* 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.

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.

Weird flex, but ok.

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.

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.

4:21 Ah, yes, The Parker Card Trick!

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.

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.

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.

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.

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.

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.

Weird flex but ok.

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

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

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.

"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"'

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!

What are those toys called?

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.

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

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!

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

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

I guess Euler wasn’t so smart after all

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

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.

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

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!

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

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

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

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.

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

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

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

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!

The shape in geometry test :

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

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

Fun fact, the test for a structure to be not infinitesimally flexible (isostatic or iperstatic) is at the base of all structural mechanics jobs

Yay, Matt easter-egg!

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?

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

"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 wonder how much the manufacturing tolerances play into this

4:18 😂

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

matt parker cameo pulling the parker trick, enlightening

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

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

6:41 feels a little bit like Stephen Hawking in Futurama - "I almost fell into that freezer" - "I call it the Hawking chamber"

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.

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

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

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.

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

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.

12 seconds in, damn good quality already!

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.

Throwing shade at Matt Parker's card tricks, delightful

Thank you for existing, Steve Mould

Wow I've never been so early

sprite

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

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

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

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

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

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!

Polyhedron: **literally flexes and moves air in real world** mathematicians: “nope, not flexible”

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.

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 love the chain fountain standing in the background like a trophy

mould conjecture sounding as good as a parker square

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

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.

"...a mathmatician's bad slight of hand....". 😂😂 Poor Matt, great oblivious cameo 😅

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

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.

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.

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.

I love that you used Matt as a silent 1 second punchline

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.

It's awful that we have to pay for such a thing as incogni. Our data was SOLD, someone profited from it, and to take it back WE have to pay. It's like having to buy back a stolen car from a thief. Where are the police in such matters? I guess it's our fault for not reading all the terms and conditions when signing up to things, it still feels wrong though.

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

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

I Love this channel. I also love robust "Description" sections on KZfaq 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.) .

Matt Parker's dad magic is pure gold

I’m a first grade teacher and I have polydrons in my classroom for exploration, play and 3D math skills! I can’t wait to explore them more with my students!