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Steffen's polyhedron is a flexible concave polyhedron. Euler thought such a shape was impossible. I also show infinitesimally flexible polyhedrons and bistable polyhedrons.
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Additional filming by Nicole Jacobus
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* polyhedrons - it's a valid plural and I'm taking it out for a spin. The sponsor is Incogni: the first 100 people to use code SCIENCE at the link below will get 60% off: incogni.com/science
It might be valid (inasmuch as English doesn't have any official rules so anything's valid as long as more than one person agrees) but it's still weird to hear. It feels like when someone says vertexes, matrixes (unless they're referring to the movies), or phenomenons.
i NEED A Candle
It's "polyhedra", and that's the hill I'm prepared to die on.
Did you say Stephens polyhedron? Edit: Sorry, I looked at the description and you said it's called Steffan's polyhedron.
@@BruceElliott No, you may not die on that hill. Only after you've fought over each and every Latin and Greek word being formed as plurals in English according to the rules of their origin language, when you've reddened the craggy landscape with your lifeblood, at last uttering your final grammatical gasp, do you have my permission to die on that hill.
Every neuron in my brain is screaming "IT'S JUST FLEXING WITHIN THE TOLERANCE OF THE IMPERFECT PRINT" which I know isn't the case, but I can't NOT see it that way
exactlyyy!
Same!
Or in the rigidity of the material.
That's the infinitesimal one later on!
Same here... in my limited mind the tolerances play a part, but at the same time, material flex must also play a part... instant head ache
Weird flex, but ok.
Lol
Legendary comment
😂
Badum tss
The little stretchiness in the triangle you were talking about reminds me of illegal Lego builds where people combine many small Lego pieces in patterns so they bend and create curved surfaces
Yes!
"illegal lego builds" i love it 😂❤
@@retro4711That's what the Lego company calls them! It means they won't use these techniques in an official set, usually because they aren't stable or can get stuck.
@@laureng2110 i didn't know that, thanks! When I read "illegal builds" i couldn't help but imagine the lego police busting through my door because I built something using a forbidden technique :D
@@retro4711 this will be a B-story in the Lego Movie 7
I always love it when you and Matt pop up in each other's videos :D
Magic!
@@standupmaths was that a Parker card trick?
"Mathematician's bad sleight of hand," sounded entirely reasonable. I didn't suspect it was a set up at all. Very funny.
@@gorden2500Parker card illusion.
Spoilers!!!
I remember making "hexa-flexagons" in school. They're technically six tetrahedrons attached to each other, but are pretty fun to play with.
*Memories of Vihart*
@@The_Moth1I just showed my dad the vihart hexaflexagon video yesterday. It's kind of funny seeing it brought up a decade later.
Weird "flex" but OK. ;-)
@@sophiedowney1077 strange... I didn't realize there was a 2D-ish version. The ones we made are always 3D with regular tetrahedrons.
im glad im not the only one@@The_Moth1
I wasn't convinced until I saw the simulation. This feels like tolerance problems in the 3D printed joints. It only makes sense in my head when it's a simulation with rigid definitions that aren't allowed to flex or stretch.
I was thinking the same thing at first, but you gotta realize that they probably proved this stuff mathematically a while ago. Making it physically is just a fun bonus step.
“They probably proved” is not “There’s a proof over here they are referencing”. If I know Steve he will realize he has to show the proof. *I don’t know Steve at all. 😅
it slides though
@@jasond4084 The actual proof is probably really long and opaque, not worth referencing in full in a quick, 9 minute, general audience video. But Steve does give enough information in the video to look it up for yourself if you were so inclined: 2:48 - the polyhedron in question was discovered by Klaus Steffen in 1978 and is known as Steffen's polyhedron.
