Showing posts with label Meaning. Show all posts
Showing posts with label Meaning. Show all posts

16.5.12

Humanity

"Men are so necessarily mad, that not to be mad would amount to another form of madness." --Blaise Pascal

Since the Space Ship Building Mission is temporarily suspended until otherwise informed, and yet still feeling the pull-- out of force of habit-- to visit and update this blog, I have gone through and peppered it with relevant quotes having to do with Madness and Insanity.

Note: 'Peppering' implies having excercised some level of restraint since I could just as easily have Salted it; but given the News that Your WIFE Is A Lunatic-- not that I suffer from lunacy but have come to enjoy it very much--I thought YOU would appreciate the nexus.

BTW...Insanity in English means Madness and sounds just like the Arabic word for "My Humanity",  إنسانيتي  and if One happens to be afflicted with a lithp it may just as well be given as "inthanity" which means "My Woman"  or "Bend/Fold" or just as likely to be mistaken for,  if not listening carefully,  انفنتي  "Infinity".

11.5.12

Alright, All In

...and just as She suspected, He is All American! (ALL BETS ARE OFF IF HE INSISTS ON LOOKING LIKE MY EX-MOTHER-IN-LAW!)

Because the minute He finally does something that sets Her off, He says, "I'm Sorry--about that."

That being she discovers Her erstwhile 'Husband' is a Cross-Dresser!

Since the Revelation takes place in a public venue, She sucks it up (along with half a pack of cigarettes (American Spirits, organic, no added preservatives, pesticides, or accelerants--so they must be good for you) and says, "It's Okay."

And tries Her best to look on the bright side, after all, Her wardrobe had instantly doubled (if the 70's street fashion should ever take hold again in Her so-called-Lifetime), and She settles quickly on the idea that He's raiding Her panty drawer and doesn't mind it at all because He is so Clean He Squeaks.

And the cigarettes help interim with any oral fixation issues She may have that He is too busy exploring His femme side to satisfy.

{Entered 06_11_2011 I get it, 'It's just a phase He's going through'--It took a whole month to figure that one out and it's so funny--but I finally got it in a public venue so I can't crack up, or double over, or whatever the appropriate response is expected to be...}

And He finds some solace in Knowing that She takes the News so well, because She is not likely to let 'Beautiful' and 'Sexy' go to Her head since She doesn't trade in that , and where She comes from that and $2.50 gets Her a ride on the NYC MTA.

Level headed to the bitter end She was never One to confuse Love with Pain or Brave with Stupid, either.

What else Ya got? (I Heard You Say 'Not another word,' but since this is off-topic, I felt it would be permissible)

{Added Monday May 14, 2012...The Identity Crisis continues, but She has come across this type of behavior before---it all falls under the umbrella of MisUnderstanding--like when He Miss Takes for a Smile Her pained grimace (Her Old Age Talking) and what He calls that deep thoughtful introspective look on Her face for--- je ne c'est quoi--gravitas?, is simply Her trying to remember where She parked the car.}

"There is no point in driving yourself mad trying to stop yourself going mad. You might just as well give in and save your sanity for later."
--Douglas Adams, The Restaurant at the End of the Universe

11.4.12

You Go To Church And I Run For The Ship

Tough nut to crack!
Tundra Comic Panel 04.13.2012
Unrealized/UnRecognized Discovery?!!
(You and I both know that not to be the case; here are some caveats Quran 2:41, 2:79, 2:174, 3:77, 3:187, 3:199, 5:44, 9:9, and 16:95...However, that black Audi R8 that keeps popping up sure is tempting,
but like everything else You've thrown my way i have to walk away.)

{Aside: What's up with the 2 Mooning Moscovites kissing on the sofa across from me on 04.11.2012 at Webb Way?--all I have to say about it is, "Only the devil whispers, the rest of us speak aloud.--I do, however, wish to thank her for the help."}

وَالأَنْعَامَ خَلَقَهَا لَكُمْ فِيهَا دِفْءٌ وَمَنَافِعُ وَمِنْهَا تَأْكُلُونَ (16:5
وَتَحْمِلُ أَثْقَالَكُمْ إِلَى بَلَدٍ لَّمْ تَكُونُواْ بَالِغِيهِ إِلاَّ بِشِقِّ الأَنفُسِ
إِنَّ رَبَّكُمْ لَرَؤُوفٌ رَّحِيمٌ(16:7
وَالْخَيْلَ وَالْبِغَالَ وَالْحَمِيرَ لِتَرْكَبُوهَا وَزِينَةً وَيَخْلُقُ مَا لاَ تَعْلَمُونَ (16:8
16:5 And (God) Creates cattle for you....
16:6...to carry your loads..without hardship....Indeed Your Sustainer is Most Compassionate, Beneficent.
16:7...and horses...for you to ride and in which you find beauty...and He Creates things that you have no knowledge yet.

