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!

I Think I Hear My Mother Calling...

...and, Boy! Does He Sound Mad!

A dog comes when he is called, but a cat takes a message and will get back to you later.--Mary Bly

{PS: Things could not have gone any worse for me up until 6 months ago, then they absolutely plummeted and now I have to put up with more angst inducing weirdness and take it on the chin?
--nuh-uh!
I hear these guys mumbling "beautiful" and "who is she" and some say "she's my mom" and some say "she's my wife". 

And sometimes to actually hear my name called!--one guy in particular seems to have lost his wife and is searching high and low for her--the bus, the train, any restaurant I happen to randomly pop into--so much stress to have to hear about it and hold down a meal or catch the next bus while all this is going on...
(I am not 'the wife' in the biblical sense, since I see him go home to his real wife, he just flies in for the day, does some work and goes back home to her; I am a 'working girl', the worst kind ever, the type that actually does not get paid for a job well done! ergo~GOD, and the need for One!)

Still my second rate life; where most girls get to age to perfection and 'beautiful' to them means 'timeless', for me that kind of remarkable beauty translates to 'dateless'--or am I mis.hearing 'beautiful' for 'pitiful'?
Besides, there's no shortage of beauty in the world, just a lack of clean public restrooms and frequent LA Metro buses that run on time.
What's more, as I read this entry to edit it, I am watching this really pretty young woman (a classic beauty, like Venus) outside a coffee shop shedding a stream of tears and it just reinforces what I have known all along, Beauty is not an Armor impervious to being pierced by Sorrow. 

And do You mean I don't get to kick back and play "Can You Tag (Finger) Malcolm"  (the adult version of "Where's Waldo") forever hereafter--I thought I was on some level of the death plane (heaven, hell, earth2,3, 4...); or tell jokes to myself once I get home (i.e., haunt the house; a step up for me-- that type of behavior was limited to the kitchen in real life)?

I know this is not any kind of love story, the last thing I would be looking for, assuming I do get to the threshold of heaven, would be a 'husband', so that begs the question, 'Have we caught the culprit(s), yet?' I  soo need a vacation! (and at this rate the interest alone you all owe me for backpay must be astronomical--so the sooner we settle this the better).

Great (10/18/2012/..so that's how it is Dad, 'hello' n 'good-bye' n 'you ditch, too?!' it's not like I can't hear You telling bro # 2 and bro #3 'it's your sister'!! If You all insist on this routine, then make it so I don't have to hear yous guys.}

On The Subject of Name Calling

Using functional magnetic resonance imaging (fMRI), brain activation patterns were examined in response to hearing one’s own first name in contrast to hearing the names of others. There are several regions in the left hemisphere that show greater activation to one’s own name, including middle frontal cortex, middle and superior temporal cortex, and cuneus. 

These findings provide evidence that hearing one’s own name has unique brain functioning activation specific to one’s own name in relation to the names of others. 

(Hi, Norman--You wouldn't be Swedish, by any chance?)

On Hearing Your Name Called 
& Putting The Conspiracy Theories To Rest

There are several instances of reputable people (of the non-psychotic variety) hearing voices and responding to them prior to the historical development of electronic PsyOps type weapons (microwave, brain enhancement (ie. air freshener), etc.) development.

In recent history, there is such an instance described by Mother Teresa which led Her to what became Her life's mission in India.

She, being a faithful person, did not question the Voice as anything other than being patently genuine and the Voice of some Agency of God, (God, an Angel, my Mom in her kitchen?), and did not run around trying to locate the source like the rest of us would by first typing in some keywords into our favorite online search engine.

And prior to the advent of such emergent technology, there is evidence that this sort of thing has been going on for quite some time, and Say What!--no modern technology required?

A Smattering of Selected Case Studies In The Bible

Isaiah 43:1
And now, thus said Jehovah, Thy Creator, O Jacob, and thy Fashioner, O Israel, Be not afraid, for I have redeemed thee, I have called on thy name -- thou art Mine.


Isaiah 45:3
I will give you the treasures of darkness, riches stored in secret places, so that you may know that I am the LORD, the God of Israel, Who summons you by name.

Yet, with the advent of technology (the Internet) and information from questionable sources (like bloggers and Fox News) for unverified reasons (disinformation and to hang on to their money) it becomes necessary to figure out what may or may not be a viable call from a legitimate caller (Holy God and Company).

How to Suss Out Whether It's A Higher Power 

or 
Steve at the US Army Signal Corps Doing the (Name) Calling...

1. God is Responsive to the caller and can help

Surah 2 The Calf

 وَإِذَا سَأَلَكَ عِبَادِي عَنِّي فَإِنِّي قَرِيبٌ 

أُجِيبُ دَعْوَةَ الدَّاعِ إِذَا دَعَانِ 

فَلْيَسْتَجِيبُواْ لِي وَلْيُؤْمِنُواْ بِي

 لَعَلَّهُمْ يَرْشُدُونَ  (2:186 

2:186 And if My Servants Ask You (the Prophet, pbuh) about Me (God), Truly, I Am Near; I Respond to the call of the caller, whenever they call. So, have them answer (respond) to Me and Follow Me, that they may be guided aright.

2. The Caller Calls you by your Name or Gives you a name of some significance (non-derogatory). 

(Figure on a planet of xxx billion people, what are the odds? What makes you so famous to attract Steve's attention?)

• Examples of those hearing the call are called by name and sometimes even Named by the Caller:

Surah 3 The Family Al.3'mran

فَنَادَتْهُ الْمَلآئِكَةُ وَهُوَ قَائِمٌ يُصَلِّي فِي الْمِحْرَابِ

 أَنَّ اللّهَ يُبَشِّرُكَ بِيَحْيَـى مُصَدِّقًا بِكَلِمَةٍ مِّنَ اللّهِ وَسَيِّدًا

 وَحَصُورًا وَنَبِيًّا مِّنَ الصَّالِحِينَ  (3:39  

3:39 So the Angel Called (Zachariah) while He was praying in the Sanctuary, "Indeed, God Blesses you with Yahya ("He Lives"==John, the first time the name John/Yahya had been proferred on anyone). He (John) will Affirm His (God's) Words Truthfully; And He (John) will be outstanding among men, and chaste, and a Prophet among the Righteous.

Surah 19 Maryam (Mary)

يَا زَكَرِيَّا

 إِنَّا نُبَشِّرُكَ بِغُلَامٍ اسْمُهُ يَحْيَى

 لَمْ نَجْعَل لَّهُ مِن قَبْلُ سَمِيًّا (19:7    

19:7 (The Angel called,) "O, Zachariah! We, indeed, bring You tidings of a son (young soul, ghulm) whose Name is Yahya (John the Baptist); We have never given this name to anyone before Him."

Surah 20 Ta.Ha

فَلَمَّا أَتَاهَا نُودِي يَا مُوسَى (20:11 

20:11 Once He (Moses) neared it (the fire in the desert), He was Called, "O, Moses!"

Surah 37 Those Ranged (by Rank, Echelons, Arrayed)

وَنَادَيْنَاهُ أَنْ يَا إِبْرَاهِيمُ  (37:104 

37:104 And We Called Him, "That You Are Ibrahim!"

