---
title: 'New shape discovered!'
source: 'https://youtube.com/watch?v=eeVaUNPxXy8'
video_id: 'eeVaUNPxXy8'
date: 2026-07-01
duration_sec: 1390
---

# New shape discovered!

> Source: [New shape discovered!](https://youtube.com/watch?v=eeVaUNPxXy8)

## Summary

This video explains the concept of 'Nopert'—a shape that cannot pass through itself—contrasting it with 'Rupert' shapes, which can. The presenter discusses recent discoveries by Tom 7 and other mathematicians, highlighting the search for a provably Nopert convex polyhedron.

### Key Points

- **Breaking Maths News** [0:00] — New shape discovered that cannot pass through itself, known as a Noperthedron.
- **Personal Nopert** [0:34] — The presenter's preferred Nopert is a pentagon-based polyhedron with 20 vertices and 27 faces, discovered by Tom 7.
- **Rupert Property Explained** [1:25] — A shape is Rupert if one projection fits entirely inside another projection of the same shape, using a cube as an example.
- **Shift in Perspective** [4:09] — From 'amazing that shapes can fit through themselves' to 'surprising that some cannot'—a Nopert is a convex polyhedron that is not Rupert.
- **Tom 7's Research** [4:29] — Tom 7 developed a numerical solver to find Noperts, leading to candidate Nopert 214 with 20 vertices and 27 faces.
- **Historical Context** [5:52] — The Platonic solids all have the Rupert property, but some Archimedean and Catalan solids remain unknown.
- **Conjecture About Convex Solids** [6:16] — A conjecture claimed every convex solid has the Rupert property; Tom 7's work challenges this.
- **Numerical Solver Performance** [7:25] — Tom 7's solver found Rupert solutions for known shapes in milliseconds but could not find any for some shapes after trillions of attempts.
- **Candidate Nopert Generation** [8:28] — By pairing a solver with a random polyhedron generator, Tom 7 discovered many candidate Noperts, with number 214 being the smallest.
- **Prism and Churro Analysis** [9:13] — Tom 7 investigated prisms (manhole covers to churros) and anti-churros, finding continuous solution trends but no Nopert gaps.
- **Unexpected Spikes in Solutions** [11:25] — Even-‘n’ churros show multiple solution spikes, while odd ‘n’ have a classic handover, indicating complex orientation effects.
- **Anti-Churro Complexity** [12:40] — Anti-churros (triangular side faces) show even more complex solution patterns, suggesting much remains unknown.

### Conclusion

While no provably Nopert convex polyhedron has been officially confirmed yet, Tom 7’s work and the independent discovery of a Noperthedron by other mathematicians suggest such shapes exist. The video encourages viewers to watch Tom 7’s full video for deeper understanding.

