Embedding a Torus (John Nash) - Numberphile

This videos features James Grime with a little bit of Edward Crane.
More links & stuff in full description below ↓↓↓
Ed's full discussion of Nash and embedding: • Nash Embedding Theorem...
Nash shared both the Nobel and Abel Prizes. Full details.
Nobel: www.nobelprize....
Abel: www.abelprize.n...
Other mathematicians have done much work in this field (we have focused on Nash in these videos).
Great summary here: math.univ-lyon1...
And top paper on the corrugated torus: www.pnas.org/co...
James Grime: singingbanana.com
Edward Crane at the University of Bristol: www.maths.bris....
Twisted Torus video: • Topology of a Twisted ...
Torus Balloon at Maths Gear: bit.ly/TorusBal...
Support us on Patreon: / numberphile
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Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumb...
Videos by Brady Haran
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanb...
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Numberphile T-Shirts: teespring.com/...
Other merchandise: store.dftba.co...

[@NowhereManForever]

Grime has to be my favorite Numberphile speaker.

[@The_Aleph_Null]

Yeah. I really like him and the two guys from the old Graham's number video. So fun to hear them talk.

[@CalvinHikes]

NowhereManForever He's the best explainer of things. And his voice is calming.

[@metalhusky]

NowhereManForever Professors Grime and Moriarty, for me.

[@nandafprado]

NowhereManForever Did you know he has his own channel? Look at singingbanana

[@NowhereManForever]

nandafprado Did you read the other comments in this thread?

[@TheMaplestrip]

James Grime is so awesome, probably my favorite Numberphile professor.

[@CanariasCanariass]

Same here. Love his enthusiasm when he explains stuff!

[@devistnathan730]

Same. He's just the most fun to see. You can really tell he loves his job.

[@fade6827]

***** He's awesome

[@TheMaplestrip]

I am now telling people about how I got 59 likes on a KZbin comment by saying that a specific mathematician is an amazing person. What's with all the confused looks?

[@General12th]

***** Most people agree with you, methinks? I agree that James Grime is an excellent professor, although I prefer Dr. Simon Singh and wish he would make more videos.

[@BroadcastBro]

Big thumbs up for Dr James Grime, he's superb in his communication technique

[@IceMetalPunk]

BroadcastBro And for his next trick, here's a poodle XD

[@Lil.ian.anzac.12]

Yes, even I almost understood!

[@Mallyhubz]

You really should get Sharpie to sponsor your vids.

[@numberphile]

Mal Hubert do you know anyone in their marketing department!?

[@DrGlickenstine]

Mal Hubert That is an amazing idea

[@chromatosechannel]

Mal Hubert true! i support that.

[@TheSentientCloud]

Numberphile Are you sponsored by the people that provide your brown paper?

[@johnlapage599]

+Numberphile I know someone who handles sponsorship for 3M, who have a rival range of markers pens. How brand loyal are you?

[@PhilBagels]

Much more important is the question: Where do you get toric balloons?

[@papayaman123]

PhilBagels I want to stick my nob init

[@anonomusanonomus1589]

Blow torical breaths

[@pupyfan69]

Maths Gear

[@eoagr1780]

Look for donut ballons

[@Triantalex]

false.

[@PotatoChip1993]

Prof. Nash and his wife died in a car accident when coming back from receiving the Abel prize. It's weird to see this wasn't mentioned in the video...

[@numberphile]

PotatoChip1993 that is true, and I think many people watching the video probably know that - but in a year or two I hope people still watch this video, and the fact he "died recently" might seem less important than his accomplishments. I think this video can be watched now in the context of his death, but later can just be a discussion of his work.

[@rentzepopoulos]

Numberphile I salute your way of thinking!

[@KrakenTheKode]

It mentions it somewhat at 12:30 when it says John Forbes Nash, Jr 1928 - 2015

[@LiviuGelea]

Numberphile , perhaps so, but the video still sais "John Nash is..." instead of "was"

[@johngalmann9579]

PotatoChip1993 It might have been filmed before his death I saw his lecture in Oslo, really weird when i heard he died just a few days later

[@KTC88]

RIP John Nash. Your work helped inspire so many mathematicians and economists. May your legacy continue on for many generations to come.

