Coffee Cup Vibrations - Numberphile

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  • Опубліковано 27 вер 2024
  • Tadashi explores coffee and tea cups to new depths...
    More links & stuff in full description below ↓↓↓
    Playlist of Tadashi Tokieda videos: bit.ly/tadashi_...
    Support us on Patreon: / numberphile
    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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    A NOTE ON THIS VIDEO:
    A few of our Tadashi videos blur the already blurry line between mathematics and physics... Some people suggest they may be a better fit on Brady's dedicated physics channel (called Sixty Symbols).
    In response, the reasons they are on Numberphile are:
    1) Tadashi, while certainly a modern polymath, is based in a mathematics department.
    2) This sub series (with its animation and extra production work) has been supported by the Mathematical Sciences Research Institute, a Numberphile collaborator.
    3) Some of the previous and future Tadashi videos are more math-oriented than physics - and it just makes sense to put them on one channel rather than dividing the series between two channels on an ad hoc basis.
    In a multi-disciplinary world, it is challenging to run UA-cam channels which people come to associate with just one discipline.
    That said, feel free to go over and watch hundreds of physics videos on Sixty Symbols --- / sixtysymbols

КОМЕНТАРІ • 858

  • @Robin_Nixon
    @Robin_Nixon 8 років тому +1273

    I love how just as the Professor explains something you immediately see where he's going and a whole load of stuff you never thought about before suddenly makes perfect sense.

    • @slendy9600
      @slendy9600 8 років тому +90

      same. excellent teacher he is

    • @bgezal
      @bgezal 8 років тому +69

      He was very elegant about the "what is sound?" sidetrack. A concise explanation and then swiftly back on track again.

    • @Louis-fh4sh
      @Louis-fh4sh 8 років тому +59

      Yeah he basically explained the whole concept of sound in around 20 seconds, it was impressive x)

    • @fossilfighters101
      @fossilfighters101 8 років тому +1

      +

    • @OhSoUnicornly
      @OhSoUnicornly 8 років тому +6

      I don't find him as interesting as the other people on this channel, but you're right... His explanation of sound had me going, "wow, sound is just how we perceive tiny, fast movements in the air, how weird is that? What if we could do that with all air movements? We could 'see' everything going on around us, without our eyes".

  • @willfreese
    @willfreese 6 років тому +193

    Because of Tadashi, I am now aware of the very low musical quality of my mug. It is almost as if the manufacturer wasn't thinking about tone at all.

    • @samuelthecamel
      @samuelthecamel 4 роки тому +7

      What manufacturer would forget about tone?

    • @aurelia8028
      @aurelia8028 3 роки тому +3

      How dare they! You should never buy a cup from them again xD

    • @unrealnews
      @unrealnews 3 роки тому +2

      How would apple design a coffee mug?

    • @Triantalex
      @Triantalex 11 місяців тому

      ??.

  • @leckam
    @leckam 8 років тому +466

    it was inevitably, I immediately went to my kitchen for a cup and try it...
    and after hearing it! I was smiling like a kid!

  • @mikosoft
    @mikosoft 8 років тому +592

    Whoooom Whooom vs heehooheehoo - that just made me roll on my office floor laughing loudly

    • @baranxlr
      @baranxlr 8 років тому +11

      Japan...

    • @baranxlr
      @baranxlr 8 років тому +5

      *insert nuke joke*

    • @somberyu
      @somberyu 8 років тому +6

      +Baran Hekimoglu Oh no, not the nuke jokes! Must... resist...
      This is why you don't nuke a country twice.
      Goddamnit!

    • @CaseyShontz
      @CaseyShontz 6 років тому +10

      mikosoft this one goes W H O O M M M M M M M W H U M M M M M M M W H O O M M and the other one goes *heehawheehaw*

    • @Chewychaca
      @Chewychaca 5 років тому +6

      Full 6:00 . Whoom 6:03 . Heehaw 6:09

  • @jacobh1995
    @jacobh1995 8 років тому +617

    6:00 "Heehaw" Take notes, guys, this some scientific stuff.

    • @goldjoinery
      @goldjoinery 8 років тому +119

      Wom, wom, wom.

    • @fossilfighters101
      @fossilfighters101 8 років тому +1

      +

    • @Schobbish
      @Schobbish 8 років тому +43

      This is why we love him.

