I have been obsessed with learning this box for a little while now. I lost the original book that I had learned it from, and only recently recovered a working sample to relearn it from.
I apologize for not acknowledging the original origami master that created this design. If I knew who it was, I wouldn't be posting this. Thank you for taking the time to read this.
This instructable was created with some diagrams using Google Sketchup and photographs.
Step 1: Start With a Square Piece of Paper.
You will need to begin with a square paper.
If you only have regular 8.5"x 11" or rectangular paper, I would suggest that you measure a piece of paper, and cut it square rather than folding it diagonally. This pattern does best without the long half diagonal folds, as you will see.
You may want to fold it horizontally and vertically to begin. I have included in the visual the first two folds you'll want to make on the square as solid lines.
Step 2: Make Horizontal and Vertical Folds
Keep folding parallel lines as shown above, so that you have four more folds in total. Open it all flat again.
Then fold up the four corners so that they meet in the center as shown in the second image above.
There are a few more pre-folds here in the third image that will help set up the final corners. You can match up the outside corner to the inside as shown in the fourth image above to get some of these folds. Just be careful not to fold inside the squares that don't need folds, which will become the exterior of this cube.
Step 3: Create the First Corner
Here comes the first challenging step.
Open two of the four corners, so that two corners are still folded with their edges touching as shown above.
There are four prefolds you want to make as shown in the image above. The third and fourth images show which ones will be valleys and which ones are mountain folds.
Tuck the tiny triangle up and away so that the other part of the paper can fold around it, creating the first cornered edge.
Shape at two right angles, folding the top over and tucking the corner A into corner B (see 6th image).
The final photo here shows the completed first corner with the rest of that flap tucked under so that the edges line up with the against the inside of the cube.
Step 4: Final Steps
The first diagram shows the cube with new prefolds which are mountain folds diagonal to the sides of the paper.
This is probably the most intriguing fold, which both conceals the inside of the box, while creating the other two corner walls, for a cut out pyramid effect. As you can see in the photographs, it's like a tongue that folds upwards and then tucks neatly inside the box.
Corners A and B right in the center of the 2nd image fold up and together into the middle of the floor of the cube.
Straighten up the two other walls of the cube and angle the bottom corner up and into the cube so that it butts right up against the inside floor of the cube. The shape is a bit lumpy in certain spots where corners have been tucked away, and it takes some finesse to make the square look just right.
In the last image, corners A & B will meet together in the middle of the floor of the cube, hidden under the flap.
I hope this is clear, and that this has been helpful in some way.