Church Nave vs Naval v Ship's Hull vs Coracle vs Walnut vs Kernel

The ‘nave’ of a church derives from the Latin ‘navis’ for ‘ship', which also gives us the word ‘naval’--having to do with maritime, nautical, seafaring and navigational activities (ie., applications that include seabasing, projective power/force, etc.). While ships primarily refer to seagoing vessels, a ship may also refer to any vessel that traverses a fluid medium, be it water, air/space or 'time'. In the case of a 'space ship' it would be something which flies, ie, 'tayer' refer to 105:3.


Notes: Projective geometry, like affine and Euclidean geometry, can also be developed from the Erlangen program of Felix Klein; projective geometry is characterized by invariants under transformations of the projective group. (re projective power/projective force)
The incidence structure and the cross-ratio are fundamental invariants under projective transformations. Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". 
An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates. On the other hand axiomatic studies revealed the existence of non-Desarguesian planes, examples to show that the axioms of incidence can be modeled (in two dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems. 
In a foundational sense, projective geometry and ordered geometry are elementary since they involve a minimum of axioms and either can be used as the foundation for affine and Euclidean geometry. Projective geometry is not "ordered" and so it is a distinct foundation for geometry. 
There are many projective geometries, which may be divided into discrete and continuous: a discrete geometry comprises a set of points, which may or may not be finite in number, while a continuous geometry has infinitely many points with no gaps in between. 
The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 (1 D) consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be carried out in either of these cases. 
For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem. The simplest 2-dimensional projective geometry is the Fano plane, which has 3 points on every line, with 7 points and lines in all arranged with the following schedule of co-linearity...
The simplest 2-dimensional projective geometry is the Fano plane, which has 3 points on every line, with 7 points and lines in all.

"Projective geometry" is sometimes used to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyze through the use of homogeneous coordinates, and in which Euclidean geometry may be embedded (hence its name,Extended Euclidean plane).
The fundamental property that singles out all projective geometries is the elliptic incidence property that any two distinct lines L and M in the projective plane intersect at exactly one point P.
The special case in analytic geometry of parallel lines is subsumed in the smoother form of a line at infinity on which P lies. The line at infinity is thus a line like any other in the theory: it is in no way special or distinguished.
(In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity).

The elliptic parallel property is the key idea which leads to the principle of projective duality, possibly the most important property which all projective geometries have in common.



Presumably the nave of a church came to be known as such from a fancied resemblance to a ship shape, but the shared characteristics between a church nave and ship shape is more intricately bound than mere imagination might lead us to believe.

Winchester Cathedral is a good example (images below; Winchester Cathedral's nave and its reflection (reflected across the horizontal x-axis) reminiscent of a ship's hull, compare to a Viking ship's ribs).









Winchester Cathedral (Gothic Architectural Influence)
See Sacred Geometry, Reuleaux Triangle

Upside Down Image of Winchester Cathedral's Nave
Compare with Ship's Hull Planking/Ribs vs
Hull Sections













Viking Vessel Hull Ribs
(See Jonah vs Fish vs Whale vs 'knitted' paths vs.
W-curve-Throw-Woof-Wales-Knots, etc.)

The secondary meaning of nave derives from an Old English word ‘nafu’ (relates to the Arabic, nafs) having to do with the idea of ‘centrality’, like in a ‘hub of a wheel’ (refer to the Hindu concept of 'Chakras'), or connected to the root for the word ‘navel’ from the Latin for umbilicus (relating to womb/matrix/cervix/neck/belly), and/or the Greek omphalos, relating to ‘end of a scroll’ (refer to 105:4 'sijil') and ‘the boss of a shield’, respectively.

Other roots for which describe 'disk, sphere' , 'whirlpool,' and 'bud' --considered a feminine sexual center since ancient times and still in parts of the Middle East, India, and Japan (see Hindu concept of yoni, the Reuleuax triangle, Freemason Symbols/Architecture/Sacred Geometry). 

And that the general layout or plans of a church mimics the shape of a seafaring vessel is more than just a passing fancy. The similarities between a church and ship may even be extended to that of the human form, most conspicously reflected in Native American Kachina dolls.










Amiens Cathedral, France vs Kingston Vessel vs Kachina Doll



(Note the west end/nave/aisle/crossing/transept/choir (lattice)/altar/east end of a typical cruciform church vs stern/transom/aft/engine room/port and starboard/bridge/pilot house/fore (bow or prow) of a ship vs the legs (ground contact/ actuators,--68:42 refers to  سَاقٍ 'saqiin' , refering to the legs or shins of the human body as well as bole/shaft/trunk and in verb form 'to drive') /pelvic region/torso/arms (actuators)/head (sky) of the kachina doll representation of the human form).

