Frieze Patterns - Numberphile

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  • Опубліковано 5 сер 2019
  • A surprising link discussed by Professor Sergei Tabachnikov.
    Extra footage at: • Frieze Patterns (extra...
    More links & stuff in full description below ↓↓↓
    Sergei Tabachnikov's homepage at Pennsylvania State University : www.math.psu.edu/tabachni/
    David Eisenbud discussed MSRI during his Numberphile podcast: • A Proof in the Drawer ...
    Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
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  • Наука та технологія

КОМЕНТАРІ • 428

  • @N.I.R.A.T.I.A.S.
    @N.I.R.A.T.I.A.S. 4 роки тому +1165

    1:51 Stuff you hear on Numberphile: "This is a big one - seven."
    Also on Numberphile: *TREE(3)*

    • @numberphile
      @numberphile  4 роки тому +158

      Ha ha. It’s all relative.

    • @proximacentauri8038
      @proximacentauri8038 4 роки тому +26

      TREE(TREE^(TREE(TREE)))

    • @persereikanen6518
      @persereikanen6518 4 роки тому +13

      @@proximacentauri8038 +1

    • @KohuGaly
      @KohuGaly 4 роки тому +12

      @@persereikanen6518 +ω

    • @egilsandnes9637
      @egilsandnes9637 4 роки тому +16

      The number of triangulations of a TREE(3)-gon is a tad bigger than TREE(3) though.

  • @Xormac2
    @Xormac2 4 роки тому +413

    *ACCORDION NOISE INTENSIFIES*

    • @sarysa
      @sarysa 4 роки тому +7

      At first I thought my office printer was malfunctioning...

    • @haskell_cat
      @haskell_cat 4 роки тому +11

      I don't like it. How about a subtle "woosh" sound instead?

    • @alexbartoszek7348
      @alexbartoszek7348 4 роки тому

      I’m now extremely aware of every accordion noise

    • @dan-gy4vu
      @dan-gy4vu 4 роки тому +3

      I beg to differ. It sounds like an old counting machine and I honestly love that.

  • @Zwijger
    @Zwijger 4 роки тому +316

    Now I understand why Conway is sick of the game of life, another mathematician talks so highly about him, so he probably has done some brilliant stuff, but most people know him only about that one game.

    • @dominiquelaurain6427
      @dominiquelaurain6427 4 роки тому +46

      Yes, he has done a BIG work..in tiling, arithmetics and so on..that's why he is not so keen to enjoy publicity about a so small part of his lifetime masterwork. I guess Fermat would have not enjoyed to be known only by his famous conjecture.

    • @NatePrawdzik
      @NatePrawdzik 4 роки тому +12

      First world problems.

    • @BlakeMiller
      @BlakeMiller 3 роки тому +2

      Like Tchaikovsky

    • @genericusername4206
      @genericusername4206 3 роки тому +4

      @@BlakeMiller Tchaikovsky is known for a lot of pieces though

    • @PC_Simo
      @PC_Simo Рік тому

      It’s like Christopher Lee only being known for his role as Dracula.

  • @jonathanbeeson8614
    @jonathanbeeson8614 4 роки тому +263

    It seems that Brady has become over time much more active as an interlocutor in these Numberphile videos, and for me as a mathematical amateur that makes them much better. Thank you !

    • @MoPoppins
      @MoPoppins 4 роки тому +4

      I thoroughly enjoy the Numberphile podcast. Every episode has been riveting, and I’m not even strong in math...just curious about useful things and interesting people that I don’t yet know about.
      Anyone who hasn’t subbed the podcast yet should DEFINITELY check it out. 👍

    • @Triantalex
      @Triantalex 6 місяців тому

      ??

