The Forgotten Flexagon - Numberphile

[@ryPish]

Ah yes, the slightly less magnificent *Parker Flexagon*

[@Sam_on_YouTube]

It is also square. And, as explained in the second video, the diagram doesn't work as well as it looks like it should. That makes this the Parker Square Flexagon.

[@LordHonkInc]

I love that even after three years that meme's still going strong

[@ABurntMuffin]

sonofabitch beat me to it

[@G.Aaron.Fisher]

@@LordHonkInc I finally bought the t-shirt. No regrets.

[@Klaevin]

@@Sam_on_KZbin it's only been 3 years?

[@Koisheep]

I was about to say "where my monomonoflexagons at" until I realised That's a Möbius strip

[@electromorphous]

*Duuuuudddeee!* STOP BLOWING OUR MINDS LIKE THAT!!

[@bbgun061]

And a plain sheet of paper is a bitetraflexagon.

[@muchozolf]

@@bbgun061 or bi-n-flexagon, it can have any number of sides.

[@zmaj12321]

duuuude

[@1DerSiedler]

@@muchozolf Than it's no longer a sheet, at least I wouldn't call it so.

[@Sam_on_YouTube]

My 8 year old twin girls love ViHart's hexaflexagons. I often help them make them out of straw wrappers at restaurants. Thursdays are half days at their school. You just gave them an afternoon project. Thank you.

[@duckrutt]

I use wrappers to make jumping frogs. The game is to land it in someones drink (preferably someone at your table but I don't judge) while not launching it somewhere it would be socially awkward to retrieve.

[@mvl71]

Thank you for directing me to hexaflexagon madness. I'll finish my current affairs and retreat to my room where I will happily waste away, folding hexaflexagons like there's no tomorrow. Gollum 2.0: my hexioussss

[@dragoncurveenthusiast]

Can you provide links to crease patterns for the jumping frog?

[@BloodyHaemorrhoids83]

I found a way to edit the sides on a hexaflreagon without ungluing it! me and my friends did a study on them a few years ago on top of the normal typically found studies, focusing particularly on editing. there are editing states with certain numbers of holes that determine the availability of editing and more. i’ve tried to bring light to this but most things about flexagons are multiple years old.

[@kimberlypichardo6884]

I think I've seen u on comment on ViHarts channel

[@incription]

Weird flexagon, but ok

[@r-prime]

Weird flex, but ok

[@rawovunlapin8201]

@@B----------------------------D :o

[@kiverismusic]

Hhhahahahahahahahahah

[@VeritasQuasitor]

👏👏👏👏

[@KINGLADUDU]

It's a Parker flexagon

[@OrangeC7]

Doodlephile is just Vi Hart's channel

[@brianmiller1077]

and 12Tone is doodlephile with elephants

[@GrapefruitGecko]

Haha this is so true

[@want-diversecontent3887]

Sem Zem is Vi Hart minus the Doodlephile

[@liquidkey8204]

I mean yes, but like yes.

[@shim64]

and my channel is doodlephile without the math

[@_sine_]

The only time where "weird flex but okay" is valid

[@mrmonster3434]

"You could draw pictures - I'm not very creative, I used numbers. This is NUMBERphile, not Doodlephile!" 70 seconds later... "We're gonna use colours, cos that's how we roll."

[@OrangeC7]

To be fair, Doodlephile is kinda what Vi Hart's channel is for

[@jones1618]

"Dammit, Jim. I'm a doctor, not a bricklayer!"

[@woodfur00]

I loved how defensive he got about that

[@NortheastGamer]

And then proceeds to not use those colors, or the circle on the flexagon at all for the whole demonstration.

[@lyrimetacurl0]

That would be colourphile.

[@awsomeabacus9674]

I generally don't like pick up lines, but "You wanna see some new flexagons" is definitely a winner.

[@mcnichollsdj]

I wanna get flexagons with you... - sorry, didn't mean it to sound creepy!

[@maxhaibara8828]

next video will be "N-dimensional Tetraflexagon"

[@matthewellisor5835]

He IS the Kwisatz Tesseract!

