Showing posts with label Helix. Show all posts
Showing posts with label Helix. Show all posts



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 

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 isometrya 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.
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. 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 axisThen 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!


Seeing Doubles

The women pictured above are not any direct relation to me, however, their uncanny resemblence to my birth Mother and my Daughter are more than just a passing interest.

I don't even know how I chanced on the image of Nevada Stoody (Duchess of Oporto), I wasn't searching for her, and I remember thinking it odd that in that particular image she looked alot like my mom, although not so much in other pictures of her.

The image of the Paris Woman, by Dennis Hopper, I had come across when looking into the retrospective of his art exhibit soon after he passed away last year, and it floored me how much His muse looks like my daughter, born 5 years prior to the 1994 date of this particular artwork.

On seeing Nevada's image and my mentioning the likeness to my daughter's grandmother, my daughter said, 'Is that why you're so pretty, Mom?'

And it being a mutual admiration society, I countered, 'That's why You're so pretty, daughter; and then look at Dennis Hopper's muse!'

Then she said, 'Yeah, that one is weird, she does look just like me.'

And then she asks how that could even be possible and answers her own question, 'Maybe it's because there is something to us all coming from just the one pair. Or maybe, it's just some kind of weird dream we are all having.'

She goes on to say, 'Whenever I say things like that to Dad, he says, 'There you go, sounding just like your mother!''

And then she says, 'You mean, I sound like a Scientist?'

And then I think-- more like, sober.

To get to the Science behind these Dopplegangers (convincing doubles, anyway), we need to take a closer look.

Duck Season_Rabbit Season
خَلَقَكُم مِّن نَّفْسٍ وَاحِدَةٍ ثُمَّ جَعَلَ مِنْهَا زَوْجَهَا
 وَأَنزَلَ لَكُم مِّنْ الْأَنْعَامِ ثَمَانِيَةَ أَزْوَاجٍ
 يَخْلُقُكُمْ فِي بُطُونِ أُمَّهَاتِكُمْ خَلْقًا مِن بَعْدِ خَلْقٍ فِي ظُلُمَاتٍ ثَلَاثٍ
 ذَلِكُمُ اللَّهُ رَبُّكُمْ لَهُ الْمُلْكُ لَا إِلَهَ إِلَّا هُوَ فَأَنَّى تُصْرَفُونَ (39:6

39:6 He Created you (all) from One (singular/united) Soul, then Brought forth from Her (that Soul) Her Mate/Spouse/Dual; and He Propagated/Revealed/Sent Down to you (all) of/from Blessings/Affirmations (usually al.'an3am'  الْأَنْعَام     is interpreted as 'Cattle', but does not make sense in this context, so here rendered as secondary meanings having to do with 'affirmation/yes/aye/agreement?') 8 Pairs/duals/mates; He Creates you (all) in your mothers' stomachs (wombs) (one) Creation after another Creation, in a threefold darkness/mystery; This Is Allah, Your Sustainer, To Whom Belongs The Dominion; There is No God Except for Him; How is it that you are (all) so misguided/lost/wasted?

يَا أَيُّهَا النَّاسُ اتَّقُواْ رَبَّكُمُ الَّذِي خَلَقَكُم مِّن نَّفْسٍ وَاحِدَةٍ
 وَخَلَقَ مِنْهَا زَوْجَهَا وَبَثَّ مِنْهُمَا رِجَالاً كَثِيرًا وَنِسَاء
 وَاتَّقُواْ اللّهَ الَّذِي تَسَاءلُونَ بِهِ وَالأَرْحَامَ
 إِنَّ اللّهَ كَانَ عَلَيْكُمْ رَقِيبًا4:1
(4:1) Harken! All You People! Pay careful attention to your Sustainer Who Created You (all) from One Soul (singular/united) and Created from Her ('soul' is feminine gender in Arabic) Her Mate/Dual/Spouse and broadcast from them Men in multitudes and Women; and heed God (by whom) you demand mutual rights (from each other) and (respect) the wombs (that bore you); indeed, God Is Ever Watchful over you. (رَقِيبًا  raqib, means Observes/Watchful but also implies a 'sergeancy' (serves a protective duty))

Sign 4:1, from Sura 4, The Women, makes mention of our common ancestry from one 'integrated' soul. The theme of having been created out of one 'living entity' is repeated in Signs 7:189 and 6:98; that the 'living entity' is a 'man+woman' is mentioned in 49:13.