@@iout it wasn’t clear in the video that the printed version and the proven version were the same. I thought this was a new find. But okeeee. Thanks
The infinitesimally rigid polyhedrons which flex in the real world remind me of (I think) a practical application of this, which is "negative stiffness isolators". The object to be isolated from vibration is mounted to metal flexures (at the centre of the polyhedron that "pops" in and out like the fresh seal on a jam jar lid). This means that the deflection can actually increase as the force decreases, over a portion of the stiffness curve. They are very useful for extreme sensitivity environments where vibration on the order of 0.1 micrometres/s RMS velocity can be detrimental, and for high frequency vibration that active isolation can't respond to.
Oh thats interesting man !
I can’t completely understand wtf u just said but the parts I do sound neat. Ima need to see this for myself now lol
Polyhedra*
Replying to @chrisburn7178: SARZHERFLURGERFLARRBZHSHAR?
Heaps interesting cheers
6:44 i'm surprised you didn't think of the dodecahedron. any pentagonal face, when removed, if it permits flexibility will permit two degrees of freedom.
This makes intuitive sense: the pentagonal face can be broken up into multiple independent triangles, which thus can easily have their own flexibility. Since they do not share an unconstrained edge. I'm not sure if this is necessarily true independence, since the flexibility likely transfers through the rest of the body, but in the real world with the amount of flex in models the amount of movement transfer may be negligible. We can rephrase the question, then: does there exist any polyhedron where the removal of two faces results in only a single degree of freedom introduced? If not, then the polygonal face question becomes irrelevant, since any polygonal face can be divided into triangular faces: structurally the polygonal version and the triangulated version are equivalent when the faces constituting the polygon are removed.
@@haphazard1342 A cube with two opposite faces removed has 1 degree of freedom
I was thinking along similar lines, although I didn't work toward a minimal example - I just thought "OK, cut an icosahedron in half such that one face is much larger than the others and has a bunch of vertices, then remove it and there must be a way to get multiple degrees of freedom out of this".
A pyramid, but with penta-, hexa- or more-gon as a base instead of square would become a flappy umbrella with increasingly more degrees of freedom (as the number of vertices increases) when the base is removed, wouldn't it ?
@@krzysztofsuchecki4967 Doesn't that approach the top of a cone as the number of sides of the base increases? Intuitively I imagine a cone being rigid though I don't know if that is true. Anyways perhaps something like a pentagon base would be flexible anyways, its an interesting idea.
You mentioned polyhedra that are bi-stable, and it made me realize that the phenomenon of bi-stability is actually quite common - it's just that in most cases, the stable points are so far from each other that we can't really flex between then even with real-life, "rigid" pieces. Take the icosahedron for example - imagine applying enough pressure to one vertex that it gets "punched in", and the vertex now points inward rather than out. What you're left with is a structure with 20 perfect equilateral triangles, it's just concave now. Maybe the interesting problem regarding bi-stability is to find bi-stable shapes (or "multi-stable", it shouldn't have to be just 2) whose stable positions are as close together as possible. And I suppose a flexible polyhedron is the infinite limit of multi-stability, where its stable points are so infinitely close together that they become continuous.
I hate that I understand this run on ass sentence regardless of how many of the words I literally couldn't define given half a chance
@@fabulousflufferbum2051you should probably just embrace it
@@fabulousflufferbum2051these are completely normal sentences
Lmao this comment section is funny af Though OP you do a good job creating a picture
Huh
4:21 Ah, yes, The Parker Card Trick!
I have to wonder what Euler's reaction would be if you took this back through time and showed it to him.
He'd be like "holy shit time travel is possible?"
"Huh."
"Oh come ONNNN!"
Euler was blind if remeber correctly so it would be hard to show him that lol
@@bluelemon243 He'd still be able to feel the shape and hold it in his hand
I guess Euler wasn’t so smart after all
If he was so smart, why aren’t more things named after him? QED.
What a poser
When I was a kid, back in my old school Maryetta, we'd compete in trying to build 3D shapes strong enough not to shatter when thrown on the ground. Those were the days.
