Off The Page

'Unknown and Undefined do not mean Non-Existent'.--That would be Me that said that.

Divide by Zero, or the Step Function, or a Cusp...

I tried not to bring G-Ma into it, but that I managed to avoid it for this long is to be commended.

{You Know, Dad, You can just make it easy for me at this point and just come pick me up, instead of listening to what Rich has to say about it and have to go by His indefinite timeline--He will never figure out the James Dean angle!!--and I don't need to do any more bad math! BTW I take it not everyone else can hear You and they think I am talking to my inner voice, which sounds suspiciously a lot like my Dad and not Malcolm.

Seeing how everyone is so focused on what 'she looks like' I take it they are less than impressed with my arithmetic skills so I get the booby prize of being a real beauty! {I'm playing witch you; I know You toss a real looker their way every time He asks 'how does she look' --nothing says we can't have fun, t00; after all what r friends for? 

Addendum 11.03.2012   But after the incident at the bus stop yesterday 11.03.2012 at around 4:30 pm (NW corner Olympic/WestWood blvd); He can't toss enough red heads my way to cheer me up--a Gross misinterpretation of 'bring the car around!' and flagrant abuse of what we are on about here 
(; _ ;)
}

And seeing how it's Ramadan and all, I actually had a brief  tête-à-tête  with G_D and after Seeing how banged up my knees are from millennia of praying, all He did was make sure I got a good night's sleep and assured me next time I would get a proper burial---How when the earth spits me back out faster than they can toss me in--are We thinking burial at sea, just so we can watch'm boil over?--Why would I think It was G_D in the First Place? Well, all We had to talk about was cars and the weather; and He actually liked my cooking, which means He had to be around long enough (like an Eternity) to have tasted worse!}

Surah 19 Mariam 

وَاذْكُرْ فِي الْكِتَابِ مَرْيَمَ إِذِ انتَبَذَتْ مِنْ أَهْلِهَا مَكَانًا شَرْقِيًّا (19:16   
19: 16 And Remember In the Scripture, Mariam (PBUH), when She withdrew from Her family (community) to an Eastern Place.

فَاتَّخَذَتْ مِن دُونِهِمْ حِجَابًا
 فَأَرْسَلْنَا إِلَيْهَا رُوحَنَا 
فَتَمَثَّلَ لَهَا بَشَرًا سَوِيًّا (19:17
19:17 And secluded Herself from them (lit. She Veiled Herself exclusive of them//screened Herself from them) and We Sent Her Our Spirit, and He Appeared to Her as a Perfected//Flawless Man.

فَأَجَاءهَا الْمَخَاضُ إِلَى جِذْعِ النَّخْلَةِ
 قَالَتْ يَا لَيْتَنِي مِتُّ قَبْلَ هَذَا
 وَكُنتُ نَسْيًا مَّنسِيًّا (19:23 
19:23 When the throes of childbirth drove Her to the trunk of palm tree, She said, "Would that I Died before this and were Nothing! Forgotten!" 
(Aww, Gramma! You all didn't Know She was the First Dark Matter/Dark Energy Physicist? Because we all know Nothing Last Forever!)

The reference to the 'heavens' being rife with interwoven/knitted /meshed paths and similar entangled/juxtaposed tracts on Earth occurs with some regularity in the Quran. It was mentioned in passing in earlier posts and the Signs were tabulated in 'Is the Sky a Hologram' and expounded in 'My Turn to Say...' and most recently in 'Ah- A Question', and specifically Sign 13:4.

The precept is that Divine Knowledge is often Revealed using Metaphors, Similes, or Allegories as the most accessible Method to Impart 'New Information' among a populace that has yet to develop the language for understanding the 'News.' {Like, I still can't wrap my head around how I got into the Ka3'ba.}

So it stands to Reason, that the Source (read G_O_D--Who apparently has more advanced degrees than I have) is choosing the metaphors and Language of the Revelation very carefully. 

{A new first novel, Alif,  by Willow Wilson is a very funny fast-paced story that seamlessly incorporates this concept into the narrative--best not to read it in public unless you don't mind astonished onlookers' attention when they hear you laughing in fits and starts.}

Having touched on the subject of fabrics, knitting, and the very 'construction' of the Sky, some more insights relating to the 'close-knit' relationship are worthy of further investigation.

The Orbit of the Moon around the Sun as Arithmetically Described in Sign 75:9 resulted in a degenerate conic that was not 'closed', in math parlance, it appeared to have a 'discontinuity' (i.e., per posets, the ends superma and infima do not join or meet, respectively) although we are all familiar with the fact that the Moon travels in a complete orbit around the Earth and around the Sun.

ends

1. the last part or extremity, lengthwise, of something that is longer than it is wide or broad;  
2. a point, line or limitation that indicates the full extent or degree of something. 
3. To terminate, kill;
4. indicate the ultimate example of something.

Moon's Orbit Described in Sign 75:9 vs. Knitting

Compare this to the shape a knit (or purl) loop takes when first formed. Prior to  the yarn (thread) being added to the 'stitches' along the knitting needle the 'open loop' structure of the knit (or purl) stitches is very much akin to the Orbit of the Moon as described just before they are 'stacked' (fall through) to the lower rows once  the 'new stitch' is added to the row on the 'needle'. {Note that the knit (or purl) stitches are stacked in 2's to form the knitted row beneath the new stitch (2 levels/tiers under==past/ago), in a conventional knit pattern (or purl pattern on the backside of the same fabric--the 'purl stitch is essentially the 'dual' or 'reverse' of a knit stitch)}.

And the associated Helix is clearly defined by the row of stitches 3 Tiers below (3 layers/levels under) the new stitches riding the needle.

Sighting along an imaginary central axis of this helix gives a sense of what exactly is happening along the 'undefined' region or 'step' formed by the open end of the yarn of one loop as it merges to form the beginning of adjacent loop, and how this 'step', 'gap', or 'discontinuity', is 'distorted' by the amount of 'tension' the yarn is under. 

Pulling the needles apart, joining them when knitting, or simply pulling the yarn, serves to widen or reduce the 'space' between the loop ends as well as between the individual coils of yarn in a manner akin to that of a Slinky toy.

Do You Know What Time it is?

Lio by Mark Tatulli
August 27, 2012Pulling â Rabbit Out of â Hât  ^¸^

...time for Me to Pull another Rabbit out of my proverbial hat ^  by weaving a story that would enrapture the kids about needles, and flying shuttles, spinners and looms; and how they relate to Gravity and Time and Space--the very Intercalation of Higher Dimensional Objects into the Lower Dimensional Realms.