And given that ancestry and heredity are closely linked to chromosomes, it would not be a great stretch to take Sign 4:1 as the Choromosome Number 4: Region 1 (either p or q arm; but since the p arm has only 1 region for the human chromosome 4, I would assume this particular sign is pointing something out in that region that has to do with "tasa' luna bihi" which breaks down to something like "T sa' loena bihi wa al.arhham' , 'T's color is worsened by (it) and the wombs/matrices', or 'its color worsens by it and the wombs/matrices' , variably 'T-C (ar. letter seen--as in cytosin) Termination color..."

A geneticist may have some insight into what the God/controller is in this particular region or chromosome 4 that behaves in such a manner, and it also functions as a protection/something that serves to protect or discharges other duties akin to those of a 'sergeant' and further that it serves as a 'marker' ('marker', 'sergeant', 'watches/observes/control/oversee/proctor//blue-pencil (edits)' all derive from رقيبا /رقيبه  raqiba, raqibah,  and the absolute form raqibat means 'stake-holder'; and, the simple root rqb رقب means 'neck').

Sign 39:6 in Chapter 39, Az_Zumar (The Groups), can be treated the same way as 4:1 paying careful attention to the word الْأَنْعَام      al.an3am, that while it does mean cattle and points to '8- couples/pairs/espoused twosome' ثَمَانِيَةَ أَزْوَاجٍ,  something these 'cattle' have in common/differ is also associated with the secondary meaning of al.an3am's of it having to do with several variants:

  • the root n3m نْعَم  v. to refine/improve/enhance/polish/upgrade/perfect/cultivate/hone
  • نعم the same word without the diacritic marks means 'yes', an affirmative/positive/absolute/agreement
  • نعام  or نْعَام  na3am means 'ostrich'; some genetically remarkable characteristics of which include: 
acute eyesite and hearing; the fastest animal on 2 legs (run over 70 kilometres per hour (43 mph) for up to 30 minutes); have just two toes on each foot ( an adaptation that seems to aid in running, most birds have four),  the nail on the larger, inner toe resembles a hoof (found in cattle/horses/ungulates) and the outer toe has no nail; the largest eyes of any land vertebrate and they are shaded from sunlight falling from above; largest egg of any living bird; female ostrich can pick out their own eggs from communal nests. The word 'ostrich' derives from the Latin word for 'camel'.

  • لْأَنْعَام la.an3am with diacritica لانعام  seems to mean 'to fully' and without the diacritic marks لأنعم ' to softer' which would indicate making something most flexible/pliable or have the most plasticity.
  • and finally, the full word الْأَنْعَامِ   al.an3am(i) primarily meaning 'cattle'
Of which  8-pairs have been proferred to Humanity. 8-pairs that are in agreement (like matching autosomes? that are somehow associated with 'cattle'). Makes one wonder what the 8 types of cattle share chromosome-ally speaking? Cattle, or hoofed animals/ungulates is an order of mammals, of which 6 to 8 survive.

Recent sequencing of bovine genome has determined that cows have 22,000 genes, and 80% of bovine genes are shared with humans--ie., that cows are much more similar to humans than rodents. (Do the 8 pairs (matching/affirmatives) mentioned here add up to that 80%?)

The Human DNA is comprised of 46 chromosomes (22 pairs of autosomes and one pair of sex chromosomes, half are contributed by the mother and the complementary half by the father).

If counted individually, and taking the maternal chromosomes first, then autosome #39 is the maternal portion of chromosome 20, which happens to be metacentric, (no luck finding a diagram of chromosome 20, so presumably there is a region 6).

The indication being that these shared 8 pairs have characteristics that enhance/refine the human; makes for plasticity/pliability when 'matrixed' or in the womb (?is it what makes infants 'softer' or more completely 'pliant' that they are able to pass through the birth canal in humans?)

But the really interesting synchronicity is the shared human/bovine DNA and that this post is about body doubles/clones and my having read the sci-fi novel The House of the Scorpion by Francis Farmer a few years ago (a great read) in which clones were propagated using cows. I wonder how much of this information was available to the author at the time of her writing the book and how much of it she had contrived only to see the related Truth of it some time later.

The Arabic word for womb and matrix is the same, and according to the online etymology dictionary the word matrix drives from the following:

late 14c., from O.Fr. matrice, from L. matrix (gen. matricis) "pregnant animal," in L.L. "womb," also "source, origin," from mater (gen. matris) "mother." Sense of "place or medium where something is developed" is first recorded 1550s; sense of "embedding or enclosing mass" first recorded 1640s. Logical sense of "array of possible combinations of truth-values" is attested from 1914.
The logical sense is the most captivating here in that the reference in Sign 39:6 to 'affirmations'/agreement' or 'yes' for the word an3am is very closely related to 'truth-values' of which there are 2; either 0=False or 1=True, and is would appear that what was selected is the 'matching' or pairs that agreed, 0 for 0, 1 for 1; and, again, of which there are 8 pairs.