  • @eemikun
    @eemikun 4 роки тому +36

    That feeling when he says "Two famous mathematicians, one of them unfortunately not with us" and the first picture you see is of John Conway D:

  • @SubhashMirasi
    @SubhashMirasi 4 роки тому +342

    A new professor.👏👏

  • @chrismorong931
    @chrismorong931 4 роки тому +269

    9:52 He's a ventriloquist

    • @mtiman1991
      @mtiman1991 4 роки тому +37

      Plot twist: The Numberphile Mathematicians dont speak english, so the videos are translated

    • @Vaaaaadim
      @Vaaaaadim 4 роки тому +3

      Mind Freak

    • @shmunkyman33
      @shmunkyman33 4 роки тому +14

      I just assumed up until then he had been telepathically communicating the whole time and accidentally forgot to move his lips

    • @JorgetePanete
      @JorgetePanete 4 роки тому +1

      @@mtiman1991 don't*

    • @mtiman1991
      @mtiman1991 4 роки тому +1

      @@JorgetePanete really?

  • @johnmulhall5625
    @johnmulhall5625 4 роки тому +14

    Conway will always be one of my favorite mathematicians. When I heard he died from covid, I was truly bummed.

  • @kenhaley4
    @kenhaley4 4 роки тому +58

    Amazing how mathematicians can find correlations between seemingly totally unrelated concepts/phenomena. Nice video!

    • @adamfreed2291
      @adamfreed2291 4 роки тому +7

      Much of Math is figuring out how two seemingly unrelated problems are actually the same problem in a different form.

  • @JuulSimon
    @JuulSimon 4 роки тому +15

    The audio for the brown paper sections was strangely fantastic. Kinda reminded me of playing old DOS games.

    • @sashimanu
      @sashimanu 4 роки тому +1

      DOS games had a much more versatile repertoire of midi notes!

    • @XenophonSoulis
      @XenophonSoulis 4 роки тому +1

      @@sashimanu It was accordion.

  • @HalcyonSerenade
    @HalcyonSerenade 4 роки тому +10

    Brilliant choice of clip for John Conway: *"I'm not going to worry anymore! Ever. Again."*

  • @Czeckie
    @Czeckie 4 роки тому +153

    No, I want to see the proof!

  • @Abdega
    @Abdega 4 роки тому +309

    All patterns Frieze during the Russian Winters

    • @ericschuster2680
      @ericschuster2680 4 роки тому +1

      lol

    • @BobStein
      @BobStein 4 роки тому +17

      In mother Russia, patterns frieze you.

    • @riftinink
      @riftinink 4 роки тому +2

      @@BobStein I'm living in Russia. Not all regions as cold as you think. For example Krasnodarskiy region, the least temperature hear is about -5 C° (sorry if some of sentences are obscure)

    • @Ri0ee
      @Ri0ee 4 роки тому

      @@riftinink это была шутка

    • @riftinink
      @riftinink 4 роки тому

      @@Ri0ee я уверен, что некоторые думают, что это правда

  • @Petemackenshaw
    @Petemackenshaw 3 роки тому +9

    "One of whom is sadly not with us anymore." Sigh.. Now neither are.

  • @spencerarnot
    @spencerarnot 4 роки тому +302

    Not to be confused with Frieza forms. That’s a bit different.

  • @KillianDefaoite
    @KillianDefaoite 4 роки тому +9

    4:31
    Unfortunately John Conway is no longer with us either.

  • @arirahikkala
    @arirahikkala 4 роки тому +298

    I didn't like the weird electric noises in the animations at first, but they really grew on me by the end of the video. Still not as satisfying as 3blue1brown's clacks, though.

    • @alephnull4044
      @alephnull4044 4 роки тому +54

      The 3B1B nosies are therapeutic.

    • @Kaerulans
      @Kaerulans 4 роки тому +22

      I think those might be sounds of an accordion

    • @Artaresto
      @Artaresto 4 роки тому +5

      They are

    • @burtonlang
      @burtonlang 4 роки тому +27

      I suppose they chose this sound because frieze patterns are arranged sorta like an accordion's buttons.

