The object we thought was impossible
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- Опубликовано: 28 сен 2024
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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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* polyhedrons - it's a valid plural and I'm taking it out for a spin.
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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
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
Weird flex, but ok.
Lol
Legendary comment
😂
Badum tss
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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
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.
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.
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.
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 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
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.
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.
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
What are those toys called?
"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.
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
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
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.
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 was definitely quite a head scratcher indeed. Flexible polyhedron 3D printed house when?
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!
I appreciate that this is approachable and clear without in any way dumbing down the math or avoiding terminology.
This immediately made me wonder whether we could synthesize organic compounds with such structure and whether they would have aby unusual properties
The shape in geometry test :
Whenever I've had an overdose of random YT shorts, I return to this channel to regain some brain cells.
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!
Fun fact, the test for a structure to be not infinitesimally flexible (isostatic or iperstatic) is at the base of all structural mechanics jobs
I can’t help but watch your videos every time one pops up. It’s just too intellectually stimulating. It’s like brain candy.
4:18 😂
Throwing shade at Matt Parker's card tricks, delightful
I wonder how much the manufacturing tolerances play into this
Thank Phineas & Ferb for discovering this thing that doesn't exist?
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 think you can easily make as many degress of freedom as you want since it doesn't need to be a regular polyhderdon. For simplicity, start with a triangle. Now, divide each edge into 3 parts, and delete the middle one. Rotate one of the edges outwards (could be both, could be inwards, but we keeping it simple), and elongate them a little but less than the original length. Now, reconnected the two dangling vertexes with a segment, making it a polygon again (or a "triangle" with Z ish shaped edges). Now each of these trios have independent degrees of motion as a polygon, you can keep the original vertexes fixed as hinge points. Now, we move to 3D. Just pick an arbitrary height (so 1) above the figure and connect all those vertexes to it, forming a Z faced "tretrahedron". If you remove the original polygon face, you have 3 degrees of freedom. Of course, you can pull this trick with any base polygon, so you can literally have as many degrees of fredom as you'd like depending on what you start with.
In fact, you didn't even need to subdivide the middle into just 3 parts, that is just the minimum. You could have subdivided each original edge into 4 or more parts, but all it means is that each sequence of 3 of those are themselves one independent degree of freedom like in the original, so you could achieve infinity degrees of fredom that way too. Except that that is mathematically identical to the original method, so it is literally the same thing just presented differently (in the original, any sequence of 3 new edges forms that Z shaped hinge and is therefore an independent degree of freedom, it doesn't need be confined to the 3 that came from the same original edge, I lied by omission for simplicity).
This is a great video. Thank you for making it!
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
I've never gotten to one of your videos this early before!
Mould Conjecture counterexample: Make a pyramid with a many sided base (for example a regular decagon). Remove the base polygon. The remaining shape should have many degrees of freedom. As the number of sides of the base grows, so do the degrees of freedom of this shape, without limit. For even side counts N on the base, this can be shown by bringing every other vertex together, resulting in a shape with N / 2 flaps which can rotate independently along a axis from the pyramid point the where the free vertexes were brought together. Unless I visualized it wrong, which is quite possible.
3:19 Anyone who's made a waterbomb base with origami can feel that...
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.
"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."
Aliens must be looking at us like we're babies playing with blocks and just not quite getting it yet.
LMFAO @ the cut to Matt doing bad sleight of hand. That was really good 😂
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.
Remember that videogames use Triangles. So this geometry could revolutionize physics simulation in videogames down the line
oooh yeah
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.
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.
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.
Polyhedron: **literally flexes and moves air in real world** mathematicians: “nope, not flexible”
Thank you for existing, Steve Mould
My brain could not comprehend the movement of the grey, green and blue shape you had printed. For me, it was like if the walls of a house suddenly started shrinking and growing as you flexed it. Logically that is impossible and it is just moving/angling, but I genuinely could not visually comprehend what was going on, I had to take your word for it. I think it is because of how the concave and convex areas are arranged in a very unnatural looking shape I would have never encountered combined with the effects of lighting and plastic colours. The brain is neat like that.
4:15 the most important bit in the video
matt parker cameo pulling the parker trick, enlightening
3:19 "this is fun" combined with this dead unemotional voice had me cracking
It sounds a bit like it was recorded separately, so I guess this is why I get that feeling.
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.
Wow I've never been so early
That jumpscare from my childhood tensegrity toy delighted me! I always know I liked that thing- but never because it involved cool maths!
3:40 i recognize that cursor, Algodoo?
Of all the closed 3D shapes, the most amazing result comes about when you remove one side of a sphere. It is definitely worth experiencing.
And good sponsor. We need more anti "spam" services.
Another consideration with mechanical objects is that tolerances on the object itself are never consistent. Plastics shrink and stretch as they cool, filament is never precisely 3 or 1.75mm and all of those factors together can make a big difference in objects like this
Have you ever heard about hyperstatism ? It's a simple number, h, that once calculated gives the number of design constraints that should be followed to properly assemble a mechanism, the higher the number, typically the more parts are inter-connected and the overall system gets rigid.