This reminded me of origami, and how that can be used to demonstrate and illustrate mathematical concepts. I still have a copy of my favorite origami book from when I was a kid that actually contains a full chapter on "Beautiful Polyhedrons" that got little me asking my scientist mother math questions that she couldn't answer (which made little me feel very, very smart at the time.) They are mostly multi-sheet builds, but unitized in a way that you can easily assemble them into intriguing polyhedrons. I highly recommend "Origami Omnibus", by Kunihiko Kasahara if you can track down a copy of the 384pg tome as one of the few origami books printed in English that I've encountered that actually explores the mathematical beauty and concepts behind folding square sheets of paper. It covers everything from cute and simple animal models up through multipage books (no cutting) with a matching bookcase to store them in, and the method (and math) of using different sized paper (without rulers or calculators) to make interlocking 3, 4, 5, 6, 8, and 10 sided polygons of equal side length (pg 222) to build things like a rhombitruncated icosidodecahedron (pg 229) and the reversible stellate icosahedron (pg 234, which you can actually turn inside out and change it from flat sides into something starlike.) I'd love to see you explore some of the more technical stuff from that book. Even young kids can understand complicated subjects when they have real-world demonstrations in their hands.
Ivan Miranda deserves far more subscribers than he currently has. He's been building amazing machines and prints for years and he's always enthusiastic.
gigantic printers and gigantic stuff
I was struck by the passing mention of Robert Connelly. Back in the mid 90s, I made some flexible "carbon ring" models for Dr. Connelly and for a Swiss post doc named Beat Jaggi.
I love your curiosity and desire to explore the little things that many of us think are simple. The more I learn the more depth I realize there is to unlock.
It's good you printed the side with the window. Otherwise, I could have suspected it's just tolerances within the hinges allowing the thing to move.
6:34 *J O I N U S*
That was quite the nostalgia hit. Those toys were one of my favorites. I remember experimenting with this exact concept, except with no language or basis to understand it. It makes me think that people could become so much smarter if they were taught on an individual level. I was probably 2 when I had these toys and I was feel like i was ready to understand these types of concepts with the right teacher.
wow you're so smart.
Hey, do you know what those toys are called? I want to look them up on online shops.
@@ElcoCanon I'm just saying that these kinds of concepts could be learned so much earlier in life with the right teaching. This is like some late high school level stuff, but it's so easily accessible with these toys that its almost a natural progression if you play with them long enough. If you played with them as a small child all the time you would know I'm not lying. everyone does this exact thing with them but just don't develop a deeper understanding because of the lack of teaching.
These toys still exist, but they’re magnetic now. Kids love them, usually making castles.
@@abangfarhan1 Polydron
Seeing this reminds me of seeing those rocks that are flexible. So strange to see something that your mind does not expect to happen happen.
Can you tell me more about these flexible rocks?
@@bathbomber Google "itacolumite"
@@bathbomber its called Itacolumite, there are youtube videos about it. something about a solid-looking rock bending feels so unnatural (despite it being natural)
@@kirtil5177 beat me to it, thanks!
@@bathbomberbasically flexibility of an object is arguably more about an objects shape than it is about the physical properties. Think about a metal block and it’s not really flexible at all but make it thin, like a spring or foil and it can become very flexible. There’s a specific type of rock that has enough inherent flexibility that a regular looking centimeter thick or so sheet of it can flex around in a way that looks bizarre. What I haven’t seen more people talk about though is the fact you can make just about any rock flexible by shaping it correctly and making it thin and perhaps spring like. Those rocks specifically known for being flexible lose all of their flexibility too if they’re not shaped right and are too blocky
You should look into auxetic structures and or negative poisson ratio materials. It feels a little bit related to this. Basically, instead of a material getting narrower across as you stretch it length wise (like how a rubber band gets thinner as you stretch it) it instead gets wider. It also feels really unnatural but they exist!
3:27 If you have an object like this in a 3D format you can put it into software like PepakuraDesigner to get glue flaps, so you don't have to use tape to hold it together.
This reminded me of cyclohexane. Used to image how it can have various shapes (conformations).
cis and trans, but those words have taken on a somewhat different meaning these days.