Then, I write as I choke back the tears, guess who's doppleganger this is?
Complete with 3aba' عباءة
(Derives from last name having to do with Cloak)

How'd They Do That?

And here is the maternal contribution to your name:

Starting with the letter '3ayn' ع:
The letter name for which is derived from Proto-Semitic *ʿayn- "eye", and the Phoenician letter had an eye-shape, ultimately derived from the ı͗r hieroglyph.

To this day, 'ayin in Hebrew, Arabic and Maltese means "eye" (‘ayno in Assyrian).

The second syllable is taken as (ab) then it means 'water' in Persian, and it means Father in Arabic, a root that more than likely derives from the ancient word in Egyptian (nb) for Lord + Master. (In combination with the '3ayn' ع, it forms words meaning 'spring' or 'well' and 'Father's /Master's/Lord's Eye'

If the second syllable is taken to be (n.ab) then there is a root in ancient Egyptian  'nb' which means "All". Interesting in that the phoneme N in ancient Egyptian hieroglyphs is the 'water ripple' and is also used as a preposition from, in, by, to...

Taken as a whole, it is what's known as Ziziphus zizyphus , a fruiting shade tree, commonly called jujube.  The Arabic name for which sidr, and sadr, appears in the Quran Surah 53 (Najm. Revelation, The Star) and Surah 56 (The Event) as one of the trees the denizens of Paradise will find themselves surrounded by:

 عِندَ سِدْرَةِ الْمُنْتَهَى  (53:14  
عِندَهَا جَنَّةُ الْمَأْوَى (53:15) 
     إِذْ يَغْشَى السِّدْرَةَ مَا يَغْشَى (53:16  

مَا زَاغَ الْبَصَرُ وَمَا طَغَى (53:17)
53:14 (He met Him) by the Lote Tree of the Farthest Limit (Sidrat al-Muntaha)
53:15 Near it is the Garden Haven (Sanctuary, Resting Place, Abode)
53:17 His (The Prophet's) Eye (vision) did not waver/evade/avoid nor dominate/control (it)
Sign 53:14 refers to one point at which The Prophet (pbuh) met Gabriel during one of His descents, at the Sidrat al-Muntahā (Arabic: سدرة المنتهى) -- a Lote tree that marks the end of the seventh heaven, the boundary where no creation can pass (and do You wanna know why no one can pass? Because if You do manage such a feat and find Yourself at God's Doorstep, God Himself acts like no One's ever died before and hijacks Your spaceship and brings You right back down to Earth where You (or at least I) keep wishing this persistent after image would all just go away!--that's why).

 فِي سِدْرٍ مَّخْضُودٍ (56:28  

56:28 fee sadrin mkhdthoudin

56:28 (They will find themselves) among fruit-laden Lote Trees  


The Unfolding

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

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

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

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

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

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

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

The profile of such a wrapped line is given by:

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

p(t)= ct^2

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

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

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

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

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

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

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

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

Anyway, back to the present.

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

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

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

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

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

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

or x= Time*θ Csc[θ] = ; and


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

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

Does this mean that In Reality Nothing Really Ever Happens?

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

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

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


The World Turns? Screw That!

Perhaps this post is more appropriately titled , 'An (Hyperb-) Ode to Alan'

"Curvature is the Obstruction to a Flat Field.
Tidal Forces are the Obstruction to an Acceleration Field
Curvature of Space-Time is the same as Tidal Forces."

--From a link associated with "GR02 Gravity Acceleration Equivalence, Gauss’s Law 2008-09-29"