And if I were to do that, what can I look forward to as a reward for my even making such an attempt? 

A derring-do that could go one of 3 ways for me:

Baby Blues Comic Strip July 29, 2012
'Mom's Could Hear Aster*sks'

1. I succeed and make everyone finally believe in Angels, proving that I am unlike any w*ore they know (yes, I heard) in that I deserve to get paid (handsomely) without having to get Laid to do it (in my grave or otherwise)! or
2. I fail, and Firmly but Seamlessly Establish my place as the Peripatetic Comic Relief to 'You Know Whose Company I Been Keeping', (based on past experience, this may be the more likely of the 2 Outcomes, but I get used to that--ever the optimist, choosing to believe past performance is not a guarantee of future results).
3. Either of the above 2 mentions and You Tell me it's all been done before,  and I finally get to find out Who keeps asking, 'Who is She?' (Apparently I don't leave much of an impression either coming or going--indelible or otherwise).

Surah 2 Al.Baqara (The Cow)
أُحِلَّ لَكُمْ لَيْلَةَ الصِّيَامِ الرَّفَثُ إِلَى نِسَآئِكُمْ
 هُنَّ لِبَاسٌ لَّكُمْ وَأَنتُمْ لِبَاسٌ لَّهُنَّ
 عَلِمَ اللّهُ أَنَّكُمْ كُنتُمْ تَخْتانُونَ أَنفُسَكُمْ فَتَابَ عَلَيْكُمْ
 وَعَفَا عَنكُمْ
 فَالآنَ بَاشِرُوهُنَّ وَابْتَغُواْ مَا كَتَبَ اللّهُ لَكُمْ
 وَكُلُواْ وَاشْرَبُواْ حَتَّى يَتَبَيَّنَ لَكُمُ
 الْخَيْطُ الأَبْيَضُ مِنَ الْخَيْطِ الأَسْوَدِ مِنَ الْفَجْرِ
 ثُمَّ أَتِمُّواْ الصِّيَامَ إِلَى الَّليْلِ
 وَلاَ تُبَاشِرُوهُنَّ وَأَنتُمْ عَاكِفُونَ فِي الْمَسَاجِدِ
 تِلْكَ حُدُودُ اللّهِ فَلاَ تَقْرَبُوهَا
 كَذَلِكَ يُبَيِّنُ اللّهُ آيَاتِهِ لِلنَّاسِ
 لَعَلَّهُمْ يَتَّقُونَ (2:187  

2:187 Permitted to you, on the night of the fasts, is the approach to your (women). 
( from أُحِلَّ  ==parses out to "A Solution", taken together means 'lawful; permitted'; the word حِلَّ means; solve, resolve, untangle, uncoil, ravel out, unbend, unhook, unclasp, acquit, untwine, clear-cut, detach, disband, fix, loose, exonerate, solvent, solution...
from رفث  ==sexual contact (here, I'll break it down for you in case some of you don't remember where you came from== the mating of a male part with the female part, and does not mean 'obscenity' <> obscene;)
They are (as) your garments and you are (as)  their garments.  
(from لِبَاسٌ ==garments/clothing/cover/robe, etc.;  also means: endurance, responsive, perfusion==passivation/easement/paving/continuous injection, spray...)  
God Knows what you used to do secretly among yourselves, but He Turns to you and Forgives you. 
So now associate with them, and seek what God Ordains for you;
(from بَاشِرُ, explicitly stating it's okay to 'man your women' on the nights during fasting, so, there is nothing bad or obscene about it since it's human nature (the ar. word 'bashr' also means 'human'), the word also alludes to the physical movements/activity associated with sexual intercourse== 'rasping' (not raping), grating, intone/say in glowing/excitement/thrilling sound, buoyancy, 'twill' and moreover
from ابْتَغُو one should 'desire', 'wish', 'want' and 'seek'; noun means 'archer,' 'thrower,' 'shooter'...  what God has taken the trouble to write out for you (Ordains)).
and eat and drink, until it (becomes) obvious to you (apparent to you, you can distinguish between)
the white thread from the black thread of daybreak (sunrise, dawn). 
(from الْخَيْطُ ==thread, fibre, yarn, tale, story; also means: governing wire, fuse, wire, silica hair; cord, tendon, ligament; deviation, bend, distortion, trick; series (سلسلة), queue, line, row; strain, vaccine 'repository' (warehouse, depot, storage, depository,  magazine where vaccine is stored/resides, var. ammunition, 'secrets silencer', storehouse, storeroom. 
from الْفَجْرِ== daybreak, dawn, morn, sunrise, daylight; also means, rooster or its call at daybreak, cock, weapon/arm/armed, tap/faucet, rod; verb, promote, support, reinforce, enhanced, forces... ) 
Then complete (resume) your fast (abstinence) until the nightfall.
(from أَتِمُّو meaning to accomplish, complete, fulfill, consummate; also means 'compass', 'cap', 'round off,' 'perfect.')

One way to put this together (see System Analogies) is by taking the meanings  the male and female being the equivalent plug/receptacle in an electric system: allow continuous injection (or passivation) as it applies to an electric circuit (or capacitor or fibre optic, since there is a reference to a glass filament); so that 'eat and drink' means consumption (usage), and continue in the 'excitable' /pumping state until you can clearly distinguish the black wire (in North America, the convention for black-clad wires are phase wires, governing wire) from the white one (in North America the convention white clad wiring is used for neutral) of the fajr (primarily means sunrise, but also means 'onset' or even 'detonation'). (Put Your 2 cents here, Dear)

And do not associate with them while you are in retreat in the Mosques. 
(Clearly, this means do not have sexual contact when in the mosques;
 from عَاكِفُو the actual position of being 'bent' or 'stooped' down to pray, with further instruction to my fine-feathered friends 'while you are in a stoop during predation,'  going after prey or pursuit of a victim, slain or slaughter, murdered or prey. (My Sakar was so sick of a steady diet of insects-->Somthing Had To Be Done!).
and from الْمَسَاجِدِ at the human level meaning Mosques or places where one 'kneels down/prostrates' in prayer, 
the root سَجِدِ s.j.d. means 'prostrate,' 'kneeling,' 'bowing'; 
and this part is for the birds; whilst 'stretched' ((wings) outstretched/expanded, procumbent); without force; weak, feeble, atonal...) 
These are Limits (min =2, max = 187; and as above) set by God, so Do Not breach (curb, infringe, besmear, get proximate to) them-- 
In this manner, God Illuminates His Signs to people (Nass, communities),
that they may be strengthened/braced/learn self-restraint/restored/renewed/healed/recovered/become devoted...