    • @Schobbish
      @Schobbish 8 років тому

      Hasnain Hossain double comment warning

    • @HasnainHossain_h
      @HasnainHossain_h 8 років тому +1

      Nathan Adam got an error the first time. Down with the system 👎

  • @kellen_mcsmellin
    @kellen_mcsmellin 8 років тому +198

    Tadashi's sound effect for an unweighted spring kills me

    • @Nobody-Nowhere-Nothing
      @Nobody-Nowhere-Nothing 6 років тому +1

      Kellen Dooley Sounded like doodle-bob from SpongeBob. "Mihoy-NINOY"

    • @Triantalex
      @Triantalex 11 місяців тому

      sad to hear that.. RIP in peace yo..

  • @realcygnus
    @realcygnus 8 років тому +102

    this dude is always awesome.

    • @fossilfighters101
      @fossilfighters101 8 років тому +2

      +

    • @tiuk23
      @tiuk23 8 років тому +5

      6:04 Made me meditating.

    • @LaGuerre19
      @LaGuerre19 5 років тому

      Causes a lot of problems. His classes are AA (always awesome) meetings.

  • @maldoran9150
    @maldoran9150 8 років тому +26

    Prof. Tokieda always manages to find a topic that is both easy to get into and interesting to delve into.
    Fantastic teacher. And thank you Numberphile!

  • @LangKuoch
    @LangKuoch 8 років тому +477

    Tadashi is the only man who can get me interested about cup clinking

  • @Nugglashine
    @Nugglashine Рік тому +2

    Tadashi Tokieda is so charming and charismatic. Very relaxing speaking voice. I could listen to him all day.

  • @AlanKey86
    @AlanKey86 8 років тому +116

    0:41 Jaws theme!

  • @blue0eyes6
    @blue0eyes6 8 років тому +11

    I like the accompanying animations. Really helps me to understand the concepts

  • @brothas
    @brothas 8 років тому +8

    "Since we're talking about the pitch .., let's imagine ourselves in a pitch dark room"
    10/10 Tadashi

  • @noahzuniga
    @noahzuniga 8 років тому +21

    this guy is excellent at explaining things

  • @hesgrant
    @hesgrant 8 років тому +9

    Tadashi never fails to teach me something new in these videos. Awesome stuff guys!

  • @adrenaldarknes3916
    @adrenaldarknes3916 7 років тому +12

    I just about spewed out my coffee when he made the 'hee haw hee haw' sound effect for the little spring. Caught me off guard lol.

  • @InkEyes
    @InkEyes 8 років тому +10

    I didn't even know that different points would have different pitches! I was blown from the beginning! lol

  • @Conguy97
    @Conguy97 8 років тому +6

    when he made the sounds of the springs it killed me

  • @mienzillaz
    @mienzillaz 5 років тому +4

    "Tadashi, stop playing around and finish your coffee" - Tadashi's mom

  • @snubbelbuff1471
    @snubbelbuff1471 8 років тому +167

    I've wanted to understand this for so long!! I can die now! Thank you

    • @toad2390
      @toad2390 8 років тому +1

      Hi

    • @dave5194
      @dave5194 8 років тому +14

      that can be arranged....

    • @TheTechBite
      @TheTechBite 8 років тому +3

      +David -flamingsword1 The authorities are on the way to your address.

    • @Spiderboydk
      @Spiderboydk 8 років тому +5

      That escalated quickly.

    • @homerotreto9391
      @homerotreto9391 8 років тому +5

      Just give me a call.
      I know people...

  • @MrJPEzra
    @MrJPEzra 5 років тому +1

    Tadashinis flippin’ crazy. He has the weirdest yet most brilliant demonstrations. I am so enamored by this man.

  • @aedric
    @aedric 8 років тому +101

    I'm a Potter, and if you send me your PO box I'll send you a 3 handled cup, 5 handled, whatever you'd like!

    • @s0me1up
      @s0me1up 8 років тому +76

      i guess you will create those cups with your wizardry...

    • @StephenTack
      @StephenTack 8 років тому +3

      Only while foregoing visits to the barber.

