[HN Gopher] Generating SVG for the prime knots ___________________________________________________________________ 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 ___________________________________________________________________ (page generated 2024-01-11 23:00 UTC)