The Anatomy of a Church v The Ship v The Human Being


Tradionally churches are oriented facing 'east'  and in a cruciform church as depicted above the entrance is directly opposite the altar, or at the 'west' end which is situated in the church 'facade' (front of the building). In this example, the entrance at the facade of the 'church' corresponds to the  ship's aft which corresponds to the human's feet (where the Human makes contact with the ground, Earth). (This establishes how the orientations of each body correlate--see Special Orthogonal matrix, SO(n, F) )
The relevance or orientation and the Eastward direction may have its basis in navigating and piloting effectively along the Earth's Magnetic Norms , early article with respect to bird flight and overcoming the Earth's Coriolis effect; The Journal o f Applied Physics, Vol. 18, No. 12, A Preliminary Study of a Physical Basis of Bird Navigation, by Henry Yeagley, December 1947)


Chapter 2 The Calfوَإِذْ قَالَ إِبْرَاهِيمُ رَبِّ أَرِنِي كَيْفَ تُحْيِـي الْمَوْتَى قَالَ أَوَلَمْ تُؤْمِن
قَالَ بَلَى وَلَـكِن لِّيَطْمَئِنَّ قَلْبِي قَالَ فَخُذْ أَرْبَعَةً مِّنَ الطَّيْرِ فَصُرْهُنَّ إِلَيْكَ
ثُمَّ اجْعَلْ عَلَى كُلِّ جَبَلٍ مِّنْهُنَّ جُزْءًا ثُمَّ ادْعُهُنَّ يَأْتِينَكَ سَعْيًا
وَاعْلَمْ أَنَّ اللّهَ عَزِيزٌ حَكِيمٌ 2:260


2:260 And when Abraham said, "Lord, show me how you give life to the dead." (God) Said, "Do you not Believe (accept it as True)?" (Abraham) said, "Yes, (of course), but so that I am reassured (with) inner (certainty, in my heart)." (God) Said, "Take 4 birds and train them to obey you, then place a portion of them on each mountain and call them; they will seek you (come to you) at your command (in order); and know/teach that God is Wise, Powerful."

By extension, those forces which are known to effect the 6 modes of motion of a ship (ie., translations--sway, surge and heave and rotations--yaw, pitch and roll) in the water (be it ocean, river, lake or estuary) may be understood in the context of movement in the analogous human (earth) and church (spiritual or sky) 'bodies'. (In 3-D, 6 degrees of freedom refers to these 6 modes of motion, while robotics and gaming applications refer to higher 'degrees of freedom' these are merely incremental of the 6 in general and not in  fact 'additional' degrees of freedom, which may only be accessible from higher dimensions in order to allow truly greater ranges of motion, see affine matrix, projection/perspective, etc.)


Watercraft Rotation
Note Hull Front View vs. S Curve vs. Walnut Shell




The front view of a ship's hull bears a strong resemblance to a shield and in terms of the geometry is reminiscent of a 'bean curve', the smoothed out version described well by Cassini ovals. And the sides of which resemble an upside down 'ship's bell', each half of which can readily be modeled by the cumulative distribution function or S-Curve:
S-Curve Distribution Function
In terms of the mean (mu) and scale (beta, positive)

Bizarro Comics April 4, 2012
(I get it, I get it...
 There really is no need for You to rub it in...)
Coracles are keel-less, flat-bottom boats that are oval in a shape very similar to a half walnut shell (see walnut vs 2-D and 3-D Cassini Ovals, vs. Bean Curve, image below).

The design is intended to evenly distribute the weight of the boat and its load across the structure and to reduce the required depth of water — often to only a few inches, making it ideal for use on rivers (see Rivers of Time).

The coracle, called quffa in Arabic, is mentioned in the Bible, Exodus 2:3 as the water craft in which Moses's mother launched Him down the river to protect Him from Pharoah's decree to do away with the first born male children while the Israelites were dwelling in Egypt.

It was also frequently mentioned by my Dad whenever He wanted to remind any of us not to rush into anything in the form of a rhetorical question, "Must we jump from the quffa onto it's ears (handles)?" The short answer, 'tho none is called for to answer a rhetorical question is, "No, no, for the simple reason that you may find yourself in an awkward position that looks something like Dilbert in the comic panel here."
Dilbert April 15, 2012
What it looks like to jump from the coracle onto its handles.


The Welsh apparently have a saying about it, too. Since it is a light-weight flat-bottom boat, it can be carried by one man (or woman) on his (or her) back, the saying goes, "the load of a man is his coracle".

While inherently unstable by design because it sits "on" the water, rather than "in" it, a coracle can easily be carried by currents and the wind,  the coracle makes for an effective fishing vessel since it hardly disturbs the water or the fish, being easily manuevered with one arm, while the other arm tends to the net. (Generally 2 coracles are required for 1 net).

Walnut, 2-D Cassini Ovals & 3-D Cassini Ovals vs.
2009 Homework Assignment (compare with geometry of Wankel engine)& Bean Curve

Coracles vary in design, as they are tailored to the river conditions where they are intended to be used. In general there is one design per river, but this is not always the case.


Quranic Reference to Vessels, Ships, Ark, Orbits, Boats/Coracles & Orbits


Walnut Halve Shell, Vietnamese Coracle
 (Watercraft, Ship, Naval Application),
Boat Hull, Whole Walnut (note flange)

The Teifi coracle, for example, is flat-bottomed to negotiate shallow rapids, common on the river in the summer, while the Carmarthen coracle is rounder and deeper, better suited to negotiate the tidal waters on the Tywi, where there are no rapids.

Traditional Teifi coracles are made from locally harvested wood — willow for the lats (body of the boat), hazel for the weave (Y bleth in Welsh — the bit round the top) — while Tywi coracles have been made traditionally made of sawn ash.

