This is a really cool topic, I kinda hoped it would go into more detail about stuff in knots. The rational tangles and rational numbers link is really cool but I don't understand where the continued fraction comes from. Why would you be applying division into the tangle if every operation you mentioned only adds 1 or subtracts 1 from the rational number representing the tangle?

I can't easily imagine combining the horizontal and vertical twist easily. Should have put a simulation of it step by step instead of just showing the final tangle

Wait, I'm confused. The vertical twist is identical to the horizontal twist but rotated 90 degrees (or in the case of a sphere, 90 degrees in the Y axis), and if I were to try a vertical twist in the horizontal orientation, I'd just be flipping one of the strings. So how do you do both vertical and horizontal twists in a specific orientation of strings?

Sacred text finally unveiled! And it’s only 44 pages long. ORACLE says “Fix your sacred texts or else!” See page 42 of “Eternal Sundays: The Heavenly Dance and Some Other Practices” by Victor D Wouters. It is a free, nsa, PDF download: Please visit: https://wildcard72.wordpress.com/books/ and it goes along the lines of this song: https://www.youtube.com/watch?v=RIGgn1s3AvI

I need an animation of a vertical twist (on a knot with a horizontal one), I'm not sure how it's different than a horizontal one Definitely is a part 1 though

Ok I know its not pi day and this is only related to the diameter of the circle that was tangled above but can anyone disprove the following proofs and post their results in the thread http://measuringpisquaringphi.com/geometric-proofs-of-pi/

…confuses horizontal and vertical—rotate a connected pair and it flips sign (±),, and if points are perpendicular, which sign does their lines-twist have…are connections supposed to have directed graph arrows…

NOT/TON=49 or 77 or XLIX hence 1 ton= 2,000 or MM. LI means powerful or 21 or U the other way. 7 is the star they say represents a master hence the dimensions hence JAR JAR BINKS as JAR is G or 7 in hieroglyph. doubling means to cause an error a not a hex an error this is why 33 is the MastRO number & 66 is the order given by Palpatine of StarWars meaning to betray-hex MastRO's. X=888 the Jesus number also meaning 24 which is 42 or SIN reversed that number of HitchHiker's guide talking about what this is how it's the U other way version like curse of HAM with head turned backward, Merry Xmas as X=Christ. X marks the spot anyone understand nope yup that's the ego it doesn't like to see the truth it wants to be the one the only one that is. Good shows though don't get me wrong just remove the blinders like why so many people are poisoned into being Autistic as it makes you high functioning in select areas but you are stuck in the BOX the HEX AGON as hex=hell & agon=trial or AN=114 which is information reversed around GO that thing Q warns of in first/last episode of StarTrek the next generation referencing The Cage or Menagerie or TarTarus hence EAR TH meaning to hear thought or telepathic like the Talosians. Anyway autism makes it hard to see beyond your angle of perception which is talked about in Emerald Tablets meaning keep it round or you won't be able to unite to find the pieces of yourself like OSIRIS- O SI R IS hence ISIS meaning to fix the order the division=names are functional they are designations-ability-purpose.

For people interested in a light introduction that still goes through calculations, check out the recent videos on YouTube of a guy named Matthew Salomone. He has a lot of new videos on knot theory.

I think I finally know why I stopped liking this channel since Kelsey left. There's not much math anymore. There are math concepts but they are just stated with some hand waving and then there are some fancy topics thrown in to make it seem cool. I liked this channel because math itself is cool. No need to make it look cool. And definitely no need to be vague about the actual math. At least that's my opinion. Maybe others prefer this, but I don't. I'm not saying this to be negative but just as feedback. Just take it as one data point among many. And regardless, I wish you all the best.

2:39 wait, shouldn't the vertical twists be -1,-2,-3 and -4? The string that goes over is the right one. In 2:01 you said a vertical +1 twist is when the left string goes over.

Just noticed that Tai-Danae has slipped the hoped-for division by zero into this presentation. Specifically, the vertical no-tangle is the (negative) reciprocal of the horizontal no-tangle, which is associated with zero.