Surah 16 An.Nahl (Bees)
وَلاَ تَكُونُواْ كَالَّتِي نَقَضَتْ غَزْلَهَا مِن بَعْدِ قُوَّةٍ
 أَنكَاثًا تَتَّخِذُونَ أَيْمَانَكُمْ دَخَلاً بَيْنَكُمْ
 أَن تَكُونَ أُمَّةٌ هِيَ أَرْبَى مِنْ أُمَّةٍ
 إِنَّمَا يَبْلُوكُمُ اللّهُ بِهِ
 وَلَيُبَيِّنَنَّ لَكُمْ يَوْمَ الْقِيَامَةِ مَا كُنتُمْ فِيهِ تَخْتَلِفُونَ (16:92

16:92  And be not like unto her who unravels the thread, after she has made it strong, to thin filaments, making your oaths a deceit between you because of a nation being more numerous than (another) nation. God only tries you thereby, and He verily will explain to you on the Day of Resurrection that wherein you differed.


Surah 20  Ta. Ha
الَّذِي جَعَلَ لَكُمُ الْأَرْضَ مَهْدًا
 وَسَلَكَ لَكُمْ فِيهَا سُبُلًا
 وَأَنزَلَ مِنَ السَّمَاء مَاء
 فَأَخْرَجْنَا بِهِ أَزْوَاجًا مِّن نَّبَاتٍ شَتَّى 

(20:53) 

20:53 He Created the Earth for you a Cradle (starting place, as an explanation, even, level, smooth, a place to be nurtured, a surface to support you, serve as a framework)
and threaded for you throughout her (the Earth) Ways (to travel)
and Brought Down Rain from the Sky
and with it We Brought about Pairs (Couples, Twinned Mates) of Various (Diverse) Plants

Surah 71 Noah
لِتَسْلُكُوا مِنْهَا سُبُلًا فِجَاجًا (71:20  
71:20 (Surely, So That) You (can, will, may) thread your ways (travel)  throughout her (the Earth's) valleys/ravines {This being the traditional, classical interpretation}
71:20  You shall make your ways from her (on her, the Earth) as (in) gages (pledges, like a security against an unpaid debt, or a thrown gauntlet, a challenge to fight 
{ An interesting secondary meaning} or more importantly 'Carinae' (as though traveling through or in the constellation  جاجئ, a solar system in the Constellation Carina (southern part of Argo Navis)}

Surah 113 Al.Falaq
وَمِن شَرِّ النَّفَّاثَاتِ فِي الْعُقَدِ (113:4
113: 4 And (I Seek Refuge In the Lord) from the Evil (of those who cast spells) by blowing on Knots
(this being the most traditional interpretation of 113:4)

Tie-ing all together:

From 113:4 (touched on previously in Morning, Noon and Night) and since it is no Big Secret that I never learned that trick, being that I am such a poor speller and all (I think it's a genetic defect), I have to find other ways this may apply to what we are on about here.

From  شَرِّ, 'shari', a root that means evil or inequity (-tous) and can be used as prefix meaning (mal-, as in malevolent), the indication is that we seek protection or a fix for something that is 'harmful' (deliberately causes harm, injustice, sin, immorality)

Al.nafathati means jets as in aircraft, planes, things that 'spurt', the root n.f.th essentially meaning 'breath' or 'breathe' and all of the following connotations: air intake/breathing (inhale/exhale)/draining, exhausting//use up completely, let out waste// (physics) remove gas to create a vaccuum//drive out/banish//expel//propulsive//expectoration, from cough up or spit out//publish/burst//launch//erupt//volcano//utter/said


 'Fi',   فِي , is a preposition, or adverb typically indicating a state of being within something, or during, after, consequent to, etc.

And al.3uqd,  الْعُقَدِ , as a noun means anything that arches overhead, a vaulted ceiling, the Sky or in anatomy, an arched part of the body. The root 3.q.d., includes all of the following connotations: a contract/10-year lease or agreement//knots//knitting//hitch/marriage/promise/deed/work/instrument//complicate/congeal/muddy; and from vault/jump in an arithmetic sense we get a discontinuity/the point at which a function or curve undergoes a major transition.

{To the Ones saying 'Thank You,' I say, 'yer welcome' ; to the One who says 'Sorry, or 'I feel sorry,' I says, 'we don't need your pity, this is not a pity party, remember?' 
But basically what it is is that now I have developed my language skills to such an effect that He (That Means God) finds it acceptable that I can now use the 'c ' word as both subject and predicate in the same sentence, {yes, I know, 'less than charming'--but, then, I am not the one who is the Prince around here!}; and He Allows my expressing this kind of sentiment around Him about 'them', apparently I have learned to 'call'em as I sees'em!'}

The Material Facts
While much of what follows appears to be headed in a tangential direction, the goal is to establish the probability of someone committing a crime based on just what they are wearing or habitually tend to wear as far as clothing, while I continue to build a Spaceship to replace the one You downed in 1997...

Fabric Structure
Definitions:

fabric can mean and is used to refer to any and/or all of the following:
noun

1. a type of cloth made of woven or knitted threads or fibers; the actual texture or quality of the cloth;
2. the fundamental structure or makeup of something;
3. the structural material itself of what is being constructed, be it the brickwork, masonry, tiling, superstructure, roofing, etc.

thread

n.

• fine twisted cord made of two or more twisted fibers; 
• ridge on a screw that is a continuous helical ridge; 
• very thin strip or fine strand of material, liquid or gas; 
• spider’s web filament;  
• a continuous unifying element running through a story, argument, discussion or series of events; 
• course of human life; the vein of ore

v.

• trans. v.  to pass something through a hole or gap in something else (another form of intercalation); 
• to string beads or pearls on a thread or wire; 
• tans. and intransitive v to go carefully along a winding course;  
• trans. v. to disperse something, distribute it at intervals in something else; (ie, intercalation of 'Angels' or 'Demons' or 'Spirit')
• trans. v. to produce a thread on a screw or bolt, or in the material receiving a screw or bolt

yarn

n.