    • @juchemz
      @juchemz 4 роки тому +18

      I didn't like them, even by the end

  • @wmpowell8
    @wmpowell8 2 роки тому +2

    If you use a polygon to generate these patterns, you can connect a line from every vertex to a specific vertex and this creates an amusing pattern

  • @Djaian2
    @Djaian2 4 роки тому +76

    There is one thing the professor should not have done: he spoiled the fact that he would arrive at a line with only ones. Would have been better if he didn't say it early, and just, after some calculations suddenly produces a line of ones. And then, explain everything like he did.
    He could even have asked Brady: "What do you think, will this explodes to infinity with numbers getting bigger and bigger?"

    • @Lexivor
      @Lexivor 4 роки тому +20

      This would have made it more dramatic, I like your idea.

  • @GijsvanDam
    @GijsvanDam 4 роки тому +1

    The enthusiasm of the professor is contagious. Love to see more vids with him.

  • @77Chester77
    @77Chester77 4 роки тому +5

    Satisfying to see that mr Tabachnikov writes the "ones" with a hook on top :-)

  • @KipIngram
    @KipIngram 21 день тому

    I absolutely LOVE that Conway "I'm not going to worry any more, ever again" moment - as far as I'm concerned being able to come to such a point in one's life is the greatest achievement any of us could ask for, and I dearly hope he was successful in following through on that.
    As a counterpoint, I read once that a guy was interviewing Paul Dirac, fairly late in his life, and was stunned when Dirac told him that he really thought of his life's work as a failure. This is the guy who CREATED quantum field theory - our very very best theory of how nature works. And he thought of himself as a failure intellectually. That really makes me quite sad for him. A man like him should have gotten to be content with his accomplishments. Conway found the better path - that's for sure.

  • @JCW7100
    @JCW7100 4 роки тому +16

    Love your videos so much! Thanks for the great content! :)

  • @justinhoffman5339
    @justinhoffman5339 4 роки тому +1

    Another way to think about the pattern is adding triangles onto the edges of the previous shape. Adding a triangle is effectively the same as inserting a 1 into the cycle, and incrementing the adjacent numbers because you're drawing a point (1) and connecting a line to 2 existing vertices.
    Starting with the simplest case (111), you can insert a 1 in front and get 1212, insert a 1 in the second position and get 2121, insert a 1 second last and get 1212, or insert a 1 at the end at get 2121. You keep the unique cycles (in this case 1212 and 2121) and continue the pattern of inserting 1's into those new cycles.

  • @xenontesla122
    @xenontesla122 4 роки тому +2

    The sound design in the animated parts is on another level. I'm guessing the dot arrangement reminded the animator of a button accordion?

  • @cmusard3
    @cmusard3 4 роки тому +8

    Is there accordion sounds bc the frieze grid looks like the accordion bass keyboard?

  • @rudyhero1995
    @rudyhero1995 4 роки тому +84

    Like the video, didnt realy like the sound effects sounded a bit heavy or something

    • @aldasundimer
      @aldasundimer 4 роки тому +5

      The beeps were annoying to be honest. But great video as you said.

    • @emilchandran546
      @emilchandran546 4 роки тому

      Look up stradella bass system

  • @veggiet2009
    @veggiet2009 4 роки тому +2

    Whenever any number fact or theorem relate to geometry, I invariably will ask is this generizable to multiple dimensions in some way? Like if you divide a polyhedron into multiple tetrahedron, could you craft a number sequence from that and what mathematically properties would it have?

  • @pedroscoponi4905
    @pedroscoponi4905 4 роки тому

    This was really cool :) I am all for more Prof. Sergei!

  • @usualsuspect2259
    @usualsuspect2259 4 роки тому +31

    What would have happened if we get, instead of shapes in 2D space,
    Shapes in 3D space and we triangulate them, if that's possible?