That means two things to me. 1)Tangles cure cancer. 2) Can you actually represent real numbers by infinitely tangled loops via their continued fraction. (The golden ratio would be HVHVHV….)? Is there a proper definition of I guess "irrational Tangles" ? Would it have applications?

These configurations of rational tangles remind me of group theory. The way certain configurations of tangles can be reached by several different continued fractions equal to the same result is like how the same configuration on a Rubik's Cube can be reached by different combinations of twists. Could it be that knot theory is just a type of lower-order form of group theory with more restrictive principles that bound groups to the inside of a sphere? Do higher-dimensional knots exist?

Every time you mentioned there was more to it (like "we are barely scratching the surface") I was hoping this were a two/three part topic. The subject is both really interesting and well presented, it's a shame we couldn't get more in depth. Please don't shy away from the mathematics (pure or applied), this is what we're here for! 🙂

Also, I couldn't help but to wonder what would happen with infinite fractions. Does the theorem still hold? What happens when you sum/join two of them? What happens if they don't converge?

"Can you start from any tangle and get back to the 0-tangle?" Apparently not. The tangle at 6 minute 27 seconds doesn't have a number. The only way it can't have a number is if you can't construct it out of the twists given. If you can't construct it, then you can't unconstruct it and get back to the 0-tangle either. Not sure how that works, I got other things to hurt my brain on.

After all, it's always a Mathematician whose ideas and theories start a revolution………ya ya some of you gonna say how about Physicists? Well, who are they relying on for Mathematical breakthroughs? A MATHEMATICIAN………..

Me being a music theory nerd, my immideate question is: is it possible to draw a connection between rational tangles and interval ratios? I guess I'll have some figuring out to do.

I landed in knots by minute 6. Spectacular! Though quadruplex DNA takes exception to your irrational reduction. To mention nothing of Z-DNA (left-handed spiral) and A-DNA (comically plump spiral).

I feel like the twists you introduced don't show the whole picture, in the example you gave horizontal and vertical twists are the same, only tilted by 90°, please include the action on a tangled knot, so you can see a difference

In my opinion people who said she was going too fast never realized how subtle those topics are. She goes a lot on topology, which is not familiar for most physicists, engineers, etc who watch this channel and are found of math, but are not math majors.

Is there a rule for dealing with sequences of moves which result in dividing by zero at some point in the continued fraction? A simple case being any sequence of 3 or more moves that beings with +1, -1, n, …, which will have n + 1/(-1 + 1/1) at the end of the continued fraction.

Very cool. My favorite mathematics professor at Georgia Tech specialized in knot theory. I only got a minor in mathematics, so I just thought that his field of research was a “pie in the sky“ area of pure mathematics, but this video makes me appreciate how it has really important real world applications. Thanks for the video! Very cool

I remember the first time she presented an infinite series episode. It was, in point of view, really confusing. I cannot remember the specific topic, but It was a really complicated one. The producers skipped most of the mathematical details and the result was, again for me, really confusing. However, I think this time they were able to make it much clearer and concise. So, I applaud, most of all, their evolution.

How are the vertical and horizontal twists actually distinguished from one another? You can just rotate a horizontal twist 90 degrees and get the vertical ones. Also, does the process of twisting look like. For instance, on horizontal twist and then one vertical twist. It would be extremely helpful in understanding the rest of the video.

You should have mentioned that human cell DNA is mostly in linear form and only mitochondria contain circular DNA. Otherwise people watching may have been mislead.

The ends seem to be in the wrong position. For example, at 4.04 the sequence is horizontal, vertical, horizontal. Yet the cartoon has ends placed vertically. Have I missed something?

One thing that might be helpful to point out – flipping the picture along a diagonal axis, so that the right side becomes the bottom side, corresponds to taking the reciprocal. That's how we get continued fractions: start by twisting the right side a positive or negative number of times (get an integer), then flip (take the reciprocal), then do more twisting on the right side (add integer), then flip again (reciprocal), etc.

