Figure 8 bends

[enhaut]

Xarax,
I told you once and I repeat; the convention when showing a knot, a bend , a loop, a binder, a noose is to always show the tag ends.
It is a practice shown all over the WORLD and I considered good etiquette.
Of course when you do this you loose the symmetry you like so much, but we are not posting images for a photo contest, we are posting images to escape
the ambiguity of the written word.
PS (from wiki)
Together with the cutaway view the exploded view was among the many graphic inventions of the Renaissance, which were developed to clarify pictorial representation in a renewed naturalistic way. The exploded view can be traced back to the early fifteenth century notebooks of Marino Taccola (1382?1453), and were perfected by Francesco di Giorgio (1439?1502) and Leonardo da Vinci (1452?1519).
One of the first clearer examples of an exploded view was created by Leonardo in his design drawing of a reciprocating motion machine. Leonardo applied this method of presentation in several other studies, including those on human anatomy.[5]
The term "Exploded View Drawing" emerged in the 1940s, and is one of the first times defined in 1965 as "Three-dimensional (isometric) illustration that shows the mating relationships of parts, subassemblies, and higher assemblies. May also show the sequence of assembling or disassembling the detail parts."

[enhaut]

 See the clear picture of the loose knot shown by enhaut, and a less clear picture of a more symmetric form of it, at the attached pictures.

Tell my one thing Xarax at reply #31 you are showing  a 'side-by-side' fig. 8 bend, did you published this before you ever seen the "Loose-form-fig8. that I presented, if yes can you provide the link?


I presented a bend with those characteristics;
Each eye (4 total) encircle two diameter of rope.
The two standing ends are embraced together two times, once by their own fig 8 then by the other.

Have you presented such a similar bend with the same features before and  if so can you show me the loose version please?
Yes try more but with pictures.

I get your vision of an exploded form but it's impractical, any representation of a reality is a different way of cheating it.
For me the best way of presenting a knot idea is to first;
Show the thing in is final form, well dressed, recto and verso when needed, then;
Show the loose form, no need for both recto and verso.
On this forum I came to realize that the pictures with loose forms are most visited!

[Dan_Lehman]

  The exploded view of a knot should ...

... enable the tyer to understand what goes where.
As images are given in just 2 dimensions, this goal
might not be able to ...

... retain the proportions of the distances between the points
at the outer = visible shell of the compact / tight un-exploded knot.

Sometimes a compromise between the first goal
of unambiguous connections and the second of
proportional disposition can be worked by means
of numbers or letters identifying the connective
sequence of material in the knot.  (I *flow* from
SPart to tail.  With end-2-end knots, I might use
numerals on one line and letters on the other;
one could do the same with an eyeknot, putting
the SPart's passage one way and the tail's finish
the other.)

--dl*
====

[xarax]

As images are given in just 2 dimensions, this goal might not be able to ...

... retain the proportions of the distances between the points at the outer = visible shell of the compact / tight un-exploded knot.

   You mean, they will be distorted, because of perspective ? Perspective distortion destroys the accurate proportions of the parent, the compact knot, too. Our post-Renaissance minds are accustomed in taking it into account.
   An much "inflated" balloon ( = the 2-D image of the "exploded" knot ) seems more close to the real thing, the less inflated balloon ! ( = the 2-D image of the compact knot ) - and sufficiently close to the real thing, the ( compact) knot itself.

   I know one can explain ( on the image of the loose knot ) what will go where, in the finished, compact form of the knot, with many ways : arrows, letters, animation, etc. However, what I propose is much simpler, and it will not me many years later that it will be achieved, as a possibility, by the Illustrator-like programs... Just make the diameter of the rope on the image of the compact knot thinner, and you are done ! You get the equivalent of your "exploded" knot, in one stroke, because the "exploded", regarding the empty space, knot, is identical to the "imploded", regarding the filled space, knot.

[Dan_Lehman]

As images are given in just 2 dimensions, this goal might not be able to ...

... retain the proportions of the distances between the points at the outer = visible shell of the compact / tight un-exploded knot.

   You mean, they will be distorted, because of perspective ?

... Just make the diameter of the rope on the image
of the compact knot thinner, and you are done !
You get the equivalent of your "exploded" knot, in one stroke,
because the "exploded", regarding the empty space, knot, is identical to the "imploded", regarding the filled space, knot.

No, I mean that there is NO perspective that enables
the in-proportion knot (parts) to be seen unambiguously
--the "explosion" thus will pull some things out of being
hidden from the perspective view to disclose their place.
(Sometimes I have made a note such as "3-4 goes UNDER
all" in a case where that segment is, along with another,
obscured by some near part(s).)  (Making the line smaller
only makes hiding & hidden things smaller.)
.:.  It is a problem of 2D /= 3D, painting vs. sculpture
(Picasso/Braque went into distortion to show all).


--dl*
====

[xarax]

No, I mean that there is NO perspective that enables the in-proportion knot (parts) to be seen unambiguously --the "explosion" thus will pull some things out of being hidden from the perspective view to disclose their place.

Imagine the usual mathematical knots representation, in a 2-D surface.

1. All "crossings" are crossings of two, only lines.
2. All crossings are visible, i.e., they are apart the one from each other. There are no two crossing points in one point of the drawing, because, if this would had happened, this point would had not be a crossing point of two lines, but a crossing point of four lines.
3. In each and every crossing point, there is a "first" line going "over" a "second" = a "second" line going over a "first".

