Maths has finally discovered a self-righting tetrahedron!

0:00 [News jingle and music plays] Breaking math news.

0:02 Mathematicians have finally built a tetrahedron, just months ago,

0:05 that they've been theorising about for over 60 years.

0:08 To find out more about it, I tried to build my own (fairly unsuccessfully).

0:13 And of course, we spoke to an expert in the maths field.

0:18 I mean, an actual maths-field.

0:20 For some reason, Matt wanted to film outside

0:23 for this video despite being constantly blinded by the sun.

0:26 [News music stops] In 1966, mathematicians Conway and Guy theorised that both,

0:32 there is a tetrahedron out there which is only stable on two of its faces,

0:39 but there is no tetrahedron that's stable on one face.

0:43 And half of that was answered the very next year when Hungarian mathematician,

0:48 Heppes, came up with this tetrahedron.

0:51 Also, this video, brought to you by Bamboo Lab Printers.

0:54 Yes, all the models in this video were made on my Bamboo Lab X1C printer.

0:59 An incredibly versatile and reliable 3D printer.

1:02 So, if I get this convenient and, now, perfectly level surface, here.

1:08 There are two surfaces on the tetrahedron,

1:12 that one and that one, where it's completely stable.

1:17 However, if I try to balance it anywhere on the other two,

1:21 if I try this one here, and this is not the wind, ready?

1:25 Falls over.

1:26 And if I try this one here, falls over.

1:29 And, quite interestingly, that second one,

1:32 when I balance it on that face, it doesn't fall directly over.

1:35 It actually goes to the other unstable first and then over.

1:40 So, if I balance it there, fall.

1:43 So this is, I guess, a bi-stable tetrahedron and in their paper,

1:49 Heppes defined it with its coordinates of the corners plus or minus epsilon.

1:55 Here is the 1967 paper by Heppes and you

1:58 can see there's just the four coordinates right there.

2:02 with a minus epsilon to define where vertex "A" is.

2:05 Now the rest of the paper is Heppes showing where

2:08 the central mass moves based on the value of epsilon.

2:11 But I wanted to see this thing.

2:13 And so I actually just took these coordinates.

2:15 I assumed an epsylon of 0.1 and I made

2:18 this .stl file just defining the four faces of the tetrahedron.

2:22 That's what I 3D printed.

2:23 And that's okay.

2:24 But to get a better sense of it,

2:27 my friend Sam put the coordinates into a GeoGebra file.

2:29 So there's a tetrahedron.

2:30 There's the center of mass.

2:32 And there's where it projects down.

2:33 And if I turn on the vertex labels, they're the ones as defined by Heppes.

2:39 And the face labels,

2:40 we just give each one the letter of whatever vertex is opposite.

2:43 And you can see initially the center of mass is directly above that edge.

2:47 And that's why we have epsilon to move vertex "A" around.

2:50 So if we move it this way, look at that.

2:53 You can see now the center of mass has moved out.

2:55 It's off the base, which means this is now unstable.

2:58 It's going to tip over from this; face "D" is going

3:01 to lift up and face "A" is going to slam down.

3:03 But where's the center of mass now?

3:05 Oh, look at that.

3:06 still just a little bit out.

3:08 We can move it around more by adjusting epsilon,

3:11 but as long as it's not on the base,

3:13 we'll start to tip and face "A" will lift up and face "C" will slam down.

3:18 Boosh!

3:18 Now, this is a big face.

3:19 The center of mass is directly, look at it, it's just above the face.

3:24 That is not going anywhere.

3:25 So, to recap, face "D" was unstable,

3:28 it will always transition from there to face "A"; that's also unstable,

3:31 it would transition to face "C"- face "C" is stable.

3:35 Now face "B", if we start there, it's just stable.

3:38 So there are two stable faces.

3:40 It's a bi-stable tetrahedron.

3:43 So 1967 we know a two-stable arrangement tetrahedron is possible.

