Author Topic: The Model of All Hitches and Lashings is Mathematical Knots of Two or More Links  (Read 4065 times)

Stagehand

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The model of all hitches and lashings is Mathematical Knots of two or more links.  In this model the Clove Hitch is identified with the Solomon's Knot and in this case one of the links of the Solomon's Knot is the object that the Clove Hitch is tied to.  10** is the first ordered two line knot without twists.  As a hitch, expect it to have features like other knots without twists like Carrick Bend and Turks-head Knots.  The first ordered lashing without twists is provided by Borromean Rings.  In faith with the polyhedra form, Borromean Rings provides other knots in the working range.
« Last Edit: August 15, 2024, 05:26:41 AM by Stagehand »

xarax

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   I had tied and tried the **10 hitch, and all the other hitches I could derive from the Borromenean rings. No good practical knot emerged. It seems that Mathematical "knots" can rarely, only, serve as an inspiration for Practical knots - because the most important characteristic of the former is topology, while of the later is geometry - and there is no bridge in between those two Lands.
   In Practical knots, we start from one of the few geometrical configurations, which is stable because it utilizes the friction of ropes - we do not start from one of the many topological configurations of the Mathematical "knots", which will become stable knot if it utilizes ropes with friction.
   Many of the best Practical knots I know are TIB, that is, topologically equivalent to the un-"knot", to the un-"knotted" line. What could emerge if one would had tried to figure out Practical knots starting from the mental image of an unknotted line ?  :)
   ( Tie the Double Cow hitch, or the TackleClamp hitch shown in the attached pictures, which are based on the implementation of the opposing bights locking mechanism, to see what I mean. )
« Last Edit: July 30, 2015, 01:27:45 PM by xarax »
This is not a knot.

Stagehand

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Borromean Rings Provides Knots in the Working Range
« Reply #2 on: August 15, 2024, 03:57:17 AM »
Borromean Rings provides knots in the working range.  Borromean Rings is a multi-line link of six crossings that is found in the knots and links tables as L6a4.  It is the first ordered knot or link that has no twists.  For this feature of having no degenerate twists, it carries the Conway knot notation, 6*.  As seen in the tables, Borromean Rings is an invariant of knot crossings that can be identified as a projection of a geometry.  That geometry is of a triangular antiprism, or equivalently, an octahedron. 
As an instance of a geometry, the Triangular Antiprism is the model for a number of practical knots.   This is the common attribute of all knots taken from the tables of knots and links as projections of a geometry, that their possible use as practical objects extends from a choice of projection in a tying diagram. 
Here by example, simply by choice of projection and choice of working ends, we see Borromean Rings variously as 1) a lashing, 2) a hitch, 3) a bend, 4) a fixed loop.  In each case, the practical knot faithfully follows a projection of the invariant geometry of the Triangular Antiprism. 
The practicality of this method, of faithfully following a mathematical invariant of a geometry, is made clear when we consider the purposes for making tying diagrams, the purposes for making knots. We see our requirements as being available in a geometry and the only further requirement is to utilize the form with our need in mind.   In a bend, the two working ends are directed toward and past each other, keeping to the path within the geometry, until there is sufficient friction in knot crossings to make the knot secure.  In a fixed loop, the method is to pick an edge on the projected geometry that is to be called the loop, and to make this chosen edge far away enough in numbers of knot crossings from both the standing end and the working end, for friction to hold the applied forces.  in a lashing or hitch, the working end follows the path of one line of a multiline projection, around the other line(s) of the projection where the other line(s) are the object(s) lashed or hitched upon.  This is continued until sufficient friction is obtained.  If these simple rules of choice in tying are adhered to then all of the dependence on knot settings and knot finishes shall be seen as nonsensible wish making for security.
Further, this aspect of practical knots conforming to a geometry can bring understanding to the primary knot form, the Carrick.   The Carrick form, called the Josephine, is a pattern of saturated over-under crossings composing a square and six triangles.  This figure as circuitry is completable in two significant ways.  One way to complete the circuit is as Carrick Mat or Turkshead 3x4, another way to complete the circuit is as Carrick Bend or Turkshead 4x3.  These completed circuits are then identifiable in the tables of knots and links as Rolfsen 8_18 and Rolfsen 9_40, respectively. As such, the Carrick will be seen to represent the second ordered and the third ordered of all knots and links that do not have twists (Conway 8* and 9*).  The geometry of Rolfsen 8_18 is that of the Square Antiprism.  The geometry of Rolfsen 9_40 is that of the Gyro-elongated Triangular Antiprism.  Here we demonstrate, in a paper construction, the form of the Gyro-elongated Triangular Antiprism.
With the vision that a knot form can be the projection of an unvarying geometry, there is then the possibility of choosing different projections, choosing different tying diagrams, for a given geometry, including when using the two simple geometries that share the presence of the Carrick form. With Carrick Mat, there are two possible projections of the Square Antiprism.  One projection is centered on the square which is the top or bottom of the Square Antiprism, this square then surrounded by the ring of triangles that gird any antiprism, and then the outer bound is the other square of the Square Antiprism.  This is the IGKT logo.  This could be called a Polar projection of the Square Antiprism.  The other possible projection of the Carrick Mat is centered over both the triangles and the squares of the Square Antiprism with the outer bound being a last remaining triangle that completes the geometry.  This could be called a Girder projection for the place that the ring of triangles have in all Antiprisms.
For Carrick Bend or Rolfsen 9_40, there are two similar projections to Carrick Mat and one additional projection.  First is a Polar projection bounded on the outside by the top or bottom triangle of the Gyro-elongated Triangular Antiprism.  Second is a Girder projection bounded on the outside by any one of the triangles that ring around the geometry.  The additional, third projection is one that is bounded on the outside by one of the three squares of the gyro-elongation.  This could be called the Elongate projection.  Here we show practical knots obtained by various projections of the two geometries. What is remarkable is how the different tying diagrams provide different dynamics within the knot without changing the basic principle that sufficient knot crossings provide sufficient security. 