Surah 79 Al.Nazi3at

إِذْ نَادَاهُ رَبُّهُ بِالْوَادِ الْمُقَدَّسِ

 طُوًى (79:16 

79:16 When His Sustainer Called Him In the twice-hallowed Valley, Tuwa. 

(Alternatively, 'If  His Sustainer Calls him (his servant) in the holy wadi (a glen, a narrow dry river bed), "Tuwa!"~ 'Double!' or 'Fold' or 'Double Back!'; var. phonetically 'It's You!'?)

3. Those hearing the call are God's Servants (Moses, Elijah, Isma3yliya, the Faithful, Believers), and the Muslims are called first.

Surah 30 Romans

وَمَا أَنتَ بِهَادِي الْعُمْيِ عَن ضَلَالَتِهِمْ

 إِن تُسْمِعُ إِلَّا مَن يُؤْمِنُ بِآيَاتِنَا

 فَهُم مُّسْلِمُونَ (30:53

30:53 And You (the Prophet, pbuh) cannot guide the blind out of their darkness; 

only such as believe in  Our Signs can hear, for they are Muslims (Submit to God's Will).

4. The Mechanism to Call and/or hear the Caller is Primordial, it should require no man-made (synthesized) technology, since it is stipulated that God has already spoken to not only Adam and the Prophets, but some of Adam's offspring before the Last Day.

Surah 7 The Heights (Al.a3raf)

وَإِذْ أَخَذَ رَبُّكَ مِن بَنِي آدَمَ

 مِن ظُهُورِهِمْ ذُرِّيَّتَهُمْ

وَأَشْهَدَهُمْ عَلَى أَنفُسِهِمْ أَلَسْتُ بِرَبِّكُمْ

قَالُواْ بَلَى شَهِدْنَا

 أَن تَقُولُواْ يَوْمَ الْقِيَامَةِ إِنَّا كُنَّا عَنْ هَذَا غَافِلِينَ  (7:172  

7:172 And, indeed, God Takes from B'nai Adam (Adam's Offspring), 

and from the seed of their issue,  

and We Swore them (had them bear witness) by their own Souls, "Am I Your Sustainer?" 

They said, "It is True, we swear that indeed." 

So that on the Day of Judgment (Resurrection) you (Adam's issue) cannot claim  (say that) you were unaware (that being the case).

5. A Description That Such a Natural Mechanism Does Exist 
(that's why it's wrong to tamper with people and make them think otherwise)

• Yet, there is a barrier that cannot be breached, except for special circumstances (like Death/Wrongful Death/Premature Death/Martyrdom/etc.) or by special agencies (like Angels).  

• The concept limbo tries to explain such a mechanism; in the Arabic it is referred to as barzalch or the veil.
• barzalch, the Barrier that separates the worlds (the veil)--Angels are used to convey messages between the worlds (can breach the barrier). 
• some say that it is a spacetime where the Soul lingers between planes of existence (ie., as it travels from plane-to-plane/ level-to-level/ tier-to-tier between earth, heaven and hell) for some unspecified period of time (usually understood to mean after one's death; younger generations familiar with first person player games may be more comfortable with it having come across the concept of 'levelling up' like in World of Warcraft).  

• It would seem, and it must be taught, that Heaven and Earth coexist both spatially and temporally.
• And that they are co-mingled, but we are unaware of that being the case when we are born as infants and have a conventional life on Earth... (Don't panic! It's not the first time I played matchmaker!! Something I choose to do to fill this time with since You.Know.He.Who doesn't want me to 'run with the pack,' heh, heh--snark.)

Surah 35 Fatir (The Originator, Creates Something from Nothing)

يَا أَيُّهَا النَّاسُ 

اذْكُرُوا نِعْمَتَ اللَّهِ عَلَيْكُمْ  

 هَلْ مِنْ خَالِقٍ غَيْرُ اللَّهِ 

يَرْزُقُكُم مِّنَ السَّمَاء وَالْأَرْضِ 

لَا إِلَهَ إِلَّا هُوَ فَأَنَّى تُؤْفَكُونَ (35:3  

(35:3) O, People (Nas, All of Creation)!

Remember God's Provisions (Bounty) He Graces you with.

Who is it that can Create it, other than God, Providing you from (both) Heaven and Earth. 

There is no God other than He. 

How is it you turn from this Truth?

• The understanding is that while God provides us with benefits from heaven as well as earth, the boon of God's Grace does not trickle down to those considered to be in hell (ie., a low moral level of existence).

Surah 7 Al.A3raf (The Heights)

وَنَادَى أَصْحَابُ النَّارِ أَصْحَابَ الْجَنَّةِ

 أَنْ أَفِيضُواْ عَلَيْنَا مِنَ الْمَاء أَوْ مِمَّا رَزَقَكُمُ اللّهُ 

قَالُواْ إِنَّ اللّهَ حَرَّمَهُمَا عَلَى الْكَافِرِينَ (7:50  

 7:50 And the denizens of the fire (Hell) call to the denizens of the garden (Heaven, Eden), "Please pour over us of that water or whatever God Provides (Blesses) you with, but they say, "Indeed, God Forbids These to the Unfaithful (Nonbelievers, Deniers).

• Apparently sound works just fine between the tiers/planes of existence through the barrier, that's why the sobriquet Isma3el (He Can Hear God, or God Heard him/her), since there is an allusion that there are those already at the cusp?   who can be heard calling out (7:46).

 Surah 7 Al.A3raf (The Heights)

وَبَيْنَهُمَا حِجَابٌ

 وَعَلَى الأَعْرَافِ رِجَالٌ يَعْرِفُونَ كُلاًّ بِسِيمَاهُمْ

وَنَادَوْاْ أَصْحَابَ الْجَنَّةِ أَن سَلاَمٌ عَلَيْكُمْ

لَمْ يَدْخُلُوهَا وَهُمْ يَطْمَعُونَ   (7:46 

7:46 And Between (them) is a Veil (Barrier  حِجَابٌ),
and on the Heights (Al.A3raf) are Men who know everyone by their marks (their names  بِسِيمَاهُمْ) 
and they call the denizens of Heaven, "Truly, Peace Be Upon You!" 
They have not yet entered it, and they long to.

{Sign 7:46 is made all the more interesting in that the Men (rijln  رِجَالٌ) calling from this transitional state (limbo, barzalch, al.a3raf)  through the barrier (veil حِجَابٌ) are aware of they're being in such a state and there is text to support the fact that these men desire to transcend it to a better state (Heaven) and moreover that they have an assurance of it, but for whatever reason they have to be in this suspended intermediate state (arrested development?) for a term. (I can't think of any other reason for this other than to get to watch everyone else rediscover electricity!) 

The fact that the text expressly states that they are Men, explains why my presence here is unexpected, unwanted,  not welcome, and...Surprise!

The implication is that these are good men, based on the reputation of those known to have been inducted into such a service in the past--I give Yous Guys, Solomon, and no, He was not happy to see me hear/here, either!