3:49 But what about the monostable tetrahedron [music sting:

3:52 "dun dun duuuun!"] with a single face it always ends up on?

3:55 Well, we can't talk about monostable shapes

3:57 that always end up in the same orientation,

4:00 no matter how you put them down, without talking about the "Gömböc".

4:04 I haven't got a gömböc or Steve Mold.

4:06 Steve Mold has a gömböc.

4:08 Oh, I could ask him to show it to me.

4:10 Ah, the one time I go somewhere nice to film

4:11 and now I got to go inside, phone Steve.

4:15 [Matt] "Steve, you own a gömböc." [Steve] "I do.

4:17 Funnily enough, I have it right here on my desk." [Matt] "Oh,

4:19 that's convenient." [Steve] "Yeah.

4:21 Yeah.

4:21 So, look, the thing about the gömböc is it's got two, well,

4:25 it's got two equilibria, but one is unstable,

4:28 and so it's only got one stable equilibrium,

4:32 which means it doesn't matter how you place it on the table,

4:36 it's always going to end up rolling back to this position.

4:39 So, that's the stable equilibrium position.

4:40 There's an unstable one, right?

4:42 If you put it like that on the edge, it'll balance there.

4:44 But, as soon as you give it any kind of knock,

4:47 it'll end up back in that position." [Matt] "So

4:49 how does this differ from just like a weighted

4:52 sphere that would always end up the same way?"

4:55 [Steve] "So the difference is this is homogeneous throughout.

4:58 So the density is the same all the way through.

5:01 So you could do it like a, a weeble.

5:04 A weeble wobbles but it won't fall down.

5:07 Just to give you that.

5:09 So whereas this it can achieve the same thing.

5:12 It's a weeble except that it's not weighted inside anywhere.

5:15 This is the only object that does that.

5:17 But I mean, I'm not a maths guy.

5:19 I like I don't know.

5:20 Like it's all it's interesting that it's curved.

5:22 Would you think it would be possible if,

5:25 to do it with a shape that's all flat sides?" [Matt] It's not,

5:29 It was proven in 1969.

5:31 You cannot have a monostable tetrahedron, question answered.

5:35 In fact, even in higher dimensions,

5:38 so a higher dimensional tetrahedron is called a simplex.

5:41 All the way up to 8D, there are no monostable simplexes.

5:46 It's all over for shapes that have

5:50 a delicious homogeneous filling like the gömböc.

5:54 But if, instead of a homogeneous filling,

5:57 if the weight was unevenly distributed, suddenly it is possible.

6:02 And, apparently, we believe in 1984 Conway, John Conway,

6:07 had a sketched out sort of proof that it

6:12 would be possible to make a non-homogeneous monostable tetrahedron.

6:17 [SUM sting] Richard Guy had a conversation about

6:21 this with another mathematician in the 80s at Cambridge University,

6:24 named Robert Dawson.

6:26 and Robert Dawson tried to build with some of his friends at Cambridge,

6:32 this non-homogeneous monostable tetrahedron.

6:34 It was built out of lead and bits of bamboo.

6:38 As far as we're aware, this is the first and only previous attempt

6:43 before this year to make one of these.

6:46 And if you read the recent paper that just came out a couple months ago,

6:50 that Robert Dawson is also involved with, it seems

6:52 that that model from the 80s is long lost.

6:55 That's the only other attempt and we no longer have it.

6:58 Or so we thought until a couple weeks ago,

7:01 where it turns out it does still exist.

7:04 It's been looked after by, unbelievably, a friend of mine named Colin Wright.

7:09 He still got it.

7:11 And oh, I could give him a call.

7:14 He can show it to me.

7:16 Uh back inside.

7:17 [Matt] "So Colin, you have the missing model." [Colin] "Well,

7:20 I have a missing model.

7:22 I've spoken Yeah, I've spoken with Bob Dawson.

7:25 Um, one of the authors on the paper,

7:28 and his memory is that there were actually two models.

