[HN Gopher] Generating SVG for the prime knots
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Generating SVG for the prime knots
Author : prideout
Score : 131 points
Date : 2024-01-11 18:50 UTC (4 hours ago)
(HTM) web link (prideout.net)
(TXT) w3m dump (prideout.net)
| l0b0 wrote:
| Love the double hearts in
| https://prideout.net/blog/svg_knots/knots/8_5.svg
| gilleain wrote:
| Also known as the 'true lovers knot' I believe:
|
| https://en.wikipedia.org/wiki/True_lover%27s_knot
|
| (It isn't the smallest non-alternating knot. This page says
| 8_19 is the smallest.
| https://mathworld.wolfram.com/NonalternatingKnot.html)
| phkahler wrote:
| I've never studied knots, but understanding the algorithms used
| here seems like a really good starting point.
| OscarCunningham wrote:
| The unknot isn't a prime knot for the same reason that 1 isn't a
| prime number.
| dekhn wrote:
| There's some fun stuff here (I'm reminded of the algorithms used
| to render planar drawings of proteins, which are similar to
| knots).
|
| My real interest which I haven't seen much literature about is
| generating real-world knots that have good properties. For
| example if you look at the various knots, some knots have nice
| properties like "easy to untie" and "does not get tighter under
| load", which has huge impacts. These properties derive from the
| topology but also the physics of the knot. Would be nice to find
| a new hitch knot that worked better.
| gilleain wrote:
| > I'm reminded of the algorithms used to render planar drawings
| of proteins, which are similar to knots
|
| Yes, like the PTGL - https://bio.tools/ptgl - or, er, TOPS
| diagrams. The main relationship to knot diagrams is really the
| chirality of beta-alpha-beta motifs (the majority of which are
| right handed).
| dekhn wrote:
| Thanks! I was searching for this citation to include in my
| comment: https://www.cambridge.org/core/journals/protein-
| science/arti...
|
| (I have no idea how my brain can remember a paper from 20+
| years ago but not enough to find it in the literature)
| contingencies wrote:
| A lot of knot books and websites provide property-based
| classifications.
| dekhn wrote:
| Yes. my goal would be to identify those properties from 3d
| models of knots, and a mechanism to generate plausible 3d
| models.
| contingencies wrote:
| The general properties of knot categories are already
| known. Real world factors such as the diameter of the line,
| its mechanical properties and environmental considerations
| (eg. presence of powder/dust, oil, rain, etc.) will
| significantly affect the actual deployment properties of a
| given knot. In addition, there are infinite places within a
| given knot that forces can be applied. Lines will also
| degrade over time, and factors such as complexity, time and
| fingers/hands/tools required-to-reliably tie/untie will
| also often be practical concerns in selection for
| deployment. Therefore, while your interest in the
| algorithmic exploration is interesting, if the goal is to
| generate novel practical results then there is a lot more
| complexity to add to the modeling before a useful result
| might be obtained, and any such result would have to be
| clearly based in assumptions around deployment scenario.
| dekhn wrote:
| Sure. you're not telling me (a person who used to model
| knots in proteins using molecular dynamics, and works
| with FEA and other mechanical engineering tools) anything
| really novel.
|
| Humans discovered hundreds of knots just playing around,
| and developed excellent knots in the past 400 years; new
| knots, never before tied, were invented some ~100 years
| ago. One imagines that a bit of searching with a computer
| might find a few cases that were overlooked.
|
| For example, take a look at
| https://en.wikipedia.org/wiki/Butterfly_loop and
| https://en.wikipedia.org/wiki/Butterfly_bend and
| https://en.wikipedia.org/wiki/Hunter%27s_bend and
| https://en.wikipedia.org/wiki/The_Ashley_Book_of_Knots
| dexwiz wrote:
| Mathematics has probably already classified all knots that are
| human tieable. So from there you could iterate these knots in
| different physical positions, and perform different tests on
| them. The topological space has been investigated, now you need
| to decide between a teacup and a doughnut. This would be a
| mechanical engineering question, not a math question. Maybe
| look elsewhere?
| dekhn wrote:
| math knots and real knots aren't the same thing and I don't
| think all possible human tieable knots have been enumerated
| and classified although I am happy to be pointed to evidence.
|
| Math knots embed circles while real knots are typically made
| with free ended ropes (although some knots are not). Math
| knots ignore friction and the width of the rope; real knots
| can't ignore friction and the width (and other phyhsical
| properties) matter. See the comment in this article
| https://en.wikipedia.org/wiki/Overhand_knot
|
| While researching my answer I found out there's a term for
| what I wanted to do,
| https://en.wikipedia.org/wiki/Physical_knot_theory
|
| I've had this discussion with mathematicians a few times now
| and they don't see the difference, so maybe I'm just missing
| something important.
| whatshisface wrote:
| A knot with free ends can be turned into an embedded circle
| by connecting the ends far away from the knot. That's why
| they share a classification scheme.
| zokier wrote:
| Neat visualization. I noticed though that the images (when
| enlarged) have some awkward angles etc to them, they are not
| super rounded and smooth shapes... maybe the width modulation
| could use some easing or interpolation or something?
| mlyle wrote:
| I feel like that would be nice. It also might be nice to have a
| slight change in hue over the length of the knot, so that there
| is more contrast on crossings and things are a little clearer.
| dexwiz wrote:
| Very cool. I have been trying to do something similar with
| tilings, but there is more information to be encoded. I am
| targeting something similar to the ability to render the
| tessellation catalog [1] and perform searches. Turns out there
| are multiple notations, and most of them use some level of
| implied knowledge. Also seems like rendering of knots is more
| open ended, while tilings are essentially their rendering. But
| they all do use a similar approach of Notation > Half Edge Data
| Structure > Render.
|
| [1] https://zenorogue.github.io/tes-catalog/?c=
| lovegrenoble wrote:
| Not sure I learned anything about "Knot Theory" from playing
| this, but that was fun. Knot theory game:
| https://knots.netlify.app
| samstave wrote:
| As a knotable person myself, this is AMAZING.
|
| Some thoughts on application of this knowledge would be to look
| at the patterns as you have described as cross-sections of rope
| weaving with the circular looming as the individual
| bobbins/spinny-things in an industrial loom - so rather that just
| woven-sheeth and full spin style cabling, one might achieve some
| really incredible properties in the woven elements along a axis
| such as these represent.
|
| Especially if you further differentiate btwn material and woven
| state (Are you weaving in an already spun set of filiments? What
| are the materials for the various inputs, and even further -
| imagine you have a set of elements in the loom where youre
| certain threads are the static, more rigid scaffold - like woven
| titanium strands which then feed into another loom which is
| weaving in the kevlar or other materials including a core of
| optics which is protected by the outer woven sheath from these
| patterns of 2D knots stretched out along an axis - certain
| elements can be printed such like the articulating spine of a
| snake.
|
| It could make a machinable-high-tensile strength cable with an
| optical core with protected turn radii (titanium snake spine)
|
| See here for reference to advanced cabling:
|
| https://www.youtube.com/watch?v=AD5aAd8Oy84
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(page generated 2024-01-11 23:00 UTC)