Frieze Patterns - Numberphile

A surprising link discussed by Professor Sergei Tabachnikov.
Extra footage at: • Frieze Patterns (extra...
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Sergei Tabachnikov's homepage at Pennsylvania State University : www.math.psu.ed...
David Eisenbud discussed MSRI during his Numberphile podcast: • A Proof in the Drawer ...
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[@N.I.R.A.T.I.A.S.]

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

[@numberphile]

Ha ha. It’s all relative.

[@proximacentauri8038]

TREE(TREE^(TREE(TREE)))

[@persereikanen6518]

@@proximacentauri8038 +1

[@KohuGaly]

@@persereikanen6518 +ω

[@egilsandnes9637]

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

[@Xormac2]

*ACCORDION NOISE INTENSIFIES*

[@sarysa]

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

[@haskell_cat]

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

[@alexbartoszek7348]

I’m now extremely aware of every accordion noise

[@dan-gy4vu]

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

[@jonathanbeeson8614]

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]

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]

??

[@Zwijger]

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]

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]

First world problems.

[@BlakeMiller]

Like Tchaikovsky

[@genericusername4206]

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

[@PC_Simo]

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

[@SubhashMirasi]

A new professor.👏👏

[@persereikanen6518]

He is a student

[@martynaxyz6658]

@@persereikanen6518 *read description*

[@AleksandrMotsjonov]

Russian one, even better! =)

[@persereikanen6518]

@@martynaxyz6658 Yes, he is still a student. Repiit.

[@persereikanen6518]

@mxt mxt professor of a global warming? 😅

[@eemikun]

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:

[@cubing7276]

He is gone now :(

[@johnmulhall5625]

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

[@Petemackenshaw]

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

[@KillianDefaoite]

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

[@catchara1496]

Oh no!

[@kenhaley4]

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

[@adamfreed2291]

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

[@HalcyonSerenade]

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

[@JuulSimon]

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

[@sashimanu]

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

[@XenophonSoulis]

@@sashimanu It was accordion.

[@chrismorong931]

9:52 He's a ventriloquist

[@mtiman1991]

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

[@Vaaaaadim]

Mind Freak

[@shmunkyman33]

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

[@JorgetePanete]

@@mtiman1991 don't*

[@mtiman1991]

@@JorgetePanete really?

[@Djaian2]

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]

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

[@arirahikkala]

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]

The 3B1B nosies are therapeutic.

[@Kaerulans]

I think those might be sounds of an accordion

[@Artaresto]

They are

[@burtonlang]

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

[@juchemz]

I didn't like them, even by the end

[@Czeckie]

No, I want to see the proof!

[@spencerarnot]

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

[@kəanıncupıdo]

DB Math.

[@spencerarnot]

@Vahseline On the complex Z plane

[@randomdude9135]

I was gonna make a comment on that 😭

[@Blutsaugher]

And this ain't even its final form

[@soslunnaak]

are we there yet

[@wmpowell8]

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

[@Abdega]

All patterns Frieze during the Russian Winters

[@ericschuster2680]

lol

[@BobStein]

In mother Russia, patterns frieze you.

[@riftinink]

@@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]

@@riftinink это была шутка

[@riftinink]

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

[@KipIngram]

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.

[@GijsvanDam]

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

[@justinhoffman5339]

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.

[@77Chester77]

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

[@Sylocat]

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.

[@veggiet2009]

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?

[@lfestevao]

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.

[@JCW7100]

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

[@rudyhero1995]

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

[@aldasundimer]

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

[@emilchandran546]

Look up stradella bass system

[@АртурАбдуллин-ц4х]

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

[@jannegrey593]

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

[@АртурАбдуллин-ц4х]

@@jannegrey593 yes, you are right!)

[@djkm9558]

Artur Abdullin???

[@dmitry-dmitry]

Зато все понятно. Англичан носителей сложнее на слух воспринимать.

[@user-tk2jy8xr8b]

Zato vsyo ponyatno ;)

[@UnorthodoxSoundwave.]

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.

[@xenontesla122]

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

[@dominiquelaurain6427]

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.

[@cmusard3]

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

[@Goryllo]

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

[@usualsuspect2259]

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]

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

[@usualsuspect2259]

That's probably an approach

[@andymcl92]

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

[@JamesDavy2009]

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

[@tempestaspraefert]

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.

[@3dplanet100]

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

[@navjotsaroa2518]

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]

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.

[@krahnjp]

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?

[@msolec2000]

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

[@RunstarHomer]

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.

[@liggieep]

Gonna be honest, those sound effects are terrible and distracting, i thought my headphones were broken and getting feedback.

[@ragnkja]

liggieep I don’t mind the “blops”, but the synthetic hum is terrible.

[@XenophonSoulis]

@@ragnkja It's not synthetic, it is accordion.

[@Bigandrewm]

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]

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.

[@jasonpatterson8091]

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]

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)

[@indiarnav]

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]

The animator must've had fun with this one.

[@liamvictor]

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

[@elmo2you]

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

[@nemeczek67]

Maybe he is 63 in base-7.

[@mate_on_f7916]

45?

[@richardgratton7557]

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

[@technoguyx]

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

[@penand_paper6661]

The sound effects are really on point.