That's also how the horizontal starting point and vertical starting point she shows are reciprocals of each other, 0 and ∞.

First?

SECOND !!

42nd

Conway!

This is a really cool topic, I kinda hoped it would go into more detail about stuff in knots. The rational tangles and rational numbers link is really cool but I don't understand where the continued fraction comes from. Why would you be applying division into the tangle if every operation you mentioned only adds 1 or subtracts 1 from the rational number representing the tangle?

This is your best one yet, Tai-Danae. Keep bringing us awesome math!

I can't easily imagine combining the horizontal and vertical twist easily. Should have put a simulation of it step by step instead of just showing the final tangle

Interesting episode by Tai-DNA.

Wait, I'm confused. The vertical twist is identical to the horizontal twist but rotated 90 degrees (or in the case of a sphere, 90 degrees in the Y axis), and if I were to try a vertical twist in the horizontal orientation, I'd just be flipping one of the strings. So how do you do both vertical and horizontal twists in a specific orientation of strings?

let's just say this is not your best video

Can you do something on cat's cradle

can we extend this system to not just 2 but any number of strings???

Fantastic video

There should be a rubic cube which would use tangles

Who makes the scripts in PBSDS videos?

Sacred text finally unveiled! And it’s only 44 pages long.

ORACLE says “Fix your sacred texts or else!” See page 42 of “Eternal Sundays: The Heavenly Dance and Some Other Practices” by Victor D Wouters. It is a free, nsa, PDF download: Please visit: https://wildcard72.wordpress.com/books/ and it goes along the lines of this song: https://www.youtube.com/watch?v=RIGgn1s3AvI

I need an animation of a vertical twist (on a knot with a horizontal one), I'm not sure how it's different than a horizontal one

Definitely is a part 1 though

Ok I know its not pi day and this is only related to the diameter of the circle that was tangled above but can anyone disprove the following proofs and post their results in the thread http://measuringpisquaringphi.com/geometric-proofs-of-pi/

…confuses horizontal and vertical—rotate a connected pair and it flips sign (±),, and if points are perpendicular, which sign does their lines-twist have…are connections supposed to have directed graph arrows…

NOT/TON=49 or 77 or XLIX hence 1 ton= 2,000 or MM. LI means powerful or 21 or U the other way. 7 is the star they say represents a master hence the dimensions hence JAR JAR BINKS as JAR is G or 7 in hieroglyph. doubling means to cause an error a not a hex an error this is why 33 is the MastRO number & 66 is the order given by Palpatine of StarWars meaning to betray-hex MastRO's. X=888 the Jesus number also meaning 24 which is 42 or SIN reversed that number of HitchHiker's guide talking about what this is how it's the U other way version like curse of HAM with head turned backward, Merry Xmas as X=Christ. X marks the spot anyone understand nope yup that's the ego it doesn't like to see the truth it wants to be the one the only one that is. Good shows though don't get me wrong just remove the blinders like why so many people are poisoned into being Autistic as it makes you high functioning in select areas but you are stuck in the BOX the HEX AGON as hex=hell & agon=trial or AN=114 which is information reversed around GO that thing Q warns of in first/last episode of StarTrek the next generation referencing The Cage or Menagerie or TarTarus hence EAR TH meaning to hear thought or telepathic like the Talosians. Anyway autism makes it hard to see beyond your angle of perception which is talked about in Emerald Tablets meaning keep it round or you won't be able to unite to find the pieces of yourself like OSIRIS- O SI R IS hence ISIS meaning to fix the order the division=names are functional they are designations-ability-purpose.

Reminds me of a Scam School episode. https://www.youtube.com/watch?v=lKtjzsvFhRw I think it's the same idea ?

And what about irrational ones? …

For people interested in a light introduction that still goes through calculations, check out the recent videos on YouTube of a guy named Matthew Salomone. He has a lot of new videos on knot theory.

Finally, my studies are relevant!