    • @topstitchgirl
      @topstitchgirl 8 років тому +2

      You could sent it to his attention at Stanford's math department :)

    • @topstitchgirl
      @topstitchgirl 8 років тому +2

      Aedric Donovan Just know that his full name is Tadashi Tokieda.

    • @hsterts
      @hsterts 8 років тому +10

      From a recent video, a three handed mug was demonstrated to be homeomorphic to a torus with three holes which is homeophoric to a hole in a hole in a hole

  • @iamnobody8979
    @iamnobody8979 8 років тому

    The recreation of the logo on the cup was spot on.

  • @ckannan90
    @ckannan90 3 роки тому

    His choice of words is always so delightful!

  • @wixic111
    @wixic111 8 років тому

    I love the way the Professor talks, it is just so engaging

  • @Audiojunkk
    @Audiojunkk 8 років тому +3

    Love this dude always get excited when he pops up on the channel! Excellent video!

  • @jakejakeboom
    @jakejakeboom 8 років тому

    this guy's accent just makes me happy, it's so soothing

  • @Folko
    @Folko 8 років тому

    Please never stop making videos with Tadashi

  • @richardnixonshead
    @richardnixonshead 8 років тому

    Tadashi is amazing. Every explanation is crystal clear, and he makes any subject absolutely fascinating

  • @TheTechnoLocker
    @TheTechnoLocker 8 років тому

    Adding to everyone else's comments, the reason I love Tadashi's video is that he rarely actually uses maths equations (as far as I can remember) to explain what is happening. It's all visual with animations and practical testing, which is great!

  • @lasagnahog7695
    @lasagnahog7695 8 років тому

    Seeing a new Prof. Tadashi video makes me instantly happier.

  • @petarbul
    @petarbul 8 років тому +3

    My favourite Numberphile video so far. Please post extras (I know it's 11 minutes long and there might not be any).

    • @pmcpartlan
      @pmcpartlan 8 років тому

      Tadashi's explanations are usually really fully formed and to the point. The only major thing I cut was a short explanation of why the cup want's to maintain it's volume, short answer: it takes a lot of energy to distort the cup like that.

  • @TommiHimberg
    @TommiHimberg 8 років тому +3

    This is brilliant! A great, easy to understand demo and explanation of the inverse problem. The inverse problem is indeed everywhere in science, e.g. in brain data analysis. The electric or magnetic field pattern the EEG or MEG (respectively) measures from outside the skull can be a result of infinitely many different combinations of sources in the brain, yet we usually want to know where in the brain the acitivity comes from, which requires a solution to the inverse problem. Basically, given the measurement of pitches, we want to figure out where the handles are in the mug. Thanks, this demo will be useful!

  • @rockdog2584
    @rockdog2584 5 років тому

    Okay...I'm geeking out here.... It's kind of funny how the mathematical side of my brain can interlope upon the creative (i.e. engineering/physical building) side and start to understand the subtle nuances of just what it takes to make things do what it is that you might wish them to do.
    Take, say...the making of a brass bell. I now understand that if any one point of that bell rim where just slightly thicker than the rest (thereby giving it more mass)...you could produce a bell that would actually produce more than one tone when struck (depending on just where it was struck).
    This is exactly why I love watching videos like this. These give me insight into the world around me that I otherwise would have overlooked.

  • @robertocastillo1471
    @robertocastillo1471 8 років тому +1

    Impressive explanation skills on Brady's side as well. Amazing animations and great editing

  • @ziyadalkilic
    @ziyadalkilic 8 років тому

    I did my masters on structural health monitoring (civil eng) and i've been using the Chinese porcelain example to elaborate on what was it all about. But, i see this and say this was the way i should have been explaining it. Very neat exemplification and it covers the mode shapes and the backwards problem solving issue so well ! I absolutely love this guy's style.... Just to add a note, Dr. Tadashi may find the case of 3 handles that are not 120 degrees apart more intriguing, since it is the basis of very stable honeycomb structure after all.

  • @arnauarnauarnau
    @arnauarnauarnau 2 роки тому

    Wow this is one of the nicest explanations I've ever heard for inverse problems

  • @TheArgusPlexus
    @TheArgusPlexus 8 років тому

    When it finally clicked in my head what he was trying to tell me, I just couldn't help but smile. I love learning new shit.

