The applications of non-euclidean distance | Metric Spaces
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The chess metric is very relevant to a lot of endgame studies. They're hard to solve because it's not intuitive to us that the king can move diagonally in the same amount of time that it can orthogonally. Look up the Réti endgame study in particular.
I was about to mention the Réti endgame study since it's so famous and relevant.
Maths problem
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One time see.
Agreed
I remember knowing this would mean win/lose in a lot of games. At some point in the late game you need to mobilize your king or else all of their pawns will get to just walk right past you.
*Flashbacks to studying the Lucina Endgame*
Actually, the metric between (three letter) words is very useful in areas such as auto correct. It is called Levenshtein distance in computer science and it can be calculated through dynamic programming!
Huh, I think we used that in my computer vision class, but never knew the name. You can use it to correlate points between 2 images and generate a depth map and occlusion map. It's far from the state of the art any more, but it's useful.
Yeah, levenshtein distance can be handy sometimes but most of the time there are better algorithms for detecting similarity. For autocorrect for example it is better to check the physical distance of the pressed key with the key you'd need to press for a word and to make it extra fancy you even take this distance (and the direction) of the other keys into account for the score. That way you can find out that "gwkki" is meant to be "hello" and the user was just one key to the left, even though there is not one common letter in those two words. The same thing applies to other areas as well, so there might be certain areas where double characters should not play much of a role, e.g. when someone is using something like "whaaaaat?!" to express excitement or any other emotion in the text. Levenshtein distance is not really the best distance to use in pretty much every use case I came across so far.
Maths problem
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One time see.
Levenshtein distance is useful when comparing DNA sequences too.
@@johnrobertson4450 calm down. They just didn't tell him it had a name.
Metric, okay, but does this work in America?
System named after the function, not the function after the system, I would presume.
That said, America adopted the metric system in I believe the 1960's, before the UK. Our difference being, cultural adoption was optional, and we ended up defining imperial measures by the metric system and keeping our several million signs listing imperial measures :P
The inch is defined as 25.4mm, for example.
@@giin97 I found out recently that there are countries where non metric systems of measurement are banned. An American can't even send a tape measure to France. I was surprised.
@@CIorox_BIeach that's... Bizarre. Of course, we are talking about France, so...
@@CIorox_BIeach why is matric horrifying to you
@@puckry9686 he didn't talk about that, he said some place banned non-metric system is horrifying
Knight: I takes me 1 move to get to the white square.
King: Lucky, I takes me 2
Rook: Luck, I need 3
Black bishop:
*_you guys can get to the white square?_*
the surviving pawns on the other side of the board: *you guys can move without dying?*
@@kirtil5177 Pawns at the end of the board gets changed out for a different piece
@@9nikolai yes, thats why they are always killed or atleast threatened to be the closer they are to promoting
@@kirtil5177 The pawns are sweating profusely waiting for the end of the game like "You guys are allowed to move?! I'm just sitting here and everyone is already threatening me!"
@@chaotickreg7024 or 2 pawns locke in a tie till the end of time
I really like that you are using the technique of proof by example
Maths problem
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One time see.
@@mr.knight8967 stop spamming
I believe the mathmatically rigorous proof by example requires you to cover all possible cases, it isn't the cleanest of proofs to do but it is valid and occasionally used (knot types were done this way).
He is closer to proof by heres an example to explain it and trust me on all the cases I don't have time to cover. It does work great for teaching concepts though since the best way to learn a concept is in parallel to learning the calculations.
@@mr.knight8967 shut up
Of course, he's not really doing proofs. But TEACHING by example is a very good technique.
Definitions
5:28 2D distance formula
6:01 metric
8:00 taxicab metric
11:35 maximum metric
13:28 discrete metric
14:03 hamming distance
Wish I could bookmark a comment
@@palmberry5576 you can actually
right click on the time (x month ago) next to my name
open link in new tab
now you can bookmark my comment
wow I never knew that! thank you so much!
@@wada-wada I see a flag, is it the same thing?
@@ZalamaTheDragonGod
no. that's too far right
right click on "2 years ago" next to @wada-wada
"open link in new tab"
The chess example gets even more interesting when you consider the default bishop under normal rules. Then the sample distance you pointed out instantly becomes undefined, i.e. "You can't get there from here."
I was wondering if he was going to bring up the bishop...
Even a pawn can get there eventually, by walking up the board and promoting to something other than a bishop.