I think I finally know why I stopped liking this channel since Kelsey left. There's not much math anymore. There are math concepts but they are just stated with some hand waving and then there are some fancy topics thrown in to make it seem cool. I liked this channel because math itself is cool. No need to make it look cool. And definitely no need to be vague about the actual math. At least that's my opinion. Maybe others prefer this, but I don't. I'm not saying this to be negative but just as feedback. Just take it as one data point among many. And regardless, I wish you all the best.

2:39 wait, shouldn't the vertical twists be -1,-2,-3 and -4? The string that goes over is the right one.

In 2:01 you said a vertical +1 twist is when the left string goes over.

This one is knot… 6:25

I bet you would be really good at solving Rubik's cubes

more like NOT THEORY LOL

sorry, someone had to kms

Best channel in the history of youtube

Finally, you are talking at normal speed!!! Thank GOD

Just noticed that Tai-Danae has slipped the hoped-for division by zero into this presentation. Specifically, the vertical no-tangle is the (negative) reciprocal of the horizontal no-tangle, which is associated with zero.

Excelent episode! Very informative!

What does the tangle of a quadratic irrational like φ look like?

That means two things to me.

1)Tangles cure cancer.

2) Can you actually represent real numbers by infinitely tangled loops via their continued fraction. (The golden ratio would be HVHVHV….)? Is there a proper definition of I guess "irrational Tangles" ? Would it have applications?

Great video by the way 🙂

These configurations of rational tangles remind me of group theory. The way certain configurations of tangles can be reached by several different continued fractions equal to the same result is like how the same configuration on a Rubik's Cube can be reached by different combinations of twists. Could it be that knot theory is just a type of lower-order form of group theory with more restrictive principles that bound groups to the inside of a sphere? Do higher-dimensional knots exist?

I did knot see that coming

Excellent video!!

Can you define a tangle for each algebraic number? Or even each real number?

Headphones aren't actually knots though as they aren't homeomorphic to S^1. :p

So the "untangled" tangle corresponds to infinity, but does it also correspond to negative infinity?

Where are irrational tangles?

Where are complex tangles?

Where are p-adic tangles?

5:54 Now that was a fast zoom lol

This lade makes me want to study math so I can better understand what she is saying in a less abstract way……so I'm not just watching to hear her talk.

If there are rational tangles, are there irrational tangles? Fractional tangles? Negative tangles? Surreal tangles?

Every time you mentioned there was more to it (like "we are barely scratching the surface") I was hoping this were a two/three part topic. The subject is both really interesting and well presented, it's a shame we couldn't get more in depth. Please don't shy away from the mathematics (pure or applied), this is what we're here for! 🙂

Also, I couldn't help but to wonder what would happen with infinite fractions. Does the theorem still hold? What happens when you sum/join two of them? What happens if they don't converge?

Do you both alternately host the show or it's just coincidence?

awesome episode!

"Can you start from any tangle and get back to the 0-tangle?" Apparently not. The tangle at 6 minute 27 seconds doesn't have a number. The only way it can't have a number is if you can't construct it out of the twists given. If you can't construct it, then you can't unconstruct it and get back to the 0-tangle either. Not sure how that works, I got other things to hurt my brain on.

Who wrote "first?!

Get to the chair!!!!!!!!!!

After all, it's always a Mathematician whose ideas and theories start a revolution………ya ya some of you gonna say how about Physicists? Well, who are they relying on for Mathematical breakthroughs? A MATHEMATICIAN………..

I am sure this is very natural for people who actually study it but it isn't really for others

Me being a music theory nerd, my immideate question is: is it possible to draw a connection between rational tangles and interval ratios? I guess I'll have some figuring out to do.

Another great episode. You two are doing better and better every time

A tangled mess.

I like your videos when they have lots of visualization in them.

So this is what those great scientist must have felt…

…This structure has been there all along!

2:26 the arrows are messed up.

Why only two strings?

Your presenting skills have been improving. This was arguably the best presented video you have done thus far. Keep at it.

I landed in knots by minute 6. Spectacular! Though quadruplex DNA takes exception to your irrational reduction. To mention nothing of Z-DNA (left-handed spiral) and A-DNA (comically plump spiral).