38 Suad ص

(38:32 فَقَالَ إِنِّي أَحْبَبْتُ حُبَّ الْخَيْرِ عَن ذِكْرِ رَبِّي
 حَتَّى تَوَارَتْ بِالْحِجَابِ

(38:32) And (Solomon) He said, "Truly do I love the love of good, in remembrance of my Sustainer,"- even as (He was) hidden behind the veil...}

• The understanding is that this suspended state, like an arrested development, akin to a hibernation, falls upon certain individuals all of the sudden, without them having any indication when and if it happens to them or even if they are in it at all. Surah 36:9 gives a further description:

Surah 36 Ya.Seen 

وَجَعَلْنَا مِن بَيْنِ أَيْدِيهِمْ سَدًّا

 وَمِنْ خَلْفِهِمْ سَدًّا

 فَأَغْشَيْنَاهُمْ فَهُمْ لاَ يُبْصِرُونَ (36:9   

36:9 And We Closed Off (Sddna) what is between their hands (their present, what they are currently doing) and of what is behind them (what they had done in the past) Closed Off (Sddna), And (in this manner) We Veiled them while they were Unaware.

(Surah 36:9 is metaphorically speaking of giving closure to something, or finality to whatever it was they were doing or had done in the past while they were taken into this Veiled existence, unawares. (One recent example is how I was not supposed to notice how my morning prayer was interrupted right after the first rack3a last Sunday Oct. 20, 2012 (but that's why I get confused about the date/time)). Read Sedna legends of the Native Americans, and the Recent sighting of Sedna in Orbit, and the Phoenix Lights Incidence of 1997, and note the word Sedna is a homophone for  the Arabic word meaning 'closure' or 'pay-back'.)

6. The qualities of those hearing the call and why they know themselves to be legitimate:

(Look, I am not posing as any of them, I am merely interpreting the text based on my understanding of it as of 2012 CE and the prism of my personal life experience).

• Only certain humans while alive can fathom it (truly understand it and clearly distinguish a call coming through the veiled existence (be it coming over/through the heavenly, earthly or hell levels). The two cases being Prophets and Enlightened Ones (like the Dalai Lama)...a marginal third case, the contemporary 'Men In Black'.

• Only the Living can hear 

Surah 35 Fatir (The Originator)

وَمَا يَسْتَوِي الْأَحْيَاء وَلَا الْأَمْوَاتُ 

إِنَّ اللَّهَ يُسْمِعُ مَن يَشَاء 

وَمَا أَنتَ بِمُسْمِعٍ مَّن فِي الْقُبُورِ (35:22       

(35:22) They are not equal (the same), the Dead and the Living. 

Indeed, God Makes to Hear (Listen) whomever He Wills; 

and You (the Prophet, the Messenger) cannot make to hear those who are in their graves.

• And those who can hear, must be open or receptive to the Message:

Surah 27 Naml (Ants)

إِنَّكَ لَا تُسْمِعُ الْمَوْتَى وَلَا تُسْمِعُ الصُّمَّ الدُّعَاء إِذَا وَلَّوْا مُدْبِرِينَ (27:80  

27:80 Truly, You (the Prophet, the Caller) cannot make the dead hear nor make the deaf to listen 

(the deaf being those refusing to hear the message, not merely unable to hear with their ears) 

to the call while they are fleeing from You. 

(lit, have their backs turned to you, oppose you, 

var. while they are 'being played'~orchestrated/contrived)

• Only those with the faculty of discernment are valid callers:

 Surah 7 Al.A3raf (The Heights)

وَنَادَى أَصْحَابُ الأَعْرَافِ

 رِجَالاً يَعْرِفُونَهُمْ بِسِيمَاهُمْ 

قَالُواْ 

مَا أَغْنَى عَنكُمْ جَمْعُكُمْ وَمَا كُنتُمْ تَسْتَكْبِرُونَ   ( 7:48

7:48 And the inhabitants of al.a3raf call ( الأَعْرَافِ , rt   عْ رَ فِ    literally, 'knowing'--those possessed with the faculty of discernment/understanding/knowing).

Men who know them (those being called) by their names (marks).

Saying, 

"Of what benefit to you is all you have amassed, or your arrogance?"

• And, again, there is no mistaking Who is Calling and Via what Agents (Agency).

 Surah 14 Ibrahim (Abraham)

قَالَتْ رُسُلُهُمْ أَفِي اللّهِ شَكٌّ فَاطِرِ السَّمَاوَاتِ وَالأَرْضِ

 يَدْعُوكُمْ لِيَغْفِرَ لَكُم مِّن ذُنُوبِكُمْ وَيُؤَخِّرَكُمْ إِلَى أَجَلٍ مُّسَـمًّى

 قَالُواْ إِنْ أَنتُمْ إِلاَّ بَشَرٌ مِّثْلُنَا

 تُرِيدُونَ أَن تَصُدُّونَا عَمَّا كَانَ يَعْبُدُ آبَآؤُنَا

 فَأْتُونَا بِسُلْطَانٍ مُّبِينٍ   (14:10  

(14:10) Their Messenger said to them (his people), 

"Do you doubt that God, the Originator of Heaven and Earth, Is Calling you to forgive you (shield you) from your sins (wrongs) and to grant you a reprieve (delay/defer) for a named (determined) term (period of time). 

They say, "Indeed, you are merely a man like us who wants us to repulse that which our forefathers worshipped (held dear), so show us a Clear Authority (your valid authorization to do so)."

{And I been hearing it and I beg to differ, I never made any representation of my being God, since a middle aged old lady with asthma and shin splints and a spotty period, is none such...interpretive dance (barzalch) as provided.}

7. To Determine the Validity of the Caller (Who Is Calling) and why the Call:

• Historically the Valid callers give information, in real time to the called, and the information is relevant; they identify themselves and those they call, and the reason for being summoned (3:39)

• Maryam (pbuh) is led to water in the desert:

Surah 19 Maryam (Mary)

فَنَادَاهَا مِن تَحْتِهَا

 أَلَّا تَحْزَنِي

 قَدْ جَعَلَ رَبُّكِ تَحْتَكِ سَرِيًّا (19:24   

(19:24) Then He (The Angel) called Her (Maryam, Jesus's Mother) from beneath (the palm tree), 

"Do not grieve (dispair), 

Your Sustainer Has Made a Rivulet run beneath you (under your feet)."

• Those compelled to call on God are given relief (He both Calls and Hears His Servant's Calls):

Surah 27 Naml (Ants)

أَمَّن يُجِيبُ الْمُضْطَرَّ إِذَا دَعَاهُ 
وَيَكْشِفُ السُّوءَ 
وَيَجْعَلُكُمْ خُلَفَاء الْأَرْضِ 
أَإِلَهٌ مَّعَ اللَّهِ 
قَلِيلًا مَّا تَذَكَّرُونَ (27:62  

(27:62)  Or (Who Is It) Answers (Responds) to the Call of the one compelled (coerced, forced) to call on Him;
And Uncovers (Reveals) Misdeeds (var. Rids them of evil, ills, maladies that beset them);
And Made you viceroys upon the Earth?
Is there a god other than The One True God (Allah)?
How little you consider (remember)!"