7:31 Uh so I have one of them, which I have very cleverly put down somewhere.

7:35 Here we go.

7:35 And I have a milk bottle..." [Matt] "It's

7:37 in a milk bottle." [Colin] "with one of the models.

7:39 And if I tip it out, let me just, I'll hold it up to the camera here.

7:43 I don't know how well you can see that cuz

7:45 it's a it's a spindly little thing." [Matt] "It's

7:48 tiny." [Colin] "Yeah..." [Matt] "So it's a tetrahedron

7:50 in that it's got three edges and a vertex." [Colin] "Yeah.

7:55 So we, we created the vertex on the top and then

7:59 three arms which are made out of bamboo, splints of bamboo,

8:03 and then the other three edges on the bottom

8:06 and the, and the interior are entirely missing." [Matt] "Right." [Colin]

8:10 "So it's an implicit tetrahedron to, to reduce the weight

8:13 as much as we could and it doesn't quite work.

8:16 I mean, this is one of the reasons why, at the time,

8:19 we made the model, we thought this could be made to work,

8:23 but we need something that's got a greater density to be the weight at the top,

8:27 and we need to get the angles slightly better.

8:29 So, the angles weren't quite right.

8:31 Um, but if I put this down, uh,

8:34 on a surface here and I bend the camera down," [Matt] "Oh,

8:38 yes." [Colin] "then you may be able to see the little spider-like thing

8:42 that's sitting there.." [Matt] "Yep." [Colin] "And then if I let that go,

8:48 you'll see that apparently it rolls over" [Matt] "Oh!" [Colin] "like that.

8:53 And there..." [Matt] "Yeah." [Colin] "you go.

8:55 Now the thing is..." [Matt {simultaneously}]

8:56 "always end up face," [Colin] "Right.

8:58 So, so it's only stable, it's only, It.

9:00 Well, it, it should be only stable on that face." [Matt]

9:04 "Got it." [Colin] "But what happens is the model doesn't quite work.

9:07 So, uh, when I release it from here,

9:10 it rolls from, call this 'Face A", that it's resting on now.

9:14 It rolls from A to B to C to D.

9:17 Uh, so it rolls from A to B to C to D.

9:22 But the model doesn't quite work.

9:25 Uh, so what actually happens is, if I allow it to roll slowly,

9:31 it actually rests on face B as well.

9:34 So..." [Matt] "Right..." [Colin] "The angle here.

9:37 Oh, no.

9:37 It's very." [Matt] "There it goes" [laughs] [Colin] "Yeah.

9:41 The, the angle, the angle here, uh,

9:44 is almost exactly 90° and it needs to be more than that.

9:49 It needs to be an obtuse angle with the mass outside of it.

9:54 So if I just help that over there, which is a bit tricky to do,

9:58 but if I just help that over there, then it goes and it does the final one.

10:02 But what happens is, uh, if I release it from the initial position,

10:06 it's got enough speed as it comes over here.

10:09 To carry it over onto the next face

10:12 and then onto the next face." [Matt VO] "Interestingly,

10:15 what Colin just showed me from the 1980s bamboo model

10:18 is a system where it starts on face A, that's unstable.

10:23 So, it goes to face B, also unstable, tips to face C,

10:27 face C, unstable, all the way to face D.

10:30 That's stable.

10:31 We have a monostable tetrahedron and that is what we now call a "type two",

10:37 which as you may have noticed, implies the existence of a "type one",

10:41 because what they had in the 80s was not the only option.

10:44 Imagine a tetrahedron that starts on face A that's unstable

10:48 goes to B that's unstable onto D which is stable.

10:52 But what about face C?

10:53 Turns out face C unstable also goes to D.

10:56 So you don't have a chain going from one to the other to the other.

11:00 There's just two different ways to get to the one stable face.

11:04 And the big breakthrough, 40 years later, was when Gergő, Robert, Gábor,

11:09 and Krisztina realised you can actually build

11:13 both of those different types of monostable tetrahedron.