• strand of fiber, continuous, natural or synthetic, used in knitting or weaving;
• strand of glass or metal; 
• a long story (tall tale full of incredible events)

structure can mean and is used to refer to any and/or of the following:
noun

1. something built or erected
2. a system of interrelated parts that function as a whole
3. the way in which the system parts link or relate to each other
4. biology, part of a body or organism characterized by its shape and other properties
5. chemistry, the arrangement of atoms in a molecule
6. geology, the physical disposition of a rock or its mineral components

transitive verb

1. to organize or arrange something as a whole

weave can mean and is used to refer to any and/or all of the following:
verb

1. trans. and intrans. v.; to make cloth by interlacing threads vertically and horizontally, especially on a loom
2. trans. v.; to make something by interlacing strips or strands of material
3. trans. v.; to spin something such as a spider's web
4. trans. v.; to construct something by combining different parts, like in story
5. trans. v.; to introduce parts into something larger, like 'weaving characters into a plot'
6. intrans. v.; to move from side-to-side in a zigzag fashion

noun

1. the pattern itself in which something is woven

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

The Loom Is A Weaving Apparatus

Image and some excerpts from
Learning How to Weave with Weaving Today
©2011 Interweave Press LLC.

loom

noun

1. weaving apparatus made of a rectangular frame; the  basic purpose of any loom is to hold the warp threads aligned under tension to facilitate the interweaving of the weft threads (so that weaving can take place)
2.  middle part of an oar

int. v.

1. be about to happen, often in a threatening way; 
2. be seen as a large shape, indistinct and sometimes menacing

beam

1. breast beam 
• the horizontal crosspiece on the loom directly in front of the weaver. The woven cloth passes over the breast beam on its way to the cloth beam.
2. cloth beam
• is the (usually) cylindrical rotating crosspiece of wood (sometimes of metal) at the front of the loom on which the woven cloth is wound
3. back beam
• is the horizontal crosspiece on the back of the loom around which the warp passes on its way to the warp beam
4. warp beam
• is the (usually) cylindrical rotating crosspiece of wood (sometimes of metal) at the back of the loom on which the warp is wound

{Yet another system analogy: 

(And why not, if He can't keep His boots off the coffee table, no reason why I can't have more than one iron in the fire.) 

Much like the loom is a basic machine that holds warp threads aligned under tension and selecting treadles forms sheds to allow weaving of the weft threads (sett, weaving pattern, mesh) through the warp threads according to a sett pattern, an automobile transmission gear box may be thought of in a similar fashion where a small linear motion in the gear shift knob (analogous to selecting treadles) allows for the gear selector fork (shedding) that selects the dog gears to mesh with the selected gears on the drive shaft (the rod that governs the speed to the differential) (gear ratio/sett/weaving pattern)...

This may seem a bit of a stretch but it may help establish how to set the best 'tension' for such an operation. When the product is cloth or fabric the tension pertains to how the yarn is stretched under optimal control for a particular fiber and final product, and in machinery it is analogous to either pressure (fluids dynamics) and/or electric tension (U, or voltage in electro-motive systems).  

And just as the case for knitting and weaving getting the 'tension' just right is critical to making a good fabric or final product, establishing the proper 'tension' in analogous systems is also critical to optimizing its quality and producing a sound, stable functioning that is less prone to error. 

Since the loom is a Neolithic machine, one of the earliest machines employed by humans, it is natural that it serves as the basis for the many analogies found throughout the Quran (ie., between weaving/thread/and clothing and garments). 

The few references above include using such similes to a broad range of categories from marital relationships to delineating and making clear criteria for differentiating and developing things that humans have yet to ride. 

(Some Signs explained in detail are referenced above, a comprehensive table with as many such metaphors as I could find is shown below)...

A broad System Analogy between the loom (the Neolithic machine), the automobile (a contemporary machine) and any Machine (Machine-X) could go something like this: 

Elements of Automobile Transmission

Machine Analogies
Loom_Car_Machine-X

(I know, the smart ones figure out this is essentially an exercise in optimal matching, or match-making, and go out and find their True Love--the shortcut to getting this right-- but it only works for the one set (the Love Birds) and not the collective...that's right, that's why the directive so long ago and I paraphrase..."God's servant...don't F#%$k with Her...." but now that they have, what have We?)

And this constant background mumbling is not conducive to getting anything right: thanks for the help fellas!  

But, plugging right along...[despite the distraction where I am perched at the coffee shop bar table listening to some SteveH go on and on about the ravishing beauties he wants (all these USee LA Beauties, and saying my name for oomph to remind me not to take it personally (We should all learn a lesson here from that novel mentioned earlier, Alif, The Unseen, to 'never tell your name to a Jinn' since apparently they name drop it everywhere and make you wonder what exactly it is you are doing to make you so famous...I got bit by an ant at my desk today and didn't die from the allergic reaction or the nausea from the anxiety such a reaction may cause...did you get it on video? Is it viral? Is that why I am so famous?] 

In the machine analogies, tabulated above, comparing the loom, the car and any machine (or machination), an important part of actuation is making a pattern selection, it could be either choosing from fixed combinations such as the weaver's sett, or quasi random as one would encounter when driving under different conditions, or infinitely variable orders or sets as one would have from using random number generators. 

And making such a choice leads to predictable outcomes, or in a broader sense, provides a method to delineate or make a determination. The Quran mentions such criteria for clear-cut determinations in Sign 2:187 mentioned above, (there may be others, but this is the one that comes readily to mind. Let's have a cursory look at the numbers: 

First, each number:

2, 

187 is 2^7 + 59 and divides 67^2-1, has binary form 10111011 and in base 16 is a single repeating digit bb, and the quotient ring of the field QSqrt -187 has class 2.

Then using the contextual reference in the Sign's language that 'these are limits' to set 2 and 187 as actual max-min limits. 

Then consider the fraction 2/187:  is approximately 1.07% and has an exact form of repeating decimal period 16. 

Then consider the related 187 mod 2 (this approach was already used when setting clocks/timing in an earlier post, "Building a Mystery") is 1, and the modulo function for 187 mod 2, 3, 4, 5...etc is given by 1, 1, 3, 2, and so on. 

What this means to me (now that I am having trouble not taking it personally) is it gets my Steve to pipe down for a second. whew!

}

• Building Trades --a horizontal bearing member, as a joist or lintel.
• Engineering--a rigid member or structure supported at each end, subject to bending stresses from a direction perpendicular to its length.
• Nautical--a horizontal structural member, usually transverse, for supporting the decks and flats of a vessel; the extreme width of a vessel; the shank of an anchor.
• Aeronautics--the direction perpendicular to the plane of symmetry of an aircraft and outward from the side.
• v. to transmit rays or a signal in a particular direction or to a defined audience. 

heald
noun

1. the heald frame or harness is a rectangular frame to which a series of wires, called heddles or healds, are attached

heddles

n.