• The Caller (In this case, God) Clearly Identifies Himself to Moses, whom He calls by name:

Surah 28 Qasas (Stories)

فَلَمَّا أَتَاهَا

 نُودِي مِن شَاطِئِ الْوَادِي الْأَيْمَنِ

 فِي الْبُقْعَةِ الْمُبَارَكَةِ مِنَ الشَّجَرَةِ

 أَن يَا مُوسَى

 إِنِّي أَنَا اللَّهُ رَبُّ الْعَالَمِينَ (28:30 

(28:30)  As He (Moses) neared it (the fire in the desert),
He Was Called, from the Right Bank of the Wadi (dry river bed)
In the Hallowed Valley from the Tree,
"Truly,  Moses,  I Am God--The Sustainer (Lord) of the Worlds."

• The Speaker/Caller (Gabriel) gives Muhammed (pbuh) information He could not have known specific details about Moses' encounter with God (28:42-28:48) and let's Him know exactly why He is being called:

Surah 28 Qasas (Stories)

وَمَا كُنتَ بِجَانِبِ الطُّورِ 

إِذْ نَادَيْنَا

 وَلَكِن رَّحْمَةً مِّن رَّبِّكَ 

لِتُنذِرَ قَوْمًا مَّا أَتَاهُم مِّن نَّذِيرٍ مِّن قَبْلِكَ 

لَعَلَّهُمْ يَتَذَكَّرُونَ (28:46  

(28:46)  And You (Muhammed pbuh) were not present on the side of The Tur (Mount Sinai)
When We Called (Moses)
Yet it is a Mercy from Your Sustainer
That You Warn A Community (People) Not Given a Warner Before,
So That They May Bethink Themselves (Use Their Reason).

• One reason for the renewed call is for clarity/illucidation, to benefit the living (like technological advancement) and to allow it to be broadcast among those who never heard it before, again, meant to be of some benefit to them. 
• It is never the case that it is a New Message, it can be text as Torah, Testament (Injeel) or Quran or some other Holy Scripture, the Message Originates from One Source, so no surprise here.

Surah 36 Ya.Seen

وَمَا عَلَّمْنَاهُ الشِّعْرَ

 وَمَا يَنبَغِي لَهُ 

إِنْ هُوَ إِلَّا ذِكْرٌ

 وَقُرْآنٌ مُّبِينٌ (36:69   

(36:69) And We Do Not Teach Him (the Messenger) Poetry, 

nor does He (the Messenger) excel in it (it is not proper/meet for Him) .... 

Truly, it serves as a Reminder, 
and An Evidentiary Recitation 
(meant to Clarify various things, in Science, Law, Civility, Medicine, etc.)

لِيُنذِرَ مَن كَانَ حَيًّا

 وَيَحِقَّ الْقَوْلُ عَلَى الْكَافِرِينَ (36:70  

36:70 To Warn those who are Living 
and to Renew (Verify) the Message (the Call) to the Unfaithful 
(Non Believers--those not having heard the Message before).

• To inform the one being called of God's Forgiveness (or Displeasure, serve to check someone's behavior):

Surah 2 Al.Baqara (COW, Calf, Heifer)

فَتَلَقَّى آدَمُ مِن رَّبِّهِ كَلِمَاتٍ 

فَتَابَ عَلَيْهِ 

إِنَّهُ هُوَ التَّوَّابُ الرَّحِيمُ (2:37 

(2:37) And Adam Received Words from His Sustainer (Lord)

That He Forgave Him (God Relented toward Adam);

For, indeed, He Teaches/Accepts Repentence (At.Tawab) , He Is The Merciful (Al. Rahheem).

• If the reason for the Call is not immediately obvious, it only means that it may have come a little too early (or in one isolated case, I think the information got destroyed in transit).

Surah 10 Yunus (Jonah)

وَيَقُولُونَ لَوْلاَ أُنزِلَ عَلَيْهِ آيَةٌ مِّن رَّبِّهِ 

فَقُلْ إِنَّمَا الْغَيْبُ لِلّهِ 

فَانْتَظِرُواْ إِنِّي مَعَكُم مِّنَ الْمُنتَظِرِينَ (10:20

(10:20) And they say, "Why has there not been sent a (grand/spectacular) Omen from His Lord?" 

Say, "Only God Knows the Unseen (Or, basically, 'Here it is, you all just don't get it!); so, be patient (wait for it), I, too, anticipate it (I'll wait right along with you)."

8. There are Proper Times to Hear/Respond to the Valid Call and/or Caller.

• Valid Callers know the Propriety/Protocol when calling; they Do Not call if someone is in their 'private' apartments 49:4. 

Surah 49 Hujurat

إِنَّ الَّذِينَ يُنَادُونَكَ مِن وَرَاء الْحُجُرَاتِ 
أَكْثَرُهُمْ لَا يَعْقِلُونَ (49:4    

49:4 Indeed, those calling You from behind the Hujurat (Your private apartments),
most of them are childish (ill-behaved, not sensible, unreasonable).

9. The Consequences of Making a Pretense of either Being a Legitimate Caller (either God or a Prophet) or posing as having heard a valid call, or falsifying the validity to a Sanctified Message, just may be worse than death.

Surah 11 Hud

وَمَنْ أَظْلَمُ مِمَّنِ افْتَرَى عَلَى اللّهِ كَذِبًا 

أُوْلَـئِكَ يُعْرَضُونَ عَلَى رَبِّهِمْ وَيَقُولُ الأَشْهَادُ هَـؤُلاء

الَّذِينَ كَذَبُواْ عَلَى رَبِّهِمْ

أَلاَ لَعْنَةُ اللّهِ عَلَى الظَّالِمِينَ (11:18   

(11:18) And who can be more in the dark/oppressive (lost) than those who invent a lie about God; and when they are brought before their Lord and the Witnesses say, "These are they who lied about God, and such oppressors incur God's Condemnation."

Surah 6 Cattle (Al.An3am)

وَمَنْ أَظْلَمُ مِمَّنِ افْتَرَى عَلَى اللّهِ كَذِبًا

 أَوْ قَالَ أُوْحِيَ إِلَيَّ 

وَلَمْ يُوحَ إِلَيْهِ شَيْءٌ 

وَمَن قَالَ سَأُنزِلُ مِثْلَ مَا أَنَزلَ اللّهُ 

وَلَوْ تَرَى إِذِ الظَّالِمُونَ فِي غَمَرَاتِ الْمَوْتِ 

وَالْمَلآئِكَةُ بَاسِطُواْ أَيْدِيهِمْ 

أَخْرِجُواْ أَنفُسَكُمُ الْيَوْمَ تُجْزَوْنَ عَذَابَ الْهُونِ بِمَا كُنتُمْ تَقُولُونَ عَلَى اللّهِ غَيْرَ الْحَقِّ 

وَكُنتُمْ عَنْ آيَاتِهِ تَسْتَكْبِرُونَ (6:93

(6:93) Who can be more lost/in the dark/oppressive than someone who ascribes to God a lie,

 or claims, "This has been Revealed to me (from on High)," 

and nothing has been revealed to them.