[@DustinRodriguez1_0]

This was a really interesting video, but also a bit sad. I had not heard that John Nash died yet. Major bummer. I'm from West Virginia, where he was from, and his work in game theory has always interested me.. comes in handy when working with simulations. His life was fascinating, too, in that he suffered from schizophrenia but after years and years of it he made a conscious decision to stop listening to the voices he heard, analyzing the things they said with reason and ignoring anything irrational. He was able to, essentially, think himself sane. That is, to me, absolutely astonishing. I am very sad to hear that after all of the things he survived in his life, a stupid car accident took him from us. At least he was able to receive the Nobel prize he so deserved before he left us.

[@4mathieuj]

I can tell that the months of absence have been invested in making wooshing-sounds while drawing.

[@ahenryb1]

An excellent use of time I would say

[@SocialWalrus]

ahenryb1 I concur.

[@4mathieuj]

Social Walrus Me too, I always like the videos with James most

[@marksmithwas12]

Not just any theory, a GAME theory!

[@loicoberle6156]

***** Thanks for watching !

[@jonahmioduszewski45]

***** Beat me to it

[@unvergebeneid]

***** Pretty sure Matthew Patrick knew about the actual field of game theory before he named his channel and in fact derived the channel name from mathematical economics. Apparently for many of his viewers it's the other way around.

[@1996Pinocchio]

***** my thought :D

[@zelivira]

Penny Lane Sucked when I was looking on youtube for game theory related videos a long time ago and a bunch of the results were from that channel.

[@jOpaaD]

Dr. Grime is so good at explaining complicated things in a simple way.

[@strengthman600]

You know, I know a beautiful quote from John Nash "It's just a theory, a game theory"

[@kbking16]

liar. George Washington said that

[@MoonTheGoat]

+Kbking16 No, you're both wrong. Donald Trump said it

[@minnarewers3573]

You know it's MatPat, right?

[@strengthman600]

+Minna Rewers Psst, it was a joke, I know it was MatPat

[@godsadog]

And if the most efficient path was to treat everything a game, it would be the only theory.

[@alsoethan]

'waves' ~~~~~ 'hand action' ~~~~~

[@dharma6662013]

The torus has points of positive, zero and negative Gaussian curvature. The "outer" points are elliptic points (+'ve), the "inner" points are hyperbolic points (-'ve), and there are two circles of parabolic points (0) separating them.

[@gabrielsayers360]

James is surely the best Numberphile speaker, he explains it all really clearly, without being patronising and while maintaining the audience's interest through his own evident enthusiasm.

[@bobbysanchez6308]

James Grime is my favorite mathematician who appears on this channel.

[@CodyBenson13]

Dr. Grime is easily my favorite speaker on this channel. He is so excited to explain things. It really effects me when I watch. :D

[@hermes_logios]

This is the coastline measurement paradox. The length of a coastline is infinite if you use a small enough unit of measurement. The more irregularity you ignore by using a longer unit of measurement, the shorter the final measurement will be. In other words, distance depends on granularity. It’s what Greek mathematicians called “exhaustion” (measuring geometric curves by dividing them into smaller and smaller units), and what algebraic mathematicians call “calculus.” In topology, the granularity is called “smoothness” of a surface.

[@AdrianRowbotham]

This corrugation technique appears to be related to the so-called "π = 4 paradox" - whereby constantly cutting corners out of a square (and out of the resulting shapes each step) gets you to an approximation of a circle where the perimeter is the same as the original square.

[@vincentfiestada]

James Grimes explained it so much better than the other guy.

[@Fiendxz]

I designed this EXACT system, without being able to mathematically prove it (obviously) in my senior year of high school for a game that I was designing. Not that this had any significance, but it's really cool to see an idea that you had years ago re-appear with mathematical relevance. This is what learning is about.

[@parkerd2154]

Brilliant, one of your best. I'm working on 3D printing Nash's embedded torus at the moment.

[@elmoreglidingclub3030]

This is beautiful! I thoroughly enjoy your videos. I am 70 years old and just completed a PhD two years ago. I study AI. I wish I had years in front of me to immerse myself in maths, to hang out with guys like you.

[@Oozes_Dark]

Ugh, I'm so happy I found this channel! As a high schooler who loves math, it's so exciting to look at these complex problems and be able to understand them on some level even though I haven't gone past somewhat basic calculus.