    • @JamesDavy2009
      @JamesDavy2009 4 роки тому +11

      To look at the 3-D version, one would need to ask how many tetrahedra does the vertex in question have in common?

    • @usualsuspect2259
      @usualsuspect2259 4 роки тому +1

      That's probably an approach

    • @andymcl92
      @andymcl92 4 роки тому +10

      @@JamesDavy2009 Possibly a trivial question. Is it always possible to split a polyhedron into tetrahedra?

    • @JamesDavy2009
      @JamesDavy2009 4 роки тому +3

      @@andymcl92 There's a question for the people of Numberphile.

    • @tempestaspraefert
      @tempestaspraefert 4 роки тому +2

      There is exactly one (relevant) way to make an n-sided (convex) polygon.
      There are several possible ways to make an n-sided polyhedron (e.g. an n-1-sided pyramid or an n-2-sided prism). This makes it less likely that this also works in 3D, I think.

  • @liamvictor
    @liamvictor 4 роки тому

    I get such joy from these videos. One day I might even understand some maths.

  • @penand_paper6661
    @penand_paper6661 4 роки тому

    The sound effects are really on point.

  • @navjotsaroa2518
    @navjotsaroa2518 4 роки тому +6

    So could this be extrapolated to 3D solids and then even higher dimensions, where you would draw lines in order to make pyramids? If so, what would that look like and what difference would be made if we used triangular based pyramids or square based pyramids or one with any other base?

    • @FiniteJest
      @FiniteJest 4 роки тому +1

      Algebraically it seems related to a determinant so you would need to relate 9 numbers together instead of the 4. It might work with stacking parallelpipeds, might be a fun research project.

  • @msolec2000
    @msolec2000 4 роки тому

    Yes! More about Catalan Numbers, please! They are awesome and they are EVERYWHERE!

  • @61rmd1
    @61rmd1 4 роки тому

    Amazing, and well described...thanks a lot for this video

  • @Bigandrewm
    @Bigandrewm 4 роки тому +5

    I'm guessing that sound effect for drawing is a sampled accordion? Might be neat to modify that idea slightly by having a set of accordion notes which are chosen by some pattern referencing the video.

    • @pmcpartlan
      @pmcpartlan 4 роки тому +2

      Yes, it was an accordion that I sampled a while ago, not sure it quite worked here (or maybe there was just too much of it). But yeah, working on this has made me want to do more fun systematic things with the sound design.

  • @indiaaranv
    @indiaaranv 4 роки тому +4

    Could you go through the recent proof for the sensitivity conjecture by Hao Huang? Seems like it could be an interesting topic under graph theory.

  • @tomfryers2
    @tomfryers2 4 роки тому +4

    The animator must've had fun with this one.

  • @assasinsbear
    @assasinsbear 4 роки тому

    Good job on the sound desing in this video !

  • @GrapefruitGecko
    @GrapefruitGecko 4 роки тому +5

    I want to know what this has to do with the Catalan numbers.... also how did Conway and Coxeter think to relate these two seemingly different ideas??

    • @jaydendickson
      @jaydendickson 4 роки тому +3

      The catalan numbers are just the number of ways of partitioning the polygon into triangles.

  • @Sylocat
    @Sylocat 4 роки тому

    Something I didn't notice until I showed my mom this video and she pointed it out, was that the nontrivial rows have vertical symmetry. The first and last rows are the same, just offset, as are the 2nd and 2nd-to-last rows, and so on.

  • @Ruddigore
    @Ruddigore 4 роки тому

    A fascinating video. Thank you.

  • @3dplanet100
    @3dplanet100 4 роки тому +6

    Amazing. Math is like a logic puzzle that everything is related and connected.

  • @Kaesekuchen002
    @Kaesekuchen002 4 роки тому

    And at 6:20 I was like: "wooooooow". Great video as always. I would like to see more with Professor Tabachnikov.

