطِحَتْ (88:20) 
فَذَكِّرْ إِنَّمَا أَنتَ مُذَكِّرٌ (88:21   
....
88:17 Do they ever Consider (See) Rainclouds (الْإِبِلِ, al.abili, borrows from the root a.b.l.h, to be full/wet/forerunner and when put together= rainclouds, or even 'pregnant camel', but in context, it makes more sense if the root is أبله, abl.h, meaning vacant/vacuity/emptiness/gap/space/vapor and var. sod, adj., inanity, vacuum/senselessness, what we would call a 'void') and how they are (it is) Created?
88:18 And the Sky (Heavens) how they are Raised (Elevated)
88:19 And the Hills (Mountains) how they are Erected
88:20 And the Earth how it is spread out (ssutihhut)
88:21 So Remind (them) for you are a Reminder.

Surah 87 Al.A3la (The Most High, Ultimate Power)
سَنُقْرِؤُكَ فَلَا تَنسَى 87:6
87:6 We Recite It (Teach It) to you so that you do not forget.

• In physics, the torsional constant of an oscillator is given by κ as well as Einstein constant of gravitation.
• In structural engineering, κ is the ratio of the smaller factored moment to the larger factored moment and is used to calculate the critical elastic moment of an unbraced steel member.
• In electrical engineering, κ is the multiplication factor, a function of the R/X ratio of the equivalent power system network, which is used in calculating the peak short-circuit current of a system fault. κ is also used to notate conductivity, the reciprocal of resistivity, rho.
• In chemistry, the compressibility of a compound is given by κ.

"What's Wrong?" He wants to know. Here, here, I'll s-p-e-l-l it out for ya so we're all on the same page and we all know what' wrong.

Generalized Gas Compressibility Chart
Note: Isotherm at 20% Compressibility

The Generalized Gas Compressibility Chart above is an example of a generalized compressibility factor graph derived from hundreds of experimental PVT data points of 10 pure gases, namely methane, ethane, ethylene, propane, n-butane, i-pentane, n-hexane, nitrogen, carbon dioxide and steam. Note behavior at z=20% (compressibility factor at a reduced pressure ~1 and that these gases are associated with oil well lighters) and gases used to develop the isotherms which model near ideal gas behavior and what is in a can of one particular air freshener. 

A less harsh, read less harmful approach, would be to use a benign real world model on deteriorated brains rather than such a toxic exposure to healthy brains.

Need I say more, my math may be questionable, but I think I got the chemistry right--in a nutshell the implication is deliberate and malicious intent to cause harm. 

To which He follows with, "تأكد 085", which means 'be certain', 'ensure' or 'confirm'; Surah 85 Al. Buruj (The Zodiac)-- to the best of my ability, I confirm they are still in the Sky and I got burnt. 

and Sign 08:5--
Surah 8 Al.Anfal (The Spoils of War)

كَمَا أَخْرَجَكَ رَبُّكَ مِن بَيْتِكَ بِالْحَقِّ
 وَإِنَّ فَرِيقاً مِّنَ الْمُؤْمِنِينَ لَكَارِهُونَ (8:5
8:5 Even as Your Sustainer brought you forth from your home in Truth, although a segment of the Believers were averse to it (disliked it). 

That would be True, both instances.

• In psychology and psychiatry, kappa represents a measure of diagnostic reliability.
• In biology, kappa and kappa prime are important nucleotide motifs for a tertiary interaction of group II intron.
• In biology, kappa designates a subtype of an antibody component.
• In pharmacology, kappa represents a type of opioid receptor.
• Kappa statistics such as Cohen's kappa and Fleiss' kappa are methods for calculating inter-rater reliability.

Upper-case letter Κ is used as a symbol for:

• In set theory, kappa is often used to denote an ordinal which is also a cardinal.

--------------------------------

{Addendum: Dec. 4, 2012, so I am to have a sense or get a feel  حس  for 3:79 and 37:9, just in case I missed it. 

Surah3 Al.3'mran 

مَا كَانَ لِبَشَرٍ أَن يُؤْتِيَهُ اللّهُ الْكِتَابَ وَالْحُكْمَ وَالنُّبُوَّة 

ثُمَّ يَقُولَ لِلنَّاسِ كُونُواْ عِبَادًا لِّي مِن دُونِ اللّهِ 

وَلَـكِن كُونُواْ رَبَّانِيِّينَ بِمَا كُنتُمْ تُعَلِّمُونَ الْكِتَابَ 

وَبِمَا كُنتُمْ تَدْرُسُونَ (3:79  

 3:79 It's not likely (apt, possible) that a person whom God has Granted Scripture and Wisdom and the Prophethood that he should afterwards say to people, 'Serve me instead of god.' But (what he said was), "Be faithful servants of the Lord by virtue of your constant teaching of the Scripture and of your constant study thereof."