 
Teifi coracles use no nails, and rely on interweaving the lats for structural coherence (see 'cracking open an alien spaceship'), while the Carmarthen ones use copper nails and no interweaving.

 
Modern working coracle boats are made of fiberglass.

The general idea for integrating the conepts above:
(2-D representation from 3-D or higher dimensional equivalents involve orthonormal projection, reflection, translation, etc. to conceptualize real-world equivalents which may account for why UFO witnesses percieve the craft morphing/transfigures when it impinges on Earth's airspace).


Proportions Based on a Kernel


___________________________
Walnut Shell Cross Section vs
Wankel Engine (note Reuleux Triangle)
While the coracle resembles the shape of a half walnut shell, the cross-section of a walnut shell is characteristic of the compression chamber of the Wankel rotary engine.

In the basic single-rotor Wankel engine, the oval-like epitrochoid-shaped housing surrounds a rotor which is triangular with bow-shaped flanks (reminiscent of the Reuleaux triangle, a three-pointed curve of constant width, but with the bulge in the middle of each side a bit more flattened).

In the Wankel engine, the expansion phase of the cycle is much longer than that of the Otto cycle, and the four strokes of a typical Otto cycle occur in the space between a 3-sided symmetric rotor and inside the housing.

The central drive shaft, called the eccentric shaft or E-shaft, passes through the center of the rotor and is supported by a fixed bearing.

The shaft turns 3 times for each rotation of the rotor around the lobe and once for each orbital revolution around the eccentric shaft.


The Wankel engine cycle is shown in the diagram below.

In the diagram above:
The "A" marks one of the three apexes of the rotor
The "B" marks the eccentric shaft and
And the “white portion” is the lobe of the eccentric shaft

The Wankel engine is actually a variable-volume progressing-cavity system. There are 3 cavities per housing, all repeating the same cycle.

Eccentric shafts do not have the stress-raising internal corners of crankshafts and Wankel engines also generally have a much higher redline than a reciprocating engine of similar power output. This is in part because the smoothness inherent in circular motion, but especially because they do not have highly stressed parts such as a crankshaft or connecting rods.

Points A and B on the rotor and e-shaft turn at different speeds--point B moves 3 times faster than point A, so that one full orbit of the rotor equates to 3 turns of the e-shaft.

The shape of the rotor between the fixed corners is meant to minimize the volume of the geometric combustion chamber and a maximize the compression ratio.

The symmetric curve connecting two arbitrary apexes of the rotor is maximized in the direction of the inner housing shape with the constraint that it not touch the housing at any angle of rotation (an arc is not a solution of this optimization problem).
.
.
.
While the Wankel engine has some disadvantages compared to conventional 4-stroke engines, having to do with thermal boundry layer, apex seal failures, and slightly higher fuel consumption, it has certain performance and safety advantages that make it difficult to overlook as a viable future engine technology.
.
.
.
Wankel engines have certain advantages over conventional 4-stroke engines; they are considerably lighter, simpler, and contain far fewer moving parts than equivalent piston engines.

When operating within their original design parameters Wankel engines are almost immune to catastrophic failure.

Because valving is accomplished by simple ports cut into the walls of the rotor or side housings, they have no valves or complex valve trains; and since the rotor rides directly on a large bearing on the output shaft, there are no connecting rods and there is no crankshaft.

The elimination of reciprocating mass and the elimination of the most highly stressed and failure prone parts of piston engines result in high reliability, a smoother flow of power, and a high power-to-weight ratio.

The surface/volume-ratio  (S/V) problem is so complex that a direct comparison between a reciprocating piston engine and a Wankel engine cannot be readily assessed. The flow velocity and the heat losses behave quite differently.

Surface temperatures behave absolutely differently; the film of oil in the Wankel engine acts as insulation (and contributes to some smoking on start up). Engines with a higher compression ratio have a worse surface/volume ratio.

The surface/volume (S/V) ratio of a Diesel engine is much worse than a gasoline engine, but Diesel engines are well known for a higher efficiency factor than gasoline engines. Diesel and alternate fuel versions of Wankel engines have been developed by Deere and others, and the Wankel engine lends itself to hydrogen fuel (fuel cell) applications.

Comparing engines of equal power, a naturally aspirated 1.3-liter Wankel engine with a naturally aspirated 1.3-liter four-stroke reciprocating piston engine with equal power, such a four-stroke engine is not possible and needs twice the displacement for the same power as a Wankel engine.

The extra or "empty" stroke(s) should not be ignored, as a 4-stroke cylinder produces a power stroke only every other rotation of the crankshaft. This doubles the real surface/volume (S/V) ratio for the four-stroke reciprocating piston engine and the demand of displacement.

So the Wankel engine has higher volumetric efficiency and a lower pumping loss since it has no choke valves.

Because of the quasi-overlap of the power strokes that cause the smoothness of the engine and the avoidance of the 4-stroke cycle in a reciprocating engine, the Wankel engine is very quick to react to throttle changes and is able to quickly deliver a surge of power when the demand arises, especially at higher rpm. This difference is more pronounced when compared to four-cylinder reciprocating engines and less pronounced when compared to higher cylinder counts.