[@mathoc5273]

This is my all time favorite numberphile video! I love how such a simple concept we're all so familiar with (asteroids) creates such a stunningly complex 3 dimensional shape!

[@GaryMarriott]

I think what you just described is an Origami Torus, something whose surface is flat except for a large number of folds where curvature has no meaning. It is something tedious but no way impossible to make.

[@willdeary630]

I'd think it would be really good if you went more into the maths behind these topics for those with a higher maths level, e.g. I'd like to know how partial differential equations are applied to this situation.

[@wugsessed]

The terminology of corrugations and imagery of what that deformed torus looks like really reminded me of the process of sphere inversion. It's a fascinating topic, and there are some pretty good (but very old) KZbin videos on it.

[@diebydeath]

If you're playing the game of Asteroids, there's some interesting applications of this donut. If you're playing the game of Hemorrhoids, you're probably sitting on the donut.

[@bleis1]

what I enjoy the most of your videos is that you take the time explain with paper and numbers in a way someone who as difficulty with math can still understand very clearly thank you for all your interesting videos, I am always looking froward to the next one thank you

[@ColossalZonko]

"ta sqeekz r extra" - singing banana 2015

[@hubert6943]

5:03

[@Anonymous71575]

I wonder... why is it only Mr. Grime that can make me understand and not bored during a Numberphile video? Maybe every Numberphile video that doesn't have Mr. Grime should have a reupload with the version whose speaker is Mr. Grime.

[@Nykstudio]

understanding a complicated theory is one thing, but explaining it in a simple way takes brains

[@vimalgopal5873]

I am not a mathematician... and yet, I get so much joy out of watching these kinds of videos on KZbin. In fact, I can't stop watching them! I'm completely addicted! What is wrong with me?

[@josephhargrove4319]

Fascinating subject that appears to be able to unlock a lot of doors in applied mathematics. Glad to see James Grime back. We haven't seen him for a while and he was missed.

[@JNCressey]

When I was a kid I had a play mat that was a flat torus (although it was a slightly longer rectangle, not a square). It had an aerial representation of roads and buildings on it; where the roads went off the sides they lined up with the roads going off the opposite sides.

[@bjrnvegartorseth9028]

Grime's wave hands are top notch.

[@jpopelish]

Another way to look at this solution is that the corrugations make the flat, inelastic paper stretchy and compressible, so after you form it into a cylinder, the cylinder is elastic enough to bend round on itself, into a torus, without actually having to stretch or compress the paper along its surface. The stretching and compression only alter the shape of the waves.

[@technopoke]

Been watch Numberphile for years, and only just seen one with someone I know in it. Hi Ed!

[@Dithyrambos-h5e]

Embedding a Torus: Subtitled "Why Mercator Projection is Horribly, Horribly Wrong"

[@NothingMaster]

In pure mathematics you could theoretically define a space just about any way you like, and even pump it full of straight edges, singularities and other niceties to which you might even be able to find some cooked up, albeit internally coherent, solutions. That’s the inherent power of a purely/mathematically conceptual creation. Now, whether such a made up creation could translate into anything physically meaningful or not is subject to a philosophical debate or an empirical observation. In a Universe where pure mathematics gives you wings to fly a fantasy plane you might as well take to the skies, even if you never actually leave the ground.

[@CobaltYoshi27]

Rest In Peace John and Alicia Nash. You and your contributions will never be forgotten.

[@michaelmoran9020]

astonishingly I'm watching this because I have a very practical use for this information in computational-chemistry.

[@pietvanvliet1987]

I like numberphile video's, but normally I can hold on for a minute or two (doesn't keep me from watching the full video though). This time, I feel like I sort of got this. Which makes me conclude that Dr. Grime either did an excellent job explaining, barley scratched the surface of this topic in order to avoid scaring people like me, or a combination of the two. I'll carry on believing the first one is true. Thanks.

[@m8e]

The flat square torus can also be seen as a straight tube where the height is half the circumfence and both the inside and outside make up the surface. ie fold the square paper along the red line(the equator), then make it into a cylinder. Works in videogames...

[@mueezadam8438]

4:58 finally, some numberphile ASMR

[@CylonDorado]

Reminds me of how the distance of the borders of a country (or whatever) on a map changes depending on how much you zoom in and account for every nook and cranny.

[@zh84]

Doesn't the infinite number of corrugations form a fractal surface?