## Transcript

Breaking Maths News!
Some of you will have seen the headline, "New Shape Found That Cannot Pass Through Itself."
And I know some of you have seen it, because you've all e-mailed me.
I mean what a year to have declared, "The year of breaking maths news."
When this Noperthedron was discovered, everyone started e-mailing me;
two people walked up to me in real life to tell me about it;
one of them had printed out the paper to show me,
so thank you everyone for alerting me to the breaking maths news.
But, I've got some terrible news: this, this is not my Nopert.
This is my Nopert.
[Shopping channel music plays]
I say my Nopert, not as in I discovered it, Tom 7 discovered this.
My Nopert, just as in, this is the one I vote to be Nopert.
I believe in this Nopert.
Look at that, it's got a pentagon top and bottom.
I think, I think it's really neat, check it out.
I, I, um, added the colours myself just to make the faces more obvious.
Twenty vertices, twenty-seven faces, this shape cannot go through itself, probably...
To expand on the word "probably" and "cannot go through itself" we need to go back to the cube.
Which I also 3D printed, as, like, as the greatest misuse of 3D print-time ever.
But, you know, sometimes you just, you just need a really precise cube.
First-up, we're going to talk about what it means to be a Nopert.
Indeed, we're gonna recap what it means to be Rupert.
And some of you will already know this.
I wrote about this in a book back in 2014; the, the idea of being Rupert has been around for a while.
And the question was: Prince Rupert, centuries ago wondered, could you fit a cube through itself?
And so here, I've also 3D printed another copy of the cube, slightly more complicated this one.
And it's exactly the same size, so if I line them up: look at that, boom!
Exactly the same in every dimension, these are identical cubes,
other than this one has a removable piece.
Because we can fit this cube through a cube exactly the same size.
If I remove that section, there, there's a cube shaped hole and so, while it seems counter-intuitive,
it is possible to fit a shape through itself.
Or, at least, that used to be counter-intuitive.
Turns out there's loads of room to fit a cube through a cube.
So if I, very carefully, align this so you can look straight down the, like, the diagonal axis there.
You're looking at the projection of the cube that is a hexagon, that's kind of fun.
And it turns out that hexagon is bigger than that square.
In fact, you could fit a square 3.5% bigger by length through that hexagon.
Oh, if you went off-axis slightly, there's actually another projection that's ever-so-slightly bigger.
And you can fit a cube that's 6% bigger by length through a smaller cube.
So, um-um, the Rupert property is just:
"are there two projections of a polyhedron, convex,
such that one of them can fit entirely inside the other one?"
And it turns out - yeah, if something's not a sphere, if you move it around its projection changes.
Some of the projections are big, some are small, turns out small things can often fit inside big things.
So, I mean for the longest time I was like, "oh, would you believe it: a cube fits through a cube!"
And after a while everyone's like, "oh yeah, actually it's quite obvious that a cube fits through a cube."
A couple of years ago, 2021 I think, I did a video about this where,
for Halloween, I put a pumpkin through a pumpkin.
So I bought the two most identical pumpkins I could find in the shops
and then I cut a hole through one pumpkin that would fit the other pumpkin.
So pumpkins are Rupert, although not convex.
Anyway, the point is the story went from being,
"ooh isn't it amazing that you can fit a cube through itself, that's stunning."
And eventually kind of became, "oh turns out it's quite easy to fit shapes through themselves."
And now it's gone full cycle and everyone's like,
"oh my goodness! Can you believe there are shapes that don't fit through themselves?"
So that's the world we live in now.
I mean a decade after I wrote "Things to Make and Do in the Forth Dimension", the fact has inverted.
So after centuries of showing that shapes are Rupert,
it was only this year that humans managed to prove that there is a convex,
polyhedron that is "not Rupert" or "Nopert".
And I actually saw this one coming because I was in New York to do
"An Evening of Unnecessary Detail" last July.
And I caught up with Tom 7 and he'd already sent me the paper he was working on.
And we talked about how he was trying to research shapes that were Nopert,
and he'd found this one at that point.
And he was working on a way to prove that, definitely.
Now, I'm not gonna lie: this video, you don't need to watch this,
you can go and watch Tom 7's excellent video.
[Tom] So this one is Nopert No. 214, or potential Nopert No. 214.
[Matt] The only issue is, it's an hour and eighteen minutes long.
And I love, I love Tom's videos, they're incredible, you should go and watch that video.
But I know a lot of you aren't gonna watch that video.
And so I thought I would make a Rupert video, 'cos my video is just gonna go through Tom 7's video.
Um, so I'll recap some of it and then you should really go watch the rest.
Er, but to explain the Nopert journey,
Tom 7 first got a bit distracted when he saw an interesting sentence on Wikipedia.
[Tom] I was reading about the dodecahedron, which is my favourite, er, Platonic solid.
Maybe even my favourite solid.
