This was a really interesting project.<p>1. The community solved it really fast, within a day or three, but a single programmer solved it <i>even faster</i> [1]. You could also argue that it was the month or so [2] making the website that led to the solution.<p>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].<p>[1]: <a href="https://mathoverflow.net/questions/199097/which-unfoldings-of-the-hypercube-tile-3-space-how-to-check-for-isometric-space/392828#392828" rel="nofollow">https://mathoverflow.net/questions/199097/which-unfoldings-o...</a><p>[2] <a href="https://twitter.com/oliverdunk_/status/1393366708652548114" rel="nofollow">https://twitter.com/oliverdunk_/status/1393366708652548114</a><p>[3] <a href="https://twitter.com/standupmaths/status/1393516840232624133" rel="nofollow">https://twitter.com/standupmaths/status/1393516840232624133</a><p>[4] Just... any Brett Victor video. e.g. <a href="http://worrydream.com/MediaForThinkingTheUnthinkable/" rel="nofollow">http://worrydream.com/MediaForThinkingTheUnthinkable/</a>
It's interesting that Penrose tilings are also using several copies of hypercubes that, when projected to 2D, tile the space, aperiodically. [0]<p>I always wondered if one could do something to tile cones aperiodically.<p>[0]: <a href="https://www.gregegan.net/APPLETS/12/12.html" rel="nofollow">https://www.gregegan.net/APPLETS/12/12.html</a>
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.<p>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.
many folding solutions are Minecraft screenshots.<p>e.g. <a href="https://whuts.org/unfolding/124" rel="nofollow">https://whuts.org/unfolding/124</a><p>We are living in a very interesting time
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:<p><a href="https://en.wikipedia.org/wiki/24-cell_honeycomb#Cross-sections" rel="nofollow">https://en.wikipedia.org/wiki/24-cell_honeycomb#Cross-sectio...</a><p>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.
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.<p>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?
If all 261 unfoldings tile space, it makes me wonder if all arbitrary collections of 8 connected cubes do as well.<p>Is there an example of 8 cubes that are proven to <i>not</i> tile space?