11:17 This fantastic paper, that came out a year ago.

11:20 So, this was a big breakthrough that did not hit the news.

11:23 It was only when someone made one that it did.

11:27 And spoiler alert, they made a type one in this fantastic GeoGebra file,

11:31 also made by my friend Sam.

11:33 She's really helped me out on this.

11:35 Now, here, oh, ignore these extra lines.

11:37 They're the supports that were eventually built in the physical one.

11:40 So, spoiler, but if this was a homogeneous tetrahedron,

11:43 that's where the central mass would be.

11:45 And you can see on face A, perfectly stable.

11:48 Look at that.

11:48 Right above the base.

11:49 But this dark region, that's a different density region.

11:52 And if that was sufficiently more dense compared to the rest of it,

11:56 you can see here there's its centrid,

11:59 the center of mass, and it's just above outside the base.

12:02 So in theory, this means by making

12:04 one region a different density to another region,

12:07 we can kind of pick any arbitrary location inside

12:10 the tetrahedron and move the center of mass there.

12:12 If it just happens to be right in there,

12:14 that moves it far enough off that we can tip from face A to face D.

12:19 Now in this case face D is the only stable face and actually

12:24 B and C both tip onto A and then A tips onto D.

12:28 So it is a type one and this is the one

12:31 that was most likely to be able to be made.

12:33 You just needed to have one region here that was

12:35 a different density compared to the rest of it.

12:37 The only problem now is the ratio between the less dense part

12:41 of the tetrahedron and the dense part has to be at least a thousand times.

12:47 It's like three orders of magnitude change

12:50 in density for this tetrahedron to work.

12:52 So I thought even though I won't succeed, I'm going to give it a go.

12:58 So I have 3D printed this tetrahedron.

13:00 I had to print it in two different sections and then I've glued them together.

13:05 To make this bit heavier, I paused the print where I'd left a cavity.

13:10 Put in some metal washes, like not a thousand to one,

13:13 but it's something, and then continued the print.

13:16 And so, while this won't properly right itself,

13:19 it's still a completely adequate teaching aid.

13:21 So, I can show you that if I put the heavy face down flat,

13:26 that is definitely stable.

13:27 Now...

13:27 [Business Matt] "Actually, while you're speaking about 3D printing,

13:30 it's me, Business Matt.

13:31 This video is sponsored by Bamboo Lab.

13:34 They make the 3D printer we used

13:36 to make that tetrahedron." [Matt] "Business Matt.

13:38 Are you allowed outside?" [Business Matt] "There's

13:40 a very good point and particularly when it's windy.

13:43 It turns out I can be outside.

13:46 But the longer we sit here,

13:48 the angrier camera-person Alex gets." [Matt] "Although if they're already angry,

13:52 would you like me to just pass this to you?" [Business Matt] "Yeah,

13:55 why not?" I'm I'm sure Alex will make it work.

13:59 Thanks!

14:01 I was able to take the coordinates from the maths paper,

14:04 recreate it in Bamboo Lab,

14:06 split it very easily into two parts that I could later glue together,

14:10 and I could design the print with a hollow void

14:13 and then set it to stop printing at that point.

14:16 So, I could put in the metal washes.

14:18 I could not have made this tetrahedron, which almost works,

14:21 or this video if it wasn't for my Bamboo Lab printer.

14:25 If you already have a 3D printer and you

14:27 want to find some fun maths things to print,

14:29 without getting coordinates from a maths paper and turning them into an .stl,

14:33 which I do recommend, you can head on over to Maker World

14:36 where there's a huge range of models available.

14:39 And I always like to print something someone has requested,

14:42 which this time was from video producer Alienne,

14:45 who wanted two of these phenomenal knot prints to use as fashion accessories.

14:50 Oh, good.

14:50 You you've got it back.

14:52 That simplifies things.