And from 

Surah 37 As.Saffat (Rangers)

إِنَّا زَيَّنَّا السَّمَاء الدُّنْيَا بِزِينَةٍ الْكَوَاكِبِ (37:6) 

وَحِفْظًا مِّن كُلِّ شَيْطَانٍ مَّارِدٍ (37:7) 

لَا يَسَّمَّعُونَ إِلَى الْمَلَإِ الْأَعْلَى وَيُقْذَفُونَ مِن كُلِّ جَانِبٍ 37:8) 

دُحُورًا وَلَهُمْ عَذَابٌ وَاصِبٌ (37:9

37:6  Indeed, We Bedecked (Adorned) the Worldly Sky with the beauty of the planets 

37:7 And (they serve as a) Security from every titanic devil (rebellious demon). 

37:8 They cannot hear anything but the Higher Chiefs (Congregation), and they will be pelted/ejected/liabled/slandered/defamed/flung from every direction (side). 

37:9 Stopped (Repulsed), for theirs is a perpetual persecution. 

I don't have to defend myself here, I already covered this almost at the inception. I never took any of this lightly, and the tragedy of it is that I know I wasn't invited to the dance, and I didn't want to come along, either.  

That's why what they did is harram, and I took every measure I could to prevent it from reoccurring--to me or any one else. 

I take it my recently sampling this first hand is just to be sure, but I never contested it and hopefully this hazing will let up as quickly as it set in. I don't believe I am such a titan, but the only rebellion You'll see in me is if I am forced to wake up to that a!@#@*%s--One time through that shi1112 is enough. 

I heard my daughter on the bus today. And my brother in the locker room, and my Dad, someone calling my Mom 'Um Ahmad'. I pretty much know the whole family made it.
So, I take it this reminder is for 'them thar others' edification.

...And my friend Francis keeps cropping up everywhere I plan to be...

Harmonica Player
Black and White Charcoal Sketch
Artwork by Japie Goosen 2011

What makes me so special?

I find myself asking myself the very same question.

But my Mother answers me so I don't sound so crazy..."You know, dear, they didn't want a 'virgin', they wanted to read this version--you always had a problem with your annunciation!"

(She thinks it's part and parcel with my poor spelling).

 Here's one example:

Surah 55 Al.Rahman
مُدْهَامَّتَانِ (55:64 
Nearly all accepted translations take this to mean:
55:64 (Gardens, from the context) 'dark green' (in color or of foliage)

And this is my understanding of the same verse Sign 55:64 based on my 'annunciation' (that's right, that would make me a lousy Catholic, too!)
55:64 Its Range Extensive (the maximal range, enduring/endurance interval/or gap...)

Then use Sign 55:64 as angles or velocity/angle or some such combination to maximize the Range! That's if we're talking projectiles (or mystery of alpha).}
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TiraMissU

Torus Knots (Planar)

As long as the world is turning and spinning, we're gonna be dizzy and we're gonna make mistakes. --Mel Brooks 

Rotational Matrix Take II

Tommy's Interwoven Paths In His Past and Present Life
In His Home
Raiding His Father's Refrigerator
I See Him Standing by the Fridge
His Father Said the Pot Flew Out of a Pantry Kitty-Corner to the Fridge During a Crisis Some Time Later
His Kitchen Is Just About Square
And Tommy's Path Seems to Be On a Diagonal , His Orientation Turned Around

I take it Yous guys are less than impressed with my arithmetic derring-do, since yer all so focused on her looks (not my looks, her looks, because I'm not supposed to take it personally).

Based on the above Scenario, our brave young soldier Tommy goes into his kitchen (at his father's house) just days before he was supposed to report to boot camp for deployment to some vacation spot.

Like any other young man, he goes through the kitchen door and heads due East to raid the refrigerator.

Surely there must be something to eat!

Some time after, Tommy dies of dehydration because his drill Sergeant couldn't allow a break for water during a strenuous exercise before ever getting deployed. Which causes Tommy's father a great deal of distress, and makes his good friend wonder 'what is it about men that makes them act like accomplished women?'

And during a pivotal moment in his father's coping mechanism failure, a sauce pot comes flying out of the NW pantry of the kitchen (no earthquakes reported in that part of town on that particular occasion) and lands right in the middle of the square kitchen (6 ' x 6').

Determine the rotational matrix based on the above scenario.

 (A bad drawing to kind o' confuse the issue (the planes are offset along a central axis, much like a helix/screw threads/or augers), think 3D and not certain the time).

{"Sit on it...and Rotate!"--The Fonz (Henry Winkler), popular 1970'a sitcom Happy Days

The Rotational Matrix

In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. 

Rotations are distinguished from other isometries by two additional properties: 

1. they leave (at least) one point fixed, and 
2. they leave "handedness" unchanged. 

By contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, and a glide reflection does both.

A rotation that does not leave "handedness" unchanged is an improper rotation or a rotoinversion.