[@murrfeeling]

All you gotta do is fold a cube.

[@KuraIthys]

In 4 dimensions, isn't this essentially what a hypercube amounts to? Let's assume you 'fold' a hypercube in on itself. Now all of it's constituent 3d cubes occupy the same 3d 'space' in some sense. If you were to navigate this space by somehow passing through the walls of each cube, you'd end up in another cube, and you can keep going through the same 'wall' 4 times in row before you appear at the opposite wall of the cube you started in. However, certain sequences of such moves put you in a different set of cubes from which you cannot navigate back to the original set by going in a single direction... Actually the graph of this Hexatetraflegagon in terms of which faces you can see simultaneously has a lot in common with what you'd see if you graphed a hypercube in terms of which other cubes you can reach by passing through the walls of the cube you're currently in...

[@JNCressey]

@@KuraIthys, When you fold a square, two of the edges lay on top of each-other (same 2D space), and the other two become folded in half on themselves, and a crease that looks like an edge joins the middles of the folded edges, and the final shape is a rectangle that's half the square. So my guess for folding 3D would be that a 'folded cube' would involve some (square) faces being lain into the same 3D space as each-other, and some (square) faces being folded in half on themselves (and we know what a folded square looks like), and a crease that looks like a square joins the middles of the folded faces, and the final shape is a cuboid that is half the cube. And my guess for folding 4D would be that a 'folded hyper-cube' would involve some (cube) cells being lain into the same 4D space as each-other and some (cube) cells being folded in half on themselves (and folded cubes are described in the first bit), and a crease that looks like a cube joins the middles of the folded faces, and the final shape is a hyper-cuboid that is half the hyper-cube. But that's just a guess. Is there any math that says it behaves differently? Perhaps shapes with more dimensions than 2 are just completely rigid and can't be folded? Also there's different ways to fold a square in half. The half rectangle way, the triangle way, and some random oblique angle. So maybe something that sounds completely different is equivalent.

[@MrEugenio1994]

Do you even N-flex, bro?

[@wishiwasabear]

*And to make it, we are going to use a square-* Oh, here we go again.

[@ejnissley546]

Kuma You got an issue with that?

[@CasualMitosisCollective]

@@ejnissley546 nope. Just a Parker square flashback.

[@OsyenVyeter]

It can be a rectangle. I have one made of playing cards.

[@seeseefok7659]

ah sh** here we go again

[@Triantalex]

??.

[@GraemeMcRae]

Very nice! My granddaughter (age 7) and I made two flexagons and colored them in. The project is complex enough to be entertaining, but simple enough to stay within our attention span.

[@LeoStaley]

Numberphile: forgotten flexagon Vihart has joined the chat.

[@SuperAWaC]

@@tthung8668 it's funny how easy it is to get even very high end mathy types with party tricks

[@liquidkey8204]

@@SuperAWaC Yeah

[@Triantalex]

??.

[@Watchmedothatfor.u]

​@@Triantalex vi hart is a math channel , if you don't know of course .

[@modernkennnern]

Not gonna lie. I thought this was a Vihart video when I read the title. Had to recheck the channel name

[@TheLostSorcerer]

@Jack Could have been a colab.

[@cheyenne1309]

I literally jumped up thinking Vi uploaded 😂

[@ericstoverink6579]

It's a Parker ViHart video.

[@Triantalex]

??

[@error.418]

If you've watched Vi Hart, then you know you don't need glue to make a hexaflexagon. There are folding methods to do it.

[@BloodyHaemorrhoids83]

I found a way to edit the sides on a hexaflreagon without ungluing it! me and my friends did a study on them a few years ago on top of the normal typically found studies, focusing particularly on editing. there are editing states with certain numbers of holes that determine the availability of editing and more. i’ve tried to bring light to this but most things about flexagons are multiple years old.

[@Triantalex]

??.

[@Watchmedothatfor.u]

After vi hart I didn't even watch till the end :(

[@laurihei]

"And to make it ...we're gonna use a square of paper." Ok, queue up the memes then. It's not like we were only 20 seconds into the video already.