Due to a 50% longer stroke duration than a four-cycle engine there is more time to complete the combustion. This leads to greater suitability for direct injection.

In addition to the removal of internal reciprocating stresses by virtue of the complete removal of reciprocating internal parts typically found in a piston engine, the Wankel engine is constructed with an iron rotor within a housing made of aluminium, which has a greater coefficient of thermal expansion--ensuring that even a severely overheated Wankel engine cannot seize, as would be likely to occur in an overheated piston engine. 

This is a substantial safety benefit of use in aircraft, since there are no valves and valve trains to burn out, jam, break, or malfunction in any way, again increasing safety.

A further advantage of the Wankel engine for use in aircraft is the fact that a Wankel engine generally has a smaller frontal area than a piston engine of equivalent power, allowing a more aerodynamic nose to be designed around it.

The simplicity of design and smaller size of the Wankel engine also allows for savings in construction costs, compared to piston engines of comparable power output. 


And the Wankel engine is capable of running at very high speeds and operates almost noiselessly.


A Wankel engine that loses compression, cooling or oil pressure will lose a large amount of power and fail over a short period of time. It will, however, usually continue to produce some power during that time. Piston engines under the same circumstances are prone to seizing or breaking parts that almost certainly results in major internal damage of the engine and an instant loss of power.

For this reason, Wankel engines are very well suited to applications in remote places where a failure could result in fatalities; and aircraft, where abrupt failure is likely to lead to a crash or forced landing.
_____________________________


{ok, ok, I am more than a  little weirded out that someOne is asking just how long this has been going on--not only did I think You Knew but I thought You Planned it!}

Quran Surah 7 Al-A3Raf
  فَأَرْسَلْنَا عَلَيْهِمُ الطُّوفَانَ وَالْجَرَادَ وَالْقُمَّلَ وَالضَّفَادِعَ وَالدَّمَ
آيَاتٍ مُّفَصَّلاَتٍ فَاسْتَكْبَرُواْ وَكَانُواْ قَوْمًا مُّجْرِمِينَ (7:133

7:133 That We Sent them floods, and locusts, and lice, and frogs, and blood; Separate Signs (of severence, disentanglement, judgment, to decide...), yet they (continued) in their arrogance and were an immoral (criminal) community. (133/7==19)



























Green Frog vs Origami Frog




Green Frog vs. Walnut Halves vs.Kernel
Child Pose Yoga Position vs Walnut Kernel (See Above)

14.2.12

An Affine-ity of Sorts

The Preamble/(sick to my stomach today anyway so this is a fitting reminder):

Surah 10 Yunas (Jonah)
قُل لَّوْ شَاء اللّهُ 
مَا تَلَوْتُهُ عَلَيْكُمْ وَلاَ أَدْرَاكُم بِهِ
 فَقَدْ لَبِثْتُ فِيكُمْ عُمُرًا مِّن قَبْلِهِ أَفَلاَ تَعْقِلُونَ
(10:16)

(10:16) Say, "If God Willed I would not have brought it (The Quran) to you and you would not have known anything of it, and I dwelled amongst you a Lifetime before (I brought it). Will you not use your Reason?"

{Aside: There are some out there who think I took it upon myself to do this, when the fact is that I did not volunteer and I know it's not my place to do so and I tried my best to avoid it. Sign 10:16 is eloquent in stating how this is not just in my head, it's all in my past. Which begs the question, what am I still doing here?}

Algorithms exist which can track critical distance from a moving plane point P (ie, where You are when You visit the Orange Field) to a static smooth curve (ie, where You Originate/Are Projected from,  be it the Blue or Gray field), without resorting to global search strategies beyond initialization (ie, so You do not get lost in space and always know the starting point).

An ordinary Cusp
on the Curve
x^3–y^2=0
Such an approach involves detecting transition events associated with the point's perturbation and successfully dealing with annihilation or creation events.

Annihilation/creation is one and the same event temporally speaking.
The one You experience depends on where You happen to be 'standing, (ie., Your Point of View/Perspective), basically the position associated with the perturbations at the transitions.

In order to avoid global searches for critical distances once such transitions are encountered, and at such transitions or various curve discontinuities (ie., like if You find Yourself at the cusp of a logocycle or saddle point in space), You can travel through it because...

U.R.IBIN (ie., My 'Son'-- 'Of Me'--):

Surah 3 Al.3'mran (Jesus' Family Name,'3mran)
قَدْ خَلَتْ مِنقَبْلِكُمْ سُنَنٌ
فَسِيرُواْ فِي الأَرْضِ
 فَانْظُرُواْ كَيْفَ كَانَ عَاقِبَةُالْمُكَذَّبِينَ  (3:137

3:137 We Created Generations/Orders/Aeons before (We Created) you, so travel (course) throughout the Earth and Observe/See the outcome of the deniers (those who falsify the truth/liars)

Surah 6 Al.An3'am (Cattle, Chattle...Property)
قُلْ سِيرُواْ فِي الأَرْضِ 
ثُمَّ انظُرُواْ كَيْفَ كَانَ عَاقِبَةُ الْمُكَذِّبِينَ 
(6:11)

6:11 Say, "Travel (set a course) throughout the Earth and then look (consider) the outcome/the end of the deniers (those who falsify the truth/liars).