Surah 86 At-Tariq (الطَّارِقِ)
وَالسَّمَاء وَالطَّارِقِ 86:1وَمَا أَدْرَاكَ مَا الطَّارِقُ 86:2النَّجْمُ الثَّاقِبُ 86:3إِن كُلُّ نَفْسٍ لَّمَّا عَلَيْهَا حَافِظٌ 86:4فَلْيَنظُرِ الْإِنسَانُ مِمَّ خُلِقَ 86:5خُلِقَ مِن مَّاء دَافِقٍ 86:6يَخْرُجُ مِن بَيْنِ الصُّلْبِ وَالتَّرَائِبِ 86:7إِنَّهُ عَلَى رَجْعِهِ لَقَادِرٌ 86:8يَوْمَ تُبْلَى السَّرَائِرُ 86:9فَمَا لَهُ مِن قُوَّةٍ وَلَا نَاصِرٍ86:10وَالسَّمَاء ذَاتِ الرَّجْعِ 86:11وَالْأَرْضِ ذَاتِ الصَّدْعِ 86:12إِنَّهُ لَقَوْلٌ فَصْلٌ 86:13وَمَا هُوَ بِالْهَزْلِ 86:14إِنَّهُمْ يَكِيدُونَ كَيْدًا 86:15وَأَكِيدُ كَيْدًا 86:16فَمَهِّلِ الْكَافِرِينَ أَمْهِلْهُمْ رُوَيْدًا 86:17
{Some of 'em want to know what's driving me to do this. Simple vanity! I decided the only way to shed a few pounds around here is to get rid of Gravity.}

The diacritic marks were added to the Quran some time after it was first put down in writing, to ensure the proper pronunciation of the text to avoid any misunderstandings. However, it would appear some people go to great lengths to make sure the meaning is lost on them. So, enough about them.

For the classic Star Trek aficionado who may remember Spock and McCoy facing off in a game of multi-level chess, the Multiple nuances of the Quran are more to do with performing multiple math problems than to dig deep to find a way to screw your neighbor.

For a limited example, take the first line of Chapter 86 At-Tariq. 'The Tariq' has been variously rendered as 'the night visitor', 'the piercing star', 'Sirius', just to mention a few variations on the theme. Tabulated below, in a nod to Multi-level Chess and Math Word Problems, the first line 86:1 consisting of 14 letters is broken down to its basic elements of variables, constants and other arithmetic operators.

If the table on the left, parsing out Sign 86:1 of the Quran, does not make it immediately obvious that one of the relationships embedded in it describes a helical path or orbit, then perhaps some clarification is called for.

I am presuming no one else has stumbled on it since I have never heard anything about this particular information embedded in the Quran, or what turned out to be my very handy-dandy-field-survival-guide, in all the years I've been on this (sad forsaken) planet and considering that this information has been hiding in plain view for over 14 Centuries by our reckoning.

In fact, the majority of those I've encountered over the years only see the worse things in it (which I have always justified as a general lack of literacy in the population) and most of them can only sit around thinking of ways to desecrate it or disparage the One (pbuh) who saw fit to share it with us.

The few more widely read, or the handful following along at apageinthelife, may scoff at the discovery, because there is more vivid imagery and reference to this phenomenon than the example I selected. Not only in the Quran itself, as in Chapter 81 Takwiir (Lit. "Spiralling'/'Twisting,' moving off-center), but in other Ancient lore including the Maya's of South America and the Maori's of New Zealand.

The Maya, of course, are best noted for their 2012 so-called 'doomsday' calendar, and there is a Maori oral tradition that alludes to 'When the Snake Bites the Sun.'

The Snake or Dragon reference appears in one form or another in most world cultures, and while it essentially is a reference to new beginnings from old endings (depicted by the serpent usually eating its own tale), such myths are imparted in simplistic memorable terms akin to presenting someone with a colorful Easter Egg--something that can be easily recalled, so that when we get older and have learned enough or experienced enough, and there comes a time to do away with childish things, we can then truly understand the Reality that 'hatches' from all the fairy tales.

I am probably not as brilliant as the young man looking over my shoulder at our neighborhood Starbucks the other evening, so I can't do multi-level math; and given that I can't even do 10th grade math thanks in great part to those who for whatever reason saw to it I lose the only thing I had going for me (my mind) coupled with the lack of proper care and out-right cover-up from the so-called 'medical community', I have to resort to single-level easy-as-she-goes-so-as-not-to-blow-out-her-one-remaining-brain-cell approach. Since I was never all that great at Set Theory or Statistics, I will go with the Algebra (you can follow along and pick your forte).

The first step, as when working any Code, is to find a Key, or agree on a Convention. Since the 'bismillah' is among the most often repeated phrases on Earth, maybe it is a Key to untangling the Arithmetic embedded in the Quran?