[@laurihei]

This must either be the record or at least getting close.

[@Fra321]

@@laurihei A parker square of a record attempt.

[@Triantalex]

??

[@laurihei]

​@@TriantalexParker Square 😅

[@LaGuerre19]

Fantastic forgotten finite fractal flexagons freely folded for fun! Fie, foursquare fingerwork! (Nice work, Matt and Brady!)

[@syriuszb8611]

0:24 "Fold it in half, witch ever way you want" Are you sure?? Well then, I want to fold it in half diagonally!

[@rewrose2838]

Does it work? A triangular flexagon maybe?

[@colscoco6616]

Yeah, if you look at 4:35, #1 and 2 are the only numbers not to be in any corners (I think)

[@N.I.R.A.T.I.A.S.]

5:18 "It's Numberphile. It's not Doodlephile, is it?" Brady. *Start a channel called Doodlephile.*

[@SimonBuchanNz]

Isn't that Drawfee?

[@brianlane723]

Just use Hovah to register the domain.

[@KingJellyfishII]

No that's vi heart

[@pvic6959]

@@KingJellyfishII i support this message

[@MarcusAntonio.]

"You wana see some new flexagons?"" i'm goin, "YEESS." 12:21

[@yerwol]

It's like you imagine some guy with a long trenchcoat in the street approach you, opening up one half to reveal a vast array of flexagons pinned to the inside. The seedy underground world of flexagons

[@steveydoesglasgow]

Love the commitment to making tiny flexagons at the end, shout-out to whoever managed that!

[@DanTheStripe]

0:20 You're going to use a square of paper, are you, Mr. Parker? A square? Parker? Hmmmm....

[@bashily2844]

Parker chose option 2 when someone told him to be there or be square

[@OsyenVyeter]

It’s ok it can be rectangles. I have on made of playing cards

[@Triantalex]

??

[@wendysolomon1608]

at the behest of one of my favorite students, we once embarked upon a quest to create a dodecahexaflexagon. It works, but is quite fussy and fragile. Even graphed it out! Love them!

[@EdoTimmermans]

Thanks to your comment I just found a KZbin video on how to make an icositetrahexaflexagon named 'Awesome!! 24 sided hexaflexagon!'.

[@ffggddss]

Several decades ago I actually made a 48-faced hexaflexagon. The finished product was veeeeerrrry clumsy to operate! Fred

[@Dixavd]

When I was at school about a decade ago I remember kids in class making these Tetraflexagons whereas I hadn't heard of a hexaflexagon until today. They'd play games similar to that octahedron papercraft you put your fingers in: for instance, they might start on a given arrangement with the names of other kids on and say "fold it 3 times and that's the person you'll marry".

[@xenialafleur]

We played with these all the time with these in school.

[@charlenejo2490]

You mean what was called a “cootie catcher?”

[@PTNLemay]

Aww, I thought it was going to be a collab with Vihart.

[@rohanglenmartin]

That was fantastic! You've inspired me to use this to make a treasure map and build a D&D campaign around it.

[@the_mad_ratter]

I had totally forgotten about these until I saw this video... my dad had (and I now have somewhere) a couple of display "toys" from the 70s that did this, about the size of credit cards (but much thicker and made of plastic) the design on one side had a series of linked circles (like the olympic logo), and the other side had them all separated.

[@pherlong7]

Rubix magic?

[@justsomeone4347]

Vihart: how dare you fight my hexaflexa-skills

[@Watchmedothatfor.u]

OK. I'm late with the answer ) stopping . me : I should get out * watching vi hart *

[@TforThought]

It is wonderful that a bunch of numbers/symbols and what we can do with them. Can even explain things like rubrics cube, Hexaflexigon etc

[@guhan4606]

Weird flex, but Parker square

[@vanderengland5775]

I’ve been up to 24 faces with a hexaflexagon. It’s wild. Also I’d love to know how to make a tetraflexagon with more than 6 faces

[@Yevgen4000]

"There are 3 easy steps and one difficult step" Me, an origami master: *ARE YOU CHALLENGING ME*

[@adamweishaupt3733]

Don't each of the faces have 3 potential "partners," the one that reveals them and the ones that come after a horizontal and vertical fold? Why aren't those represented in the graph? Everything except 1 and 2 is only listed twice.