A geometric rotation transforms lines to lines, and preserves ratios of distances between points. A rotation is a linear transformation of the vectors, and can be written in matrix form, Qp. 

The fact that a rotation preserves, not just ratios, but distances themselves, it follows that...

Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, all these properties may be summarize by saying that the n×n rotation matrices form a group, which for n > 2 is non-abelian. 

Called a special orthogonal group, and denoted by SO(n), SO(n,R), SOn, or SOn(R), the group of n×n rotation matrices is isomorphic to the group of rotations in an n-dimensional space. 

This means that multiplication of rotation matrices corresponds to composition of rotations, applied in left-to-right order of their corresponding matrices.

To rotate points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system use a rotation matrix R; each point is represented by a column vector v, of each point's coordinates (position). The rotated vector is then obtained by matrix multiplication Rv. 

Rotation matrices are square matrices (nxn), with real entries; characterized as orthogonal matrices with determinant 1.

The special orthogonal group SO(n) is the set of all such matrices of size n, generally not commutative.

Improper rotations, characterized by orthogonal matrices with determinant -1 (instead of +1) combine proper rotations with reflections (which invert orientation). 

The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. -90°).

Basic rotations:

Three basic (gimbal-like) rotation matrices rotate vectors about the x, y, or z axis... 

All other rotation matrices can be obtained from these three using matrix multiplication. 

...to represent a rotation whose yaw, pitch, and roll are α, β, and γ, respectively. 

...whose Euler angles are α, β, and γ (using the y-x-z convention for Euler angles).

Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis (Euler rotation theorem).

There are several methods to compute an axis and an angle from a rotation matrix one is based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.

...which shows that  is the null space of... 

Viewed another way, [] is an eigenvector of R corresponding to the eigenvalue; every rotation matrix must have this eigenvalue.

To find the angle of a rotation, once the axis of the rotation is known, select a vector  perpendicular to the axis. Then the angle of the rotation is the angle between__and__.

It is much easier to calculate the trace (i.e. the sum of the diagonal elements of the rotation matrix)...

In three dimensions, for any rotation matrix , where a is a rotation axis and θ a rotation angle,

 (i.e.,  is an orthogonal matrix)

 (i.e, the determinant of  is 1)

 (where  is the identity matrix)

The eigenvalues...

The trace of  is  equivalent to the sum of its eigenvalues.

Some of these properties can be generalized to any number of dimensions. In other words, they hold for any rotation matrix...

Ambiguities:  Alias and alibi rotations

The interpretation of a rotation matrix can be subject to many ambiguities.

The change in a vector's coordinates can be due to a turn of the coordinate system (alias) or a turn of the vector (alibi). Any rotation can be legitimately described both ways, as vectors and coordinate systems actually rotate with respect to each other. Here, the alibi approach is used to describe rotations. 

The vector can be pre-multiplied by a rotation matrix (Rv, where v is a column vector), or post-multiplied by it (vR, where v is a row vector). Here rotations are produced by means of a pre-multiplication.

The vector space has a dual space of linear forms, and the matrix can act on either vectors or forms.

In most cases the effect of the ambiguity is equivalent to the effect of a transposition of the rotation matrix.

Two features are noteworthy: 

1. First, one of the roots (or eigenvalues) is 1, i.e., some direction is unaffected by the matrix. For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension). 
2. Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is 2 cos θ (the negated linear term). This factorization is of interest for 3×3 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any n×n rotation matrix. If the dimension, n, is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). Its a given that the characteristic polynomial will have degree n and therefore n eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized. 

Every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most n⁄2 of them...

The sum of the entries on the main diagonal of a matrix is called the trace...

The large number of options is possible because rotations in three dimensions (and higher) do not commute. 

Reversing a given sequence of rotations results in a different outcome. 

This implies that two rotations cannot be composed by simply adding their corresponding angles; i.e., Euler angles are not vectors, despite a similar representation as a triple of numbers. 

} 

(The rest of us just spin on our heels and walk a distance about 6Ö2 along the diagonal heading WNW if we think we are at the fridge or ESE if we think we are at the pantry and want something out of the fridge.

Or become really well acquainted with quaternions to get a sense for what Tommy was seen doing in his 'spare time'. )

Like I said, some people say, 'Thank You,' and some people can be quite demonstrative of it.

Just like those 'I love you' tokens I been hearing of late.

If You Love Me, you would stop this (You Know Exactly what I mean).

Surah 2 Al.Baqarah (The Calf)
وَلاَ تَقُولُواْ لِمَنْ يُقْتَلُ فِي سَبيلِ اللّهِ أَمْوَاتٌ
 بَلْ أَحْيَاء وَلَكِن لاَّ تَشْعُرُونَ (2:154

2:154 And do not say of those slain in God's Cause that they are dead;
Nay, they Live and yet you perceive it not.

I know, I know, it must be End of the World because i am the smartest girl in the room; don't anybody panic because God can make another dysfunctional one just like it!