14:53 Anyway, link in the description below." [60's

14:55 Batman TV Series scene transition music] [Matt] Huh?

14:57 Okay.

14:57 So, you've got the heavy part here.

15:00 If I put that down first, that is definitely stable.

15:04 All three other faces should be unstable.

15:06 And you can tell if like that, you can see if I line it up,

15:10 this just sticks out.

15:11 If that was heavy enough, it would cause it to flip up like that.

15:16 And likewise here, if that was heavy enough,

15:19 whoop, it would cause it to tip that way.

15:22 So, while this doesn't actually work completely properly,

15:25 having it slightly weighted meant that I could have a play with it and kind

15:30 of get a sense of the overall geometry and why in theory it could work.

15:36 Although, it was only in theory until very recently.

15:41 Building this thing to actually work was an incredible feat,

15:44 and I really wanted to see it for myself.

15:48 And the lead mathematician, Gábor,

15:50 said that he'd be very happy to jump on a call with me and show me.

15:56 Oh, here we go again.

15:58 At last, I had a phone call with Gábor and his student,

16:02 Gergő, at the Budapest University of Technology and Economics,

16:06 who started off by explaining just how precise and difficult

16:11 the engineering behind this model was proving to be.

16:15 [Gábor] "And to design this thing you needed,

16:19 uh, kind of theoretical insight which we had.

16:24 We needed some geometric ideas which we added.

16:27 You needed engineering ideas how to design the objects and that we had.

16:33 We are both engineers and but you also needed technological insight.

16:38 There are several technological tricks which we could never have

16:45 designed and the result is certainly depending also on the technology.

16:50 It's not something we are happy to disclose any part of this publicly

16:56 but I still would not recommended to try." [Matt VO] "Along the way,

17:00 they showed me how their research could

17:02 be applied to future space mission landers.

17:04 And I took the opportunity to show them what I had made." [Matt] "Here's,

17:09 this is your tetrahedron.

17:11 [Gábor] "Okay." [Matt] "But I 3D printed it in two bits.

17:15 [Gábor] "Yeah." [Matt] "And then I paused the print

17:19 and I put a whole bunch of metal just in here.

17:23 So that's the bit that should be a lot heavier.

17:27 Obviously, the center of mass is still way-way in the middle.

17:30 It's..." [Gábor] "How does it behave?" [Matt] "Almost good enough.

17:34 So, if you give it a bump, a little nudge, it'll do it.

17:39 Or if it's got some momentum, but you can you can put it down

17:44 and it will just sit there in place." [Gábor] "Matt.

17:47 Can I can I say something?" [Matt]

17:50 [Laughs] "Yes." [Gábor] "So I think that building

17:54 the actual tetrahedron was a big step

17:57 towards making it accessible to many people.

18:00 But another huge step is what you have done." [Matt] "Yeah." [Gábor] "Yes.

18:06 Yes.

18:06 This is this is an enormous step towards many

18:10 people because now they will see what is the difficulty,

18:13 because this is not a bad attempt.

18:15 This is a kind of a professional homemade stuff.

18:18 Yes." [Matt] "Thank you." [Gergő] "Yeah, that's true" [Gábor] So this is,

18:22 this elevates the whole story to a different, yeah.

18:24 Yeah.

18:25 Yeah.

18:25 Communication wise it is, uh,

18:27 it is in a different league now." [Matt VO] "But then finally it was

18:30 time to see the actual model they

18:33 had built." [Matt] "Do you have the tetrahedron,

18:35 the engineered?..." [Gábor] "I have a piece of good news." [Matt] "Good,

18:41 good-good-good." [Gábor] "Of course." [Matt] "Look at this, it's got

18:46 a special box." [Gábor] "And, um..." [Matt] "Oh wow." [Gergő] "So,

18:51 would you like to see it tip?" [Matt] "Yes, please." [Gábor] "I think Okay.