[@absalomdraconis]

The "center pair" has 4 neighboring pairs, two for each fold direction, but the other pairs only have 1 on each axis. In particular, to get to anything else you would have to unfold the flexagon into a flat sheet again. This is why you see him trying to fold it in certain directions before giving up on occasion: this flexagon does _not_ allow infinite folds along a single axis.

[@fireballme1153]

You can also flip to the other side

[@highpath4776]

Think about the start shape, it has 4 corners, those corners match with two numbers on each side of them, those numbers have only one additional number they match to this explains how a starting shape as folded has the number of axis combinations and with what.

[@wafikiri_]

This tetraflexagon has a two-square map. Once I mapped a hexaflexagon and got a central triangle touching three other triangles with its three vertices.

[@digitig]

I learned about flexagons from Martin Gardner's Mathematical Puzzles and Diversions books back in the 1960s. I got as far as making a dodecahexaflexagon. I also learned the Tuckerman Traverse, a systematic way of getting to all the faces on any flexagon.

[@robinbrowne5419]

Thanks.That's great. I showed this video to my little grand daughter and now she's busy trying to make a flexagon. However she doesn't quite get it yet. She is just coloring squares on the paper. She will probably need a bit of help from grampa. But a really fun activity for kids. :-)

[@lasamisalagne7377]

I'm gonna flex in my maths class with my newly aquired knowledge about flexagons

[@XenoghostTV]

"flex" is an understatement hahahahah

[@XeLaNoiD]

When you start calling your paper square a Parker square.

[@HrsHJ]

So Parker Square guy is back what's next?? Maybe bring back everyone's favourite James Grime??

[@lazarusnecrosis5869]

I love Matt and James both. It's hard to choose a favorite.

[@lr31415]

YES PLEASE

[@supersleepycj]

i saw this notification and the forgotten pile of hexaflexagons on my desk started calling out to me again

[@sandervanderhorst9851]

1:40 "you can get rid of those"?!? Nooo, make another one!

[@Qenton]

Wow, I didn't know that one! I always made the one with the same number of squares but outer corner squares are removed and an X in the center. You can fold it in one motion. This one only reveals 4 faces of the 6 but does cycle nicely. I guess you would call it a quad-tetra-flexagon? I did make a Dodeca-hexa-flexagon once in the 80s but had to figure that one out myself.

[@1p6Gaming]

And by using a pattern like so, you can make a five sided tetraflexagon, a pentatetraflexagon! (o's are paper and slashes are cut out squares) ooo/ o/oo oo/o /ooo

[@amoryacosta2915]

You can also make a dodecahexaflexagon. My friend and I worked on it for almost a whole semester before getting one to work properly.

[@dropintheocean7779]

*The best thing about this channel is that it contains so many languages that anyone understands what is said in the video, especially Arabic*

[@fiziwig]

I remember hexaflexagons from Martin Gardner's Mathematical Recreations column in Scientific American. That's been a long, long time ago.

[@gustavgnoettgen]

The one wanted to be a cube. The other triangular. Together they fight for their destiny. *_FLEXAGONS._* Now on VHS and DVD.

[@gudadada]

So simple and complex at the same time.

[@MrMidlandman]

Flexagons are at least 50 years old! Check out 1970's books on Recreational Maths by Martin Gardner, (though probably out of print now!)

[@gabrieldoudna6570]

this is the most satisfying numberphile video in a while

[@MusicCriticDuh]

I'm pretty sure that ViHart hadn't forgotten about the beloved,hexaflexagon!

[@FN0844]

ViHart has revived the hexaflexagon.

[@unknown_demi6902]

someone get vihart on the phone, we got fractal flexagons.

