Hans Walser, 
Cube made from six pyramids
Suggestion and idea: Walter Arn, Flawil, and Caspar Schwabe, Japan
Side faces model of the cube. The six components are made from rectangular paper with an aspect ratio of one to the square root of two (Fig. 1a). For example, a rectangle with this aspect ratio appears as a diagonal plane in the cube (Fig. 1b).
Fig. 1: One to square root of two
We can divide the cube from the center into six pyramids, the bases of which are each a side of the cube (Fig. 2). These six pyramids will be our components.
Fig. 2: Six pyramids
We fold a map or a rectangular paper with the aspect ratio one to the square root of two as shown in Figure 3. Blue lines are "mountain folds", red lines are "valley folds". For the diagonal fold lines, I scratched the lines with the point of a needle on the outside (mountain fold side) before folding them.
Fig. 3: Folding plan
Figure 4 shows a real example.
Fig. 4: Real example
We can fold the sheet into a pyramid. Figure 5 shows the inside of the pyramid, Figure 6 the outside. The model is temporarily fixed with paper clips.
Fig. 5: Inside of the pyramid
Fig. 6: Outside of the pyramid
For the cube we need six components, one for each side of the cube. We can use it to put the cube together with paper clips (Fig. 7). The inside of the pyramid comes to lie on the outside.
The side face diagonals are different. One of the side face diagonals is a fold line, the other a slit. Out of love for symmetry, we put the pieces together so that three fold lines meet at every second vertex of the cube and three slit lines at every other vertex.
Fig. 7: Assembled cube
Since the components are independent of one another, up to six different colors can be used (Fig. 8). In this model, glue stick was used to fix the individual pyramids and the stapler was used to assemble the six parts.
Fig. 8: Colors, glue stick and stapler, and my grandchildren's shoes
The following movie shows the movement of the six pyramids.
Movie: The six pyramids
Some special cases are quite interesting.
Fig. 9.1: Starting position
Fig. 9.2: Rhombic dodecahedron
The rhombic dodecahedron in Figure 9.2 has a cubic hole inside. The rhombic dodecahedron is a space filler. You can fill the room with rhombic dodecahedra without gaps and without overlapping.
Fig 9.3: Star
The star in Figure 9.3 is also a space filler.
Fig. 9.4: Cube
The cube is of course a space filler.
Fig. 9.5: Explosion of the cube
Hans Walser: Wźrfel aus sechs Bauteilen