This weekend I found an “Einstein” tile in 4 dimensions. It’s an aperiodic monotile: a single shape that can fill 4D space, but never in a repeating pattern. As far as I know, it’s the first one in 4D proven with a machine checker (Lean) Here’s what it looks... like! (check 0:24)show more

Code here I also have proved in lean that there are Einstein tiles in Every Dimension d ≥ 2

5 dimension Einstein tiles are impossible to visualize but here is an attempt

In an ubscure tweet, this was already shown! Actually, a friend of mine (@Liam06972452!), is sitting on a preprint since the 25th of september, though I will await his release if ever.

@Liam06972452 Yes! I cite Adam's ideas (and formalized them). There are some new things, eg the 4d case boundary consists of finitely many flat pieces.

@Liam06972452 would be cool to see that preprint, I wonder if there is exact overlap or if different families were found

I love that you are deeper and deeper in the trenches there, sir

there are trenches everywhere for those with the eyes to see

Here's a video that describes what these are

That is so fucking cool dude

I think so too! I think it's romantic A 2D observer living in it never sees its own limit (a whole tile). But by luck some portion will be postponed in a way that their boundary is a key, which compresses the world it lives in. Literally it contains what constructs the thing. And zoomed out the tiles magnify into themselves. So the shape that no shape can know contains a shape that knows it.

I feel so privileged to be able to understand and see this haha Also thanks for sharing the code, will be checking that out later too

Do you think it was easier to find in higher dimensions because there is more surface / degrees of freedom to create keys and locks? Or perhaps there is another aspect that becomes hard as the same time? Would be interesting to assign a cost function / complexity measure and then try to prove and find an example of the minimal complexity under that cost function.

yeah, I think the extra degrees of freedom help with encoding keys and locks. The genesis was that I was playing around with the 3D tile that was discovered a few weeks back and found Sierpiński tetrahedra in the structure. That sparked the question if local geometry force could forcetiles to assemble into larger structures, which then obey the same rules and assemble again, at every scale. The difficult part is forcing that in every possible tiling. Extra dimensions give you more room to encode constraints, but also more orientations and unintended ways things could fit together. So there’s a tradeoff: more freedom to build the mechanism, more freedom you have to constrain. I had like fleets of Astra agents grind away on this idea for like 72 hours straight and it panned out. I like the complexity idea. You’d want to fix the dimension and allowed class of tiles, then measure something like facet count or the length of an exact description. A simple-looking shape can hide lots of information in tiny features or precise coordinates.

holy fucking fuck this is so cool i was trying to find one of these with google gemini back in like 2024

Wow. This is awesome. 🥇

@penrose reminded of Penrose tiles

My man’s gone too deep for his 3D world

hope youre doing well freddie!

you are a man of culture i see

@mtbdesignworks Time for a new level

But the 3D slice is just stacks of regular Einstein tiles? And they are of different heights so how are they monotiles?

Not that I think things like this need an obvious application to be cool, but... is there one? I know 2d tiling can apply to structuring engineering. Does a 4d tiling solution step that up, or apply somewhere else?

Поздравляю с копированием моих наработок, я против 👎 голографических принцип работы ткани реальности

Thought this was an Etheria build lol

@relsunkaev