1. one of the sets of vertical cords or wires in the frame on a loom that guides the warp threads, the heddles are an integral part of a loom. Each heddle has an eye in the center through which the warp is threaded, and is used to separate the warp threads for the passage of the weft. Typically heddles are wires, nylon braid, string, or flat steel rods attached to (suspended from) the shafts of a loom. Because of the eye, heddles are sometimes called needles.

needle

1. see heddles

pick

1. textiles, a single pass of the weft  through a shed (on a loom); a pick is also called a shot 

shaft

1. a frame that holds the heddles on a loom; when a shaft is raised or lowered, all of the warp ends threaded through the heddles on that shaft are simultaneously raised or lowered. Harnass is sometimes used as a synonym for shaft although it originally was used for the name of the mounting that holds all the shafts (see heald)

shed

1. the opening made by raised and lowered warp threads; each weft pick passes through a shed (see weft; shot)

shuttle

noun

1. route taken, or vehicle used to travel frequently between places, often relatively near each other; 
2. same as Space Shuttle; 
3. going back-and-forth; 
4. weaving device that holds the weft thread and is used to pass it between the warp threads (a form of intercalation); 
5. spindle or bobbin, e.g. in tatting or netting, for the lower thread in a sewing machine

trans. and intrans. v.

1. go back and forth, or to cause something to move in this manner; 
2. to transport somebody or something by shuttle

spindle

n. 

1. a specially shaped rod for spinning thread with a notched end through which strands of natural fiber are drawn then twisted into thread and wound around the rod;  
2. thread spinning rod or device on a spinning wheel or spinning machine; 
3. a spindle-shaped piece of wood, like a table leg or baluster; 
4. spindle-shaped cell structure consisting of a network of microtubule fibers along which chromosomes are distributed and drawn apart during meiosis and mitosis; 
5. a spike for holding personal papers; 
6. a dragonfly;
7.  rotating rod or device such as a lathe, turntable, or door handle; 
8. the warning signal for boats consisting of a metal rod surmounted by a ball or lantern and attached to a rock or shoal

tans. v. 

1. to impale papers on a spindle; provide something with a spindle (s);

instr. v. 

1. botany, to grow a tall, slender, weak stem

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

{I think by now, regardless of who it is He finally finds out I am, He is pretty much convinced I plumbed the very depths of grief to get this far in my story...but I see how much fun He's having 'playing the Field' and I don't wanna wake Him up. I think He Thinks I am avoiding Him but the fact is I know to bug out when I see Him at Work--that, and the fact that 'trying to tell me' is not the same thing as actually telling me, because it leaves me so very sorry for something I am still struggling to come up with a reasonable explanation as to what exactly it could be that justifies having made me soo verryy sorry.}

...and sew it Begins...

Structure is the fundamental element in weaving and is established on the basis of the following 4 components:


 Fibre

• thread, yarn, or cloth
• botany, long narrow plant cell wall thickened with lignin, a major component in a plants supporting tissue, frequently found in the outer walls of a plant stems; a thin narrow plant root; anatomy, long, thin structure of body tissue (muscle cells or nerve cells).

 Color 

• property of objects depending on the light they reflect that is perceived as a visual sensation of red, blue, or other shades; not black or white 
• music, the quality of a particular sound
• physics, hypothetical quality of quarks that takes three forms, red, blue and green.
• mineralogy, the gold found in gravel or sand

Texture  
n.

• feel; 
• touch; 
• surface; 
• quality; 
• consistency; 
• grain

Sett (see Balanced Sett)

n.

• textiles, see sett discussion below;
• masonry, a rectangular stone paving block; 
• the burrow of a badger

The underlying weaving, the manner (pattern) in which the warp and weft are interlaced, both affects and is affected by (a reciprocating relationship, i.e., echo)  these components and serves to give the cloth its basic character. 

pattern

• a combination of designs, qualities, acts, tendencies, etc. that form a consistent characteristic arrangement. 

repeat 

• to say, utter, do or perform in the same manner again; echo

Warp threads typically run vertically and the weft runs horizontally, and the selvages of a fabric serve as important clues to its structure (by the same token consider looking at the 'non-fraying' ends of DNA and RNA strands for important clues relating to their structure).

warp 

transitive and intransitive v. 

• to become twisted or out of shape; 
• to deviate from its usual or correct course; (a similitude for achieving 'warp' drive--changing any one thing from an established course of doing business or standard operating procedure). 
• move a ship by pulling on ropes fastened to a dock or fixed buoy, or be moved this way

transitive verb

• textiles, to arrange threads to form the warp in a loom

n.

• a twist or distortion in something; 
• a deviation or perversion of mind or character;
• textiles, threads running lengthwise on a loom or in a piece of fabric (ie., all the threads aligned vertically on the loom before the weaving begins; a warp end is a single warp thread)

weft 

n.

• textiles, the horizontal threads of a woven fabric or a tapestry; (all the horizontal threads that interlace with the vertically aligned warp threads; biological analogy~the rungs of a DNA molecule (partial)); 
• the yarn used for the weft; a piece of woven fabric

selvage  

n.

• non-fraying edge of a fabric; 
• the edge or strip of material included when manufacturing objects to allow it to be handled; 
• lock plate through which the bolt of a lock passes; 
• rug fringe (on the ends of an Oriental rug)

Sett refers to the number of threads per inch of warp and weft (fiber or strands per unit length). 

Often, it is used to refer to the warp only and expressed in terms sett@14 epi (in inch) or if using a metric system, it is typically expressed over 10 cm. 

Sett is critical in establishing the quality of the cloth, whether it is richer, poorer, its performance and how it handles. {So, it makes one wonder, are most criminal offenders partial to fabrics that drape poorly (poor quality), like close-woven poly twills? An exercise in crime scene forensics that does not necessarily have to be limited to DNA evidence...You heard that right---how many criminals put on their fine tweed twills when committing an act of dubious merit?}

handle 

• textiles, the tactile quality of a fabric and how it can be managed (pleating, draping, etc.)

drape

•  the way a fabric hangs in loose folds on or over something to cover it

• to rest a part of the body casually (like drape an arm around the back of a chair); 

• noun, a curtain, a sterile cover used to protect a body during a medical procedure.

quality 

• the essential, distinctive characteristic of a thing; its fineness or grade of excellence

It is the 'Setting' that determines if the weave is Firm, Flexible, or Loose (Open).

A plain weave, one in which the warp and weft threads are interlaced over-and-under each other in an even (a ratio of 1:1), alternating, and regular pattern has the greatest number of intersections and makes for the strongest fabric. Theoretically, the weft will need a space of its own diameter in order to pass between each adjacent warp thread.

Intersection

•  mathematics, a common point or set of points common to two or more intersecting geometric figures;  a set of common elements that consists of all the elements common to two or more other sets, thus being the largest set contained in all of the others

• a  crossroads or crossing point where two or more points, roads or paths cross each other

• an overlapping between two things, like interests or positions. 

