The object we thought was impossible 

* 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

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

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

I always love it when you and Matt pop up in each other's videos :D

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.

Weird flex, but ok.

I remember making "hexa-flexagons" in school. They're technically six tetrahedrons attached to each other, but are pretty fun to play with.

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.

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.

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.

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 have to wonder what Euler's reaction would be if you took this back through time and showed it to him.

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.

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.

What are those toys called?

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.

6:34 *J O I N U S*

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.

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.

This reminded me of cyclohexane. Used to image how it can have various shapes (conformations).

"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"'

Matemathicians: "This is Impossible!" Guy with a 3D Printer: "Are you challenging me?"

How can you be sure the flexing isn't some kind of additive result of all the gaps in the hinges?

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!

I guess Euler wasn’t so smart after all

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.

Weird flex but ok.

This was definitely quite a head scratcher indeed. Flexible polyhedron 3D printed house when?

I appreciate that this is approachable and clear without in any way dumbing down the math or avoiding terminology.

I don't know, but did Euler only consider convex polyhedra to be polyhedra? What was the definition of a polyhedron at his time?

0:14 I used to play with larger versions of these back in school in the late 80s.

This immediately made me wonder whether we could synthesize organic compounds with such structure and whether they would have aby unusual properties

Thank Phineas & Ferb for discovering this thing that doesn't exist?

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 never thought that was impossible. I never knew it existed and I believe it does now.

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.

Rest of the World: Oh look! Might be a room temp/pressure supraconductor. Steeve: How weird are these solids you ask? 😂

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 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?

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.

The shape in geometry test :

Throwing shade at Matt Parker's card tricks, delightful

We had those exact same plastic shapes in primary school. Thanks for digging up a nice memory Steve!

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.

LMFAO @ the cut to Matt doing bad sleight of hand. That was really good 😂

Whenever I've had an overdose of random YT shorts, I return to this channel to regain some brain cells.

Are you sure that the flexing is not due to the mechanical backlash?

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.

"A mathematician's bad sleight of hand" gave me quite a chuckle.

I wonder how much the manufacturing tolerances play into this

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 😂

Someones upgraded their talking to camera set up, very nice.

Does it flex, because of material flex though, or is it genuinely moveable, JUST at the hinges?

3:19 Anyone who's made a waterbomb base with origami can feel that...

This is a great video. Thank you for making it!

Yay, Matt easter-egg!

I can’t help but watch your videos every time one pops up. It’s just too intellectually stimulating. It’s like brain candy.

Remember that videogames use Triangles. So this geometry could revolutionize physics simulation in videogames down the line

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!

I've never gotten to one of your videos this early before!

06:34 - Steve doing a Futurama 'Hypno Toad' 🤔😏😉 🤣🤣 😎🇬🇧

12 seconds in, damn good quality already!

I love the chain fountain standing in the background like a trophy

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.

matt parker cameo pulling the parker trick, enlightening

Aliens must be looking at us like we're babies playing with blocks and just not quite getting it yet.

I Love this channel. I also love robust "Description" sections on RU-vid 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.) .

Wow I've never been so early

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.

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.

2:47 A monster with only 18 faces? If only they'd discovered 3 more faces! Lives could have been saved.

"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 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).

I love how @stevemould look and vibe is that he just physically finished wrestling a math problem and won.

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.

Wow you just solved a problem we never knew existed and probably would have never known in our life.

The strangest part of this is Ivan printing in a color other than red.

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.

mould conjecture sounding as good as a parker square

Thank you for existing, Steve Mould

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”

Used to be, most kids were very familiar with coplanar surfaces and compliant structures from a very young age. Remember the folded-paper toy called the "cootie catcher," as well as other names?

I can always count on Steve Mould to find interesting toys I never knew I needed.

3:40 i recognize that cursor, Algodoo?

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.

Where the heck are the 3d models for those toys? I need them immediately for my granddaughters. Going to follow the channel you mentioned.

That jumpscare from my childhood tensegrity toy delighted me! I always know I liked that thing- but never because it involved cool maths!

How does an "infinitesimally flexible joint" differ from a "compliant mechanism"? Or does it?

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.

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

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.

4:15 the most important bit in the video

Love the Parker Square sleight of hand insert in this

Can you make a video about the Császár polyhedron? It's not well-knownn outside of Hungary.

I tried so hard as a kid to make a shape that would do move and never found one

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.