Like with everything else, one must Consider the Source before committing anything to a Methodology. While for convenience I use for an on-line Quran source, I know that the Allah is generally more detailed than what is shown in Sign 1:1 above, so I used wikipedia to get a more faithful representation of the Name 'Allah'.
As shown, Allah comprises an 'alif wasl' which is actually not an 'A' at all but what is called a 'hamza' and since this one appears at the beginning of the word with no prior letter it is pronounced as "Ah" and is called a 'hamza wasl'. Literally, 'hamza wasl' means 'connecting link'.
This is followed by the first L (the Arabic Lam) which in Arabic math notation is used for !factorial but is also the notation for complex numbers of the form r*e^(it). And it connects to the medial L in the Word Allah.
The Medial L (connects two letters, the leading Lam and the Ha (Arabic H that follows). What is noteworthy about this second lam is that it has both the 'shadeh' and the 'dagger aleph' harakat or diacritical marks above it. So, while the first Lam is understood to be connected with the hamza, making the word 'al' as in the definitive article 'the', there is no ignoring what we are to think of this medial Lam.
The 'aliph dagger' خنجر is used to mark or highlight the letter it is over, much like a sign (†) that is used as a reference mark, especially to a footnote (oh, A Clue!). And then we have the 'shadeh' as mentioned throughout the table above meaning, to 'stress' 'double' or 'take root' as well as things having to do with germination and the 'kernel'. And it is worth mentioning that the word 'Aliph' itself means 'compile'.
That the medial Lam is followed by the Ha (ه ), and there is no mistaking that in Arabithmatic ه stands for Euler's number, E, (or, e as in ln(e)=1---like e=2.7182...that E!) may be pointing to Lam as shorthand for complex numbers rather than Permutation, but that's just a guess on my part. (Plenty of room here for puns about 'Mary had a little lamb' and 'Abraham sacrificing a Lamb' or Ram...blah, blah, but let's try to focus).
Euler's number appears often in math and the sciences, and is associated with a range of things from expressing growth (or decay) rates, to Gaussian fields, to simplifying Calculus (like when integrating the Taylor equations).
So, we look for the Truth of the Thing by stringing along the letters. Lam complex form taken once and transcribed to equivalent English is re^(it).

There are two ways to double this; either one should work, so I tried both 2re^(it) and (re^(it))^2. And according to Anyone Who Has Ever Told The Truth of The Matter, There is Only 1 Allah. So, to test the Key, We Trust That ALLAH=1. And thank Heaven for, I was able to generate the table for each case.
The fact that each case based on this Method did not yield a 'False' statement, and that each case had roots and yielded recognizable functions, does seem to confirm that 'Allah-hu-Ahhad' ('God Is One'). (I have to insist it ain't me or any one of you, but I think We may have Met on my worse day on that 'flippin' planet'!). It is interesting to note that each curve varies in the same way given that they all have the same derivatives.

Euler's number also appears when expressing curves in polar coordinates (r,t) and is analogous to expressing a complex number.

A logarithmic, or growth spiral which often appears in nature (spiral galaxies, a hawk' s path toward its prey, the nautilus shell, or a moth' s approach to a flame) when expressed in polar coordinates (r,t) is given by:

where a & b are Real numbers

Parametrically, the curve is given by:


A 3-D spiral, or Helix, is formed when a planar spiral is pulled along a third axis, z;

z=t; 0≤t ≤2π

The Helix is not the only type of spiral found in nature, others include the 'Whorl' and the 'Vortex.' All this means is that the World 'Screwing Around' is nothing new, since not only does the Earth Orbit the Sun along the Ecliptic Plane, but it is following the Sun as it hurtles through space on its way to toward the Solar Apex-what is, essentially, a Helical orbit.

The image below is taken from the book Helical Helix : Solar System a Dynamic Process byDr. Pallathadka Keshava Bhat (comments added).

As of 2010 the concept that the entire Solar System moves in a vortex trajectory like a snake moving in 3 dimensions (the 'Solar Snake') instead of a flat plane is not the way it is being taught in our schools. (In fact, in some schools they are still arguing Creationism v Evolution, as if this is even an argument).
As the "Solar Snake" moves either above or below the Galactic Ecliptic (about 26,000 years), it produces the illusion of the 'precession of the equinoxes' or what is described as a wobble in the Earth's polar axis. (At least this is how it appears from the 'Accessible' perspective of Earth's inhabitants, or 'Lower Dimension', than when viewed from outside the Solar System, or the 'Remote' viewpoint from a 'Higher Dimension').
The 'Solar Snake' is creating the so-called wobble due to it's vortical trajectory above and below the Galactic Ecliptic. The Ancients, including but not limited to the Maya, Inca, Egyptians, Indians, Indonesians and Native Americans, knew this and represented the Solar Force as Serpents or twin opposing serpents (twin opposing vortices).

