How to fold a cube from a square sheet

Take a 1-by-1 sheet, and fold 16 squares. Then add diagonal foldings according following figure. (dotted-line: mountain-fold, straight: valley-fold).

   +---------+---------+---------+---------+
   | \      6|         |2      / |      3/1|
   |   \     .         .     /   |     .   |
   |     \   |         |   /     |   /     |
   |6      \ .3        . /      2| .       |
   +---------+---------+---------+ - --- - +
   |         |         |       / |         |
   |         |         |     .   |         |
   |         |         |   /     |         |
   |         |         | .       |         |
   +---------+---------+---------+---------+
   |       / |         |         |         |
   |     .   |         |         |         |
   |   /     |         |         |         |
   | .       |         |         |         |
   + - --- - +---------+---------+---------+
   |5      / |         | .       . \       |
   |     /   |         |   \     |   \     |
   |   /     |         |     .   .     \   |
   | /      5| 4       |      4\ |       \1|
   +---------+---------+---------+---------+

Short instructions:

Fold 1 onto 1, such that the two 2's remain visible. (Push the triangle with the two 2's down, such that you have three layers of paper there.) Flip the outer 2 on the inner 2.
Bring the tip (triangle shaped) with the 3 on it, to the 3 that is on the inside. For this you have to fold some other things as well.
Now fold 4 to 4, 5 to 5 and 6 to 6, almost in one move. The box should be closed now with three flaps (triangle shaped) pointing outward.
As you look careful you will notice that each flap can be flipped inside a layer of the box. If you do this, they will become fixed, and the box is ready.

Detailed instructions, for those who are not very experienced.

   A---------B---------C---------D---------E
   | \       |         |       / |       / |
   |   \     .         .     /   |     .   |
   |     \   |         |   /     |   /     |
   |       \ .         . /       | .       |
   F---------G---------H---------I - --- - J
   |         |         |       / |         |
   |         |         |     .   |         |
   |         |         |   /     |         |
   |         |         | .       |         |
   K---------L---------M---------N---------O
   |       / |         |         |         |
   |     .   |         |         |         |
   |   /     |         |         |         |
   | .       |         |         |         |
   P - --- - Q---------R---------S---------T
   |       / |         | .       . \       |
   |     /   |         |   \     |   \     |
   |   /     |         |     .   .     \   |
   | /       |         |       \ |       \ |
   U---------V---------W---------X---------Y

lay down the paper as above.
fold the upper-corner E on top of the corner Y, along the line M-N-O, such that E and Y are touching.
Now push point C to point O, folding the line M-I-E. The lines M-R-W, M-L-K and M-H-C should be folded in a square angle as well. Point M should now be in the corner of three sides that all have a surface of 4 unit squares. C and I should be visible.
Turn it a little such that corner Y is pointing towards you. Now fold point C on top of point I/S, along the line H-D, which at the same time folds the lines C-H and H-G-F in a square angle. If you look inside the corner, you should notice that four sides of the cube are in the right position now.
Now fold the points E/Y, D/J/T, and O upwards along the line X - C/I/S - H/N, such that the point O touches the point G, and the point D/J/T touches the point B.
Move the point E along the line C/I/S - D/J/T on the inside of the cube to the point G.
The right side of the cube is now finished. In the follong fold we are going to close the left side, along the line P-Q-R, such that point X goes up to point D/J/T. Point W goes on top of point S (outside the cube), folding the line R-X. As a by-product the line R-W will be folden completely, and the line R-S in a square angle.
Five sides of the cube are now in place, and the point Y is pointing towards you. (V and X should be touching each other now.)
Now we only have to push the top of the cube (consisting of the points F, G, L, K) in place. This is done by moving the points F and K down, folding the line G-L in a square angle. While doing this the line P-L should move inside, and the lines A-G and Q-U should move outside, while being folded. Some more lines will be folded. If everything is okay you should have a cube now, with three triangle shaped flaps sticking out. At the tips of the flaps are the points A, U, and Y.
As you look careful you will notice that each flap can be flipped inside a layer of the box. If you do this, they will become fixed, and the box is ready. The point A goes inside under the line O-T, the point U goes inside under the line F-K, and the point Y goes inside under the line V-W.

(Thanks to Ralph Dratman for his comments.)

Instruction video by Jorge Jaramillo

Jorge Jaramillo has made an instruction video showing how to fold the video including a colour diagram showing where to make all the folds before putting it together.

About the fold

This fold was found by a number of people independently, but is not a well known fold, which is rather surprising as it is a very elegant folding. I too found it by myself.

It was published in the book Origami for the Conoisseur, where it can be found on page 58. It says that it was independently invented by Kazou Haga and Kunihiko Kasahara.

Jeannine Mosely told me that she invented it in 1974 on her own.

Origami pages

For other ways to fold boxes look here in the The garden of Origami by Victoria Beatty. See also Joseph Wu's Origami Page, and a Guide to Paperfolding Instructions on the Web.

How to fold a cube from two square sheets

Take two 1-by-1 sheets, and fold them according to the following figure. (dotted-line: mountain-fold, straight: valley-fold).

   +---------A-------------------L---------+
   |       /   \               / | \       |
   |     .       .           /   |   .     |
   |   /           \       /     |     \   |
   | .               .   /       |       . |
   I-------------------H         |         G
   | \               /   \       |       . |
   |   \           /       \     |     /   |
   |     \       /           \   |   .     |
   |       \   /               \ | /       |
   |         B                   F         |
   |       / | \               /   \       |
   |     .   |   \           /       \     |
   |   /     |     \       /           \   |
   | .       |       \   /               \ |
   C         |         D-------------------K
   | .       |       /   .               . |
   |   \     |     /       \           /   |
   |     .   |   /           .       .     |
   |       \ | /               \   /       |
   +---------J-------------------E---------+

Now fold, at the same time, A to B, C to D, E to F, and G to H for both sheets. You will get two semi-boxes where the points I, J, K, and L stick out.

The two semi-boxes can be put together, such that the points I, J, K, and L of the one semi-box go inside the other semi-box touching the points A, C, E, and G.

As an alternative, you can also join one of the above with the following:

   +---------A---------+---------B---------+
   |       /   \       .       /   \       |
   |     /       \     |     /       \     |
   |   /           \   .   /           \   |
   | /               \ | /               \ |
   C                   x                   D
   | \               /   \               / |
   |   \           /       \           /   |
   |     \       /           \       /     |
   |       \   /               \   /       |
   +-.-.-.-.-x                   x-.-.-.-.-+
   |       /   \               /   \       |
   |     /       \           /       \     |
   |   /           \       /           \   |
   | /               \   /               \ |
   E                   x                   F
   | \               / | \               / |
   |   \           /   .   \           /   |
   |     \       /     |     \       /     |
   |       \   /       .       \   /       |
   +---------G---------+---------H---------+

The first tricky thing about this fold is to get the diagonal folds at the right place. Now fold A to B, C to E, D to F, and G to H. This will give you a box with one open side. The second thing is to make this one join with a paper folded according to the previous diagram. There is only one way to do it, but I cannot describe it accurately.