18:57 So, we can do this uh tipping to the stable

19:03 side." [Matt] "Yep." [Gergő] "Uh, from this place.

19:09 Oh, so it does one tipping and it's

19:13 the most." [Matt] "One more to go." [Gergő] "Yeah.

19:17 Yeah.

19:17 Yeah.

19:18 The most interesting one from the biggest opposite from the biggest face.

19:25 Uh, how can I do it?

19:28 So you see when it does two tips subsequently." [Matt] "One-two,

19:33 very fast." [Gergő] "Yeah.

19:35 Yeah.

19:35 Yeah.

19:36 Yeah.

19:36 It works very fast.

19:38 Uh, which is a little bit deceiving.

19:41 So it suggest that that it's easy to do but that's not the case.

19:46 So, um." [Matt] "I thought this one..." [Gergő] "if

19:49 it's working then it is working fast but if it,

19:53 um, err." [Gábor] "If it doesn't work..." [Gergő]

19:55 "if it doesn't work it very hasn't very fast.

19:58 So yeah so it either works very fast or..." [Matt]

20:01 "Or not at all." [Gergő {simultaneously}] "in no way in this gravity.

20:06 Yes." [Matt] "Can you put it back on the big side and turn it

20:09 around so I can see where the center of mass is on that plate.

20:13 So is that just off the base?" [Gergő] "Well,

20:17 the center of gravity is somewhere around here, I guess." [Matt] "Yeah.

20:22 And that's just, just above, just off there..." [Gergő] "Yeah,

20:27 just off the the big face.

20:29 Yes." [Matt] "Oh, wow." [Gergő] "Some, yeah,

20:32 that's the trick.." [Matt] "That's the bit

20:36 mine doesn't do." [Gábor] Just one comment,

20:39 which is probably obvious to you that the more faces a polyhedron has the easier

20:46 it is to make it monostable and this is

20:50 the most difficult problem in this category.

20:52 If you can do it with four faces you can do

20:55 it with any number of faces and it was a mathematical challenge.

21:00 It was a historical challenge which we owe to Conway.

21:04 It was a good topic for his diploma.

21:07 I think it's a lot of fun for many people.

21:11 People are, people just love it and and it does

21:15 connect to the space industry but not in a trivial way.

21:20 So this this object for example doesn't look

21:24 monostable" [Matt] "No." [Gábor] "But it is monostable.

21:26 And whether it can be made monostable it's a delicate matter of these angles.

21:32 So we even think that if you can take a lunar lander and design it

21:39 so that it is not omnistable but it is stable only on two or three faces,

21:45 you already win because you can design it

21:48 the antennas you can design it for just three

21:52 cases and not for 10 cases." [Matt VO] "So

21:55 given the model that was eventually made was type-one,

21:57 I did also ask if a type-two would be technically possible." [Gábor] "Now

22:03 our intuition and computations show that if you want to build that version,

22:10 you need a core which is heavier than

22:13 the core of the sun." [Matt] "So there you are.

22:16 I guess doesn't matter how good the bamboo and lead was.

22:18 It seems the type 2 model made by Colin

22:21 and Robert back in the 80s never could have worked.

22:24 So that is the non-homogeneous, monostable tetrahedron made real.

22:30 Will that be the biggest maths news of 2025?

22:34 We'll find out.

22:35 If anything else happens, send me an email.

22:37 I'll look into it.

22:38 So, thank you for watching the video.

22:40 If you're wondering where I've been this entire time,

22:43 this is a landscape sculpture called "The Cells of Life".

22:47 It's at Jupiter Artland,

22:48 a wonderful site with loads of art, just outside Edinburgh.

22:52 I've always wanted to come here and I thought filming

22:55 this video would be a fantastic excuse to come and visit,

22:58 uh, in a tax-deductible manner.

23:00 I'll have a link to it below.

23:03 I'll also link to all the .stl files

23:05 I use for printing things out in this video.

23:08 Thanks for watching.

23:09 Bye.