## Cutting an unfolded cube from a square of paper.

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### Cutting an unfolded cube from a square of paper.

There are 11 ways of putting 6 squares together in such a way that they fold to make a cube. But if you don't just use squares but other shapes, there are many more ways of doing it. I have shown one such way here:

Cube.png (648 Bytes) Viewed 5312 times

So my question is, is it possible to cut a cube net from a rectangular or preferably square piece of paper, using the entire square, that can be folded into a cube with no parts overlapping and no holes? And if so, can a net be made that consists only of squares and right triangles?
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ThemePark

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### Re: Cutting an unfolded cube from a square of paper.

I am quite sure (>95%) that this is not possible, but I do not have a complete proof.

The problem appears at edges: The sum of angles of the sides is just 270°, whereas your net in the plane has 360° if you do not cut something away. Using all the paper, you get some overlap as soon as you fold anything to form an egde.
Of course, you can cut your rectangle into 2 or more pieces and glue them together, without overlap and holes.

Additional evidence comes from the fact that I think I would have heard of such a construction.
mfb

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### Re: Cutting an unfolded cube from a square of paper.

Upon looking at the first image, I had an instinct that there must be an infinite number of possible nets like that one, by making small adjustments to the "triangle parts". Am I correct about that? I know that's a different question than the original one; I was just curious.

Lenoxus

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### Re: Cutting an unfolded cube from a square of paper.

You are right. You can do a similar thing with other edges as well. As long as the meeting sides match, the net is fine.
mfb

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### Re: Cutting an unfolded cube from a square of paper.

How about this: Is it possible to create a net for a cube out of more than 50% of a rectangular piece of paper.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
tomtom2357

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### Re: Cutting an unfolded cube from a square of paper.

tomtom2357 wrote:How about this: Is it possible to create a net for a cube out of more than 50% of a rectangular piece of paper.

Easy.
Spoiler:
Use the cube net that consists of a zig-zag pattern of 6 squares. The surrounding rectangle is 3x7, of which the squares have area 12.
This already works (12/21 ~ 57%), but if you cut one of the end squares in half you can get the rectangle down to 3x6 for an even better result of 12/18.

jaap

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### Re: Cutting an unfolded cube from a square of paper.

Okay. what is the highest percentage of the rectangle you can use, I have 2/3 so far.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
tomtom2357

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### Re: Cutting an unfolded cube from a square of paper.

tomtom2357 wrote:Okay. what is the highest percentage of the rectangle you can use, I have 2/3 so far.

I get 3/4 : draw crosses (connecting opposite corners) on two opposite faces, cut along the lines, then cut one of the edges linking those two faces. Other cuttings of the opposite faces get the same result.
Yat

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### Re: Cutting an unfolded cube from a square of paper.

A bit easier to imagine with a picture (remove the red parts), giving 3/4:

Spoiler:
mfb

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### Re: Cutting an unfolded cube from a square of paper.

mfb, I'm not quite sure what you mean. If I fold the cube, each side would have 4 90 degree corners, making it a total of 360 degrees, just like when it's unfolded. If you take the corners of the cube, then yes that would be 3 90 degree corners, thus 270 degrees.

And suppose overlap is allowed, but still no holes. Would it be possible then?
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ThemePark

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### Re: Cutting an unfolded cube from a square of paper.

I mean the corners of the cube with my 270°-argument (;)).

With overlap allowed, you can use the whole paper to fold a cube, but the surface of the cube will be smaller than the original area (due to overlap). No idea whether you could improve the 3/4-ratio with that rule.
mfb

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### Re: Cutting an unfolded cube from a square of paper.

mfb wrote:A bit easier to imagine with a picture (remove the red parts), giving 3/4:

Spoiler:

Elegant, very elegant. So can anyone do better than this?
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
tomtom2357

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### Re: Cutting an unfolded cube from a square of paper.

After about an hour of experimenting I have come up with this. The brown and pink faces have the smaller pieces with side lengths 1, 0.2, 0.4, 1.0198 assuming the squares have side length 1; the larger pieces have side lengths 1, 0.6, 0.8, 1.0198. The whole rectangle is 4.4322 x 1.7650, making the area of the rectangle 7.8231 with the cube cut-out taking up 76.70% of the total rectangle.
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patzer

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### Re: Cutting an unfolded cube from a square of paper.

I can give an analytic expression of the area: One side has a triangle with side-lengths (0.8,4,sqrt(16.64)). This triangle has the same side length ratios as the white triangle at the top with 1.8 as longest side and therefore I get w=1.8*4/sqrt(16.64)=9/sqrt(26) as width of the paper and sqrt(16.64)+w/5 as long side. If I multiply both, I get 1017/130=8-23/130 as area. This is consistent with your numbers.

Nicely done.

I think it is possible to improve this design a bit with the modifications shown here (the same is possible with the other side), but I don't know where the best position is at the moment.
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mfb

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### Re: Cutting an unfolded cube from a square of paper.

So, can you explain how that folds to make a cube? And, since by the looks of it, we have changed the problem to cutting a cube from a rectangular piece of paper, isn't that already a rectangle, so haven't you already solved the problem?
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
tomtom2357

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### Re: Cutting an unfolded cube from a square of paper.

Cut off all the white stuff and fold all other parts. Not THAT hard once you've read the post.

t1mm01994

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### Re: Cutting an unfolded cube from a square of paper.

Same width of w=9/sqrt(26), but with a reduced length of 16/sqrt(16.64)+0.8/sqrt(16.64)=21/sqrt(26)
The total area is 189/26=7+7/26, which is approximately 7.269 and gives about 82.540% used area.

The white lines illustrate how the brown side gets assembled from its three parts in the folding process.
Attachments
A cube net, using 82.5% percent of the paper
mfb

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### Re: Cutting an unfolded cube from a square of paper.

Okay, new challenge: can anyone cut an unfolded cube from 90% (or more, of course) of a rectangle?
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
tomtom2357

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### Re: Cutting an unfolded cube from a square of paper.

Improvements don't get easier with better designs, and 90% would require to find something which avoids the big white areas in the previous design. I doubt that it is possible, and even if, it probably requires something completely new.
Posting some numbers is not a challenge. Prove that it is possible to reach them first .
mfb

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### Re: Cutting an unfolded cube from a square of paper.

You could probably use part of the white parts that are attached to the left of the yellow square, and to the right of the green square. That might increase the proportion of paper used.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
tomtom2357

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### Re: Cutting an unfolded cube from a square of paper.

These region correspond to unproblematic parts of the brown/pink faces.
You can use parts of them, but that just opens up gaps elsewhere.
mfb

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