The precession of Earth's axis explanation by way of this 'higher dimensional' viewpoint has interesting ramifications when extending this 'as above, so below' observation to the way information propagates throughout the Creation.
One example is in the similarity between the shape traced by solar system's helical path on the Galactic level compared to sperm on the microscopic level and genetic material (DNA) itself at the atomic level. (I know, this all gives new meaning to the notion of 'panspermia').

And while the examples of the snapshot of the helical solar system, the movement of sperm flagellum and the structure of the DNA molecule are physical 'turns' like that of a screw, this type of displacement is strongly reminiscent of elelctro-magnetic wave propagation.
This 'Reverberating Pattern' transmitted across not only in magnitude (from the Galactic to the Microscopic) but also across forms of Matter and Energy (Movement) cannot be ignored; it is very much like the self-similarity seen in Fractals! The very fact that this self-similarity exists so widely in the Universe must have meaning of 'Grave' (-ity) importance! (Did I scare you?--to know You're Mom died?---You'll be okay, because it's still not as bad as your driving scares me! So, remember to drive safely...(well-timed, aptly-titled JumpStart comics appeared 01.03.2011 in the LATimes; is the 'grammar nazi' comment a personal attack?
The Helix also presents as a screw dislocation as it propagates in crystal growth. Screw dislocations are an important mechanism in crystal growth and appear as distinct spirals under high magnification. Turns out, where there is a surface “step” the atoms can more easily add to the crystal, and the surface step associated with the screw dislocation is never destroyed no matter how many atoms are added to it. (The arithmetic description of a ‘step’ is a ‘jump point’ or a ‘discontinuity’ caused by a function being ‘undefined’ at that point, like ‘divide by 0’)
Keeping all this in mind, about how matter holistically travels or imprints these helical paths, an electromagnetic vortex may have something more to add to our understanding of the phenomenon and, in turn, add to our understanding of the Gravity 'paradox' when formulating the Theory of Everything.
An optical vortex is created when light (a form of electromagnetic radiation) travels in the z direction with a phase front in the shape of a spiral. The intensity vanishes at the center (anyone think this is familiar to Gravity at the Center of the Earth where counter forces cancel each other out?) of the optical vortex where the phase is “undefined”.

The image of just such an optical vortex is
taken from a paper by Dr. Ulrich T Schwarz Research Group, Regensburg University. According to that paper;
“…’the threads of darkness’ are the optical analogue to screw dislocations in condensed matter physics. They are defined as topological entities and thus stable. A perturbation of the beam will shift the position of the vortex, but not destroy it (as it is impossible to smoothly merge a spiral staircase).”{…And I suppose this is where I am not to let my anger show and only wonder, ‘Why can’t God Imagine better people?!’} Ok, Enough with the EMO stuff.