   Now, go one step further. Imagine that this 2-D surface is not a plane, but a sphere.
   So, now, the crossing points are not arranged on a the surface of a flat plane, where every one is visible, but on the surface of a curved space. In other words, the crossing points are arranged on the surface of a "shell", so now one can be "over" or "under" another one. ( I had called it " the outer shell", for reasons I will explain in a while ).
   If that happens, we get a 'perspective" problem, like the one you probably mean. Two crossing points can fall on the same point of the 2-D image of this shell, i.e., two crossing points can be on the same line of sight, so the one is "over" and the other is "under" = the one hides the other.
The "explosion" is the means by which this problem is solved.

   When the shell is "inflated", the 2-D surface is "expanded", like the elastic surface of a balloon when it is inflated. The probability of two points which happened to be on one line of sight in the deflated state ( so the one was hiding the other ), to remain in one line of sight in the inflated state, is almost zero. Even in the rare case it happens, a small rotation of the shell around one axis going through its centre corrects the problem.

   So, we have a knot with many crossing points arranged on the surface of a sphere, and we manage to make all those crossing points visible, by inflating this sphere, so any two crossing points which happened to be in the same line of sight in the deflated sphere, now they are no more. We can see clearly all of them, and see which one of the two lines that crossing each other at each and every point is "over", and which is "under".
   In this sense, you are right, the 'explosion" solves a problem of perspective.

   Now, why I say "the outer sell" ? Because there can be complex knots, where, to keep the geometric proportions as accurate as possible, the crossing points of the deflated / compact knot can only be arranged on two, or more concentric shells, which are connected, through some "necks", to each other. However, the crossing points of most, if not all of the simple practical knots we are interested in can be arranged on one shell, so this "outer" adjective is redundant.

   Clear as mud ? Perhaps, but here it comes the images ! The whole idea I was trying to make you imagine with words, becomes transparent with images ( see the attached image ). THAT is why I tell you that you should learn to start to use less words, and more images ! 

   I am using the word "sphere" in the topological, not the geometrical sense. In fact, any continuous "closed" surface, topological equivalent to the sphere, is "spherical".  An egg-shaped surface, for example.
 
   So, here comes the cooking recipe :
   1. Take your compact knot, hold it in your palm, and start looking at it from any angle. Find a proper view that will help you do the following, more easily :
   2. Start imagining the crossing points of any pair of lines been arranged on a transparent and elastic "spherical" surface, on a soap bubble for example.
   3. Start imagining this surface been "exploded", that is, start imagining the bubble been inflated.
   4. At some point, you will "see" all the crossing points : there will be no points of three or more lines on the same line of sight.
   3. At that point, you will have a view where all the crossing points of all lines are clearly visible, and you can say if the one line goes "over" or "under" the other.
   4. That s all, folks ! 

[xarax]

2

[xarax]

   If we do not bother about the changes of the distances between the crossing points, any knot which can be represented on a plane, as a 2-D diagram, can be represented on a sphere, too. However, on a "planar" drawing , the crossing points which are located near its "centre" are much closer to each other than the crossing points which are near its "perimeter". So, when we represent a knot that way, the geometrical accuracy of the parent, compact 3_D knot is greatly compromised.
   On the contrary, on a "spherical" drawing, all crossing points are located at the same distance from the "centre" of the "sphere". So, the distance between any pair of them is smaller. Moreover, when the diameter of the sphere changes, the distances between all crossing points change in a "geometrical" way, i.e., in a proportionally accurate way.
   This means that if we chose to represent the parent, compact knot as a "spherical knot", and then "inflate" this "sphere", the initial image will change only in its scale, not in anything else. So, the "inflated" = "exploded" knot retains the geometrical characteristics of the initial, "deflated" = compact knot.
   So, the main problem is how we represent the initial compact knot as a "spherical knot" in the first place. My reply is the usual : by trial and error, and often with much difficulty !  However, doing this, we would not change its geometry as much as we would had done, had we represented it as a "planar" knot.
   The images of "planar", "flattened" knots correspond, more or less, to the usual images of "loose knots". In a "loose knot", the number of crossing points is not minimal - and the distortion of its shape, in relation to the shape of the parent, compact knot is not geometrically accurate / proportional everywhere, to allow us retain the mental picture of this parent knot as much as possible. At a "spherical knot", the number of the crossing points on the surface of the "sphere" is minimal, like it happens in the representations of the mathematical knots, shown in the tables of knots and links. Moreover, the differences between the initial, at the un-exploded knot, and the final, at the "exploded knot", locations of any pair of crossing points is minimal. The "inflation"="explosion" of the "sphere" changes those differences only in scale, proportionally.
  The only problem is that is the problem of "perspective". We can not watch all the points on the surface of a sphere at once, can we ? Oh yes, we can - if this sphere is transparent !  In the rare case two crossing points, one at the "front" side and one the "rear" side of the sphere, happen to be on the same line of sight, so the one hides the other, we can just rotate the sphere a little bit : the one point will move to the "left" and the other to the "right", so they will not remain on one line of sight any more. With modern, fast , interactive, "virtual reality" computer programs, we can rotate a "spherical knot" easily - and even "animate" a rotation, so we get a better "feeling" of the "real", the 3-D object.