There are some pawn cases where a distance is only literal under very specific circumstances, otherwise they are imaginary. A pawn can only move diagonally if there is a piece there to capture which means distance varies by board position. A pawn can also promote and reach any square as a knight or queen depending on need.
There are also some distances that are impossible like bishops switching color or moving into check.
it would be very interesting as it will act as a infinite distance point in the metric space
@@siddharthvikram3054 A rook can move to any point on a coordinate plane in only 2 moves. A bishop can only reach half the squares in just as many moves. A queen is slightly more efficient than the rook as it can move to diagonal squares in one turn.
This geometry gets weird really quick.
I must have missed this one. Changing the background to a city backdrop to demonstrate the different metric is absolutely genius.
Came here when the like to dislike ratio was still infinity
You mean not defined right?
x/0 =/= infinity
No bro, 1/0 is not defined. 1/(value tending to 0) = infinite.
@@Leyrann 1/x is undefined *at* 0, but the right hand limit to 0+ is +infinity
@@pbj4184 Given that 'likes' are a discrete and not a continuous data set, there is no result between x/1 and x/0. No infinitesimally small decimal to generate near infinity. Original correction holds true.
(Unless we start chopping people into pieces and getting individual's sub-sections to submit fractured code semi-like impossibilities. But one should never have to involve cthulhu-esque rituals except on a Monday)
15:41 It seems pretty applicable for catching misspellings or word suggestions.
11:00 It's interesting to note that if we drop this requirement from metrics, to get a "pseudometric", we can generate new topological spaces. For example, the function d(x,y) = 0 for all x and y generates the indiscrete space, which is not metrizable.
These uses for rather more complex topics in the realm of mathematics again shows just how important they are to our everyday and abnormal lives. So interesting as always!
Maths problem
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One time see.
This absolutely made me go nuts, I've been just introduced to this topic and seeing everything that I used to see come alive is just awesome, if possible can you do a playlist for it, or can you suggest some books that talk more about the intuition that you gave us a gist of...😅
Maths problem
czcams.com/video/j9wY3hso1WQ/video.html
One time see.
14:15 I have actually created a simply autocorrect tool that finds the "distance" between 2 words in a similar way to how you defined it in this video.
However, my distance function is more complex than yours:
= It considers vowels to be very close to eachother
= It considers keys to be close if they are adjacent on a US ascii keybaord; *this part would vary based on keyboard, and should definitely allow users to specif their specific keybaord*
= Plus, it allows for letter duplication, and omitting
Thank you for ur video. I'm a Vietnamese student who can't understand lectures from my teacher. Then I came here and this video clear my brain 100%. Hope u will make more useful videos like this
1:30 3rd to Bedford seems the shorter distance; plus, it has fewer turns.
A jet pack is also faster.
Maths problem
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One time see.
This is a nice topic to discuss with special friends. Many of them were suprised by the shape of a "circle" using different metrics!
Maths problem
czcams.com/video/j9wY3hso1WQ/video.html
One time see.
Yeah, it turns out you can have a square circle!
In the discrete metric a circle would be the entire plane
It’s crazy to see a serious video from one of my favorite comedy skit CZcamsrs, my mind is blown.
Distance is super fun and important.
In robotics, we commonly use heuristic functions to guess at the actual distance between two robot positions. Those have some fun properties too!
Another fun thing that popped into my head: sometimes, in robotics, distance (we actually tend to call it cost, because it behaves ever so slightly differently) *isn't* symmetric, and that actually forms the basis of my grad research :)
Love metrics!
Your “shortest distance using streets”, near the start, was not the shortest distance using streets.
i noticed this as well, can't say it fills me with confidence
Who cares
You aren’t taking into account expected traffic
@Jeb
Could be you are confusing distance with travel time.
Years ago I was scolded by an Englishman for telling him travel times when he asked about distance. I thought it was only we antipodeans who do that. Maybe he was just a pedant.
The max(dx, dy) metric is used, for instance, in Doom, when calculating the distance to apply splash damage (e.g. from a exploding rocket or barrel). It gives square-shaped damage areas.
Great, now criminals seeing this video will use the equation to minimize their probability of getting caught.
This is the first video of yours that I've watched, and i think it's really good. It's very relaxing to watch since you explain everything in such great detail.
Then there are two distances.
1: absolute distance (units of space between two points)
2: traversable distance (units of space that must be traversed for an entity/object to get from where it is to some other unique point)
They can also define the P-adic numbers... Can't believe you didn't mention that!
It was crucial in proving Fermats last theorem
Good lord, this title makes me think of college!