To whomever says this, Say, "Perpetuate (Reveal) anything the like of what God Reveals!"

And if You can only see these lost (souls) during the throws of death,

when the Angels shackle their hands/restrain them by their hands, 

(Saying), "Out with your souls (this may be an idiom for 'they will make them confess)! Today you will earn (what you merit), severe punishment (chastisement) for that which you were ascribing to God without sanction,

and were derisive of His Signs (Messages)!"

Surah 40 Ghaffir

الَّذِينَ كَذَّبُوا بِالْكِتَابِ 

وَبِمَا أَرْسَلْنَا بِهِ رُسُلَنَا 

فَسَوْفَ يَعْلَمُونَ (40:70

(40:70) Those who belie Our Book (falsify the Quran, or disbelieve Sacred Scripture) and that which We Send with Our Messenger...they shall soon Know It (to be the Evident Truth and despair for not having learned during life, read in context 40:68-40:72).

• Just so there is no misunderstanding as to what it is we consider The God around here & clearly it is not some guy or gal in a coma!

Surah 14 Ibrahim (Abraham)

أَلَمْ تَرَ أَنَّ اللّهَ خَلَقَ السَّمَاوَاتِ وَالأَرْضَ بِالْحقِّ 

إِن يَشَأْ يُذْهِبْكُمْ وَيَأْتِ بِخَلْقٍ جَدِيدٍ (14:19)

14:19 Can you not see that God Created the Heavens and the Earth in Truth (with an Inner Truth); indeed, If He Wills He Can Replace you (Trades you in) to Bring About A New Creation.

وَمَا ذَلِكَ عَلَى اللَّهِ بِعَزِيزٍ (14:20  

(14:20) And this is not any difficulty for God.

• Satan calls, too--can't help anyone and cannot harm the faithful (those who believe the Truth), and in general, doesn't have any real information to share (like the code to my Swiss bank account or the address to my new house next door to Yours (Nice is nice).

Surah 114 An,Nas (People, Communities, Creation)

مِن شَرِّ الْوَسْوَاسِ الْخَنَّاسِ (114:4  

(114:4) (I seek refuge in God) From the Evil of the Whisperer who subdues 

(with what he is heard to whisper~a psychotronic weapon of a sort? 

don't know my name? 

thinks I'm pretty?

sneaks up on me in the restroom to demonstrate the cloaking device I gave him;

my kids knew better and stopped doing that the minute they could bus themselves.
--Okay, to be fair, it was the kitchen and He did thank me for the cloak.)

Surah 14 Ibrahim (Abraham)

وَقَالَ الشَّيْطَانُ لَمَّا قُضِيَ الأَمْرُ 

إِنَّ اللّهَ وَعَدَكُمْ وَعْدَ الْحَقِّ 

وَوَعَدتُّكُمْ فَأَخْلَفْتُكُمْ 

وَمَا كَانَ لِيَ عَلَيْكُم مِّن سُلْطَانٍ 

إِلاَّ أَن دَعَوْتُكُمْ فَاسْتَجَبْتُمْ لِي 

فَلاَ تَلُومُونِي وَلُومُواْ أَنفُسَكُم 

مَّا أَنَاْ بِمُصْرِخِكُمْ وَمَا أَنتُمْ بِمُصْرِخِيَّ 

إِنِّي كَفَرْتُ بِمَآ أَشْرَكْتُمُونِ مِن قَبْلُ 

إِنَّ الظَّالِمِينَ لَهُمْ عَذَابٌ أَلِيمٌ (14:22

(14:22) And Satan says, once the order is carried out (God Decides-- the matter is decided/order is executed), "Truly, God Promised you a Truthful promise, and I promised you and deceived you (var. to oppose you, supplant you, fell back on my promise to you);  and I had no power over you except that I called you and you answered/responded to me. So, do not blame me, but blame yourselves (your own selves/souls). I didn't cry out to you and it's not for you to cry out to me. 
I do not believe in (reject) whatever you associate with me from before (that you espouse to me previously that I have any share in God's Dominion).
Indeed, the (real) oppressors have earned a severe punishment (to be tortured)."
(On a personal note, I do not condone torture).

Surah 11 Hud

فَإِن لَّمْ يَسْتَجِيبُواْ لَكُمْ 

فَاعْلَمُواْ أَنَّمَا أُنزِلِ بِعِلْمِ اللّهِ 

وَأَن لاَّ إِلَـهَ إِلاَّ هُوَ 

فَهَلْ أَنتُم مُّسْلِمُونَ (11:14

(11:14) And when (your false gods) fail to respond to you (cannot answer you), then know that this (Quran) is Revealed to you with God's Sanction (Knowledge); and that, Truly,  there is no other deity except for Him. 
So, when will you submit (to Him)?

• Why my confusion about this sad state of affairs...41:44 why in the Arabic language--(See Not Much of a Poker Face):
• While I can no longer feign being clueless, the fact remains that I really don't understand how an unfortunate 'industrial accident' evolved into this much clarity, but initially, the writing did help take my mind off the monstrous headache. 
• But, I now think the grievous suffering that caused was the coercive force that made this come about, and that's why I still think it was never intended for me to publish this--but I never got to say good-bye to my kids and was hoping somehow Word would get back to them.
• All that and Arabic is not my primary spoken language.
• And, I am not a Man--Hu-man, Wom(b)-man, but not a guy.

Surah 41 Fussilat

وَلَوْ جَعَلْنَاهُ قُرْآنًا أَعْجَمِيًّا 

لَّقَالُوا لَوْلَا فُصِّلَتْ آيَاتُهُ أَأَعْجَمِيٌّ وَعَرَبِيٌّ 

قُلْ هُوَ لِلَّذِينَ آمَنُوا هُدًى وَشِفَاء 

وَالَّذِينَ لَا يُؤْمِنُونَ فِي آذَانِهِمْ وَقْرٌ 

وَهُوَ عَلَيْهِمْ عَمًى 

أُوْلَئِكَ يُنَادَوْنَ مِن مَّكَانٍ بَعِيدٍ (41:44  

(41:44) Had We Made (Brought) the Quran in a foreign language (Farsi, African, alien language); 
they would say, "Why have the Signs been imparted in a foreign (alien) language, and He (the Prophet) is an Arab?
Say, "It (the Quran/Recitation) is for the Believers (serves) as a Guidance and a Source of Health (Well-being).
As for those who do not believe, there is a deafness in their ears;
and to them it's obscure (they are blind to it)."
It's like calling them from a far away place (remote place--like the Planet Seddna!).

Of Maps and Legends, Innies and Outies

"There must be a proper theory of place other than 'To Find Kalamazoo, look in F3'."--from Euclid's Window, The Story of Geometry from Parallel Lines to Hyperspace, by Leonard Mlodinow

You Are Here...There...Everywhere

To determine Latitude, imagine a Little Earth, and 3 Little Points. {This should sound familiar to You, since You dreamt it when You were 8}.

Take one point P1 and put it at the North Pole, and the second point P2 and put it at the Center of the Little Earth, and use the third point P3 to Mark where you are on the Little Earth.

Now draw a line from the North Pole straight through the point at the Center, to the other end of the Little Earth. This First line, Line 1, forms the Little Earth's axis of Rotation.