  • @sophia6861
    @sophia6861 8 років тому +1

    I love how the professor always explains things so nicely

  • @TAP7a
    @TAP7a 8 років тому

    The vocalisation of the heavy and light springs was just... so perfect

  • @Math_oma
    @Math_oma 8 років тому +4

    The difference in those pitches sounds approximately a half-step apart for both the coffee mug and tiny cup. Sounds approximately E and F for the coffee mug and then A and B-flat for the tiny cup.

    • @dragoncurveenthusiast
      @dragoncurveenthusiast 8 років тому +1

      wouldn't the pitch difference depend on the mass of the handle(s) that need to be dragged along?
      or maybe the mass of the handle in relation to the mass of the moving walls of the cup?
      if that's true it would be a coincidence that both are a half-step apart. unless it's because the handle-to-wall mass ratio is what looks nice for cups.

    • @goldjoinery
      @goldjoinery 8 років тому +1

      The tiny cup pitch difference is definitely microtonal. Probably 30-40 cents.

    • @mikosoft
      @mikosoft 8 років тому

      The tiny cup actually sounds like it's difference is only in upper harmonics. Seems like base frequency is the same

    • @goldjoinery
      @goldjoinery 8 років тому

      mikosoft Well, the sound of the tiny cup ringing is chock full of overtones anyway.

  • @bouncingbeebles
    @bouncingbeebles 7 років тому

    I love Tadashi's explanations! His voice is so calming too!

  • @LOogt
    @LOogt 8 років тому

    I like these videos where this professor explains things in a tactile way paired with animation

  • @atorrance
    @atorrance 8 років тому

    Man Tadashi has always been my favorite guest on Numberphile and he never disappoints. More Tadashi!

  • @c2ashman
    @c2ashman 8 років тому

    Could watch and listen to this guy all day. Please make more of these videos. Very interesting topics.

  • @DragonicRaven
    @DragonicRaven 8 років тому

    The best way to visualize this is using a head massager the wire kind where there is usually a set of longer prongs and shorter prongs all arranged in a circle. If you vibrate the longer prongs all the other prongs will move and the smaller prongs will stay motionless and vice versa for the shorter prongs.

  • @noblevi3623
    @noblevi3623 8 років тому +17

    Gotta find your cup with three handles? Big Rigs has you covered.

    • @noblevi3623
      @noblevi3623 8 років тому +6

      *****
      You're winner!

    • @ThePaull3d
      @ThePaull3d 8 років тому +7

      +Noble VI late for work? drive backwards through the buildings!

    • @unreal-the-ethan
      @unreal-the-ethan 6 років тому +1

      So has Cliff Stoll.

  • @nicolewestercamp6949
    @nicolewestercamp6949 7 років тому

    Ive never been so interested in the cup that holds the coffee, just usually the liquid inside! Thank you!

  • @chadisbad6
    @chadisbad6 8 років тому

    I went into this expecting to still not understand by the end, because that is how it usually is with these kinds of videos, but I got it really quick. well done video

  • @ToTouchAnEmu
    @ToTouchAnEmu 8 років тому

    I'm really enjoying this Tadashi fellow

  • @Multihuntr0
    @Multihuntr0 8 років тому

    I really like how Tadashi's stuff all seems to come from a child-like interest in the most mundane things along with enough intelligence to go through with figuring things out.
    When I grow up, I hope I'm able to do that.

  • @phaZZi6461
    @phaZZi6461 7 років тому

    i love how he sais "higher pitch" with a higher voice and "lower pitch" with a lower voice

  • @tannerpittman
    @tannerpittman 8 років тому

    His English is so excellent and learned but his Japanese accent so authentic. I love listening to this man.

  • @fierydino9402
    @fierydino9402 Рік тому

    The animations are also awesome. The last chime is indeed beautiful 😊

  • @manuelpena3988
    @manuelpena3988 8 років тому

    This guy is amazing!
    By the way, I don't agree with that if the cup were moving as a whole it wont produce sound. It would, is just that things don't remain vibrating as a whole if you don't maintaing a force for that, but they do keep compressing and expanding, without any force :)
    Thank you for all the videos.

  • @oiytd5wugho
    @oiytd5wugho 8 років тому +4

    That's my favorite kind of science right there!