Surah 9 Tauba (Repentance)
فَسِيحُواْ فِي الأَرْضِ أَرْبَعَةَ أَشْهُرٍ 
وَاعْلَمُواْ أَنَّكُمْ غَيْرُ مُعْجِزِي اللّهِ وَأَنَّ اللّهَ 
مُخْزِي الْكَافِرِينَ
(9:2)

(9:2) Travel freely (fluidly/flux) throughout the Earth four months and know (learn) that you cannot escape (frustrate) God and that God confounds the unfaithful.

Surah 16 Nahl (Bees)
وَلَقَدْ بَعَثْنَا فِي كُلِّ أُمَّةٍ رَّسُولاً أَنِ اعْبُدُواْ اللّهَ وَاجْتَنِبُواْ الطَّاغُوتَ
 فَمِنْهُم مَّنْ هَدَى اللّهُ وَمِنْهُم مَّنْ حَقَّتْ عَلَيْهِ الضَّلالَةُ 
فَسِيرُواْ فِي الأَرْضِ فَانظُرُواْ كَيْفَ كَانَ عَاقِبَةُ الْمُكَذِّبِينَ (16:36)

16:36  And in the past We Sent/Raised In every nation a Messenger (Saying): "Follow/Worship God and avoid idolization (false imagery)." Some of them were Guided unto God and some of them increased in their misguidance/straying, so travel freely (fluidly/flux) throughout the Earth and see/observe how the deniers/liars ended up.

Surah 27 Naml (Ants)
قُلْ سِيرُوا فِي الْأَرْضِ 
فَانظُرُوا كَيْفَ كَانَ عَاقِبَةُ الْمُجْرِمِينَ   
(27:69)

27:69 Say, "Travel (course) through the Earth and see/observe what became of the criminals/miscreants/guilty.

Surah 29 3'ankabut (Spider)
قُلْ سِيرُوا فِي الْأَرْضِ فَانظُرُوا كَيْفَ بَدَأَ الْخَلْقَ
 ثُمَّ اللَّهُ يُنشِئُ النَّشْأَةَ الْآخِرَةَ 
إِنَّ اللَّهَ عَلَى كُلِّ شَيْءٍ قَدِيرٌ
(29:20)

29:20 Say, "Travel (course) throught the Earth and see/observe how God originates the Creation and then Brings Forth the Emergence of the Hereafter (Its Consequence/Maturation). Indeed, God Has Power Over Everything (Is Capable of Anything)."

Surah 30 Romans (Ar.rum)
قُلْ سِيرُوا فِي الْأَرْضِ 
فَانظُرُوا كَيْفَ كَانَ عَاقِبَةُ الَّذِينَ مِن قَبْلُ
 كَانَ أَكْثَرُهُم مُّشْرِكِينَ
(30:42)

30:42 Say, "Travel (course) through the land (throughout the Earth) and look/discern (be able to make a distinction/differentiate) the final outcome of those who before were mostly heathens (idolators/pagans/those who associate anything else with the One True God).

It is important to point out the emphasis is for the Believers to travel freely (set a course + flux) and make careful observation (look to see) in order to be able to even distinguish or discern that the Creation and Recreation (Hereafter) is indeed taking place in situ and that the Creation is not as static as we may be duped into thinking.

Apparently, there is little chance of encountering or being able to gauge a fold in space/time if restricted within a narrow physical existence. So, for some who do not venture more than 20 miles beyond where they were first born, this may not be a concept they can readily assimilate, let alone detect given their circumspect movement on the Earth.

And as such, what we are on about here (tay-el-ard), may seem like weird science.

The more widely-travelled set, however, may be able to relate to at least some of these concepts having merit beyond the occasional honorable mention in fairy tales.

Given Vectors defined by Chapter:Sign:Word Location  سِيرُوا   'seeru'  'Travel':

3, 137, 6  (alif, lam, mim)
6, 11, 2  (non muquati'at)
16, 36, 24 (non muqati'at)
27, 69, 2  (tta, sin)
29, 20, 2 (alif, lam, mim)
30, 42, 2 (alif, lam, mim)
and
9, 2, 1 (non muqati'at)  ('Travel' given as سِيحُواْ   ' seehhu' so this particular Sign may or may not be part of the vector space considered here) 

First square matrix taken in chapter ascending order where the word 'Travel' appears in the same location, ie, word number==2:
27, 69, 2 (tta, sin)
29, 20, 2 (alif, lam, mim)
30, 42, 2 (alif, lam, mim)

Matrix: {{27,69,2},{29,20,2},{30,42,2}}
Determinant: 186
Eigenvalues: 70.4771; -21.3535; -0.12359
Trace: 49 =[1 + cos (t)] = [cos (t) = 24]


Second square matrix taken in chapter ascending order and share the same muqati'3at == alif.lam.mim.:
3, 137, 6 (alif, lam, mim)
29, 20, 2 (alif, lam, mim)
30, 42, 2 (alif, lam, mim)