  • @PPYTAO
    @PPYTAO 4 роки тому

    Absolutely fascinating!

  • @Goryllo
    @Goryllo 4 роки тому +2

    The sound effects during the animations are amazing! Great sound design as always, not to mention the interesting subject and the cool graphics...

  • @flumbofrommelkont6863
    @flumbofrommelkont6863 4 роки тому +11

    For you see frieze, you're not dealing with your average mathematician anymore...

  • @Dudleymiddleton
    @Dudleymiddleton 4 роки тому +1

    Like the sound effects!

  • @krahnjp
    @krahnjp 4 роки тому +1

    I might have missed it, but I didn't hear mention about the fact that the last row of numbers (above the ones) seems to always be the same sequence as you entered, and the too middle rows are the same sequence of numbers as well. Does that mirroring of sequences across the board always hold true for all polygons?

  • @dominiquelaurain6427
    @dominiquelaurain6427 4 роки тому +1

    I like very much to read Tabachnikov's papers about geometry and mathematical billiards (I am recently interested in that "mathematic dynamics). Theory he works about are really deep bridges between big parts of mathematics. ...if you can interview the others (Richard Evan Schwarz, ..) it would be great. Billiards are deeply linked with physics and some math modeling.

  • @enderwiggins8248
    @enderwiggins8248 4 роки тому +6

    Kinda random, but I really appreciate your sound design, like the harpsichord when you’re transforming the polygons

  • @jasonpatterson8091
    @jasonpatterson8091 4 роки тому +4

    It's not really strange that the first row the professor determined was entirely made of integers. If the value is (WE-1) / N, and N is always 1, then of course it would be.

    • @skyjoe55
      @skyjoe55 2 роки тому

      And positive because if W and E are positive then WE is positive and a positive minus 1 is either positive or zero
      (This only works if zero is not considered a positive number)

  • @Ojisan642
    @Ojisan642 4 роки тому

    What a nice ending. They recognized the beauty of it first, and then later it became important.

  • @Ecl1psed276
    @Ecl1psed276 4 роки тому +1

    The sound effects in this one are on point :D Props to your editor!

  • @user-ee9de6np4c
    @user-ee9de6np4c 4 роки тому +48

    Найс рашен аксент. Гуд, намберфайл, вэри гуд!)

    • @jannegrey593
      @jannegrey593 4 роки тому +8

      Did you just wrote English phonetically in Bukwa's? Sorry my Cyrillic is VERY slow.

    • @user-ee9de6np4c
      @user-ee9de6np4c 4 роки тому

      @@jannegrey593 yes, you are right!)

    • @djkm9558
      @djkm9558 4 роки тому

      Artur Abdullin???

    • @dmitry-dmitry
      @dmitry-dmitry 4 роки тому

      Зато все понятно. Англичан носителей сложнее на слух воспринимать.

    • @user-tk2jy8xr8b
      @user-tk2jy8xr8b 4 роки тому +1

      Zato vsyo ponyatno ;)

  • @RunstarHomer
    @RunstarHomer Рік тому +1

    I'm curious why the triangulations are considered different even if they're identical up to rotation. If you rotate the polygon, you still get the same frieze pattern, since they are periodic.

  • @technoguyx
    @technoguyx 4 роки тому +7

    Beautiful and totally unexpected. That's how I like my mathematics :D

  • @richardgratton7557
    @richardgratton7557 4 роки тому +8

    Best hand-written numbers ever, not like Grimes! :)

  • @davidwilkie9551
    @davidwilkie9551 4 роки тому +3

    There's a link with coordinate systems similar to 3D?

  • @meve5918
    @meve5918 4 роки тому +1

    Is it significant that row 1 and row n contain the same numbers (with starting point shifted), as do rows 2 and n-1, 3 and n-2 etc?

  • @CCarrMcMahon
    @CCarrMcMahon 4 роки тому +3

    Can you expand it infinitely to the right or left as long as you repeat the sequence?

