Ttdfjt

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`-` in tar xzf -

I found a password with hashcat, but it doesn't work

Why don't countries like Japan just print more money?

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Boss wants someone else to lead a project based on the idea I presented to him

Paralleling mosfets reduce Rds On?

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CircuiTikZ: Start ground relative to the closest component

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Trainee keeps passing deadlines for independent learning

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Term or phrase for simply moving a problem from one area to another

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How can this shape perfectly cover a cube?

Polygon wrapping a cubePoints on a cubeWhich 3D shape can you make out of this?Folding a piece of paper in halfCutting from a cube (visualization test)How many cylinders in a cubeA tricky little 3D shape1 Cube 1 Square 1 LineHow many $2 times 2 times 1$ blocks can you put in a $3 times 3 times 3$ cube?Can you cover a cube with copies of this shape?

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;

90

17

$begingroup$

The following shape:

can be folded onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps.

How can this be done?

geometry three-dimensional paper-folding

share|improve this question

edited Jun 14 at 17:08

plasticinsect

asked Jun 12 at 6:06

plasticinsectplasticinsect

914215

$endgroup$

• 
  
  
  

  
  10
  

  
  
  
  
  
  

  
  $begingroup$
  Holy grail, this is a masterpiece of a cube-net!
  $endgroup$
  – athin
  Jun 12 at 10:18
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @athin your comment reminded me of Avigrail... puzzling.stackexchange.com/users/5106/avigrail
  $endgroup$
  – Omega Krypton
  Jun 14 at 17:40
  

  
  
  
  

add a comment | 

90

17

$begingroup$

The following shape:

can be folded onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps.

How can this be done?

geometry three-dimensional paper-folding

share|improve this question

edited Jun 14 at 17:08

plasticinsect

asked Jun 12 at 6:06

plasticinsectplasticinsect

914215

$endgroup$

• 
  
  
  

  
  10
  

  
  
  
  
  
  

  
  $begingroup$
  Holy grail, this is a masterpiece of a cube-net!
  $endgroup$
  – athin
  Jun 12 at 10:18
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @athin your comment reminded me of Avigrail... puzzling.stackexchange.com/users/5106/avigrail
  $endgroup$
  – Omega Krypton
  Jun 14 at 17:40
  

  
  
  
  

add a comment | 

$begingroup$

The following shape:

can be folded onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps.

How can this be done?

geometry three-dimensional paper-folding

share|improve this question

edited Jun 14 at 17:08

plasticinsect

asked Jun 12 at 6:06

plasticinsectplasticinsect

914215

$endgroup$

share|improve this question

edited Jun 14 at 17:08

plasticinsect

asked Jun 12 at 6:06

plasticinsectplasticinsect

914215

share|improve this question

edited Jun 14 at 17:08

plasticinsect

asked Jun 12 at 6:06

plasticinsectplasticinsect

914215

asked Jun 12 at 6:06

plasticinsectplasticinsect

914215

asked Jun 12 at 6:06

plasticinsectplasticinsect

914215

1 Answer
1

active

oldest

votes

147

$begingroup$

The shape

has an area of 30, if each square is taken to be 1 unit. So one face of the cube must have area 5: the easiest way to make an area-5 square on a lattice is by using a knight's move as your side.

Using this, we can make a guess for how the cube might be folded:

Overlaying this net looks nice: the top of the cube is nearly done, the four sides around it look pretty good, and the back is mostly uncovered. The black shapes are good, but the holes in the middle are a problem. We can solve this problem by taking the gray sections along the border, and folding them along the lines between the four sides, as shown by the blue arrows. (Remember, the four sides really connect to each other! The blue arrows are legal folds, even though they might not seem like it at first.)

Once that fold is done, the shape looks more like this:

This is the same shape in 3D, and it's still connected as it was before, but I've moved some of the pieces across the seams of the net. Now it should be pretty clear how the rest folds up: each piece still hanging off the cube will be folded to cover 1/4 of the bottom face (as the lower right piece already shows), and then wrapped around to fill the small triangular hole in a different side of the cube.

A drawing of the finished product:

Here, one "arm" of the original shape has been colored dark gray to show how it folds up.