[@michellehay3639]

1:55 When I saw this I was like *MENGER*

[@kevinmartin7760]

It bothers me that the hexaflexagons must be glued for assembly but the hexatetraflexagon does not. It also bothers me that the 1/2 configuration can be transformed to four other configurations, while all the others can only transform to two. The operations from 1/2 are the combinations of the choice of which face to close (hide) and whether you close it vertically or horizontally. From the other configurations you still get a choice of which face to close, but (I assume) one way of closing each face cannot be opened in a manner different than it was closed. The closed configurations represent the arcs on your chart; perhaps you could follow up with a chart identifying the arcs and showing these dead-end arcs.

[@justinhoffman5339]

The video introduced me to "a section, long ways" aka The Parker Column.

[@BlessedForever888]

I love the fractal flexagons at the end - super adorable

[@generationedge6699]

12:12 Tetraflexagons are the bestagons

[@cobaltbluesky2276]

“Do you want to see my new flexagons?” Best pick up line

[@WDCallahan]

I was immediately wondering why 1 and 2 have three states, but the rest only have two? These are things I noted: The ones and twos are separated on the unfolded paper. Everything else is in pairs. All even numbers are on one side, and odd numbers are on the other. Numbers with only two states take up corners and edges. Numbers with three states only take up edges. All two state numbers are written in the same orientation, whereas I can see half the 2s are upside down. Presumably two of the 1s are also upside down, but it's not possible to tell. I want to know how it an fits together!

[@c4oufi]

When I was into flexagons, I got bored of hexa-hexa-flexagons, so I made a dodeca-hexa-flexagon. I ended up making myself a Feynman diagram and man, was it a tough one.

[@dougr.2398]

Václav Coufal there are too many bubble diagrams involved

[@retromatt2281]

Vihart actually found a way to make it without gluing

[@tothm129]

do we know when a n-m-flexagon exists.

[@ivantheczar]

I learnt this version of square flexagon back in an education TV program some 30 years ago XD

[@beeble2003]

"Fold it in half. Whichever way you want -- they're both the same." Well, that's a bad start... There are eight ways to fold a square of paper in half. Two parallel to the edges, two diagonals each either ridge folded or valley folded. Alternatively, if you ignore symmetries, there are two ways but they're not both the same: parallel or diagonal.

[@ScottyUtHome]

Instead of that diamond shape drawing, you.could have drawn the lines horizontally & vertically as a map for which fold gets you to the next number combo.

[@jeromevictor5182]

you can make a smaller Flexagon from that cut off square, and you make another of its remaining square and another, and another, and another, and another to get an infinite sided Flexagon.

[@angelaphsiao]

Highkey hoping you and Vihart collabed on this

[@frankhooper7871]

I was taught how to make both hexaflexagons and tetraflexagons in my highschool geometry class in 1966/67 🤓

[@chriscraven9572]

I was given a wedding congratulations card in the form of a hexatetraflexagon last weekend. Brought back fond memories of Martin Gardner's articles and books.

[@baksatibi]

I like that Matt keeps an actual Parker square on his desk.

[@munumun]

will vihart accept that "forgatten flexagon is best flexagon" ?

[@IntergalacticPotato]

Dunno

[@piotrmecht2500]

I remember this flexagon from my chilhood. You could find it as a promotional item in some crisp packages (brand owned by frito-lay). It used images instead of numbers

[@mmmmmmmmmmmmm]

I learned this as the hexaflexagon, never heard of a hexagonal one before.

[@nathanderhake839]

I love Matt Parker so much

[@Emma-rw8yo]

Can we take a moment to appreciate the person who made all these flexagons for this video

[@paulalisauskaite7719]

2:34 his words say tetraflexagon but his expression says the superior flexagon 👀

[@ebi_tempura]

"If I make this fold *very* good, very *sharp*" Proceeds to barely crease paper

[@Hootkins.]

You must've missed the part where he ran his finger nails along the creases to sharpen them.

[@unvergebeneid]

How very bold of Parker to do a video on a _square_ shaped flexagon, made from a _square_ piece of paper. He _must've_ known what the comments would look like.