Maths problem
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One time see.
This reminds me of my Data Mining course at university. Learning about the Lp metric, cosine distances etc was really fun.
Great topic and great video :)
Great! I recently went through the first chapter of Barnsley's 'Fractals everywhere' where he introduces metric spaces.
When studying in the book I really thought that I need a tutor and more visuals, to speed up the process of understanding this topic. So this is exactly what I needed!
In the book I spend a lot of time looking at formulas and reading definitions over and over again, while this video does it in a fraction of that time!
I don't have anyone to talk to about math, so I really enjoy watching your videos. I know it's a lot of work to make them and maybe sometimes you question your sanity and if it's worth it. Please know that many of us really enjoy them to a great extent. You don't know it, but you're our closest math buddy, for all of us.
Thanks for this brilliant and fun explanation - and yes, Lumberjack Feinman's lectures are good too :)
Literally so many of your comedy videos have been recommended to me, and watched by me. Now here I am in a non euclidean rabbit hole and youre about to teach me some real shit. MY MAN!!!!!!!!
Interestingly, "it's symmetric" sounds similar to "it's a metric"
i really like the formulation of metric space at the beginning before you start examining diff functions/metrics . thank you for making this video , i really have learned alot !
Briliantly explained!!! Thank You.
This is exactly what i was looking for 👌
Great timing. My real analysis homework on metric spaces is due today lol
Before you said "discrete metric" I was thinking "teleportation metric".
I'm still in the 12th grade , and I really like your videos , it basically helps me alot and also give me some of encouragement that university is gonna be good , is gonna be different, is gonna be challenging, is gonna be fun , and I also really appreciate all your work in these informative and useful videos , specially the videos where you talk about university, the applications of what you study etc, your own experience in it
I'm searching for a good niche in engineering and your videos are always helping!
Keep up your awesome and noble work Zach! Goodluck!
His videos inspire you while money keeps flowing to his money account heheh
First rule of life: Nothing is free. But some things are delicious, like Zach Star
Metrics in spacetime work a little differently, though. The distance between two points (or the "interval" between two "events", to use the correct jargon) can be zero, or even negative. There is an equivalent of the triangle inequality, but it's a little bit more complicated than the version used in these examples.
Then it is a topological space, not a metric space.
Great video, it covers so many concepts I learned in different lectures in a very applicable way, which is really refreshing
Thank you for the amazing videos zach!
Was nice for me to see this video since I'm currently taking the course 'metric and topological spaces'
Maths problem
czcams.com/video/j9wY3hso1WQ/video.html
One time see.
Deep respect, Zach Star!
was curious and decided to check this out. wasn’t disappointed, it was very interesting 👍🏼
I’ve learned a lot from this video! Thanks!
bruh nice upload time i'm literally working on a DG assignment right now
let's fucking go
1111111111
@@PapaFlammy69 u r a meme virus
DG ?
DG?
Awesome as always. Thanks.
Excellent Video!
Zach I'm planning to get a PhD in Industrial Engineering & start a business.
I think you should get a PhD in math. You clearly love it and keep learning. That's how I found my love of engineering statistics and manufacturing.
Keep it up, I love your videos bro.
Thanks for this amazing video that made my day, it helped to my homework
Great video! I would like to see a continuation of this topic talking about pseudo metrics. :)
You should do more videos on hyperbolic/spherical space they’re rly good
This was awesome!!
1:26 "Instead, the useful distance would be the shortest path, using streets"
>doesn't draw the shortest path
The fact that d_maps is less than d_euclidean for some spaces is revolutionary!
Great video. Thanks Zach. Love all your videos. What is the software you use to animate points on the graph?
Great video
Good work 👍
Thank a lot
I LITERALLY WENT OVER THIS IS CLASS YESTERDAY!
Amazing thanx alot
I'm not sure if you've covered this or not, but I would love to see a video on convex sets and their applications to concepts such as optimization.
Dmaps is called Manhattan distance BTW. (Geographer here.) In a universe as hologram or 2D surface on the event horizon distance is a bit of an illusion. Well our conventional 3D distances anyway!
An Inspiring video indeed!
Dude you’ve really outdone yourself with this video, probably my favorite ever of yours. I’ve been wondering what a metric is and you explained it so clearly. I’d love more videos explaining fancy math terms.
Maths problem
czcams.com/video/j9wY3hso1WQ/video.html
One time see..
I don't know about you guys, but just a few of days ago I was thinking about how to create non-euclidean spaces and how to calculate distances from point to point.