[@elijahgardi7501]

zh84 As I commented, it reminds me of a Fourier series. By that I mean, if a particle were represented as an EM toroidal vortex, the corrugations (Fourier series) begins to define the size of the torus. Maybe..

[@louis1001]

But the point in fractals isn't defining an infinite number of corrugations, is it? The torus example probes that a finite number and depth of corrugations would eventually get the lengths in the lines to be equal sized. That was my guess, though.

[@markcarey67]

Yes, it was, like Weistrauss's "pathological" function an example of a fractal before that concept entered mathematics

[@cyr-9564]

Luis González is correct, a fractal means that there is infinite perimeter. In this case, a finite number of grooves works because you have a finite distance set in mind.

[@Twentydragon]

I didn't notice any part of the rules here stating you couldn't "crease" your embedded torus, so to preserve distances in every direction (not just those two), I would "fold" the flat torus along the red line and then attach the "ends" together. I am, of course, also assuming that you could fold it in such a way that the thickness is 0, but that each "side" of the ring was still separate from the other.

[@urbanninjaknight]

The torus has negative Gaussian curvature as well. Also, the curve which seems to be a quarter-arc of a circle plus a straight line does have curvature defined everywhere but it's discontinuous.

[@jingalls9142]

"The squeaks are extra" That's what i was told when i bought my apartment in Hells Kitchen...i dont think the Russian dude meant Torus'...

[@thinkstorm]

The problem of curvature around 9:30 in the video is very obvious in model trains: if you have fixed track pieces and join a straight with a curve your train goes from infinite radius (=straight) to 20cm radius in an instant. It results in a jerking of the cars going around the curve at exactly that point, very annoying looking. That's why we use "flex track" and add curve "easements" to avoid that visible jerking.

[@lucidlactose]

It's been quite a while since I have seen Grime in a recent Numberphile video. I was actually shocked when I first saw a video without him when I first found this channel with every video with him and then suddenly without. Now that I think about it, why have I not subbed to the Singing Banana yet?

[@SamuelLiebermann]

On a graph where one dimension is time and the other is speed, no single point describes acceleration (not only the points where speed starts/stops changing. Acceleration is the change of speed between TWO points on the graph.

[@ira6133]

Imagine if this is the actual shape of our universe. That would mean it's endless and confined at the same time.

[@emewyn]

I feel like I may be missing something, but wouldnt just making the torus taller also make the two distances equal?

[@smoosq9501]

This is absolutely genuines idea, really really impressive

[@jopaki]

I now have a much fonder view of John Nash. what incredible things to think about.

[@sumdumbmick]

when I looked at his work in game theory I was frankly just surprised that anyone ever had to say that. it all seemed absurdly obvious to me. the other stuff he did was absolutely more important. but that's also why it was possible to make a film around the game theory stuff, because it's so simple that you can sort of explain it a bit to movie goers in the midst of an entertaining story about people and their relationships and dramas. can't do that with anything of real substance. so if you think I'm just bragging by saying his game theory work was trivial, no... it's a demonstrable fact that it was trivial. the movie is that demonstration.

[@Richard_is_cool]

Well, the colors of green and red were switched, but otherwise: Dr Grime again the BEST!

[@prathameshjoshi9199]

What if we just take a cuboid shape like a Book a very thick book and cut a cuboid hole at its centre & just make sure that the thickness of book is large enough to make sure that the length of green line on flat torus is equal to the length of red line ?

[@shambosaha9727]

6:27 Doesn't it have positive curvature only on the outside? And negative on the inside? Or is it positive everywhere? Can someone help?

[@IanFarias00]

Yeah, by curvature I also understood "Gaussian Curvature" and there are different points in the torus that have positive, zero and negative curvature.

[@omikronweapon]

I'd advise you to watch Cliff Stole's videos on gaussian curvature, or topology in general. he explains a torus.

[@seanlegge3854]

It's not just Asteroids! Games like Chrono Trigger, Secret of Mana, and Final Fantasy VII are also played on a torus. In fact, prior to last year's Geometry Wars 3: Dimensions, I had only encountered ONE game that is played on a topologically-correct sphere: E.T. the Extra-Terrestrial for the Atari 2600!

[@NoriMori1992]

I can't remember if I've watched this one before. No matter! I just finished watching Cédric Villani's RI lecture on Nash's work in geometry and partial differential equations, so this should be easy to grasp!