Moving right along, we take a second look at where we started:
وَالسَّمَاء وَالطَّارِقِ 86:1
86:1 By the Sky (Heavens) and the Tar(i)q(i)
Which is not to say we are right back where we started because all this has happened in the meantime, so it's not exactly the exact same beginning although it looks strangely familiar (Déjà vu, anyone?--Oh, the French, we'll get to them later...).
The reason the word Tariq has so many associations is probably because when it was first introduced into the Arabic vernacular people thought it quite foreign and they tried to make sense of it through context of what followed, which seems to be about the Sky and the Stars. But the words for Stars and Revelation are homonyms, so 86:3 may be just as likely talking about a Piercing Star as it is a Groundbreaking Revelation. Which brings us to the French.
French is a Romance language and derives from the ancient Latin. It is also known as one of the group of "oïl languages" which share the common word oïl (oui) for yes.
{This is where I have to digress and relate a story about a traveller who shared a (conventional) flight with me and was seated next to me on a business trip some time in the long distant past. As we were flying over New Mexico he became quite chatty about how beautiful it looked from the air and he said that travelling widely for his work had taught him to appreciate all kinds of beauty that is not so apparent to the rest of us just walking around because of the special vantage he had when hearing people speak. He was a Linguist and his work required that he find ways to preserve human language and discover how it evolved, which gave him a wider lexicon than the rest of us and an entirely different way of understanding the World.}
Consider the word الطَّارِقِ (at-tariq) first introduced in 86:1 and then followed by the rhetorical question:
86:2 "And How Would You Know (Who Taught You, or, How Can You Be Able to get the Concept of ) what is the Tariq?"
To someone like my travel buddy, sometime in the '80's, it was probably a no-brainer; but to the rest of us, (or me anyway), it's not incredibly obvious. The nearest Arabic word to it would be طرق "method(s)" similar words mean 'through' or 'way' and secondary meanings are 'knocking, beating, pounding, enter, etc.' It is interesting that an alternative word for 'methods' is الأساليب "al-asaleeb" which sounds a great deal like "the Crosses" ( الصليب the Cross, Crucifixion).
A great many of us know about the story of Jesus and the Resurrection and associate His time on 'the Cross' as a time of 'Persecution' or great 'Torture'. So, we would not be surprised to find that the Latin word torquere, ('to torment'), can remind some of us of the word 'Torque', ('to twist'). And we find even more associations to the Jesus story in similar Latin words like 'Torques", (meaning; 'a chain of,' 'collar,' 'yoke', 'garland', 'necklace', 'gorget', 'chain') and the closely associated 'Torquis' (meaning; 'twisted necklace').
All this is made more interesting when we find that the word 'tarch' or 'arch' is 'Ark' in French, and sounds like 'tark' in Estonian for 'wise, astute, sapient, intelligent...' Which brings us around full circle to the word in Farsi (Persian) for torque, where it's taken directly from the contemporary word and its modern-day meaning; 'torque' transcribed in the Arabic/Persian alphabet as 'تورک' (tork), which looks and sounds a lot like the definitive طَّارِقِ (allowing for variations due to trascription from one laguage to another).
The Answer to the rhetorical Question posed in 86:2 (although none is needed) is:
"Certainly Not Google language tools!"
Since a recent translation of 86:2 using Google Language Tools yielded the following interpretation:
"Everybody Knows What Tariq (86:2)" (is).
{I would be convinced otherwise if every government agency had not at one time or another visited this site, even by way of Malaysia and Siberia, ok? I know yousguys tamper with it, because I would not expect those visitors' default language to be English.}
Essentially, this is what it is:
86:3 An-najm al-thaqib86:3 The Piercing Star; A Penetrating/Acute/Sagacious Revelation (acute/keen/boring (like with a hand-drill/perforator Insight/Awakening)
86:4 Inna Kulla Nafsitin llamma 3aliha hhafizhu86:4 Indeed (Truly) Every Being (Soul) Has Overseeing it (Over It) A Saviour (Keeper, Guardian, Protector, Watcher)
86:5 Fayuntthiri al-insannu mimma khulliq86:5 So let (such that) the Human Being See (Understands) Of What (they) Are Created
86:6 Khuliqin min ma'in daffiqin86:6 Created From/Of Watery Effluent (stream, race, influx, afflux, outpouring, inflow/outflow, pouring, pumping, effluence...)
86:7 Yakhruju min bayina al-ssalbi wa al-ttara'ibi86:7 Coming out (emerging, exiting, surfacing) from/of between the 'al-ssalbi' (n.; steel, Crux, crucifixion, loin, ramrod, cross-cut, betterment. v.; crucify, harden, consolidate, stiffen, temper, call out, sear, castigate. adj.; solid, firm, rigid, tough, stiff, concrete, robust, metallic, adamant, unyielding, inflexible, unkind, stark, horny, stubborn, tenacious, resistant, solidity, cruel, corneous, persistent, callous, stabile, out-and-out) and 'al-'ttaraa'ibi' (Alterae (Altair); var. رأب ra'b, bridge, garage; al-tara ibi (ForeFathers' + White Flyers).