  • @drewdurant3835
    @drewdurant3835 4 роки тому

    I love your channel!!

  • @elmo2you
    @elmo2you 4 роки тому +4

    What a charming man. Also looks quite a bit younger than the 63 years he has.

  • @Vaaaaadim
    @Vaaaaadim 4 роки тому

    This is just absolutely crazy, how on earth would anyone even see a connection like this!

  • @n00dle_king
    @n00dle_king 4 роки тому

    Sound design on point today.

  • @ramansb8924
    @ramansb8924 4 роки тому +7

    But i don't understand how it works?? Please explain

  • @jakistam1000
    @jakistam1000 4 роки тому +1

    Finally someone that writes the numbers the way I do! :D

  • @RaymondJerome
    @RaymondJerome 4 роки тому +2

    why is it n-3

  • @madanisihamdi2653
    @madanisihamdi2653 4 роки тому

    Thank you MSRI

  • @SocksWithSandals
    @SocksWithSandals 3 роки тому +1

    Amazing.
    Has this phenomenon found any real-world use, like computing or encryption?

  • @scottmuck
    @scottmuck 4 роки тому

    Well of COURSE I’m going to head over to Numberphile2 now.

  • @francomiranda706
    @francomiranda706 4 роки тому +1

    that equation S(N,E,W)=(NE+1)/W looks convieniently like a more general version of the triangle formula A(b,h)=(b+h)/2.
    Considering that in order to find these non-integer solutions, we have to solve for n iterations of S, something like S(S(N,E,W),E,W), could this be the connection to the trianglization?

  • @ionutradulazar8984
    @ionutradulazar8984 4 роки тому

    You can also notice that the k-th and (n-k)-th row are the same but shifted by an amount

  • @Cernoise
    @Cernoise 4 роки тому

    I came into this expecting a refresher on what I learnt about frieze groups in algebraic geometry, but this seems quite different! (And yet, is probably isomorphic to it somehow.)

  • @UnorthodoxSoundwave.
    @UnorthodoxSoundwave. 4 роки тому +1

    I'm amazed that he didn't mention the patterns in the rows are mirrored on the grid:
    1 1 1 1 1 1 1 1 1 1 (X - 1)
    _________________ (X)
    _________________ (X + 1)
    _________________ (X + 2)
    ...
    _________________ (N - 1)
    _________________ (N)
    1 1 1 1 1 1 1 1 1 1 (N + 1)
    Like how X - 1 and N + 1 are the same pattern of 1 1 1, N and X would also follow the same sequence, as well as X + 1 and N - 1, and so on. Though the sequences don't start in the same column every time, they always shared the same one across the row.

  • @Henrix1998
    @Henrix1998 4 роки тому +26

    How about WE-NS=a? I feel like there was so much he didn't touch at all

    • @YellowBunny
      @YellowBunny 4 роки тому +18

      What about sin(W)*e^(E-N)+S^(W+E²*i)=a?

    • @evanmurphy4850
      @evanmurphy4850 4 роки тому +18

      @@YellowBunny Trivial Obviously

    • @agentstache135
      @agentstache135 4 роки тому +15

      Evan Murphy thus it is left as an exercise for the reader

    • @thejelambar82
      @thejelambar82 4 роки тому +1

      Just multiply all of the number by a

    • @P21_c
      @P21_c 4 роки тому

      @@thejelambar82 by the square root of a

  • @SupriyoChowdhury5201
    @SupriyoChowdhury5201 4 роки тому +3

    Please make a video on Robert langlands program

  • @banjofries
    @banjofries 4 роки тому +1

    huh, I remember seeing a few of those hexagon patterns in media in reference to things like "magic runes". Funny what people came up with without maths...