Er, and it came, the Wikipedia page, at least at the time, contained a puzzle.
[Matt] On the Platonic solid page,
Tom had come across the statement that all five Platonic solids have the Rupert property.
And on the page for Archimedean solids, which arguably contains the Platonic solids,
It has the mysterious, puzzling sentence that,
"at least ten of the Archimedean solids have the Rupert property."
Why some? Which got Tom wondering, are there shapes that don't have the Rupert property?
[Tom] And I found a claim, a conjecture, that every convex solid has the Rupert property.
Which would be amazing.
We can do this for eleven of the thirteen Platonic/Archimedean, er, Platonic and Archimedean solids.
But there's like two that we, somehow, didn't check.
[Matt] It is weird that we have shapes of unknown Rupertness that are Archimedean solids or Catalan solids.
After Platonic solids, with their regular faces and vertices, Archimedean solids, they're up there as well;
they've got identical vertices and every face is a regular polygon, but you can mix and match the polygons.
Then you've got the Catalan solids, the duals to the Archimedean solids; they all have identical faces.
These are blockbuster shapes.
And yet there are three Archimedean solids and two Catalan solids for which we just don't know.
And that seems strange.
So Tom 7 put together a numerical solver to take a polyhedron and work out a Rupert solution for it.
And for all the solutions that we already have, he found them in milliseconds, just was churning them out.
And the ones we couldn't find, he couldn't find with billions of attempts, trillions of attempts.
Which means, either the solutions are really obscure or just there aren't any.
[Tom] Sometimes this happens, there's a phase change between, like, two cl-two classes of problem.
And so, mathematically you know this, it's not enough to have that empirical evidence and say,
"well, it can't be solved."
But it is suggestive and it kind of taunts you because y-you kinda know.
So, err, yes that was part of the suspicion, is that it's very easy to solve them.
[Matt] This is a shape that came out of Tom 7's code.
'Cos once he had a solver he could then pair that up with a generator that's producing random polyhedra
and then see if he could produce or evolve a Nopert.
And he found loads of candidate Noperts via this method,
the best of which is candidate number 2 1 4, the 214th Nopert.
It's the best in that its only got twenty vertices and 27 faces
and that's the fewest of all, candidate Noperts out there.
And I like it because it's got this cool, kind-of pentagonal, rotational symmetry.
I think it's a really cool shape.
And Tom 7 tried literally trillions of different possible alignments of this shape with itself
and none of them went through, his solver could not find a solution to this.
And the solver's very good.
Now that doesn't prove that is is Nopert, but it makes it exceedingly likely that it is.
But in maths, we want a proof.
One avenue Tom 7 explored to find a provably Nopert polyhedron
was to look at the family of prisms that go from "manhole cover" to "churros".
[Tom] This really simple case.
So take an n-gon and extrude it.
And if you extrude it a very shallow amount, then you get a "manhole cover", as I call it.
And we know that, you know, "manhole covers" that aren't round can fall through themselves;
it's just, it's proof by idiom.
And then there, er, if you extrude it a lot, then you get a, a "churro", as I call it, which is a long one;
and you can stick that through its own side.
And, possibly, there's a s-, a simple point at which they s-, they don't have solutions.
And if that's the case, then this could be a really good route to
proving the existence of simple Noperts that no one ever thought to check.
[Matt] Here are Tom 7's solutions for the 5-churro.
And on the horizontal axis, we have how long the churro is, and each of these lines is an integer.
So this is a length of 1, or a 'd' of 1, where 'd' is the length of the sides of, in this case, the pentagon.
And, as you can see, as the length goes up to 2, to 3, to 4,
what you have here is the quality of the solutions.
So, sometimes the numerical solver is finding a very high-quality solution.
That means that the residue, what's left after the shape's gone through itself,
has reasonably, ahh, for some definition of "thick" connectors.
Down here, they're not as good, but still connected.
And, because it's a numerical solution, sometimes it finds one of these,
sometimes it finds one of these, depends how it randomly orientates the two shapes
and occasionally they're a little bit better or worse.
Overall you can see two, very clear trends:
it often find the "drain-hole cover" solution here;
Particularly when the churro is very thin, the drain-cover arrangement.
And it sometimes still finds solutions like that when it's much-much longer,
but once it does get long, it's way more likely to come in through the side.
You can see here, these are the churro solutions and they take over.
And Tom 7's theory was maybe there's a gap in between.
But no, they cross here and, in this case exactly (well, pretty much) on one,
and there's a seamless handover.
So there's always a solution.
And you may have noticed this extra spike, here.
That's a different class of solution; that, I don't think it's been investigated yet.
So if you want to go down another maths-hole, you can start to look at this.