And an animation of the whole process:

share|improve this answer

edited Jun 14 at 6:11

Wossname

1,3201611

answered Jun 12 at 7:21

Deusovi♦Deusovi

67.2k7232293

$endgroup$

• 
  
  
  

  
  41
  

  
  
  
  
  
  

  
  $begingroup$
  +1, nice diagrams
  $endgroup$
  – micsthepick
  Jun 12 at 7:54
  

  
  
  
  


• 
  
  
  

  
  19
  

  
  
  
  
  
  

  
  $begingroup$
  Let's say I knew the answer right off the bat (I didn't). There's no way I could construct an answer like this within 1 hour of the question being asked. How much of that CPH4 did you actually take?
  $endgroup$
  – Strawberry
  Jun 12 at 16:32
  
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Impressive. This is a much better explanation than I could have come up with in twice the time. (and you presumably had to spend some of that time actually solving the problem!)
  $endgroup$
  – plasticinsect
  Jun 12 at 16:44
  

  
  
  
  


• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  $begingroup$
  @Wossname Whoa, thanks for adding that animation! That's really cool (and I wish there was some way to give you a bounty for it, because you deserve credit for it).
  $endgroup$
  – Deusovi♦
  Jun 14 at 6:22
  

  
  
  
  


• 
  
  
  

  
  8
  

  
  
  
  
  
  

  
  $begingroup$
  @Wossname Wow, that animation is beautiful. May I ask what software you did it on?
  $endgroup$
  – plasticinsect
  Jun 14 at 7:14
  

  
  
  
  

 | 
show 10 more comments

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147

$begingroup$

The shape

has an area of 30, if each square is taken to be 1 unit. So one face of the cube must have area 5: the easiest way to make an area-5 square on a lattice is by using a knight's move as your side.

Using this, we can make a guess for how the cube might be folded:

Overlaying this net looks nice: the top of the cube is nearly done, the four sides around it look pretty good, and the back is mostly uncovered. The black shapes are good, but the holes in the middle are a problem. We can solve this problem by taking the gray sections along the border, and folding them along the lines between the four sides, as shown by the blue arrows. (Remember, the four sides really connect to each other! The blue arrows are legal folds, even though they might not seem like it at first.)

Once that fold is done, the shape looks more like this:

This is the same shape in 3D, and it's still connected as it was before, but I've moved some of the pieces across the seams of the net. Now it should be pretty clear how the rest folds up: each piece still hanging off the cube will be folded to cover 1/4 of the bottom face (as the lower right piece already shows), and then wrapped around to fill the small triangular hole in a different side of the cube.

A drawing of the finished product:

Here, one "arm" of the original shape has been colored dark gray to show how it folds up.

And an animation of the whole process:

share|improve this answer

edited Jun 14 at 6:11

Wossname

1,3201611

answered Jun 12 at 7:21

Deusovi♦Deusovi

67.2k7232293

$endgroup$

• 
  
  
  

  
  41
  

  
  
  
  
  
  

  
  $begingroup$
  +1, nice diagrams
  $endgroup$
  – micsthepick
  Jun 12 at 7:54
  

  
  
  
  


• 
  
  
  

  
  19
  

  
  
  
  
  
  

  
  $begingroup$
  Let's say I knew the answer right off the bat (I didn't). There's no way I could construct an answer like this within 1 hour of the question being asked. How much of that CPH4 did you actually take?
  $endgroup$
  – Strawberry
  Jun 12 at 16:32
  
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Impressive. This is a much better explanation than I could have come up with in twice the time. (and you presumably had to spend some of that time actually solving the problem!)
  $endgroup$
  – plasticinsect
  Jun 12 at 16:44
  

  
  
  
  


• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  $begingroup$
  @Wossname Whoa, thanks for adding that animation! That's really cool (and I wish there was some way to give you a bounty for it, because you deserve credit for it).
  $endgroup$
  – Deusovi♦
  Jun 14 at 6:22
  

  
  
  
  


• 
  
  
  

  
  8
  

  
  
  
  
  
  

  
  $begingroup$
  @Wossname Wow, that animation is beautiful. May I ask what software you did it on?
  $endgroup$
  – plasticinsect
  Jun 14 at 7:14
  

  
  
  
  

 | 
show 10 more comments

147

$begingroup$

The shape

has an area of 30, if each square is taken to be 1 unit. So one face of the cube must have area 5: the easiest way to make an area-5 square on a lattice is by using a knight's move as your side.