  • @lfestevao
    @lfestevao 4 роки тому +2

    I really digged this. The Polygon explanation shows why the sequence repeats to the right.
    Now I imagine it like the drawing is in the top of a Cylinder and the numbers are on the side. Then we go down filling the values like in the paper.
    In the end we go back to the trivial 1s row and can start over. This reflects as the Cylinder bending to make both bases meet, like a Thorus.
    This way I was able to see that the pattern repeats it self ALSO there's no orientation, so we can read clockwise OR counterclockwise.
    Going back to the paper examples on the video, this holds up, as it can be read and filled bottom to top.
    Furthermore the sequences repeat BEFORE reaching the trivial 1s. Maybe there is a Klein Bottle interpretation for this, but this was too much for me to imagine without doodling it up.

  • @Narokkurai
    @Narokkurai 4 роки тому +1

    Interesting. So it's a way to numerically describe the construction of any polygon using triangles? I wonder if it has any applications in 3D graphics.

  • @isaactfa
    @isaactfa 4 роки тому

    I love these theorems that deal with natural number patterns. They seem the likeliest (from a complete layman's point of view) to crop up in nature and be useful someday.

  •  4 роки тому +1

    Can you link the proof in the description, please (and tell me if/when you did)?

  • @megusta9268
    @megusta9268 3 роки тому +7

    rip john connoway

  • @pierremarcotte6299
    @pierremarcotte6299 4 роки тому

    I love how he says: "pedioric" instead of "periodic".
    0:59

  • @kevinjackson745
    @kevinjackson745 4 роки тому +1

    I didn't understand why we count the different rotations of triangulations of the n-gon as different friezes. They seem identical to me. Did anyone understand that?

    • @nemeczek67
      @nemeczek67 4 роки тому +1

      To keep the relationship with the Catalan numbers.

  • @KatzRool
    @KatzRool 4 роки тому +1

    What's with the adjacent 1s in the diagonals?

  • @tsbwarden5383
    @tsbwarden5383 4 роки тому +2

    Why the gap between videos?

  • @DJejbarros
    @DJejbarros 4 роки тому

    Is it possible to reverse it? Create a sequence, check how long it takes to get to 1111... And get the polygon?

  • @carlosuzaier5858
    @carlosuzaier5858 4 роки тому +2

    Vid is cool as always, but the guy here really takes the cake. His accent is so cool and his general vibe is nice

  • @harmidis
    @harmidis 4 роки тому

    amazing! thanks!

  • @MrApplefreaker
    @MrApplefreaker 4 роки тому

    Curious to know if such patterns are used anywhere in application.

  • @Yoshiyosh
    @Yoshiyosh 4 роки тому

    What an interesting link

  • @uweperschke6799
    @uweperschke6799 4 роки тому

    Has anyone noticed that the last non-trivial row represents another triangulation?
    I wonder if one can eventually retrieve all triangulations if that row is used as new seed row.

  • @BobStein
    @BobStein 4 роки тому

    So I get the geometric interpretation of the 2nd row. That's the number of triangles touching each vertex. Is there some geometric thing going on with the other rows?

  • @morismateljan6458
    @morismateljan6458 4 роки тому +8

    Does the second row correspond to triangulation of some other n-gon?

  • @asbestosrecuperation
    @asbestosrecuperation 4 роки тому +1

    How on earth did they come up with this?

  • @brachypelmasmith
    @brachypelmasmith 4 роки тому

    so why are both solutions for square considered separate? If the thinf is periodic then its the same where you start (starting corner is not explicitly given) numbering so 1212 is the same as 2121, the same goes for several patterns for hexagons and higher?

  • @aameen951
    @aameen951 4 роки тому +1

    love the 'r'

  • @venkatbabu186
    @venkatbabu186 4 роки тому

    This is the basic pattern of metals and that's why they conduct electricity. Magnetic polarity works similar. Special pattern of surface symmetry.

  • @tamirerez2547
    @tamirerez2547 4 роки тому

    Please raise the salary of the voice man. He deserves it.