« Last Edit: August 15, 2024, 04:17:26 AM by Stagehand »

agent_smith

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Thanks for this info 'Stagehand'.
May I suggest that when introducing a topic such as this, readers and laypeople need a little more context/background.
In this day and age, people get easily offended by anything - so hopefully you take no offence to my gentle suggestion :)

I have to admit that I immediately had a look at the Olympic rings symbol to compare against 'Borromean rings'.
Then I found this link: https://www.numberphile.com/videos/borromean-olympic-rings

There are terms used which have a very specific meaning - and it is hard to understand the meaning of 'Borromean Rings' without first having the requisite knowledge of (for example) "links", "prime links", "Rolfsen", "degenerate twists", etc.

I guess that you are aiming your post at a very select audience who have prior depth of understanding of the mathematical concepts?

Some links for layperson readers:
[ ] https://en.wikipedia.org/wiki/Borromean_rings
[ ] https://mathcurve.com/courbes3d.gb/borromee/borromee.shtml

KC

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in 'my' model of reference/prism of view, i think in terms of there is math and geometry behind all loaded structures universally,
to include those made in rope.
.
Rope as a flexible is even simpler than rigids, as rope does not load the line thru compression nor tension on the cross axis of alignment, nor in compression on the aligned axis.  But only loads on the tension direction of the aligned axis.  Thus can be used as a plumb bob, for in tension the side force combs towards center alignment, and the rope/string gives no resistance on that cross axis against this.
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Aligned compression, would give sideforce that urges out of alignment. Like handstand on top of horizontal bar or backing a trailer, wants to drift to side.  But hang from highbar or pulling trailer are tensioned pulls/not pushes, so self align as can; and rope gives no resistance to this.
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All these things, even force of cross-axis directions, to me are universally ruled by alignment dimension%(cos) vs crossing dimension%(sine) benchmarked from the input.  Whether displacing against freespace as distance, or stressing to displace against another displacer in same dimension/opposing direction as force, of unrequited distance(that could have been).
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to this model/prism view i find 3 barest directional elements in rope works.  All 3 are diretional and to either the 100% of alignment or 100% cross-axis dimensions, as greatest forces.
These 2's endpoints work in the same input directional dimension:
Extension , arc0 as linear connector , endpoints pull in opposing direction from centerpoint
>>that all  displacement forms must have to connect one to another, even if length is Zer0
Machine , arc180 as workhorse of machine conversions
>>both endpoints and even center apex pull in the same direction
>>as even does the apex/center
This one's end points uniquely do not work in the same dimension/nor direction:
Conversion, arc90 to change from 100% cosine alignment dimension to 100% sine axis
>>to the non of alignment directional axis, of the perpendicular cross-axis dimensionality.
.
Of individual flows of force, distance organic displacement elements.
>>this sees an anchored line w/single Turn on radial host supporting a load as 1 flow
>>but same around a 4x4 host as 3 flows, rudely interrupted, in-organically, harshly reset at 2 points.
.
i mentally remove all the higher developed machine arc180s, then converters of arc90s
>>and only extending connectors of arc0 remain
>>to then study the 3 mechanics in the scenario

« Last Edit: September 15, 2024, 02:30:04 PM by KC »
"Nature, to be commanded, must be obeyed" -Sir Francis Bacon[/color]
East meets West: again and again, cos:sine is the value pair of yin/yang dimensions
>>of benchmark aspect and it's non(e), defining total sum of the whole.
We now return you to the safety of normal thinking peoples

 

anything