And, we can look at the other churros.
If we go up one, to a 6-churro: ahh, hey, look at this!
We get all these extra spikes, we still got the "drain-hole cover" solution here,
we still got the "churro solution" there.
But we now got, I mean in this case, three new bonus ones.
And that's way more pronounced on an even-'n' "churro".
So when we go up to seven, this is much closer to the five,
there's still a spike there but now not as clear.
And we got the classic handover.
Ooh, look, there are other, small, sub-spikes down here, and we went back to five,
you can see they're down there as well; isn't that interesting?
But all the even ones have a whole bunch of spikes.
So eight, oh, it's always half as many, so eight has these four extra spikes here.
The nine looks like the classic odd-case.
And then ten, five spikes appearing here.
So I'm imagining these are when you've got the "drain-hole solution",
these are some orientation effects that we're seeing, but again, I'm purely speculating.
Oh, what's going on down here?
And as we go up to more and more complicated churros, approximating a cylinder,
you can see these patterns continue but start to get, you know, whoa - so messy!
As well as prism churros, Tom 7 also looked at anti-churros.
These are like a prism, but instead of rectangles on the side, you use triangles.
And check out the 5-anti-churro, look at all of this!
There's so many more complex solutions happening down there.
Is this still the "drain-cover solution"?
I mean, we don't know, what's going on here?
There's still, you know, a thing's crossing at about 1, in this case.
The even ones, now, look at this!
Now the fact that this is so covered means, because it's numerical we're not getting exact coverage.
I would say, in theory, this should be block covered;
which means there's a continuous series of solutions for any given 'd'
up to some threshold, which is the top of this.
And just the number of numerical solutions that Tom 7 ran means that we've got this kind of spotty coverage,
but it should be a lot, erm, more thorough.
But you can see there's another one here, we get these extra spikes appearing there.
Look at these: seven, eight, nine.
Ooh, there's a lot we don't know about how you intersect anti-churros.
So, if you want to look into this, go for it!
But the point is, for all our original churros, there's always something going on,
there's no point where it's gonna hit zero.
So, while there's other interesting stuff going on,
we're gonna need a different method to try and find a Nopert.
[Matt] So when you, ooh-err!
[Business Matt] Mind if I borrow this?
[Matt] Yeah, fine...
[Business Matt] Excellent.
I meant the desk.
[Matt] Oh, unbelievable.
[Business Matt] Because this video is brought to you by Bambu Lab,
the 3D printer we used to print all these fantastic objects to explain the Nopert property.
Yes, I use my Bambu Lab X1C printer, which is so quick and easy to use, to print the cube you saw before.
Of course, the Rupertable cube where the middle bit comes out.
And, because it can take multiple filament colours at once,
I could print the Nopert with different colours for each of the faces.
And if you're looking for inspiration of what to print, you can head over to Maker World;
look at all of these fantastic models and, of course, Tom 7 has put up their full set of .pac models.
You can open them up in Bambu Studio, oh look at this!
There are Platonic, and Archimedean and Catalan solids and you can print them all out.
I mean, what home is complete without a full set of these wonderful shapes?
Fun fact, as well as these, I did try to print out the churro example, with a very long pentagonal prism.
And I added in these support structures, because I knew the walls would be so thin.
But I, I totally printed it too small and so there're little tiny gaps there.
But you can, you kind of get the idea, like that will like up there, er, and slide through.
And I had a lot of fun in Bambu Studio
arranging the negative one of these to get the correct hole and add that in.
And it were, it was a fun learning experience, n-not quite good enough to be in the main video, though.
So do check out Bambu Lab printers!
Hey, I am gonna balance these, it will really annoy Regular Matt.
Erm, they are a trouble-free, 3D printing experience and if I can use it to print this, you can too.
Link in the description.
[Matt] Ok, back to the, what happened here?
Oh my goodness, those are fragile.
Back to the video.
Given there are so many ways to arrange a shape in 3D space;
I mean, uncountably, infinitely many ways.
How can we search them all, how would we prove that something is Nopert?
Well, if you look at all the orientations of a shape,
all we really care about are the 2D projection and the convex hull.
Like, the collection of points and edges that are around the outside.
And as I move this around, if I was to move it like a very-very small amount;
the same edges from your point of view, would always be on the outside.
And if I do a big enough movement, one of the edges might pop behind and so you can no longer see it.
And so you could classify every possible projection
by unique sets of which vertices are on the convex hull in a specific order.
[Tom] I was tying to visualise what is the solution-space like here.
And, so what that, erm, what the snub-cube, the textured snub-cube that's in the thumbnail of the video.
What that is, is I take all the different ways of looking at the snub-cube.
