Weave three strips of paper with the same over-and-under pattern as you would use to braid hair and you will create a cube. The model is surprisingly solid and doesn't require any tape or glue.
Video Instructions
Materials
This model works well using either card stock or normal weight paper.
Steps
Cut out the three strips from the pattern.
Score and crease mountain folds along the dot-dashed lines.
Place the strips on top of each other as shown.
Braid the strips together alternately passing over and under the adjacent strip. The faces with underlined numbers go under the face with the corresponding number.
Notes
John Gorham developed the technique of weaving polyhedra to make models that are useful when studying crystallography.
Links
John Gorham's book is now in the public domain and is available online.