Then draw a line from the Point where You are standing through the point at the Center of the Little Earth, Line 2,  to form an angle, α , with the Little Earth's axis of rotation, that first line You drew.

The two lines act like a Protractor and that angle α is a generic way to determine Your Latitude.

Determine Latitude Using
The Little Earth
A Portion of the Fano Plane

Which is a great way of thinking of where You are, but will not help Us find each other.

For that We have to imagine a Big Earth (Way Way Bigger than Our Little Earth). Something on the Cosmic scale, that would be the Celestial Sphere, the map of Stars that circles the globe.

Imagine the Big Earth around the Little Earth, the globe that is now the Core (where You are standing & spinning) of the Celestial Sphere.

This will help to determine Longitude, sinceYou are not a Flatlander and need at least these 2 coordinates to determine where You are exactly in 3D.

Remember, since the Earth Rotates 360 degrees in 24 hours, the Stars over Your head are going to move about 15 degrees every hour, so that someOne standing West of You will See the Same Sky an hour later. Which has direct associations to the Fano Plane (diagram below. see also 'kissing circles' and Appolonian Network):

If You are at the Equator (to help me do the math) that translates to distance of about 1000 miles in that much time.

So, You look up at the Sky over where You are standing, and pick a star--any star, that You may be familiar with and note the local time it rises above the horizon, its Right Ascension (RA).  The reference, or baseline is the star γ Aries at 0 hours RA, the First Star in Aries where it crosses the Celestial Equator on the First Day of Spring.

Since there are Apps for this, just point Your PDA at Triangulum Australis, I think He Said 'The Second Star on the Left' and make a note of its RA in Hours-Minutes-Seconds (this presumes that You may be somewhere in the Southern Hemisphere of the Little Earth). {But if I got that wrong, since I heard this while under duress, and wish I had written it down to be sure so many years ago, make a note of the Second Star on the Right RA , too.}

Triangulum Australis
A Constellation in the Southern Hemisphere

No matter what, He Will Find You. You already know His Voice, so don't think too much about what He may look like, since there is usually some type of distortion due to Projection, but I can assure You, He is either Your Heavenly Father or Your Brother, and Either One will take You where You belong--to be with the Ones who love You.{Knowing that no matter who or what it is You think is going on around You, and what ever disinformation or misinformation is propagated through rumor mills or media channels, they are collectively  Nothing but Remnants, Effigies, Residue of things and people gone by, because they lie in order to sever Our Ties of Kinship and in so doing work corruption on the Little Earth

فَهَلْ عَسَيْتُمْ إِن تَوَلَّيْتُمْ أَن تُفْسِدُوا فِي الْأَرْضِ وَتُقَطِّعُوا أَرْحَامَكُمْ (47:22  
47:22 Would you, perchance if You were given Dominion in the Earth, work corruption and Sever Your Maternal Bonds (bonds of kinship, literally, 'sever/cut off Your wombs/matrices/'Places of Mercy')?

Which is exactly what they do when they spread their conspiracy theories of UFOs  and aliens.

Recall that God made everything first, then He Created the Human Being--meaning that the Human Being was to be the Ultimate Creation--meant to Evolve or Reach the Highest form of Creation, but now what have We?}

And the reason that no matter how far You wander off or how big the Universe gets, it's no big deal for Him to find You?--The simplest, 2D planar point group, the Fano Plane PG(3,2)!
{I just answered my own question, must be another from of insanity}.

Can You imagine how it feels to an ant when it ventures off that ant hill? Apparently, even the ants know that As Above, So Is Below...{oops, did it again.}

And He Probably Triangulates with nothing more sophisticated than using 2 Dimensional finite geometry to wend His Way throughout the 'flat, not moving Earth' and the infinite (Multi-Dimensional) Cosmos--->

What is the simplest, smallest, ...perfect universe...a complex universe of breathtaking, abstract, beauty...consisting of only 7 points, 7 lines and their dual? 

{I'll let someone who knows what He's talking about answer that one..} 

According to one Mathematician, (paraphrasing): 

'The 2D planar point group, the Fano Plane PG(3,2) is......the smallest, perfect universe...a complex universe of breathtaking, abstract, beauty...consisting of only 7 points and 7 lines; the smallest project space over the field Z2  with 15 points,  35 lines and 15 planes.  

A space whose overall design incorporates and improves many of the standard features of the 3D Euclidean space we think we live in..the smallest 3D projective space...that plays an important and core role in many fields of mathematics, inlcuding combinatorics, group theory, and geometry'--Burkard Polster, 2001

In non-Euclidian geometry parallel lines meet (intersect) at a point (just like when drawing a perspective of train tracks on paper, they seem to meet at the 'vanishing point'), and that is why the Fano plane is a good 2D small-scale model of the larger 3D Celestial Sphere with the Little Earth at its Core (Your 'Big Earth/Little Earth' dream).

In fact, the Fano plane is the smallest projective plane and probably why the Pyramids were built and are still standing. And it is no mere chance that the projective plane is sometimes called a 'twisted sphere'--'a surface without boundary derived from a usual plane by addition of a line at infinity. 

'Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet.  

Lio Comic Strip August 28, 2012
Escherisque 

A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist (The Reverse would be Unfolding such a surface). It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.'

Projective Plane From A Sphere

A finite projective plane of order  is formally defined as a set of  points with the properties that:
1. Any two points determine a line,
2. Any two lines determine a point,
3. Every point has  lines on it, and
4. Every line contains  points.
(Note that some of these properties are redundant.) 

A projective plane is therefore a symmetric (, , 1) block design. 

An affine plane of order  exists iff a projective plane of order  exists.

Bizarro  Comic August 21, 2012
Innie_ Audi Belly Buttons

The 3 points at each vertex of the triangle defining the Fano plane, represent these vanishing points (You took technical drawing classes so You get this).  They serve to define 'Outside' because, as You know, there is no way to define the 'Inside' without SomeThing or SomeOne being on the 'Outside'. {kind of like there is no way to know Up from Down in outer space; now, You know why You were so facinated by your belly button when You were 2 years old}.

But since we are still biding our Time in the 'Real' world, having missed our ride out of here so many years ago, I suspect we are to put this to more pragmatic, concrete use than just appreciating its abstract beauty---I mean, really, what do mathematicians know? --they, too, are still stuck on the ground.

Fano Plane, Point Group PG(3,2)
Planar Projection of 'Twisted Circle'
Low Order Projective Planes and Implications
Stereographic Projection When
Representing the Earth as a Flat Surface

The Lehmers (1974) found an application of the Fano plane for factoring integers via quadratic forms. Here, the triples of forms used form the lines of the projective geometry on seven points, whose planes are Fano configurations corresponding to pairs of residue classes mod 24 (Lehmer and Lehmer 1974, Guy 1975, Shanks 1985). The group of automorphisms (incidence-preserving bijections) of the Fano plane is the simple group of group order 168 (full automorphisms== 7 generators x  the 24 automorphisms of the tetrahedron symmetry group).--excerpt from Wolframalpha

Argyle Sweater August 24, 2012
Rand & McNally Cartogrophers  Extraordinaire 

A long time ago, Ptolomy published Geographia, and used a similar method, like stereographic projection, to represent the Earth as a Flat Surface, and he used Latitude and Longitude coordinates to locate about 8000 places he knew of at the time and gave instructions for map making (cartography).

