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…
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…
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"
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 hig…
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.
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].
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 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.