The Northern Cross, or Cygnus (The Swan)__Backbone of the Milky Way
On one level 86:6 and 86:7 may very well be a lesson in biology and procreation, but the text is troubling when we have to try and figure out what the exact body part the 'effluent' or 'flood, stream' is emerging from between given that the words 'al-ssalbi' and 'al-ttara'ibi' don't, as far as I can find, correspond to any widely accepted body parts as such. They have been variously interpreted to mean the ribs, loins, or rib cage and the pelvic arch, but this seems contrived.
So, for now, we have to look elsewhere in our Universe to make a more Immediate Connection.
The Galactic Plane (Equator) runs right through the Northern Cross (Cygus, The Swan) where the asterism Fixed by Deneb and Albiero forms the long ‘Staff’ of The Cross; and Delta Cygni, Sadr, and Gibeah form its outstretched 'arms' (cross-bars).
Sighting the Northern Cross is the same as looking directly into the Milky Way disk since it floats like a backbone of the Milky Way – the luminous River of Stars passing through the Northern Cross and stretching across the sky. On clear, dark Winter nights this “milky” swath of sky appears as a hazy band and is really an edgewise view of the Galactic Disk.
The "Summer Triangle" likewise fixes the orientation of the Milky Way Galaxy as it flows out through one of the 'legs' formed by Vega and Altair, with Deneb at its western Apex.
Each of these stars in the Northern Cross and the Summer Triangle are associated with avian Constellations: Deneb and Albiero and Delta Cygni mark the Tail, Beak, and Wing of The Swan, respectively; Gieneh, means 'wing' in Arabic and it represents a wing of Corvus, the Crow/Raven (Gamma Corvi), and the eastern outstretched 'wing' of Cygnus the Swan (Epsilon Cygni); Vega is the brightest star in Lyra associated with the mythic Eagle/Vulture that swoops between Heaven and Earth; and Altair is a main star in the Aquila and derives its name from al-nasr-al-ta'er, 'The Flying Eagle', of legend; and finally, Sadr at the center of the crossbar in the Northern Cross is from the 'Breast' of the Hen.
So, while anatomically we may yet have a few more things to learn about human physiology, it would seem what we know about the Cosmos provides a more fitting depiction of what Signs 86:6 and 86:7 are on about. Since it would seem to indicate that the flow of the Milky Way through these fixed points has something to do with Birth or Resurrection. A Birth that our Ancient Forefathers noticed on a Galactic Time Scale in its streaming through what is alluded to as a 'bridge' associated with these 'White Flyers'. It's not surprising that Deneb, Vega and Altair are all white Class A stars related to Birds.
A 'bridge' connects or links places and it also enables communication (a span across a physical barrier or Time) and the secondary meaning of the word رأب ra'b being 'garage' may lead to the inference that the destination is a place for rest and repair, whether it be for cars, stars, UFOs, or people.
86:8 Innahu 3alla raj3ihi laqadirun
86:8 (Indeed) He Is Able (Has Power Over, Can Cause) him (the Human) to Return (Revert, Come Back, Repeat, Reinstate, Snap back, pass back, reestablish, send back, report back, resound) (aside, is that like a threat?)
86:9 Yawma tubla assara'iru
86:9 On A Day Secrets Are Displayed (laid bare, put forth, the understanding is that they are 'worn out' for all to see, like clothing, with all connotations including 'bearing them' and their being 'erased' and/or 'worn down' in that there will be no way to make excuses for them)
86:10 Fammaluhu min quwatin wala nassirin
86:10 He will have no other Helper/Power/Force or Friend/Patron/Benefactor
86:11 Wa al.sama' tthati alraj3i
86:11 And the Sky (Heaven) Relates the Feedback
86:12 Wa al.ardi tthati alssud3i
86:12 And the Earth Relates the Rift/Flaw/Fissure/Crack/Separation/Segregation...
86:13 Innahu laqawln fasln
86:13 (Indeed) He Can Say (Utter, Call, View, Determine) the Decision (separation season segregation division disconnection isolation severance seclusion demarcation disjunction parting dissociation breakthrough displacement; v., separate disconnect detach break dismiss fire segregate remove adjudicate cut isolate determine decide settle cut off detail sever resolve divide disentangle dissociate sack discharge judge partition lay off set apart expel part itemize wean displace adjudge dissociate oneself from disunite disjoin arrive at a resolution cashier
render a judgment particularize elaborate seclude make a decision sunder have a final word)

86:14 Wammahua b'al hazli
86:14 And He Is Not Kidding (not joking); He Is Serious
86:15 Anahum yakeydoo'na kaeidana
86:15 They are scheming a plot (They plot a plot; plan an artifice, machination, scheme, trick)
86:16 Wa'akeydo kaeidana
86:16 And I Plot a Plot...
86:17 Famahali al.kafireen amhalhum ru'wayidana
86:17 Allow Time for the Disbelievers (Unfaithful)--Take It Easy On Them (Deal Gently with them, Allow (them) Time for A Reprieve).
...I can't believe You; they have the supercomputers, a nuclear arsenal, all the national labs, and their sanity & You Tell me to take it easy on them?! Fine. Here, I Plot a Plot!

PS Henry, I was kind o'hope'n you took off with the quantitative maths while I perigrinated the qualitative aspects. The Chapter:Verse may be the actual parameters for the steps. For example, Sign(1:1) would indicate use variables=1 and result=1 ; and test for True. The example given for الله above yields:
; ; ...
True; True; True...