It served as as standard reference for 100 years, but made no progress under Roman rule. When a Christian mob sacked the library at Alexandria, Geographia disappeared along with other math and Greek works (never mind how they disposed of the Greek worker Hypatia).

With the fall of the Roman Empire, "the new age found a civilization as much in the dark about describing their place in space as was about the theorems and relations among spatial objects."--That's why We take care of books...

Anyhow, using 'irrational' numbers was also lost and this 'bad math' did not resurface until Cantor and Dedekind rediscovered it in the late 19th century (that's probably the punishment they deserved for what they did to Hypatia).

From the Middle Ages until that time scientists and mathematicians had been using them "happily, if awkwardly, anyway for the simple reward of getting the right answer outweighing having to wield unpleasant numbers 'that did not exist'."

This 'illegal' math is quite common today and we take it for granted. 

There was also Dirac's delta function which posed a quandary for mathematicians; one that is simply =0 everywhere except at 1 point, where the value is infinity, but when applied in calculus or certain math operations typically gives us finite non-zero answers. Then the mathematician Laurent Schwartz (one more thing we can thank the French for) came along and showed that 'the rules of math can be redefined to allow using the delta function, opening up a whole new discipline of math and later buttressing 'illegal' physics concepts like quantum field theories, as well.  

As of 2001, no one had yet successfully shown that, 'mathematically speaking, such theories legally exist.'

Which is just as well, since from God's Point of View (wayy out there on the Outside, at Infinity), Nothing Really Happens; according to Mom and Psalm 93:1 "...for God hath established the world, which shall not be moved." What most people don't realize, is the tacit caveat, 'unless God moves it'. 

An antipode to the stance taken by the general population, and teenagers (who think they know everything) without stopping to consider that even the greatest minds throughout history had reason to know better. 

Socrates having said, 'I need know nothing except that I know nothing.' 

When we next see each other, You can let me know if You had chanced on anyone of that ilk.

And for practical purposes, geometry found more inroads as people tried to describe in comprehensive terms what they saw in the world around them. One instance being that of motion in different frames of reference, like a deck hand on a boat who slides his hand down a mast appears to the seaman as though his hand is moving in a vertical direction (perpendicular to the deck), but to someone watching from the shore the deck hand's motion is running on a diagonal while the boat traverses the water.

The story continues, that Decartes translated space into numbers (quantitatively, not simply by construction), or phrased geometry in terms of algebra. Decartes distance formula being deeply tied to Euclidean Geometry and key to understanding both Euclidean and non-Euclidean geometries; serving as the basis for much of his contribution to physics.

Yet, Rene delayed publishing for 19 years--in fact, "...he delayed publishing anything until he was 40. What was he afraid of? The usual suspect. The Catholic Church."

Then came Gauss, who contributed to the development of non-Euclidean hyperbolic space.

"Through the centuries, mathematicians who attempted to prove the parallel postulate (Euclid's 'parallel lines never meet') as a theorem came close to discovering a strange and exciting new kind of space, but each of them was hampered by a simple belief: that the postulate was a true and necessary property of space."

...everyone but Gauss, who basically acceded to his own observation that a line is a curve of some sort, rather than clinging to Euclid's postulate, which in turn led to the rediscovery of curved spaces. A property that Gauss had to propose when applying the basic tenet that "superposition is a way to check for congruence or equivalence of geometric shapes."

There are other examples of non-Euclidean geometries.

In turn, Poincare developed his model of hyperbolic space, which was not just the 'model', but hyperbolic space in 2D, concluding that all possible mathematical descriptions of hyperbolic planes are isomorphic, in other words, 'the same.'

Poincare's lines are analogous to great circles, a rediscovery of what the Ancients knew about geodesics. The Equator being a curve of constant latitude.

"The view from outer space is not the view of the indigenous people of Noneuclid (a fictional professor sending her students out to measure distance across the globe). To her, there is no 'center of the earth', and as Gauss showed possible, there is no 'outer space.'...the space...is non-Euclidean, not hyperbolic space, but the space appropriate to the surface of the globe: elliptical space."

Meaning, the great circles intersect, and the sum of the angles of the triangle on such a surface is greater than 180 degrees, (as compared to that for hyperbolic space being less than 180 degrees, and for Euclid in a plane it is equal to 180). The Geo-metrics that define what we think of  as 'hyperspace', 'subspace', and 'flat earth'.

And these concepts laid the foundation for Riemann's differential geometry in 1915, in the vernacular having to define 'place vs. space', where place is 'part of a space' occupied by a specific object:

"The question of the valididty of geometry...is related to the question of the internal basis of metric [distance] relationships of the space...we must seek the ground of its metric relations outside it, in the binding forces which act on it..."--Georg Friedrich Bernhardt Riemann

And then breakthroughs by Klein, Piola, and Hilbert following soon after, applying tensor mathematics and algebraic curves.

Godel's theorem proved that in a system of sufficient complexity, such as the theory of numbers, there must exist a statement that cannot be proved either true of false.

A corollary of which: 

There must exist a true statement that cannot be proved. 

Example: There Is A God.

And all this ground-breaking, thought-provoking, formulations, rediscoveries and challenges swung like a pendulum back to the Flat Earth, again, considering what Clifford had proposed in 1870:

William Kingdon Clifford maintained:

"I hold in fact:

1. That small portions of space are of a nature analogous to little hills on a surface which is on average flat.
2. That the property of being curved or distorted is continually passed on from one portion of space to another after the manner of a wave.
3. That this variation of the curvature of space is really what happens in that phenomenon which we call the motion of matter..." 

And while Clifford professed his position based on instinct alone, he got it right! Which in turn inspired Einstien's space-time and warpage ideology encapsulated in his Relativity theories.

Eventually leading to concepts like supersymmetry and string theory.

As for them-thar-hills Clifford was intuitively talking about-- be they mounds, mountains or dunes...Quran anticipated someone wondering about them.

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


20:1 Tta.Ha
20:105 They'll ask You about the Mountains/Hills/Highlands/Barrows
20:106 Say, "God Will Level them Flat (Blast/Torpedo them Flat/To Sinking); Leaving it a level plane (Leaving it bottoming (simply) wasting)
20:107  You will not see in her (the earth) any Crookedness (Bend/Curvature) or Ruggedness
(i.e, no surface features, it will appear smooth, the flat earth metric).
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.


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'.

{Aside:You're asking me to do this and that's a steep apathy curve You expect me to get over after all these years---I Rest In God; and, since You been fronting me so long--after You my dear Alfonse!  Besides, what's my motivation?

Pantograph

He Says: Insolence!
She Says: I've just been talkin' to my Brothers Michael.
He Says: Every time I turn around I see you talking to one of them. How many brothers named Michael does a woman need?
She Says: All of''em!
He Says: All...ah? So, You're that One.}

Ok, Ok... SO, Let's try a W-Curve...