@@kempshott well, they're not words, they're prefixes
@@kempshott They took on a different meaning when they were adopted into chemistry as formal terms, too. I don't think the Romans had a significant amount of knowledge on cis and trans isomers
@@kempshottthe conformations of cyclohexane would be boat, chair, etc. maybe brush up on your ochem lol
@@Gakulonand yet ultimately, or etymologically, they still mean exactly what they did back then. Understand the general meaning, understand every special meaning
For all those saying it's just imperfection that allows it to flex, please look up en.wikipedia.org/wiki/Steffen%27s_polyhedron and its citations. I was also surprised.
This is a great video. Thank you for making it!
The shape in geometry test :
"Proofs and Refutations" by Imre Lakatos, which examines the nature of mathematical progress and discovery (check it out, it's got its own Wikipedia page*) is based around a discussion of polyhedra, specifically the Euler Characteristic. *From which I learn: 'The MAA has included this book on a list of books that they consider to be "essential for undergraduate mathematics libraries"'
I wasn’t looking for this comment but I’m glad i’ve found it. Ty.
Thank Phineas & Ferb for discovering this thing that doesn't exist?
Fun fact, the test for a structure to be not infinitesimally flexible (isostatic or iperstatic) is at the base of all structural mechanics jobs
Actually good to keep the infinitesimal flexibility when designing for 3d printing, had the intuition for it but having a name for things is always better for clarity of thought and communication.
This immediately made me wonder whether we could synthesize organic compounds with such structure and whether they would have aby unusual properties
I sped it up by 1.5 and I thought it was a different language
This was definitely quite a head scratcher indeed. Flexible polyhedron 3D printed house when?
How can you be sure the flexing isn't some kind of additive result of all the gaps in the hinges?
they proved it mathematically
Maths.
Looks to me like the perfect wavebreaker, put in chains as bantons in tsunami-endagered coastlines, for example as anchored-chain-boeys as well. Might be a way to divert vibrations as given in shocks of an earthquake, too. In any case, thx for sharing!
I appreciate that this is approachable and clear without in any way dumbing down the math or avoiding terminology.
Whenever I've had an overdose of random YT shorts, I return to this channel to regain some brain cells.
Yay, Matt easter-egg!
It seems like you'd get much more wobble if the single removed face had more sides. I think you're probably right that the degrees of freedom are limited for squares or triangles. If you instead imagine two regular octahedrons as the ends of something like a prisim, but with the sides replaced triangles (like the "ring" around the middle of a regular icosohedron), then it would likely be pretty wobbly with just one face removed.
Indeed, that would give more wobble and moreover ease of flexing, by making more sides you are decreasing the length of each side meaning that you are also decreasing the length you'd have to flex in order to get back to a stable position.
Always learn something here. Thanks.
This is so visually stimulating and satisfying ❤
We had those exact same plastic shapes in primary school. Thanks for digging up a nice memory Steve!
I want to get my hands on these, do you know what they're called?
@@cheeseburgermonkey7104 IIRC polydon was/is the original though there are certainly other brands.
OH MY WORD thank you! I've wondered for years what that rod-and-strings contraption is, ever since I saw it on someone's desk in some movie! I even modelled it in 2D with different colours and transparencies to figure it out! (Then I didn't make one because I have neither woodworking skills nor 3D printer access but ah well.) Now that I know what it's called (Skwish!) I could actually get one. The one in the film had a big sphere in the centre, though, and none of the endcap/sliding balls. I will google this later!
I’ve seen it too and was curious… I can’t find one on google, if you have better luck let me know! Edit: I got it… expanded octahedron model. There is also a double expanded which is pretty awesome too!
Your videos are so good in so many dimensions
I never thought that was impossible. I never knew it existed and I believe it does now.
12 seconds in, damn good quality already!
I wonder how much the manufacturing tolerances play into this
Throwing shade at Matt Parker's card tricks, delightful
I must say, that additional filming by Nicole was magnificent.
I don't know, but did Euler only consider convex polyhedra to be polyhedra? What was the definition of a polyhedron at his time?
