No, that can't be done.... WHAM!!!
December 3, 2022 12:03 AM   Subscribe

Matt Parker contemplates the question: can the same net fold into two shapes?
posted by Pendragon (11 comments total) 19 users marked this as a favorite
 
Oh wow!

Now if only there were a net that could fold into 3 different cuboids.
posted by ambrosen at 5:24 AM on December 3, 2022 [1 favorite]


Just in case: when Matt says the video is done, he is joking. There is an entire second half where he shows things like nets that can fold into 3 different cuboids.

He leaves off with very nice open questions: what is the smallest net that can fold into 3 different cuboids? (conjectured: 46) Is there any limit to the number of cuboids a single (arbitrarily large) net can fold into (restricting folds to lie on grid lines)? That raises the questions: what is the formula for the smallest size of a net that can fold into n cuboids, and can they always be found to tile the plane?
posted by TreeRooster at 6:16 AM on December 3, 2022 [2 favorites]


Wow, indeed!!

I'm lousy at math, and watching this alternately actually makes sense and makes my head want to asplode.

So what happens if it gets folded into a fourth dimension....
posted by BlueHorse at 8:04 AM on December 3, 2022


Just in case: when Matt says the video is done, he is joking. There is an entire second half where he shows things like nets that can fold into 3 different cuboids.

Yep, that bit was great (both the false ending and the 12 minutes following it), especially as the video would have been fine just with the nets that fold into 2 different cuboids.
posted by ambrosen at 8:11 AM on December 3, 2022


This is very neat! I had seen some animations, maybe related directly to the 2017 animation he shows, but for non-convex cubeplexes (cuboidoids?) and had thought that was neat, but the fact that this is doing different convex shapes is neater still by a lot.

Like, if you don't require the different shapes you fold from the same net to be convex, there's a lot of relatively trivial solutions available. The triangular net of a regular icosahedron is a good example: you can take five adjacent triangles that describe a pentagonal perimeter and just invert that pyramid so that it's a divot in the face of the icosa instead, and, bam: two different solids from the same net. Which is neat but not really remarkable.
posted by cortex at 9:00 AM on December 3, 2022


Personally I thought folding not along the grid and producing a perfect cube of side √5 was inspired and beautiful!
posted by phliar at 10:20 AM on December 3, 2022 [8 favorites]


I was waiting for non-orthogonal folds and was extremely satisfied when it showed up. The appearance of √5, which is a great number because of how closely related it is to the golden ratio, was icing on the cake.

Matt Parker is great.
posted by biogeo at 11:42 AM on December 3, 2022 [1 favorite]


That was fun. I love it when an expert like a mathematician is so sure something is impossible only to have their mind blown by some new approach. Remember the airplane and the conveyor belt?
posted by BlackLeotardFront at 1:50 PM on December 3, 2022


Other than surface area, are there any mathematical relationships between the cuboids made from the same net? I’m trying to see if there is a way to go from a box with surface area A to another different shaped box with the same surface area such that they both can be decomposed into the same net. Or… Is there a relationship between the number of squares in the net, it’s “surface area” and the surface area of the cuboids made from it? Questions from a person who regrets not paying a lot of attention in high school math classes, my last formal math education.
posted by njohnson23 at 2:14 PM on December 3, 2022


I watched it last night and loved it. Math was not my best subject and I haven't done any since college calc. But watching his channel and some others (mathologer, numberphile) has gotten me interested in it again.

my name is somewhere in the ending credits
posted by kathrynm at 4:01 PM on December 3, 2022


Speaking of Mathologer, he has a brand new one out that relates the Fibonacci sequence to Pythagorean triples, based on some fairly recent discoveries that are surprisingly accessible. Sometimes elementary results just sit there for millennia until someone notices them.
posted by sjswitzer at 4:14 PM on December 3, 2022 [6 favorites]


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