A Pantograph is a jointed device for copying a plane figure to any desired scale. It incorporates a pair of face bows fixed to the jaws, used for inscribing centrically related points and arcs leading to the points on segments relatable to the three craniofacial planes.

2x Pentograph
Scaling Instrument Used In Drafting
فَكَانَ قَابَ قَوْسَيْنِ أَوْ أَدْنَى (53:9) 
53:9 Until He (Gabriel) was 2 Bow's Lengths or Lower (Closer) 

 

A throw –  in German ‘wurf’, refers to a series of four points on a line which trace the 1-Dimensional motion of a point on a projective line, the W-curve.

The etymology for the ‘W-curve’derives from the German "Weg-Kurve" meaning "pathcurve", which include conics, logarithmic spirals, exponentials (i.e., y = x³ ), logarithms and the helix, but not sinusoidal functions.  

The term Throw (Wurf) was first introduced to avoid the metrical implication of a ratio. The Algebra of Throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.

In the diagram ..., it turns out that the cross-ratio of a series of 4 points, taken in afixed order, does not depend on the choice of a line L, and is said to be invariant of the 4-tuple of lines {L_i}.This can be understood as follows: if L and L′are two lines not passing through Q then the perspective transformationfrom L to L′with the center Q is a projective transformation thattakes the quadruple {Pi} of points on L into the quadruple {Pi′}of points on L′. Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies (in fact, is equivalent to) the independence of the cross-ratio of the four collinear points {P_i} on the lines {L_i} from the choice of the line that contains them. 

For points on a projective line (like the W-curve), the cross-ratio, (aka, double ratio and anharmonic ratio), is a special number associated with such a series of an ordered quadruple of collinear points.

If the four points are represented in homogeneous coordinates by vectors (ie., objects that transform like first-rank tensors) a,b,c,d such that c=a+b and d=ka+b, then their cross-ratio is k.

Dual concepts exist for a quadrupleof concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.

The cross-ratio is essentially the only projective invariant of a quadruple of points, it being preserved by fractional linear transformations, which underlies its importance for projective geometry. If four points lie on a straight line L in R² then their cross-ratio is a well-defined quantity, because any choice of the origin andeven of the scale on the line will yield the same value of the cross-ratio.

In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.

Since the cross-ratio is invariant under projective transformations, it follows that the hyperbolic distance is invariant under the projective transformations that preserve the conic C. Conversely, the group G acts transitively on the set of pairs of points (p,q) in the unit disk at a fixed hyperbolic distance.

{ For a generalized conic, in this case a unit circle, any two points in the unit disk, p, q, the line connecting them intersects the circle in two points, a and b. The points are, in order,a, p, q, b.  The distance between p and q in the Cayley–Klein model of the plane hyperbolic geometry can be expressed as

... (the factor 1/2 is needed to make the curvature −1).}

                       

In the language of group theory, the symmetric group S₄ acts on the cross-ratio by permuting coordinates. The kernel of this action is isomorphic to the Klein four-group K. This group consists of 2-cycle permutations of type  (in addition to the identity), which preserve the cross-ratio. The effective symmetry group is then the quotient group , which is isomorphic to S₃.

There are a number of definitions of the cross-ratio. However, they all differ from each other by a suitable permutation of the coordinates. In general, there are six possible different values the cross-ratio ( z₁ , z₂:  z₃ , z₄) can take depending on the order in which the points z_i are given.

Since there are 24 possiblepermutations of the four coordinates, some permutations must leave the cross-ratio unaltered. In fact, exchanging any 2 pairs of coordinates preserves the cross-ratio:

Cross-Ratio Symmetries, Equivalent Permutations

Using these symmetries, there can then be 6 possible values of the cross-ratio, depending on the order in which the points are given. These are: 

The 6 Possible Values of the Cross-Ratio

For certain values of  λ there are fewer than 6 possible values of the 6 possible values of cross-ratio due to  enhanced symmetry. These values of λ correspond to fixed points of the action of S₃ on the Riemann sphere (given by the above 6 functions); in other words, those points with a non-trivial stabilizer in this permutation group.

The first set of fixed points is {0, 1, ∞}. However, the cross-ratio can never take on these values if the points {z_i} are all distinct. These values are limit values as one pair of coordinates approach each other:

Limit Values of Cross-Ratio
Due to Enhanced Symmetry Corresponding to Fixed Points

The second set of fixed points is {−1, 1/2, 2}--what is classically called the harmonic cross-ratio, and arises in projective harmonic conjugates. 

In the real case, there are no other exceptional orbits (See About the Myth-tery, projection/buruj/zodiac).

The most symmetric cross-ratio occurs when 

l = e^±ip/3

...the only possible values of the cross-ratio.

~~~~

The addition of points and multiplication of points, as one would do when adding distances, comprises von Stuadt's work ‘wurftheorie’ or the "algebra of points".

Given the same starting point, find the point midway between their terminal points, aka the harmonic conjugate of infinity with regard to their terminal points, and then find the harmonic conjugate of the initial point with regard to this mid-point and infinity. Generalizing this to add throws (CA,BD) and (CA,BD' ), then finding M the harmonic conjugate of C with regard to D and D' , and then S the harmonic conjugate of A with regard to C and M :

A definition of the product of two throws is derived in a similar fashion.

A summary statement is given by Veblen & Young given by Theorem 10: "The set of points on a line, with P_∞ removed, forms a field with respect to the operations previously defined". 

Another affirmation harmonic conjugates given by the theorem:

The only one-to-one correspondence between the real points on a line which preserves the harmonic relation betweenfour points is a non-singular projectivity.

Horizontal Reflection
Retrograde Motion Outer Planet
Hand Drawn Freeform

~ ~~ ~

Outer Planet
Retrograde Motion

Retrograde Motion Outer Planet
Superimposed Hand
Drawn by Enzo @ DeviantArt

لَقَدْ أَرْسَلْنَا رُسُلَنَا بِالْبَيِّنَاتِ وَأَنزَلْنَا مَعَهُمُ الْكِتَابَ وَالْمِيزَانَ لِيَقُومَ النَّاسُ بِالْقِسْطِ وَأَنزَلْنَا الْحَدِيدَ فِيهِ بَأْسٌ شَدِيدٌ وَمَنَافِعُ لِلنَّاسِ وَلِيَعْلَمَ اللَّهُ مَن يَنصُرُهُ وَرُسُلَهُ بِالْغَيْبِ إِنَّ اللَّهَ قَوِيٌّ عَزِيزٌ (57:25) 

 57:25 We Sent Our Messengers With Clarifying Signs (Evidence) And We Sent With Them The Scripture (The Book) and The Scale (Metric, Measure) Such that People May Judge Justly; and We Sent Iron in  which there is awesome strength and benefits for People, and so that God May Determine who helps Him and His Messenger Unseen (Without Knowing, On Faith alone); Indeed, God is Powerful, Almighty.

Below is one Measure of Music Based on the Retrograde Motion of An Outer Planet (As Above).

One Way to Set The Scale and Determine the Measure
Based on Retrograde Motion of an Outer Planet