To determine an average sett, the thread is usually wrapped around a ruler over a span of 1 inch with very little deformation (neutral, little or no tension or slack), the windings touching, and a percentage (%) of the number of windings is taken depending on the weave. 

An average of 50% establishes a 'plain weave' (meaning 'simple' sett).

To Determine Sett(ing)

For example, if 8-10 windings (coils) of yarn wrap 1 inch of ruler, then 4 epi (50% of 8, nominally) establishes a 'plain sett'. This simple sett weave makes the cloth flexible and makes it strongest, since it has the greatest number of intersections--sometimes referred to as 'tabby'.

flexible  able to bend repeatedly without damage or breaking; adaptable; subject to influence

strong robust, sturdy, not easily damaged or broken; chem. producing ions freely in solution; physically powerful, able to exert or withstand considerable force

A higher percentage establishes a tighter weave, so that in this example 12 epi sett (125% of 8) would make the cloth stiffer, what's called a 'firm sett'. 

firm  unyielding to touch, able to withstand contact without deformation, compact and solid when pressed; secure; trustworthy; steady, non-fluctuating; a commercial group or organization.

And a lower setting like 0.25*(8), or 2 epi for this example, establishes an 'open sett', with more space between the warp and weft and produces what is essentially a loose fabric.

loose (open)

•  not firmly attached, fastened, or fixed in place; slack, not pulled tight; free to move around with restraint; not compact or dense in texture or arrangement; imprecise, not exact, literal or tightly controlled or contained; to fire a missile; to untie a knot; sexually promiscuous.  
• (open) textiles, having small gaps or intervals between stitches or threads; public; unfolded or apart; mathematics, a mathematical interval that contains neight of a sets endpoints, a set that has at least one neighborhood of every point with the set, a set that is in a  complimentary relation to a set;  medicine, unblocked allowing free unobstructed passage; oceanography, free of ice or hazards; music, not closed off at the end, stopped by a finger or muted; meteorology, mild and free of frost; unlocked, unsecured, freely accessible; ready to do business; unlimited; to unfold or spread apart to reveal the inner part of something, to uncurl in order to spread apart or become fully developed.

Balance refers to the relationship between setts of warp and weft, and using the same yarn in both the warp and weft having the plain sett (i.e., the same number of warp and weft per inch, 2/2, 3/3, etc, See Figure below), produces a balanced cloth.

Point Paper Depicting a Plain Sett Weaving Pattern and Repeat

Point Paper (refer to the Figure above) is used to plan the weave pattern and a color convention or marking in the squares is used to show which thread is 'uppermost' (Top) to enable 'seeing' the interlacing, otherwise the point paper would simply show the warp and weft threads as lines in the boxes simply crossing without any way of know-how they interlace. (Since Uppermost and bottom (lowest) are used, they are convenient analogs to setting Limits). [An accessible analogy is that a route plan for a road trip in a car on serves the same purpose as point paper for a cloth.]

Top (Uppermost)

• the highest, uppermost, or loftiest point or part of anything; apex

Bottom (Lowest)

• the base or deepest part of anything, distinguished from the top; 
• pedestal, underside; 
• nautical,  the part of the hull between the bilges including the keel, the part of the hull that is immersed at all times; the cargo space in a vessel, or a cargo vessel; 
• geology, low alluvial land next to a river.

If the number of warp threads that float over and under the weft threads is not one-to-one, the setts are said to be unbalanced,  which affects the behavior of the cloth, and most often the color. 

float to move or rest on a surface (air, liquid, solid, or plasma); biology, an air bladder.

The weave is: warp-faced if there are more warp threads than weft threads per inch; weft-faced if the weave is 'widely-spaced' so that the weft predominates.

It is interesting to note that 'plain weave', while being simple to weave, is the most difficult to do by hand, or manually, since the yarn must be sett carefully, beamed with perfect tension, and woven with a constant beat. 

tension 

• the act of stretching or straining; physics, a force that pulls or stretches something, the stress resulting from a force of tension or a measure of it; electricity, the voltage or EMF (more on this since the idea is to try to set the ideal tension (voltage) for the purposes of this mystery object under consideration.)

beat 

• to pulsate, move or strike in natural short rhythmic patterns; to outperform; the usual route; the act or striking or the sound made when rhythmically striking one thing against another 

slack

•  not tight or pulled taught, having a great deal of give; not busy, unused, lax; moving slowly; not showing a great deal of care; unused potential; a stretch of still or slow moving water.

Usually, it requires at least 2 shafts on the loom, and in practice, finer yarns (smaller diameter fiber, strands) need 4 shafts to reduce friction on the threads and in order to avoid crowding the heddles.

friction physics, the resistance encountered when two object or bodies contact or rub against each other.

The Basketweave is a useful variation on the plain weave and when used in conjunction with other weaves, since it enables the same yarn to be used but with fewer intersections than plain weave, thereby reducing the risk of distortion.

To make a 'Cell' the warp and weft floats are arranged around a small plain weave area and form 'raised squares' ((s^2)^n) with 'sunken centers' (umbilics, see 'navel') once the fabric is taken off the loom and the yarn is allowed to 'contract.'

cell

notes:A separable Hausdorff space has cardinality at most 2^n , where n is the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases: if Y is a subset of X and z is a point of X, then z is in the closure of Y if and only if there exists a filter base B consisting of subsets of Y which converges to z. The cardinality of the set  of such filter bases is at most (2^2)^|Y|.  

Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection S(Y)-->X when  

The same arguments establish a more general result: suppose that a Hausdorff topological space X contains a dense subset of cardinality k. X has cardinality at most (2^2)^k and cardinality at most  2^k if it is first countable.

The product of at most continuum many separable spaces is a separable space (Willard 1970, p. 109, Th 16.4c). In particular the space 2^n of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality  2^n . More generally, if κ is any infinite cardinal, then a product of at most 2^κ spaces with dense subsets of size at most κ has itself a dense subset of size at most κ .(from wikipedia, see 'projective spaces' 'metric space' 'quaternions' cardinality...

Quaternions are a handy way to compute spatial rotations affording a more compact, quicker computation method than their more contemporary rival, matrix representations. 

They have enjoyed a reprisal for their utility in number theory due to their relationship with quadratic forms, and, since they are not susceptible to gimbal lock (unlike Euler angles), they are the preferred algorithms in computer control systems and robotics, including calculating orbital mechanics and space-craft attitude control systems.  