I wont say why but I will say I think it's crazy how sometimes I think about something and a few days or even weeks later I'm recommended a video about it here on youtube.
15:00 fun fact: detour can only increase the distance by an even number
I'm enjoying contemplating the discrete metric unit circle.
Please make a video on what is mechatronics and it’s future
Even I want to know
The Taxi Cab geometry is great for the game of Snakes! (The cellphone one where the snake gets a treat and grow long)
Stop making good videos show up in my recommended while
I’m doing homework, I can’t focus
Cat-->Cot-->Dog is also like the detour point being on the line segment for Euclidean distance because on a line segment AB we will encounter every point on the line segment going from A to B just like how Cot (or any other detour word than keeps the distance tge same) comes up in the process of converting Cat to Dog (for a particular order of changing letters).
I just realized the "French railway metric" isn't as popular outside of Germany as I thought it was.
Imagine a railway system where every single train is going through the main station, but the only place to change trains is the main station.
Either your destination happens to be on the same track going through the main station as your starting point, or you have to take one train to the main station and then another train from the main station your destination.
With the French railway metric, the distance between any given point a and the origin is calculated using the euclidean metric, but for two points a and b, the distance d(a,b) is either
||a-b||
or
||a-origin|| + ||b-origin||,
depending on whether a and b are on the same straight line through the origin.
"Most of you would say 5"
Yes i definatly thought that and wasnt immediatly thrust into a distant stare at the sight of numbers.
I think I saw the use of distance in position encoders as well. By making sure the 'distance' between any two adjacent positions is only 1, they avoid imperfect transitions where the encoded position momentarily 'jump's because not all the bits change state exactly the same moment. Grey codes I believe they are called. Read a book years ago on 'taxi cab geometry' this video triggered me to go dig that out and read it again. :)
I see a Zach Star video, I watch.
Simple.
The simple complexities of the obvious, greatly explained and interesting !
Sounds good for a sub =] Thank you
I like the visual aspects of maths but not the numbers and such, so this kind of stuff really hits that sweet spot
Nice vid 👍 , can you make another part exploring metrics as H-norms and L-norms
Reminds me of what was used for movement with the board game Star Trek Ascendancy.
I think everybody is correct because you need to explain more. Like Knight is 3 spaces away, or Knight is 1 turn away, or the distance between knight and (space) is (idk).
could you do a video on how time would be effected by a negative gravity well?
The min function at 11:22's metricness can also be disproved with the "point c" requirement. If point a is (0,0) and point b is (3,3), the distance=3(for both x and y). If you place a point c at (2,1), the distance a to c=1 (change in y) and the distance c to b=1 (change in x).
*triangle method
A nice application is in (admissible) heuristics, in particular solving the 15 number sliding puzzle.
Hey Zach Star,
Please upload a video about the application of set theory.
15:43
Good timing, just watched 3b1b's Video on Hamming codes a couple of days ago :D
fun fact: Dungeons and Dragons uses taxicab distance for spell ranges, movement, etc. D&D by necessity is played in a non-euclidean world.
At 13:34 Well, that's a very appropriate name for that!😆
it is useful to say the rook travels 3 or 7 squares in the chess example because, unlike the knight, the rook can be obstructed. It's just that the rule is the rook can travel an infinite horizontal/vertical line.
what is the significant of
d(a,b) = d(b,a) and d(a,b) = 0 iff a=b?
That’s awesome
OH HE'LL YEAH, AWESOME!!!
12:50 fun fact: Magnus Carlsen, currently the #1 ranked chess player, has never been able to remember this rule, and has through his entire chess career resolved to calculating end games when he could've easily seen the outcomes using this rule.
Super video! I applauded for $2.00 👏
3:20 it depends what your anser is in, if you said "its 3 squares away" then the distance is 3 but if you say "its 2 moves away" then the distance is 2
hamming distance is quite a useful metric and something that is rarely thought of as a "distance".
Edit: nevermind 15:50
@Zach Star are Van der broeck, Alcubierre, Natario and Morris-Thorne-Kuhfittig metrics also the kinds of metrics you described here?
Fun fact: The map at 1:09 is at (40.7145380, -73.9612940) in New York City
Was doing an assignment, now I’m doing this video
I found this channel by watching a bunch of his comedy skits first, so I thought this was a setup for another sketch. Imagine my bafflement when I'm 5 minutes in, there haven't been any jokes yet, and I realize "hang on a fucking second this is just an actual lecture"