Just popping in to get this in my watch history, will watch properly in the evening. I love geometry and this looks really interesting!
You are aware of the "Watch Later" playlist, right? ;)
@@examplewastaken or even just the subscription box
@@tigrafale4610 now imagine even using it 😲😂
@@tigrafale4610 (regarding this, I have several hundred subscribed channels now so it's actually even less useful than even just the homepage for finding what I want. Imo, situationally useful if you don't have a lot of subscribed channels.)
When making a polyhedron flexible, you have to count the number of edges, not faces, to remove. Removing one face of a polyhedron does not change the number of edges, nor their connections, so it is actually still the same shape. That is why you observe that at least two faces has to be removed to make the shape flexible.
If you remove the base of a square pyramid, it becomes flexible. So that's a counterexample to your claim. The point is that the faces remain congruent through the whole flex, but the angles between faces change. So the removed square base can be flexed into any rhombus with that same side length.
Oh I guess you are right. That would probably also be the case for some polyhedrons where the faces are not a triangle.
I Love this channel. I also love robust "Description" sections on KZhead as it allows the user to find specific content, follow suggested links to other content we might like, etc. But I have one SUGGESTION: When propagating the Description section, if this is possible, put an additional "Show Less" right next the "More" on top (as well as the one at the bottom). This would allow someone to collapse it without having to scroll all the way to the bottom to do so. (I have no idea if this is possible.) .
That's a suggestion for KZhead
Are you sure that the flexing is not due to the mechanical backlash?
The physical model should be thought of as a demonstration - not a proof. Steffen's Polyhedron has been proven mathematically to be flexible, but obviously you can't built a perfect mathematical shape in the real world.
What are those toys called?
I can’t help but watch your videos every time one pops up. It’s just too intellectually stimulating. It’s like brain candy.
The strangest part of this is Ivan printing in a color other than red.
LMFAO @ the cut to Matt doing bad sleight of hand. That was really good 😂
I think its impossible unless removed wall has 5 sides. 6:00 you can move them independently when there are at least 5 free edges icosahedron with 5 sides removed is the same as if there was originally pentagon. Is icosahedron with pentagonal side a proof then since it fits definition of polyhedron 2:17?
yea
Wouldn't the dodecahedron's much better
@@koharaisevo3666 they already have pentagonal walls that are rigid on its own when 3 of them are connected
Every antiprizm with top and bottom wall that have 5 or more edges can do
Remember that videogames use Triangles. So this geometry could revolutionize physics simulation in videogames down the line
oooh yeah
Someones upgraded their talking to camera set up, very nice.
If the shape is already flexible in one degree such as the Steffens polyhedron than removing one of its faces should open a new degree of freedom. The thing is when you remove one face of a convex shape it is inherently going to remain rigid as the number of edges is the same. Until you remove one of the edges by taking off a second face you dont have a new degree of freedom.
4:18 😂
Does it flex, because of material flex though, or is it genuinely moveable, JUST at the hinges?
it works even if all faces are perfectly rigid.
I love problems like this. that are extremely simple in asking but complicated in solving, yet the solution is something you can literally hold and not only see but literally feel in your hands. It takes away a lot of the esoteric nature from modern math and gives the feeling we’re still continuing the work of ancient mathematicians.
I can always count on Steve Mould to find interesting toys I never knew I needed.
I've never gotten to one of your videos this early before!
Hi Steve. You had me at "this is a valley fold, this is a mountain fold." Some of this can be proven via origami. There's an American origami artist called Steve Biddle who made a rotating tetrahedron. I have a book with the fold pattern in it.
This is another level of nerdiness that I've never seen before. I'm glad you all can geek out over this. I find it interesting though.
Amazing. I wish I had a teacher like you in school.
Wow I've never been so early
Weird flex but ok.
I think it's only flexible because of the inaccuracy of the construction of the sides. the play in the hinges is what allows the flexibility.
Or the sum of "slop" in all of the joints. Watch closely, you can see the joints stretching.