  • @julianjimenez4046
    @julianjimenez4046 8 років тому +1

    Great animations! They really helps to understand how the things work. :)

  • @yaerius
    @yaerius 8 років тому +3

    I love professor Tadashi's sound immitations :)

  • @myrobotfish
    @myrobotfish 8 років тому

    Don't think I could've understood this without the fantastic animations ... and Tadashi's wonderful sound effects ;)

  • @YingwuUsagiri
    @YingwuUsagiri 8 років тому +5

    A funny thing I noticed is. If you have a full cup of a warm liquid. Tap the bottom. It has a singular note. Now stir the liquid and start tapping the bottom once more and it's not a singular note anymore. It rises or drops in pitch.

    • @YingwuUsagiri
      @YingwuUsagiri 8 років тому +2

      *****
      I noticed it with tea and that has no powder nor residue in it at all.

    • @MrAntieMatter
      @MrAntieMatter 8 років тому +4

      +Niels Schellekens The reason tea is darker than water is because it had tiny tea particles in it.

    • @YingwuUsagiri
      @YingwuUsagiri 8 років тому

      SadEugene
      That's my guess too but it doesn't do both. It's *either* up or down. Not up and down like a wave.

    • @Slade93
      @Slade93 8 років тому +1

      hey, I noticed it too during a coffee break and wanted to share that as well. You can tap continuously after stirring and the pitch will go up. Because I noticed it with coffee that has a significant amount of foam, my hypothesis was that it was related to bubbles in the stirred coffee slowly coming to the surface.

  • @trevorgrover5619
    @trevorgrover5619 8 років тому

    I could listen to Tadashi for hours

  • @tgwnn
    @tgwnn 8 років тому +1

    I don't know if I love the explanation more or the animations!

  • @HunterJE
    @HunterJE 10 місяців тому

    The Stanford mug was ringing about a minor second between the two, which means it could be used to play the most familiar part of the Jaws theme

  • @samslara
    @samslara 8 років тому

    I'm loving all of the Tadashi videos. Brings out the math in everyday things. In some ways more physics oriented than many other physics based videos.

  • @TheBigBigBlues
    @TheBigBigBlues 8 років тому +3

    Ok who can fire china, make Tadashi his 3 handled sonorous cup!

  • @jbrowsingj
    @jbrowsingj 8 років тому

    What a charming presenter. I would watch any video he's in!

  • @pacedelacruz4913
    @pacedelacruz4913 5 років тому

    Riveting! I so much appreciate all of your videos. Thank you so much for ALL of these.

  • @DerNesor
    @DerNesor 7 років тому +2

    voice - noises
    bird - light chirping
    cup - SMASH CUP WITH SPOON
    I love it

  • @nightangel7239
    @nightangel7239 8 років тому +1

    Oh man, I was able to see where this was going as soon as he confirmed the handle was related. I've been teaching myself guitar and noticed things like this when I found out about harmonics.

  • @angledcoathanger
    @angledcoathanger 8 років тому +1

    My life would be complete if you started an Audiophile channel, or even a Physics channel with a sound and audio sub-channel.

  • @geocarey
    @geocarey 8 років тому

    On a related theme, I have often been fascinated by how the pitch of the note rises as you add liquid to the cup. Give it a try!

  • @Arqideus1988
    @Arqideus1988 8 років тому

    We can also look at the effect of making the "lip" of the mug a different shape, other than the traditional circle.

  • @a.thiren2459
    @a.thiren2459 7 років тому

    I can't belive i haven't noticed this before! Thanks!!

  • @xerkules2851
    @xerkules2851 8 років тому

    Loved the part about the "inverse problem".

  • @idontwanttopickone
    @idontwanttopickone 5 років тому +1

    I can definitely hear a different sound on the side with a handle.

  • @LaGuerre19
    @LaGuerre19 5 років тому

    6:36 anyone who watches the show _Silicon Valley_ will recognize that green and orange ball on the shelf behind the professor! those things are so much fun. i have the same green and orange one. my kids and i love it!

  • @tiuk23
    @tiuk23 8 років тому

    I'm a simple man, I see Tadashi's video, I like it.

  • @weshard1
    @weshard1 8 років тому

    This was far more interesting than I thought it would be.