[@thrillscience]

These videos are fantastic. Thanks, Drs. Grime and Crane.

[@e7540]

Nice sound effects, James Grime.

[@finaltheorygames1781]

Actually curvature is defined for that second graph. As long as the line is not vertical then there is acceleration.

[@TheGodlessGuitarist]

So glad Nash did this. I don't know what I would do without it

[@JoshuaWiedekopf]

Just a quick side note to those who still think that Brady should have made it clearer that Mr Nash sadly died about a week before this video was released, I'd like to point out that he actually explained his reasoning for not doing so in Episode #39 of his podcast "Hello Internet", just after the half-hour mark. Not talking about his death was a conscious decision. I encourage you to listen to the podcast if you are interested in Brady's decision.

[@jeffreyhowarth7850]

How are the Nash torus and regular torus related to each other? Are they homeomorphisms?

[@whoeveriam0iam14222]

new video right as I checked the channel!

[@SocialWalrus]

whoeveriam0iam14222 You know you could just subscribe, right?

[@whoeveriam0iam14222]

Social Walrus I am subscribed.. but I came looking for the video on hyperbolic stuff and I saw this video 18 seconds old

[@ayasaki.pb_787]

For a long time a haven't seen you. It's good to see you again. =) I had watched the movie and it was really beautiful and motivated for math-lover.

[@maxisjaisi400]

Ah, back to the old feel of Numberphile videos which made me fall in love with them in the first place. I enjoyed the previous videos, especially with James Simons, but baseball just didn't go well with hyperbolic geometry.

[@SimpleThomasse]

HOOOOOW ! that's one of my math teacher that actually build this torus ! Vincent Borrelli if you look at the links in the descrption ! It nice to see a french guy here =) (even though he isn't mentioned in this video)

[@deblaze666]

The torus has positive curvature in some parts negative in others and 0 in others.

[@jonathonsanders1844]

Wow! So much easier to understand this video over the other one! Better explanation!

[@finthegeek]

But that's just a theory, a game theory, thanks for watching!

[@lewisbrearley2725]

As soon as he said Game Theory I instantly thought of Matt!

[@Coffe789]

+lewis brearley Matt actually took the name 'game theory' because he thought it'd be funny if it meant video games. (Which in some cases I guess it could..)

[@kdawg3484]

The difference between this video and the Edward Crane video (linked in the description) on this same topic is a perfect microcosm of the sad state of university education, particularly for beginning students: My reaction to Edward Crane, representing the vast majority of professors: "I know he's knows what he's talking about, but I sure don't. Is this what I'm going to have try to make sense of at some point if I continue doing this? Is that possible? Is it worth my time to see if I ever can?" My reaction to James Grime, representing a select few among professors: "I know he knows what he's talking about, and while I don't understand it completely, I kind of get it. I bet if I studied this a little more with the right tutelage, it would make sense and eventually I'd even understand more like a pure mathematician." My experiences with CS, EE, ChemE, physics, math, and many other technical professors when I was in school bore this out. I've heard plenty of other stories of the same thing and seen lots of smart people ditch courses of study simply because they couldn't find any connection to it to keep them going. James Grime seems like the kind of guy that actually wants to help more people become mathematicians. Most professors seem like they don't care or actively want to discourage students from continuing, and it shows in their pitiful communication of ideas to beginners.

[@nicolaesclarke1488]

I HAVE NO IDEA IF ANYONE HAS SAID THIS......BUT THE EPISODE IN THE SIMPSONS WITH FERMANT LAST, THEIR WERE A SET OF NONCHALANTLY PLACED DONUTS. WHICH MATHEMATICALLY SPEAKING, THEY'RE TUROS SPIRALS.

[@JonTheGeek]

"Pioneered game theory..." *cue science blaster song*

[@TheSentientCloud]

Ah now THIS is straightforward, unlike the other video. Now I see what he was trying to say. Now with the ripples on the torus, you're essentially creating a fractal surface. After an infinite number of iterations of the waves, wouldn't the dimension of the torus be a hair less than 3 (or rather, less than 2 since we're talking about the surface of a torus)? Why wouldn't this affect the theorem, or is the theorem referring to any generically higher dimensional space embedding, not a specific dimension?