[@dataunknown]

I could have sworn James Grimes used to have this flexagon in video on singingbanana. I tried looking for it about a year ago and couldn't find it.

[@sander_bouwhuis]

I would simply create a transition matrix. The nice thing about that, is that for a transition matrix M you can do M^n to determine where you can get to in n steps.

[@konstantinkh]

The center square should be used to make a smaller flexagon by cutting a smaller square out of it. And then you can use that smaller square...

[@AB-Prince]

I remember when frankie and bennys had a template version of this as a sort of toy

[@harish1928]

If you fold the numbered paper again in an anti clockwise way, you will get all the combinations of numbers that would not appear in clockwise fold

[@oafkad]

"The first half and half splits it..." That pause to figure out what a half of a half is was very relatable.

[@mychairmadeafartnois]

Some time in the mid 2000s I got a flexagon out of a package of something. I think it was string cheese. I’m pretty sure it was spongebob themed. It must have been this sort of flexagon because it was a square. I spent a lot of time mesmerized by it. Thanks for reminding me about that, this was great.

[@justaman9564]

As soon as he Matt cut the hole out of the square it became a Parker Square

[@Crosshill]

i thought the title referred to how flexagons were being forgotten and i thought back to way long ago where i first found viharts flexagon videos and was sad for a moment that i'd forgotten. i never did make my own flexagon. what if i died without having made on? terrible. i think theres a certain charm to a square flexagon, you dont think of squares as being flexy. i'll try and make one

[@doodelay]

The minimal upload frequency disturbs me

[@JennaGetsCreative]

And now I need to make a flexagon art piece!

[@peachyjin846]

You fooled me with the thumbnail. I thought vhart was bringing back hexa flexagons

[@AHBelt]

I think I saw these first on a KZbin video where someone had one made for several pictures of Norm McDonald. The other kind would baffle me when I was a kid and they were 'fortune tellers'.

[@ionobelisk]

Back in the '60s, ICL (British computer giant of the time) had promotional items that came in the form of a garishly printed A4(?) Sheet of thin card and instructions to cut it into strips that had marked triangles that you could fold into a splendid hexahexaflexagon. I wish I still had mine: it was an object of kitsch psychedelic beauty. One wonders if any still exist.

[@shearnotspear]

I’m curious as to what the various Numberphiles think of Doctor Who’s Block Transfer Computations, the mathematics that create reality, first appearing in 1980’s Logopolis.

[@Keallei]

ViHart would/is be so proud.

[@pasunurusaivineeth3739]

This video is a Parker Square of a Parker Flexagon

[@StefanLopuszanski]

Feel like this could be used for both a magic trick and an interesting board game. I've seen something similar done with board games, but it wasn't 6 faces and was only like 4 or something.

[@CristiNeagu]

This almost makes up for the Parker Square...

[@OsyenVyeter]

The wrinkle in time board game comes with three plastic tri hexaflexagons.

[@Maninawig]

I dunno if Matt Parker realized what he did in this video.... He's created a new channel for Bradey (sorry if your name is misspelled) Drawingphile. He's created a new form of puzzling. And he's created, after I am done with it, the Parker Flex (his approval pending)

[@xnick_uy]

10:22 Isn't it possible to extend the 3 and 4 lines so they meet at the bottom, and the 5 and 6 lines so they meet at the top? By symmetry I would say it *has* to be possible (but I need to fold the paper to make sure), and then the network would form another square. And this square can still be further extended to form a square grid, out of which you can form many new flexagons!!!

[@wolfsurvival2009]

Maybe this will finally pull @ViHart out the woodwork to make a new video? One can only hope...

[@emul596]

Hi Matt, is there a relation between the order of the numbers on the unfolded Flexagon and your diagram?

[@EdoTimmermans]

Not Matt here, but yes: the numbers 1 and 2 are all in the middle and their lines go through the middle of the diagram, whereas the numbers 3, 4, 5 and 6 are both in the middle and the corners of the paper while their lines go through the corners of the diagram.