Dang, I just watched the video and was going to contribute. Blink and you miss it these days...
Which hypercube unfoldings tile space?
11–20 of 34 posts
Re: Which hypercube unfoldings tile space?
#12Wow, when the video came out only what, 11 were solved? And it looks like all were found.
Re: Which hypercube unfoldings tile space?
#13So it turns out to be all 261 of them? Suspicious.
Why is it suspicious? "All of them" is a common enough answer to interesting maths problems.
Re: Which hypercube unfoldings tile space?
#14It'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.
Re: Which hypercube unfoldings tile space?
#15So 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…
Do all the 2-nets of the 3-cube tile the plane?
Re: Which hypercube unfoldings tile space?
#16On 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.
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.
Re: Which hypercube unfoldings tile space?
#17What about combining the tiles?
Re: Which hypercube unfoldings tile space?
#18It 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?
Re: Which hypercube unfoldings tile space?
#19Earlier quoted context omitted.
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…
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...
#/media/File:The_11_cubic_nets.svg){kind=link}
Re: Which hypercube unfoldings tile space?
#20So 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 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.