Which hypercube unfoldings tile space? | Hacker News Reader

Which hypercube unfoldings tile space?

whuts.org

95 pointskvnhn33 comments

This was a really interesting project.

1. The community solved it really fast, within a day or three, but a single programmer solved it even faster [1]. You could also argue that it was the month or so [2] making the website that led to the solution.

2. It was really common to use Minecraft to visualize the solution [3]. I think this speaks to the benefit of tools that make it very easy to manipulate and visualize a system [4].

[1]: https://mathoverflow.net/questions/199097/which-unfoldings-o...

[2] https://twitter.com/oliverdunk_/status/1393366708652548114

[3] https://twitter.com/standupmaths/status/1393516840232624133

[4] Just... any Brett Victor video. e.g. http://worrydream.com/MediaForThinkingTheUnthinkable/

It's interesting that Penrose tilings are also using several copies of hypercubes that, when projected to 2D, tile the space, aperiodically. [0]

I always wondered if one could do something to tile cones aperiodically.

[0]: https://www.gregegan.net/APPLETS/12/12.html

I just watched this video yesterday, and wow so surprised that they all tile! Not a math person at all (aside from a casual interest), but there was something about this that just made me intuit that it wouldn't always work.

I also love that the community was asked to solve it and it happened so fast!

Once, while at a hockey game in Colorado I had this idea that I wish the stadium could be filled with people all thinking about the same problem and trying to solve it. This result is astounding, and shows the power of mobilizing many people to think.

I suspect the trick is harnessing them in the right configuration. Otherwise you've got the old (n^2 – n) / 2 problem.

I knew it had to be this. Number of edges in complete graph of order n, K_n.

https://oeis.org/A000217

There's no way for everyone to talk to everyone. The solution I came up with at the time was to have a rudimentary chat bot connect parts of conversations intelligently. Now, of course just throw an AI at it.

A well designed website with different conversations going on in different places might work almost as well. This site does an adequate job actually.

That's an intriguing approach (AI summaries).

I've got an intense dislike for the scrum-of-scrum approach. If you've ever happened to read A Fire Upon The Deep there's a cool alien species in there that has, essentially, a configurable brain topology. Interesting musings there on how those configurations modify their thinking.

I enjoy Vernor Vinge's books. Intelligence augmentation plays a role in many of his books and short stories.

I have recently become aware that the brain topology of the Tines may be more similar to our own than I thought.

A brain structure mediates the communication between our hemispheres, but just like individuals in a Tine pack, they process things independently and contribute their specific value to the whole. I think it is possible that they even have their own personalities and moods.

People having a conversation use language to collaboratively create shared meaning between all the lobes available to them.

many folding solutions are Minecraft screenshots.

e.g. https://whuts.org/unfolding/124

We are living in a very interesting time

What other tools are good for this sort of representation? Geogebra??

There are a number of examples using Blender. Finally the default cube is useful.

I've only done a couple of things with Blender, if that's the next option I could understand why one might use Minecraft; blender's learning curve seemed pretty steep.

I was thinking Mathematica or similar might have more useful 3D shape representations?

On the topic of 4D and tiling stuff, the 24-cell is a platonic solid that tiles 4D space, with some interesting cross sections that tile 3D space:

https://en.wikipedia.org/wiki/24-cell_honeycomb#Cross-sectio...

There is no 3D equivalent of the 24-cell, but two of the cross sections (rhombic dodecahedron and bitruncated cube) hint at what that "missing" platonic solid would look like.

Its so hard to grasp that in all the pictures its basically the same object, just viewed from different angle.

It’s (a little) easier when angles are not discretely jumping. For a 2d person it would also be hard to get the idea of a drill bit from few random cross sections.

Btw, this drill bit example (basically a twisted ribbon) made me think that we could represent not only straight sections, but also rotating ones, to get a better feel of what goes on.

Here's a visualisation that helped me think about the 4-cube:

Cut the 3-cube with a plane which is diagonal to all axes, eg x + y + z = c. Start at a corner and take sequential sections. First you get a small equilateral triangle, then a bigger and bigger one, until the cut goes between 3 vertices of the cube. Next you get truncated equilateral triangles, with bigger and bigger truncations. In the center of the cube the size of the truncations matches the remaining edges and you get an regular hexagon. Then the whole thing in reverse as half the sides get smaller and smaller, until you're back to triangles.