They were at one time used rigorously in geometry and physics since they enabled such topics as kinematics in space and Maxwell's equations to be described by vectors. When quaternions are used in geometry it is convenient to define them as a scalar (an object that transforms like a zero-order tensor) plus a vector (an object that transforms like a first-order tensor). 

a + bi + cj + dk = a + v 

Hamilton was first to literally 'carve out the formula' for quaternions. The story goes that he had the breakthrough as he was walking along a towpath with his wife on his way to a council meeting of the Royal Irish Academy. 

 

A formula he carved, (a 4-tuple with rules for multiplication and addition) into the stone of the Broughman Bridge along the way. Hamilton had written to friend describing his thoughts during the occasion, "And here there dawned on me the notion that ...in some sense (is) a fourth dimension of space for the purpose ofcalculating with triples..." 

The square root of the product of a quaternion with its conjugate is called its norm, denoted ||q||,  always a non-negative real number, and it is the same as the Euclidean norm on H considered as the vector space R⁴. 

Multiplying a quaternion by a real number,  α, scales its norm by the absolute value of the number; 

This is a special case of the fact that the norm is multiplicative, for any two quaternions p and q, meaning; 

Multiplicativity is a consequence of the formula for the conjugate of a product and it follows directly from the corresponding property of determinants of square matrices and the formula;
 

where i denotes the usual imaginary unit. 

This norm makes it possible to define the distance d(p, q) between p and qas the norm of their difference; 

which makes the Hamiltonian, H, into a metric space.  

Addition and multiplication are continuous in the metric topology. ref wikipedia.org)

This 'cellular structure', aptly called a 'honeycomb' is both 'visually effective' and offers a high degree of 'insulation' (depending on the fiber content, the fabric can trap either air or moisture). Ordinarily, honeycomb weaves require at least 4 shafts.

Grouping and Bundling...(read, Partially Ordered Sets, Poset)

group
noun

1. collection of people or things considered belonging together
2. military, a military formation made up of two or more battalions and a headquarters
3. air force, an air force formation made up of two or more squadrons, but smaller than a wing (I think the implication here is do not have sexual intercourse if ever your air force wing is out-stretched, or deployed, probably some added measure of caution?)
4. chemistry, a collection of atoms that is a distinct chemical unit, e.g. the hydroxy group; a set of chemical elements classified according to the vertical column they occupy in the periodic table. There are 18 such groups, and elements in the same group have similar properties.
5. geology, a collection of rock formations that date from the same geologic era and are considered as a stratigraphic unit
6. mathematics, a set of mathematical entities that are related by a particular operation. For example, consecutive numbers are a group under addition but not under multiplication.

trans. and instrans. v

1. to come together to form a group

adj.

1. relating to or forming a group

To form 'thicker' thread using the same yarn (fiber, string), the yarn is grouped or bundled in either one of 2 ways:

1. the yarn is entered on separate heddles on the same shaft and then sleyed normally to give a 'flat' band of warp threads or
2. the yarn is used as 1 thick thread on a heald which produces a 'rounded' cord, or 'rib', in the warp

Grouping adds 'weight' to the fabric, and is used for decorative elements as well as affecting the material's 'draping' qualities (never too late to simply just let me go shopping).

Grouping the thread in the weft makes for a stiffer 'fold' to the fabric when hung, and grouping the warp threads improves 'draping.'

Ribs, or cords, ride prominently on the surface of the fabric in either warp- or weft-ways.

cord

noun

1. thick string or thin rope
2. flexible, insulated, electrical cable
3. a unit of volume for cut timber, equal to 128 cu. ft (approximately 3.6 cu. m)
4. anatomy, a body part resembling a cord, e.g. spinal cord, umbilical cord

ribs
noun

1. anatomy, any of the curved bones extending from the vertebrae and in some cases meeting the sternum, forming a cavity housing vital organs in many vertebrates
2. handicraft, a portion of knitted material with raised vertical lines of stitches, made by alternating purl stitches with plain stitches
3. botany, a raised vein on a leaf
4. architecture, a ridge or molding on the underside of a vault or arched ceiling
5. nautical, a beam extending from the keel to the top of the hull of a ship, giving it its shape
6. aviation, a part of an aircraft's wing crossing from the leading to the trailing edge of the wing
7. a teasing comment or joke

transitive and intransitive verb 

1. to make playful teasing remarks to somebody

Weft-way (horizontal) cords, ie., from selvage-to-selvage, are also referred to as 'welts'. The structure differs for cords in several ways:

1. 2 warps may be used so that the one for 'floats' at the back of the cord (or rib) and kept at high tension on a separate loom, the other remaining 'slacker' and when woven emerges (appears) on the ribs' surface.
2. pooling 'picks' (the yarn is gathered together, take a 'subset') and then inserted so that both the Top and Bottom warps weave together to form an 'indentation' between the ribs
3. the 2 warps can be of the same quality so either can 'surface' or the backing can be a 'stronger' or 'finer' yarn that remains on the back of the fabric except between the ribs.

Warp-way (vertical) cords run 'parallel' to the warp and are formed by 'pairs' of weft  (horizontal) threads weaving into the warp ends of the 1st cord and then floating right under the next cord (rib), then the 2nd pair of 'picks' float under the first cord (rib) and weave into the next one. For a 'finer structure' the pairs of ends working in a plain weave usually 'lie' between the ribs, however, cords can also be woven without this pair of plain weave ends (read, superma and infima).

lie to recline, stretch out on a surface that is slanted or horizontal; deliberately say something untrue; a falsehood.

parallel  mathematics, lines, planes, or curves always the same distance apart and therefore never meet; computing, using several pieces of information simultaneously, example, 3CB calls subroutine [ReturnAssetValue]//returns the value of a function or object; music, keeping the same interval length throughout.

While 'plain weaves' need only 2 shafts, twills need at least 3 shafts. Simple interlacing is 'regular' when the same number of threads weave over and under each other, and is called 'even' (ie., 2/2, 3/3, etc.).

twill
noun

1. is a type of textile weave with a pattern of diagonal parallel ribs (in contrast with a satin and plain weave). This is done by passing the weft thread over one or more warp threads and then under two or more warp threads and so on, with a "step" or offset between rows to create the characteristic diagonal pattern.

'Simple' twill weaves have points of intersection that move sideways and upwards on every pick, giving twills their characteristic diagonal lines. 

diagonal
noun

1. a line joining two nonconsecutive vertices of a polygon or polyhedron; can refer to any sloping line (derives from the ancient Greek διαγώνιος diagonios, "from angle to angle" (from διά- dia-, "through", "across" and γωνίαgonia, "angle", related to gony "knee").
2. mathematics, in addition to its geometric meaning, a diagonal is also used in matrices to refer to a set of entries along a diagonal line

The direction of the diagonal on one side of the cloth always runs in the opposite direction on the reverse side ( 'S' direction vs. 'Z' direction).