I feel like the twists you introduced don't show the whole picture, in the example you gave horizontal and vertical twists are the same, only tilted by 90°, please include the action on a tangled knot, so you can see a difference

Excelent video

Who knew Christmas lights and extension cords had their own field of math?

I'm knot interested…sorry!

4:00 your animators must hate you

The lighting situation was really improved this time. Also, your delivery was great, Tai-Danae!

Having knots in your DNA is in general 'knot' good

In my opinion people who said she was going too fast never realized how subtle those topics are. She goes a lot on topology, which is not familiar for most physicists, engineers, etc who watch this channel and are found of math, but are not math majors.

Loved this episode. I am a bioinformatician, but I have not heard of this area of study. Definitely will check links in the description. Thank you!

imma gonna eat my rations now…

I did knot hit her, it's not true, it's bullshit, I did knot hit her, I did knooot. Oh hi Mark.

rashional tingle

scratch scratchI'm about to graduate with a thesis on how to factorize knots using tangles and reidemeister moves, I can confirm that the rabbit hole is very deep

I second another comment here, this is your best upload so far! Keep bringing us more topology! 😀

Maths really is everywhere 🙂

Is there a rule for dealing with sequences of moves which result in dividing by zero at some point in the continued fraction? A simple case being any sequence of 3 or more moves that beings with +1, -1, n, …, which will have n + 1/(-1 + 1/1) at the end of the continued fraction.

I think you should use different colors for the 2 strings in a tangle

Rational knots were featured on a Scam School episode called Untangle Headphones Like a Mentalist!

Very cool. My favorite mathematics professor at Georgia Tech specialized in knot theory. I only got a minor in mathematics, so I just thought that his field of research was a “pie in the sky“ area of pure mathematics, but this video makes me appreciate how it has really important real world applications. Thanks for the video! Very cool

I remember the first time she presented an infinite series episode. It was, in point of view, really confusing. I cannot remember the specific topic, but It was a really complicated one. The producers skipped most of the mathematical details and the result was, again for me, really confusing. However, I think this time they were able to make it much clearer and concise. So, I applaud, most of all, their evolution.

How are the vertical and horizontal twists actually distinguished from one another? You can just rotate a horizontal twist 90 degrees and get the vertical ones. Also, does the process of twisting look like. For instance, on horizontal twist and then one vertical twist. It would be extremely helpful in understanding the rest of the video.

Love this channel btw

So what about knots and braids?

Not a good title for the video. I might even go further and say that this video was more of an advertisement to the topic than useful stuff

Looking forward to follow-up episodes on the topic. I'm excited, but just as I'm getting ready to dig in, the video ended!

Tai-D you're knocking them out of the park 👌

How are the horizontal and vertical twists not the same under rotational symmetry? This seems like a simple U(1) group

Eureka! Thanks!

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gabe is better than this pathetic excuse for a presenter/comentator.

go steal a bike

The pair that came up with knot theory were at my school a few weeks ago. So cool!

This dovetails nicely with the recent team-up from PBS SpaceTime, PBS Eons, and It's Okay to be Smart.

You should have mentioned that human cell DNA is mostly in linear form and only mitochondria contain circular DNA. Otherwise people watching may have been mislead.

The ends seem to be in the wrong position. For example, at 4.04 the sequence is horizontal, vertical, horizontal. Yet the cartoon has ends placed vertically. Have I missed something?

A channel that describes mathematics in a very interesting way

One thing that might be helpful to point out – flipping the picture along a diagonal axis, so that the right side becomes the bottom side, corresponds to taking the reciprocal. That's how we get continued fractions: start by twisting the right side a positive or negative number of times (get an integer), then flip (take the reciprocal), then do more twisting on the right side (add integer), then flip again (reciprocal), etc.

That's also how the horizontal starting point and vertical starting point she shows are reciprocals of each other, 0 and ∞.

7:16 Left-handed DNA!

That was fascinating.