And a different way is basically, like, what is the shape of its shadow, erm, topologically?
So which points from the snub-cube are on the shadow and in what order, that convex hull.
And there's, oh I forgot the number,
32 or so different ways of looking at the snub-cube that are fundamentally different.
[Matt] And he coloured in a snub-cube to show that.
There are 36 different colours on this snub-cube and each one,
if you're looking straight at it towards the very centre of the snub-cube,
that's the particular collection of vertices that are around the outside of its 2D projection.
Now I just think it's an amazing visualisation.
And that's a way to kind of tame and contain the collections of possible projections.
So what do we do with these 36 different classifications
of types of arrangements or projections of the snub-cube?
Well Tom 7 realised, if you've got your kind of target "outer" snub-cube and then you've got the moving,
hopefully "inner" snub-cube, you've now got 36^2 pairings.
So that's 1296 ways you can arrange the pair of them together.
And then within that, I mean you can fix one and the other one relative.
You've still got all its orientations and you can translate it.
So, actually there's still seven different variables within each of those 1296 combinations
and you've gotta check all of them, even though they're continuous.
If you've got seven different continuous variables and you want to search that entire space;
you can think of it as a 7D cube.
I mean, thinking about it that way is not helpful.
But if you want to divide it up, it is kind of helpful.
What Tom 7 did was realise if he could take different chunks of 7D volume
and within them cover the entirety of the 7D cube.
He can show that the entire search place does not contain a solution
and each of individual chunks he just had to be able to show,
instead of doing individual points within them, that the entire sub-volume wasn't possible.
It was a very clever way to go about it and if you want it explained much better than I just did,
check out Tom 7's video, link below.
[Tom] The main thing to realise is that the extreme values of the output,
which is what we are trying to compute, will happen at extreme values of the inputs.
But not necessarily in the same order.
If I have the interval 2 to 3 and I multiply it by an interval that's just -1, then I get -3 to -2,
so they get swapped around.
So I basically compute all the end points and do a min and a max over those.
[Matt] Spoiler for Tom 7's video, he was beaten.
When I spoke to Tom in New York, he was still working on it, wasn't aware anyone else was doing it.
Him and his buddies were crunching away to get the whole proof together.
And then I got an e-mail from him in September this year saying, "someone else had found a Nopert."
And they were doing exactly the same style of proof,
by pairing up the different possible orientations and going through it all.
However, whereas Tom 7 was trying to prove that a snub-cube, an already existing famous shape was Nopert.
These other mathematicians, this is perfectly valid,
came up with their own, custom shape that made the proof easier to show that it was Nopert.
And that's the Noperthedron that you saw reported in the media.
And if you read any of the articles, at least the ones I saw, they didn't go into a lot of depth.
Now, you can look up their paper on The ArXiv if you want to go through in excruciating detail what they did.
But, because their technique was so close to what Tom 7 was doing,
if you watch Tom 7's video it's the best explanation I've seen
to how you'd go about proving something like this.
Oh and the reason that their Noperthedron was better,
was they got the space down to a 5D hypercube
and they tackled the issue of the diagonal where you have very similar orientations.
So the shapes coming at each other almost exactly match and that's just very hard to deal with.
Thanks for watching my video that does just go through Tom 7's video.
So I guess in the analogy, this is my video here, and it's gone through Tom 7's video.
But now we've got what's called "the residue", the bit that wasn't part of the whole.
These are all the bits in Tom 7's video that I didn't talk about,
there's some really good stuff in there, I'm gonna link to it.
I know it's long, you should watch it and enjoy it.
Like a lot of maths videos on YouTube, I mean due the nature of maths, can be a bit of lacking in personality.
Let's say, Tom 7's videos have the opposite problem.
If you want to watch a different one, I mean his video about harder drives,
that makes the most sense in terms of audience overlap with people watching this one.
But I like his, his ridiculous chess algorithm video
is the one that got me into his channel in the first place, so I highly recommend that one.
Erm, I'll link to it below.
And please, if someone can prove that this is definitely Nopert, I really want.
For me, I mean I know I said this is "my Nopert",
but I really want this to be the smallest provably-Nopert, convex polyhedron.
So, if we could prove that, that would be incredible.
Er, I will link to everything I've mentioned in this video, er, below.
Er, please do check it out and go watch Tom 7's videos.
Thanks for watching my summary of one of his.
That in there.
It's just a little bit rough on that one edge, there.
Look at that!
Push.
Slides right through.
Woo.
And I'll do a bit of this.