Matrix: {{3, 137,6},{29,20,2},{30,42,2}}
Determinant =3850 (positive, eigenvectors have same orientation, product of the eigenvalues have same sign as the Determinant; the higher the value of the Determinant, the more valuable the information)
Eignevalues: 78.3513; -52.4138; -0.93749 (positive eigenvalues (real part) mean solution spirals away from the origin; negative eigenvalues (real part) mean solution spirals towards the origin; purely imaginary eigenvalues mean the solution neither spirals in or out from the origin but circles around it)
Trace = 25 = [1+ 2 cos (t) ]== [cos (t) = 12]


           sin^-1 (12) =  ~  3.17631i (r= 3.17631, theta = 90degrees) ~1.011 pi


Matrix Trace is related to Group Character; the Trace of Marix A, TrA, is the sum of the diagonal elements of a matrix; and the Trace is the sum of the eigenvalues of the matrix. The Determinant, detA, is the product of the matrix eigenvalues. For the case where TrA=0, detA=0, or (TrA)^2= 4detA the system is not structurally stable, ie., a small change in the entries of matrix A will destroy this property. Other cases are structurally stable, i.e., if the system has a sink, any nearby system will have a sink.

The Matrix A can be considered to be a Point in the
Determinant-Trace Plane
If the Determinant is Negative it is A Saddle point (one positive and one negative solution).
If the Determinant is Positive, eigenvalues have the same sign as the Trace
For the degenerate case where the detA=0, solutions are constants.
Comprehensive eigenvector ray paths explanation and movie.

Arithmetically, transition events are related to singularity and catastrophe theories.
The Swallow Tail's - Series on  Catastrophes
Oil on Canvas 1983 by Salvador Dali

(apparently some of ewes guys have already tried this vector math and that's why you've been popping in and out of public restrooms...but I'll do the math exercise just the same...)


Diagram of cusp catastrophe
Brown and Red curves show  satisfying dV/dx = 0 for parameters (a,b), 
are drawn for parameter continuously varied over several values of parameter a
Outside the cusp locus of bifurcations (blue), 
for each point (a,b) in parameter space there is only one extremising value of x
Two different values of x, giving local minima of V(x) for each (a,b), 
are possible inside the cusp, separated by a value of x giving a local maximum.
René Thom's catastrophe theory:
Thom suggested that in four-dimensional phenomena, there are seven possible equilibrium surfaces, and therefore seven possible discontinuities, or "elementary catastrophes": fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic
"The shape of Dalí’s Swallow’s Tail is taken directly from Thom’s four-dimensional graph of the same title, combined with a second catastrophe graph, the s-curve that Thom dubbed, "the cusp". 
Thom’s model is presented alongside the elegant curves of a cello and the instrument’s f-holes, which, especially as they lack the small pointed side-cuts of a traditional f-hole, equally connote the mathematical symbol for an integral in calculus.--wikipedia 
This may not seem of any valuable interest in real-world dynamics, unless you are truly observant.

For example, to the uninitiated, a drive along the freeway is a chance to go fast and burn some gas. A side benefit to which is that it is the closest a human being can get to actually flying. More so than in airplanes since the travel compartment is pressurized so the actual sensation of flight is insular at best.

In a moving car and with the windows down the feel of the wind rushing by and the traction against the road causes the experience to be more proximate to the real thing our fine feathered friends take for granted.

What has all this to do with cusps and bifurcations?

The beauty of the pragmatic application is in its subtlety. Because not only is God the Devil in the Details, but God is Imperceptible/Subtle, leaving the bludgeon approach to people like Me.

The fortunate few, who think 'forwardup' is a direction, are attuned to this happenstance occurring more commonly than one would think; to everyone else 'it's just another stack of hay.'

Those aware of the details of how such a catastrophic cusp event occurs in practice can see that while maintaining a constant speed along with the flow of other highway traffic, and all other parameters being the same, they begin to notice their vehicle either pulling away quickly from the field behind them or that they may see themselves dropping back away from the field of traffic ahead of them inexplicably fast, despite no changes in acceleration.

Magic! Or you happen to hit a fold in the road, not a bump, not a pot hole.

For the majority of us, we tend to notice what song is on the radio more than any such phenomenon.

And then we act vacant, obtuse and clueless when the same phenomena occur on a much larger scale like that of the Bermuda Triangle 'disappearances.'


An ordinary Cusp formed by the Optical Caustic
formed by light rays at the bottom of a tea cup.
The plane curve cusps are all diffeomorphic to one of the following forms: x2 − y2k+1 = 0,
where k ≥ 1 is an integer.

Algorithms have also been established that track point-curve critical distances exactly on piecewise smooth curves, deriving algebraically the unfolding formulas for degenerate critical distances (ie., to make it easier for me to find You because when it comes to Math and Science, I am a little retarded--not necessarily a self-depricating bad thing since by the time it occurs to me just how angry I should be, the problem is resolved and I just have to convince myself all there's left to do is wait for the dust to settle).