[@WillToWinvlog]

So much calculus involved here!

[@SoulShak3nLIVE]

I don't understand this... Maybe I'm crazy but you could surely stretch the Torus into a shape that allows for both line to be the same distance. Wouldn't inflating the top and bottom but keeping the middle circumference the same allow both lines to be the same? Consistently?

[@Tekar0]

This is probably wrong but I don't know why: If you are allowed to make discontinuities in your surface, why don't you just first fold the paper in half along the green line. That way you join the edges and thus join the red line. Granted, it curves infinitely much in two points but I gather that is allowed. Finally you make the cylinder like shown in the video, joining the green line.

[@frxstrem]

***** If I have understood the concept of the video correctly: First of all, we're not allowed to make discontinuities in the surface. Having discontinuities would mean that there were part of the surfaces that were not connected, which is not the case here. Secondly, sharp edges, while not being discontinuities, are still not allowed, since we want to keep the surface differentiable at every point, while the points along the edges have no derivative. The curvature of the surface is connected to the second derivative of the surface, or rather, the radius of the sphere that approximates the surface best around a particular point. At the sharp edges, there is no well-defined second derivative, so there's no well-defined curvature either. So there's no such thing as "infinite curvature".

[@YCLP]

Do all equal-length lines on the flat square surface have equal length on the 'bumpy' torus? Or does this only hold for the green and red line?

[@Strafprozessordnung]

James Grime needs his own TV show. Might help 'Murica to get out of its educational problems.

[@xcry0541]

He came to my school!

[@unvergebeneid]

undjetzh He has his own KZbin channel and he sometimes writes for the newspaper. Does this count?

[@ragnkja]

Penny Lane The KZbin channel definitely counts. I just wish he'd upload stuff there more often. I'm not sure I'd want him to waste his time on such an outdated medium as TV.

[@unvergebeneid]

Nillie As long as the videos were also posted online I couldn't care less. Far too few shows do this, however.

[@jasondoe2596]

***** *Highly debatable* (but this might not be the place to discuss it).

[@xCorvus7x]

As far as I understood, the equal lengths are preserved by making ripples, which decreases the width of the torus while the circumference along the green line is as long as along the red line since the green line is not the direct way (it goes in serpentines). What has this to do with curvature or maintaining a curvature of zero, or differentiation?

[@Djole0]

This is so much better explained, then the other video, I loved it, thanks.

[@Drachenbauer]

cool torus balloon! I have some flower-shaped ones, which also have such a hole in the center.

[@ChinaPrincessDoll]

You better feel special that I watched the whole thing! ☺️

[@atkmachinei]

Much better explanation to this concept.

[@MrSafeTCam]

It seems a little fractal, so this may not work, but doesn't this imply that there's some way to fold that piece of paper such that it has a shape very close to that one and, because it's paper, the distances would all match.

[@BrynSCat]

A Eternal wormhole-in-wormhole event horizons have undefined curvature so its not a fold as such its an instantaneous change.

[@ru-nt8wi]

I think numberphile should also discuss game theory and its impact on oligipoly market analysis and dgse.

[@reubenfrench6288]

Welcome back, James Grime!

[@mecemodzada8315]

John Nash died in car accident a week ago...

[@cai6972]

umm

[@sevi1547]

Mećem Odzada That's ... probably why they're releasing this video?

[@hermaeus_jackson]

Mećem Odzada clap clap clap.

[@NylePudding]

Mećem Odzada Why it says 1928-2015 at the end.

[@evalsoftserver]

Maybe a CIA HIT

[@PureAwesome33]

Yay, James Grime is my favourite numberphile contributor! :D

[@jochemvanoosten9350]

When you were doing the thing about acceleration, you need to keep in mind that if you want to have accelerarion than you always need a change in velocity and in order to have a change in something you need two points. So there is no point talking about a single point and its significance

[@elijahgardi7501]

The infinite corrugations reminds me of the Fourier series used to describe a particles location/momentum in QM. If a particle were described as an EM toroidal vortex, these infinite corrugations should correspond to... Something. Maybe the size of the torus, which would mean affecting the location and momentum of the particle requires the same energy. I might do some trial and error work to find out if that's true.

[@juliasmith1182]

Even with the speed example, isn't this where Fourier Analysis comes in? When you speak of compounded waves as well - what is the connection here?