[@scottmuck]

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

[@carlosuzaier5858]

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

[@Ojisan642]

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

[@enderwiggins8248]

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

[@XenophonSoulis]

It's accordion.

[@Ecl1psed276]

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

[@Kaesekuchen002]

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

[@GrapefruitGecko]

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]

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

[@Dudleymiddleton]

Like the sound effects!

[@francomiranda706]

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?

[@megusta9268]

rip john connoway

[@jakistam1000]

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

[@atrumluminarium]

Conway is fucking everywhere. Even though he's one of the most famous mathematicians, he's still pretty underrated.

[@RaymondJerome]

why is it n-3

[@venkatbabu186]

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

[@nialltracey2599]

A few thoughts. Why is he treating the rotational symmetry as different solutions? The pattern produced is recurring and periodical -- a rotation of the polygon is just a "phase shift" of the wave periodicity of the function... n-3 is the number of lines required to triangulate the polygon. Surely no coincidence. Certainly worth noting I didn't notice any explicit mention of the fact that there's a sort of symmetry in the result, with the second last row being a rotation/phase shift of the second row and the 3rd last row being a rotation/phase shift of the 3rd row. Trivially, this is a necessary condition of their being only one solution (if the 2nd and 2nd last rows were different, this would mean there were at least two solutions by flipping it upside down, which would mean the link with the vertex numbering was broken). Again worth mentioning.

[@flumbofrommelkont6863]

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

[@Vaaaaadim]

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

[@n00dle_king]

Sound design on point today.

[@kevinjackson745]

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]

To keep the relationship with the Catalan numbers.

[@pedroscoponi4905]

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

[@meve5918]

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?

[@isaactfa]

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.

[@constantinedejong9025]

The video is very interesting but after the first few times the sound effects got really annoying. I think you should try using some less dissonant sound; that buzz sound from the harmonica(?) was getting really eerie.

[@ramansb8924]

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

[@jimothyjimothy1]

math

[@ramansb8924]

@@jimothyjimothy1 thank you so much that helped me alot

[@tamirerez2547]

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

[@Sgrunterundt]

2:34 What a miracle that all those fractions with denominator one turned out to be integers. I mean that the rest of the pattern holds is super interesting, but for the first calculated row it is hardly surprising that they are integers.

[@NoobLord98]

Why does rotating the n-gon yield a new frieze pattern? Aren't you simply choosing a different cutoff point on the left side of the frieze pattern than another rotation would yield?

[@pierremarcotte6299]

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

[@georgemissailidis1504]

The sequence could also be the sequence for a recursive f(n)=(3^n+1)/2, unless you _reallY_ check if the heptagon has 42 solutions ;)

[@EmergentSea1]

When I saw “frieze patterns,” I was thinking “oh, I remember Vi Hart mentioning those, that’s a repeating glide reflection, right?” Then the video started, and was purely numbers, so I thought it was unrelated with just a coincidental name, but then I realized that, sure enough, the numbers were arranged in a repeating glide reflection.

[@SocksWithSandals]

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

[@anonsensename5101]

2:53 That's not strange, it's because N is always 1 and you divide by N. Dividing an integer by 1 never gives a fraction.

[@61rmd1]

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

[@ffggddss]

Frieze (leads to) number sandwich (leads to) cluster algebras. Cool!! A whole new mathematical world to explore! Fred

[@BeCurieUs]

All the little sound effects were fun, btw :D

[@empty5013]

can you guys tone down the sfx it was really annoying and distracting otherwise very interesting

[@Henrix1998]

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

[@YellowBunny]

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

[@evanmurphy4850]

@@YellowBunny Trivial Obviously

[@agentstache135]

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

[@thejelambar82]

Just multiply all of the number by a

[@P21_c]

@@thejelambar82 by the square root of a

[@Wout680]

4:27 Top and bottom are the same, second and second last are the same & the two middle ones are the same. Coincidence?

[@aameen951]

love the 'r'

[@brachypelmasmith]

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?

[@natheniel]

The Russian/Slavic accent makes me [ r e d a c t e d ] and strangely patriotic.

[@partygirl0101]

hard?

[@ciarfah]

I tend to be drawn to slavic mathematicians for some reason

[@U014B]

Which SCP does it make you?

[@banjofries]

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...

[@assasinsbear]

Good job on the sound desing in this video !

[@ionutradulazar8984]

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

[@PPYTAO]

Absolutely fascinating!

[@Xeridanus]

There's a really annoying buzzing sound when some parts of the animation moves, but not all. Same with the other video.

[@CCarrMcMahon]

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

[@pukkandan]

Yes

[@Cernoise]

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.)

[@Schixotica]

Some questions one of you smarties might be able to answer for me 1. Does the ngon have to be a parallelogram? Would it work with an ngon with an odd number of sides? 2. Will the same ngon always produce the same sequence? 3. Are there any rules to how the diagonals are drawn? I.e. I assume they can’t cross each other Thanks in advance to anyone in the know

[@GrapefruitGecko]

1) You would get the same triangulations regardless whether the polygon is regular or not. 2) yes, any polygon with n sides will always produce the same sequences. But depending on how many sides, it potentially produce multiple sequences. 3) not the expert on triangulating polygons but I’m assuming you can’t have the lines cross...

[@Yoshiyosh]

What an interesting link