  • @Scottage_Man
    @Scottage_Man 8 років тому

    Awww man, Tadashi is the best

  • @MichaelFrancisRay
    @MichaelFrancisRay 8 років тому

    Tadashi Tokieda is tied with James Grime as my favorite Numberphile Contributor.

  • @Diecastclassicist
    @Diecastclassicist 8 років тому

    I would love to see a video of a conversation between Professors Tadashi and Copeland!

  • @smileyourock
    @smileyourock 8 років тому +70

    Stanford cup? Stop with the product placement, Tadashi.

    • @ultimaetsolder
      @ultimaetsolder 8 років тому +2

      I thought the same.

    • @Qman621
      @Qman621 8 років тому +1

      Guys, I think he works there... Maybe put away your tin foil hats for a moment and enjoy the video.

    • @TheHuntermj
      @TheHuntermj 8 років тому +3

      product placement is illegal in the UK...

    • @magniloquentlypuncturedkey2524
      @magniloquentlypuncturedkey2524 8 років тому

      Really?

    • @StephenTack
      @StephenTack 8 років тому

      MagniloquentlyPuncturedKeyboard or MPK Yeah @10:35

  • @himanshujain4789
    @himanshujain4789 8 років тому

    I used to wonder about this effect when i was little. Thanks a lot.

  • @RanEncounter
    @RanEncounter 8 років тому

    Nice video! Something that I noticed with cups also is that you get a different sound if you have cold or warm liquid inside.

  • @kevalan1042
    @kevalan1042 4 роки тому +1

    this guy is awesome

  • @gtjames77
    @gtjames77 8 років тому

    I literally had this set as a problem in one of my university exams! I remember scratching my head for hours over it, but this explains it so well! :)

  • @chairwood
    @chairwood 8 років тому

    Tadashi Tokieda has been my favorite person on numberphile since his first video and he never disappoints. Although there's a lot of other really awesome presenters too who I

  • @dwietr
    @dwietr 8 років тому

    Hmm, a larger cup with 3 handles might be "De pot van Olen", this is a cup commonly used to serve a beer named "Keizer Karel" in Belgium.

  • @Meddlmoe
    @Meddlmoe 8 років тому

    It is not retaining a constant volume, but a constant circumference

  • @arturzathas499
    @arturzathas499 7 років тому

    if only any of my teachers were as amazing as this guy

  • @strongside4565
    @strongside4565 8 років тому

    Every time you hear someone like this explain something, you understand the difference between someone who teaches at one of the best universities on Earth and someone that teaches at your everyday, high acceptance rate university. I feel like I would have been much, much more interested in mathematics with someone like this teaching, I might have even majored in it.

  • @tiagomarques9822
    @tiagomarques9822 8 років тому

    For a moment, when I read the title of this video, I thought it was going to be about another strange phenomenon you get with espresso cups. First you make the coffee swirl fast inside the cup using the spoon; then you start tapping repeatedly with the spoon on the center of the cup's bottom, and listen how the sound changes as the coffee slows down to a halt. There is an interesting article about that in the Wikipedia: it's called the Hot Chocolate Effect, if I'm not mistaken.

  • @PC_Simo
    @PC_Simo Рік тому +1

    Gotta love, how Tadashi kept slipping in puns to his explanation, very casually 😅.

  • @EvilAcidUK
    @EvilAcidUK 8 років тому

    Tadashi-San videos are always amazing!

  • @lakejizzio7777
    @lakejizzio7777 4 роки тому

    0:14 he definitely nailed that note.

  • @suave319
    @suave319 8 років тому +41

    I want a Tadashi sensei waifu pillow

    • @243263843
      @243263843 8 років тому +17

      its called husbando

    • @PhilerinoBTW
      @PhilerinoBTW 6 років тому +3

      It's called a Dakimakura actually, that's what the pillows are called in Japanese, but yes, husbando would be more appropriate than waifu.

  • @CillBinton
    @CillBinton 8 років тому

    This will make me a star at my next tea party!

  • @dft0011101000101001
    @dft0011101000101001 8 років тому

    Love seeing cool phenomena like this emerge from every day objects and experiences. Great stuff!

  • @GuildOfCalamity
    @GuildOfCalamity 7 років тому

    This guy is a master of everything

  • @ssiltane
    @ssiltane 7 років тому

    This is so cool! The best introduction ever to inverse spectral problems! Thanks!