For a mechanism of n solids (except for a supposed fixed base) and l links, we pose γ = l-n
m is the mobility of the mechanism (number of independent outer and inner movements theoretically possible without deformation)
Nk is the sum of the kinematic unknown of each link (number of moovement allowed by each link a pivot : 1, a slider : 1, a ball joint : 3,...)
h = m + 6*γ - Nk
Just an example of it's use :
A three-legged stool is isostatic, h=0, a four-legged one is hyperstatic, h=1, and a reinforced one with cross bars gets a lot hyperstatic h>10.
In case of the bars example with one straight, h=4 and with two pivoting, h=3. As the hyperstatism degree decreases, the flexibility due to deformation and plays increases.
Your videos are so good in so many dimensions
The strangest part of this is Ivan printing in a color other than red.
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
@3:17 your gonna love it when you find out about pop fidgets! 🙃
I love the chain fountain standing in the background like a trophy
Thanks for recommending Ivan, I follow a bunch of similar channels but had no idea about him.
Just made me think how generally regular things can be a cause of so much internal ideas I would have no idea about without people who explain all the intricacies of this world
thx
I love that you used Matt as a silent 1 second punchline
GNSS (GPS) trilateration is also infinitesimally flexible, especially for elevation estimates.
I'm challenging the Mould conjecture-I think it is possible to get two degrees of freedom by removing just one polygon. But I think that polygon needs to have at least five sides. Notice how you didn't gain a degree of freedom when you removed a triangle, but you did gain a degree of freedom when you removed a square? That's because the triangle's shape is given by the side lengths, which are already in the other polygons of the polyhedra, while the shape of a polygon with four sides has one degree of freedom if all you know are the side lengths, and that degree of freedom is given to the hole that is formed when you remove the polygon. A polygon with five sides has two degrees of freedom if all you know are the side lengths.
In general, the shape of a polygon with n sides has 2n - 3 degrees of freedom if you don't know anything about it, because each corner has 2 degrees of freedom, but then we ignore translations (2 degrees of freedom) and rotations (1 degree of freedom) of the polygon. And if you know the lengths of all its sides, that puts n constraints on it, so we end up with n-3 degrees of freedom.
In Synergetics, Bickminster Fuller goes into this extensively. He calls some of these movements "jitterbug."
Wow you just solved a problem we never knew existed and probably would have never known in our life.
What camera do you use? It's like looking through a window.
this is a good way to explain the vibrations in organic chemicals
"...a mathmatician's bad slight of hand....". 😂😂 Poor Matt, great oblivious cameo 😅
0:07 The name of those coloured polygon toys is Polydron, if you're interested.
There are analogues of the truncated icosahedron that have many hexagonal faces along with the 12 pentagons. It seems to me that given one of these with sufficiently many faces and hinged edges, and insufficient rigidity, it would be possible to invert part of it by using any specified force.
Discrete Math and Geometry are fascinating.
Oh, I remember gluing myself this one several years ago, it’s so neat!
0:10 i made an entire suit of armour + sword out of those little plastic tiles when I was in school, I may of had to raid other classrooms for enough pieces but I got it done.
that sounds freakin awesome
@@gibble75310 it was.
this is what aliens in movies should be like, geometrical optical illusions.
Running this video in the background really caught me offguard with "I've got a big one here. And what's interesting is when I squeeze it I can feel air coming out."
Had to doublecheck the video.
could the polyhedron at 2:36 be constructed in 3d with holes through the faces?
6:32 "link to the invite in the description"
Steve, you can't do this to us.
Makes me happy to see that algodoo cursor in a video once again
I used to play around in algodoo on my dads pc whem i was like 7 or 8 so this is nostalgia for me
Are those hinged toys that collapse from a large ball to a small one flexible polyhedra, or are they something else?
I love this shape! We made this from paper in an discussion of important math proofs class at uni
4:20 “it’s flexible as long as it doesn’t flex”
Isn't the movement because of the play/tolerance in the hinges? ;)
*me half way thru the video
"well, my brains fried for today".
I have an unsatisfying answer to the flexible degrees of freedom problem: Consider a 16-sided polyhedron which is basically an icosahedron with 5 triangles removed and replaced with a single pentagon. Then, removing that pentagon is the same as an icosahedron missing 5 whole triangles. Which as you demonstrated before is *VERY* wobbly.
It does feel like I'm cheating the question on a technicality though.
My intuition (and I have absolutely no proof to back this up other than "it feels right") is that the degrees of freedom is basically E-3 -- that is a triangle gap is 0 degrees of freedom, a quadrilateral is 1, a pentagon is 2, etc... Would be curious if that's actually true or not.
Wow! Math and engineering getting all kinds a freaky up in here. That was hawt!
Amazing. I wish I had a teacher like you in school.
This brings back memories of playing with those at primary school on rained out lunch breaks when we had to stay inside.
5:45 ! I love this kid's toy ! My children got bored of it pretty quick but me ? Never !
Take two congruent star shapes and place them parallel to each other. Space them apart such that the length of any edge of the star equals the distance between. Now, connect each matching pair of edges between the stars using two identical right triangles, connected on their hypotenuses to form a square.
I don’t have the means to test my theory, but I think if you remove one of the stars, you get a number of degrees of freedom that equals the number of points of the star.
I’m mostly sure that the use of a star-shaped face doesn’t disqualify it as a polyhedron, but I last studied this stuff decades ago….