Using this, we can make a guess for how the cube might be folded:

Overlaying this net looks nice: the top of the cube is nearly done, the four sides around it look pretty good, and the back is mostly uncovered. The black shapes are good, but the holes in the middle are a problem. We can solve this problem by taking the gray sections along the border, and folding them along the lines between the four sides, as shown by the blue arrows. (Remember, the four sides really connect to each other! The blue arrows are legal folds, even though they might not seem like it at first.)

Once that fold is done, the shape looks more like this:

This is the same shape in 3D, and it's still connected as it was before, but I've moved some of the pieces across the seams of the net. Now it should be pretty clear how the rest folds up: each piece still hanging off the cube will be folded to cover 1/4 of the bottom face (as the lower right piece already shows), and then wrapped around to fill the small triangular hole in a different side of the cube.

A drawing of the finished product:

Here, one "arm" of the original shape has been colored dark gray to show how it folds up.

And an animation of the whole process:

share|improve this answer

edited Jun 14 at 6:11

Wossname

1,3201611

answered Jun 12 at 7:21

Deusovi♦Deusovi

67.2k7232293

$endgroup$

• 
  
  
  

  
  41
  

  
  
  
  
  
  

  
  $begingroup$
  +1, nice diagrams
  $endgroup$
  – micsthepick
  Jun 12 at 7:54
  

  
  
  
  


• 
  
  
  

  
  19
  

  
  
  
  
  
  

  
  $begingroup$
  Let's say I knew the answer right off the bat (I didn't). There's no way I could construct an answer like this within 1 hour of the question being asked. How much of that CPH4 did you actually take?
  $endgroup$
  – Strawberry
  Jun 12 at 16:32
  
  

  
  
  
  


• 
  
  
  

  
  4
  

  
  
  
  
  
  

  
  $begingroup$
  Impressive. This is a much better explanation than I could have come up with in twice the time. (and you presumably had to spend some of that time actually solving the problem!)
  $endgroup$
  – plasticinsect
  Jun 12 at 16:44
  

  
  
  
  


• 
  
  
  

  
  11
  

  
  
  
  
  
  

  
  $begingroup$
  @Wossname Whoa, thanks for adding that animation! That's really cool (and I wish there was some way to give you a bounty for it, because you deserve credit for it).
  $endgroup$
  – Deusovi♦
  Jun 14 at 6:22
  

  
  
  
  


• 
  
  
  

  
  8
  

  
  
  
  
  
  

  
  $begingroup$
  @Wossname Wow, that animation is beautiful. May I ask what software you did it on?
  $endgroup$
  – plasticinsect
  Jun 14 at 7:14
  

  
  
  
  

 | 
show 10 more comments

$begingroup$

The shape

has an area of 30, if each square is taken to be 1 unit. So one face of the cube must have area 5: the easiest way to make an area-5 square on a lattice is by using a knight's move as your side.

Using this, we can make a guess for how the cube might be folded:

Overlaying this net looks nice: the top of the cube is nearly done, the four sides around it look pretty good, and the back is mostly uncovered. The black shapes are good, but the holes in the middle are a problem. We can solve this problem by taking the gray sections along the border, and folding them along the lines between the four sides, as shown by the blue arrows. (Remember, the four sides really connect to each other! The blue arrows are legal folds, even though they might not seem like it at first.)

Once that fold is done, the shape looks more like this:

This is the same shape in 3D, and it's still connected as it was before, but I've moved some of the pieces across the seams of the net. Now it should be pretty clear how the rest folds up: each piece still hanging off the cube will be folded to cover 1/4 of the bottom face (as the lower right piece already shows), and then wrapped around to fill the small triangular hole in a different side of the cube.

A drawing of the finished product:

Here, one "arm" of the original shape has been colored dark gray to show how it folds up.

And an animation of the whole process:

share|improve this answer

edited Jun 14 at 6:11

Wossname

1,3201611

answered Jun 12 at 7:21

Deusovi♦Deusovi

67.2k7232293

$endgroup$

share|improve this answer

edited Jun 14 at 6:11

Wossname

1,3201611

answered Jun 12 at 7:21

Deusovi♦Deusovi

67.2k7232293

edited Jun 14 at 6:11

Wossname

1,3201611

edited Jun 14 at 6:11

Wossname

1,3201611

answered Jun 12 at 7:21

Deusovi♦Deusovi

67.2k7232293

answered Jun 12 at 7:21

Deusovi♦Deusovi

67.2k7232293