"s" direction

1. refers to the orientation of at twisted yarn when it is held vertically, the diagonal formed by the individual filaments appear to be on the same diagonal as the letter 'S'.

"z" direction

1. refers to the orientation of at twisted yarn when it is held vertically, the diagonal formed by the individual filaments appear to be on the same diagonal as the letter 'Z'.

And typically 8 shafts is a convenient number that allows a large enough group to work through permutations systematically, and then 'weed' out the 'duplicates.'

permutation

n.

1. transformation; 
2. an arrangement created by moving and reordering items; 
3. rearranging items in a group; 
4. math, an ordered arrangement of elements in a set;

There is an established arithmetic basis for using 8 shafts to maximize the number of permutations and then de-selecting the duplicates. It has everything to do with geometry and the linear fractional group LF(4,2) and its importance in computer applications, combinatorics and group theory.   

But, since the thrust of this post has to do with construction by weaving and structure, and having learned there is really nothing new under the Sun, W.L. Edge touched on its applicability to the topic here in a paper published in 1954 by the London Mathematical Society, The Geometry of the Linear Fractional Group LF(4,2) [paraphrasing]: 

'It is natural to ask, What are the 8 objects which undergo permutation?...The question is primarily a geometric one...'  

...which suits our purpose here addressing structure analysis.

Pictograph
Permutations Linear Fractional Group LF(4,2)

Permutations Linear Fractional Group LF(4,2)

So to weed out the duplicates according to the pictograph representing the 35 permutations of LF(4,2) by visual inspection we get Binary Representations of the 35 decimal numbers relate as follows:

• 7 Unique, Singular patterns in this field: 0, 5, 10, 15, 20, 25, 30
• 6 Twin pairs, 'the same' or equivalent by superposition:  [6, 9], [7, 8], [16, 18],  [17, 19], [23, 24], [21, 22]
• --Of the 6 Twin pairs, the Quad [16, 17, 18, 19] form Negative Images of Each Other, Dual Opposites
• 8 Mirror Image Pairs: (1, 4);  (2, 3); (11, 13); (12, 14); (23, 24); (27, 29); (31, 32); (33, 34)
• --of the 8 Mirror Image Pairs, 4 Sets form 2 Sets of Negatives, i.e, when superimposed, they 'cancel': {11, 12, 13, 14} and {31, 32, 33, 34}

Twills include a family of tweeds that include 'herringbone, 'chevron', 'point', and 'reverse' (think Snell's law and exotic or meta-material), and twills are popular for 'rug' weaving (floor covering) because they pack down well in weft-face structure.

On some fancy tweeds, regular twilling on the diagonal presents as 'shaded' areas (analog for 'shadowing') ;

shade/shadow

and alternating crammed-and-spaced warp gives 'undulating' twills when double or treble groups of warp ends are used;

and the curve back can be controlled by doubling or trebling the ends in conjunction with single ends. While this may be applied using 8 shafts, only 4 shafts are in fact required.


'Broken' twills obscure the diagonal, and can even make the persistent diagonal pattern on a twill almost unobtrusive (inconspicuous, discreet).  

break/broken

1. to become suddenly discontinuous or abruptly stop; 
2. in twills to obscure the characteristic parallel lines/ribs that run along the diagonal

Breaking a twill (effacing the diagonal) is done by simply 'removing' ends on the twill draft, ie., a 2/2 twill can have every 3rd end removed, and then it must be 'repeated' often enough to create a 'selective' weave. The number of ends necessary to do this is typically the Lowest Common Multiple so that for a 2/2 twill, 1 end must be removed of the original 4 ends and 3 'selected', in which case the LCM=12 ends.

This 'picking up the end' or 'moving a point' at the end of the weave is an operation that 'maps' directly to arithmetic functions such as bijection, transformation, reflection, and other ideas to do with set theory and vector space (geometry).

For example (see System Analogies):  

When referring to a planar object (the point at the end of a thread), "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. 

As a result, mirror images are equivalent for free objects. (just like in Alice Through the Looking Glass)  

The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.--wolframalpha.com

So, by this arithmetic definition for 'free' objects, the Binary Representations of the of decimal numbers 21 and 22 are free since they are 'equivalent mirror images of each other', and so are the 22_23; 31_32; 33_34 pairs; and the quad 16_17_18_19.

While weakening the diagonal's appearance, the fabric itself retains the 'stability' and 'flexibility' of the original twill.

There are certain ways to 'break' a twill to make 'satin'--a fabric that has a smooth, and slippery surface that shows no apparent weave and warp-ways is very strong. 

For a True satin (smooth, slippery surface with no apparent interlacing) a minimum of 5 shafts are required.

'Sateen' is a 'satin' that is turned around weft-ways.

surface outermost or uppermost part of a thing that is observed and/or tangible to the outside world; mathematics, a 2D extent, flat or curved continous area in 2 dimensions.

slippery

adj.

1. causing sliding; 
2. hard to hold firmly; 
3. unstable and liable to change, precarious; 
4. untrustworthy.

smooth

1. free from projections or unevenness of surface; not rough; 
2. flat, unruffled; free from or proceeding without abrupt curves, bends, etc.

http://apageinthelife.blogspot.com/2012/08/of-maps-and-legends-innies-and-outies.html  truncated solids, nature of the sky

merge

seam n. the join where two or more fabric layers meet (are stitched together); classified by type and position: plain, lapped, abutted/flat or French seam and back seam, side seam, inseam

Arithmetic: set theory-- A join on a set is defined as the unique suprenum (least upper bound) with respect to a partial order on the set, provided a suprenum exists (aside: thanks, but mish jayeen injedid al.sahra) and the dual for which is the meet , the unique infinum (greatest lower bound) with respect to a partial order on the set, provided an infinum exists.  

If the join of two elements with respect to a given partial order exists then it is always the meet of the two elements in the inverse order, and vice versa. 

Usually, the join of two elements x and y is denoted x ∨ y, and the meet of x and y is denoted x ∧ y.  

A partially ordered set where the join of any two elements always exists is a join-semilattice.  

A partially ordered set where the meet of any two elements always exists is a meet-semilattice. 

A partially ordered set where both the join and the meet of any two elements always exist is a lattice. 

Lattices provide the most common context in which to find join and meet.   

In the study of complete lattices, the join and meet operations are extended to return the least upper bound and greatest lower bound of an arbitrary set of elements. 

Thanks to duality,  discussing joins becomes the same as meets when considering the reverse partial order.

 seamless free from awkward or apparent transitions; having no seams; perfectly smooth; fully integrated