Sort...Cycle...Fold...and other Possible Outcomes at Catastrophic Events 
(and we're not talking just laundry, here):


Surah 15 Hijr (Rocky Tracts, have many sharp edges or Cusps)
إِنَّ رَبَّكَ هُوَ الْخَلاَّقُ الْعَلِيمُ (15:86

15:86 Indeed, Your Lord Is The Omniscient Creator!

Argyle Sweater
Forgotten Tools
وَلَقَدْ آتَيْنَاكَ سَبْعًا مِّنَ الْمَثَانِي وَالْقُرْآنَ الْعَظِيمَ (15:87

15:87 And We Have Given You 7 of the mathanee (traditionally interpreted to mean Surah 1 Al Fathha (The Opening) which is 7 of the most often repeated verses; literally,   الْمَثَانِي  means 'the folded, enfolded, pleated, flexions, folds and/or 'that which folds or causes to fold/bend') and The Quran Al.3'attheem (Magnificent, Sublime, while  عَظِمَ   the root  3.tth.m  means major, great; it also means bone and the word الْعَظِيمَ is an adjective that by this reasoning means 'that which skeletonizes', therefore, it is meant as an instrument of Resurrection)

لاَ تَمُدَّنَّ عَيْنَيْكَ إِلَى مَا مَتَّعْنَا بِهِ أَزْوَاجًا مِّنْهُمْ
 وَلاَ تَحْزَنْ عَلَيْهِمْ
وَاخْفِضْ جَنَاحَكَ لِلْمُؤْمِنِينَ (15:88

15:88 Do not extend Your sights (eyes, look longingly, envy) to that which We Endowed pairs (couples) of them,
and Do Not Grieve for them.
And lower Your Wings (of Mercy) over/to the Believers.


Sign 15:88 is important to the discussion on catastrophic cusp events in it's allusion to the Prophet (pbuh) 'dipping or lowering His 'wings of Mercy'' like a mother bird would shield her hatchlings by encircling them with her wings. 
But the actual intended imagery has to do with 'folding and bending' space and how to inspect for visual clues in our common everyday experience. 
Here, it draws attention to when looking up at birds flying high overhead they may seem to dip in and out of view (disappear) either due to spherical aberrations in the atmosphere or astigmatism on the part of the observer--or we're all plain giddy to distraction thinking ornithology is a branch of UFOlogy. 
But in a very similar fashion as the astute driver notices such subtle interactions with tay-al-ard or space-fold encounters on the anthropomorphic scale, the phenomenon has equivalence in electromagnetism and light propagation (a compelling argument for the variable speed of light in a vacuum). 

وَقُلْ إِنِّي أَنَا النَّذِيرُ الْمُبِينُ (15:89

15:89 And Say, "Indeed, I Am the Evident (Apparent, Obvious, Conspicuous, Clear) Warner!"(...that's why I keep wondering what happened to my front man?!)

كَمَا أَنزَلْنَا عَلَى المُقْتَسِمِينَ (15:90

15:90 Whom We've Sent to those in disputation (divided, separated, not united)

الَّذِينَ جَعَلُوا الْقُرْآنَ عِضِينَ (15:91

15:91 Those who take the Quran in jest (quibble about, pick over, make biting remarks, take it apart-- idiomatic expression)

فَوَرَبِّكَ لَنَسْأَلَنَّهُمْ أَجْمَعِيْنَ (15:92

15:92 To Your Lord they shall Answer (We Will Question them) Altogether. (Wholey, In the Collective, All of Them Gathered)

عَمَّا كَانُوا يَعْمَلُونَ (15:93

15:93 Of that which (they thought) they were doing.

فَاصْدَعْ بِمَا تُؤْمَرُ وَأَعْرِضْ عَنِ الْمُشْرِكِينَ (15:94

15:94 So proclaim (lit. split, fissure, crack/crevice, rift) that which You are commanded and withdraw from the idolators. (Alternatively, Split/rift/separate by an act of fissuring away (see the driving experience above for similar experience of what it means to 'break away') from that which they command or want of you, and avoid those who join other deities with God, worship other things along with God).

إِنَّا كَفَيْنَاكَ الْمُسْتَهْزِئِينَ (15:95

15:95 Indeed, We spared You from those who mock (blame) you (lit. making a stir, make waves, etc. idiomatic for rabble-rousers).

الَّذِينَ يَجْعَلُونَ مَعَ اللّهِ إِلـهًا آخَرَ فَسَوْفَ يَعْلَمُونَ (15:96

15:96 Those who join with God other gods, they will (come) to know (better).

------------------------------------------------------
Neurotransmission (see gears) means "I love you, but I don't need the same."

Yeah, Right! I think Malcolm is an Old Man, like I go around thinking I am the Queen of the Main!
I just wanted to have everybody change their concept of pre-cursor!

But since it's all been carefully arranged that I can't get off the ground, and I am as of today completely dispossessed, I would like to finally get paid so I can afford to pack my suitcase and move on down the line.

Oh, and I'd really like to see my Late (aka Delinquent) Father right about now.(Thanks for visiting August 5, 2013--let's both try not to be so surprised next time)
----------------------------------------------------

from Tracking Point-Curve Critical Distances, X. Chen, E. Cohen and R. Riesenfeld