There's a mathematical proof that Steffen's Polyhedron is flexible
@@SteveMouldOf course there is. The mathematical model will be perfect. Such a thing doesn't exist in the real world. Or at least approaching it would cost more than your budget would allow.
@@Alacritousanything imperfect will flex more. It works in both perfect maths world and the imperfect human world
I love that you used Matt as a silent 1 second punchline
sprite
I love how @stevemould look and vibe is that he just physically finished wrestling a math problem and won.
0:14 I used to play with larger versions of these back in school in the late 80s.
"A mathematician's bad sleight of hand" gave me quite a chuckle.
Polyhedron: **literally flexes and moves air in real world** mathematicians: “nope, not flexible”
Thanks for recommending Ivan, I follow a bunch of similar channels but had no idea about him.
So psyched to have discovered your channel!
I love the chain fountain standing in the background like a trophy
We used to have these toys at kindergarten, iirc it was called Jovo. I would always construct the shape in a plane first before folding it into 3D. Teachers couldn't wrap their heads around that as the other kids never built anything more complex than a cube.
I remember that I made this or of cardboard when I was teenager, almost 40 years ago, based on one article in polish mathematical magazine "Mała Delta" (Little Delta). That was fun.
Thank you sir I really needed to see this this was the last obstacle in my way of creating a time travel device that actually works
I remember being in school learning physics, free body diagrams and stuff like that. (Pulleys, strings, weights, etc). In this context, I remember struggling so much with a "made up" exercise of mine, imagine 2 bodies joined by a string, and then another string joined at the middle of this string pulling perpendicularly... Pretty much what you explained about the colinear hinge... In the constraints of idealized freebody diagrams this just wouldn't move, which is obviously not what happens in the real world... And 13 y/o me struggled for a while until I realized that in the real world the strings would stretch slightly, therefore you'd have a small component of the force actually pulling the bodies together... It was an important formative moment for me I think... realizing that the ideal models and simplyfications made while you are being taught should not be forgotten about and in the future should be referred to if something didn't make intuitive sense... Like it made clear to me some limitations of how we are taught... Haha, that was cool... it's always cool finding out about the more formal explanations for stuff like this and to remember that pretty much always someone thought our same same thoughts a long time ago and went way more in depth and actually formalized them...
I don’t understand this but I’m super appreciative this absolute mad lad took the time to tell me about them.
Matemathicians: "This is Impossible!" Guy with a 3D Printer: "Are you challenging me?"
I tried so hard as a kid to make a shape that would do move and never found one
I have read something about flexible polyhedra, and I wondered, why in seemingly all of Wikipedia, they can’t show me a single flexible one. And now I’m angry, because the simplest ones aren’t even complicated. Thank you.
Interesting, this reminds me alot about Flexagons in Origami, where you have a shape that you can basically turn inside itself to create a different shape. I think thats pretty cool, maybe look into these as well.
Thats such a random subject and im here for it, so interesting
Thank you for existing, Steve Mould
Wow you just solved a problem we never knew existed and probably would have never known in our life.
"Hey Matt Parker, I need you to do a slight of hand trick, but make it really bad." "It's the only way I know how."
I have a long time relationship with this plastic toy. I get it out sometimes and just make interesting solids, like stellated and truncated platonic solids. They are just so nice to hold in your hand and contemplate. Also straight prisms and "screwthread" prisms and their chiral partners. You can spend (waste) hundreds of hours just enjoying making nice shapes!
Do you remember what they are called or if you can still buy them? I have been looking for them / trying to remember what they were called for years now. I used to play with them as a kid in elementary school.
@@SephJoe I'm very sorry, but the original cardboard box disintegrated decades ago, and we just keep them in an old bucket now. I tried to find them with an internet search and failed. There are lots of kits with magnets but I couldn't find the old type which click together like in this video. If you do find a seller, I would be interested in buying some more just to make even bigger shapes!
@@SephJoe Polydron
matt parker cameo pulling the parker trick, enlightening
this is a good way to explain the vibrations in organic chemicals