Which hypercube unfoldings tile space?
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Re: Which hypercube unfoldings tile space?
#2Once, 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.
Re: Which hypercube unfoldings tile space?
#3I 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!
Re: Which hypercube unfoldings tile space?
#4many folding solutions are Minecraft screenshots.
e.g. https://whuts.org/unfolding/124
We are living in a very interesting time
Re: Which hypercube unfoldings tile space?
#5On 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.
Re: Which hypercube unfoldings tile space?
#6So it turns out to be all 261 of them? Suspicious.
Re: Which hypercube unfoldings tile space?
#7On 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.
Re: Which hypercube unfoldings tile space?
#8Once, 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.
Re: Which hypercube unfoldings tile space?
#9So 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.
Re: Which hypercube unfoldings tile space?
#10On 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.