If you're not sure what it looks like at any point, you can easily solve the intersection of x + y + z = c and the equation of one face of the cube (x or y or z = 0 or 1).

Now do it for the 4-cube. Important observations: 1. again you can solve algebraically, either for the 3-cubes which bound the 4-cube or the 2-squares which bound them 2. you can also just try and imagine the intersection with the 3-cubes, since it will be one of the shapes you thought about in the previous exercise (x + y + z + w = c && w = 1 => x + y + z = c - 1) 3. c goes between 0 and 4, with [0, 2] symmetrical to [2, 4]. There are two 'regions' of behavior, c in [0, 1] and c in [1, 2], with the type of shape only changing when the plane intersects with vertices.

It's possible to view arbitrary slices of the 24-cell using the demo version of Stella4D, but I recall it being very difficult to find the right combination of menu items.

It looks like many of these show a construction which you have to follow the details of to check that there aren't gaps or overlaps.

Is there a way of checking these automatically? Eg if you can tile a certain amount of space without gaps then it must be able to continue forever? Or can you write down a vector expression for the location of each shape and show finitely that you have exactly covered all lattice points?

I think the "answer to q2" here (https://mathoverflow.net/questions/199097/which-unfoldings-o...) is what you want!

It does seem like there's an underlying combinatorial nature to all of these proofs that might make them amenable to a computer based proof. A brute force approach might even work here since everything is based on a lattice. Enumerate amalgamations with a bounded # of pieces, check for tile-ability either by brute force or some clever trick. Rinse and repeat on a breadth first search basis across all shapes, and eventually you'll either prove that all shapes work or the program will run for an unbounded amount of time and you'll prove that "no tiling exists consisting of less than X pieces for this shape Y"

If all 261 unfoldings tile space, it makes me wonder if all arbitrary collections of 8 connected cubes do as well.

Is there an example of 8 cubes that are proven to not tile space?

I think a ring of cubes wouldn't:

    xxx
    x x
    xxx

To fill the hole they'd need to interlock, and I think that might make it too hard to tile

Wow, when the video came out only what, 11 were solved? And it looks like all were found.

So it turns out to be all 261 of them? Suspicious.

Yes; I wonder if the same result applies in even higher dimensions? Do all nets of the 5-cube tile 4-space, etc.

There is a proof on WHUTS that all 261 nets of the 4-cube can, but is a proof by exhaustion (i.e. try every net in an automated manner and see if it is possible), this may be possible to do for the 5-cube nets (9694 possibilities), and 6-cube nets (502110 possibilities), but starts to look difficult in higher dimensions.

Do all the 2-nets of the 3-cube tile the plane?

They do!

I haven't been able to find a good diagram of it in a few minutes of searching, but there is a video at: https://etudes.ru/etudes/cubic-parquet/

I can imagine it might be a fun task for a school maths class to try and find all the tilings; you can find the nets of the cube here: https://en.wikipedia.org/wiki/Net_(polyhedron)#/media/File:T...

It appears that someone has posted this as a MO question: https://mathoverflow.net/questions/392890/which-unfoldings-o...

As of yet (17.05.2021) there is not an accepted answer.

It would be really nice if we could see a reason (and hence generic proof) why this works. Even more so if you could say why (or why not) the n-cube's (n-1) nets can tile R^(n-1).

I'd like to set a notification for when some hero of future-geometry solves this in ten years.

Why is it suspicious? "All of them" is a common enough answer to interesting maths problems.

Dang, I just watched the video and was going to contribute. Blink and you miss it these days...

What about combining the tiles?

There is at least one way to tile 3D space using two different nets -

The Math overflow answer which provides tilings for all possible nets (https://mathoverflow.net/questions/199097/which-unfoldings-o...) gives two which fit exactly into a 4 by 4 by 2 box - https://mo271.github.io/mo/198722/tilings/plots/72.html and https://mo271.github.io/mo/198722/tilings/plots/159.html. Since a 4 by 3 by 2 box can fill 3D space on it's own, you could alternate boxes using different nets.

It's a bit of a cheat though. I would strongly suspect that there are better ways to do it!

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