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The top row shows the triangular prism. Then in each row some of the pieces are rearranged until in the bottom row we have the cube. The right hand views are the same as the left but rotated by 180° to show the back of the pieces.
The top row shows the pentagonal prism. Then in each row some of the pieces are rearranged until in the bottom row we have the cube.
The dissection is translational.
Starting with the prism, this dissection can be performed with just five straight cuts. The top left diagram shows the initial hexagonal prism. We cut this in half and rearrange the two slices as in the next diagram below. Now two more diagonal cuts add another three pieces. Again we rearrange the slices as in the next diagram below. Finally another two cuts add three more pieces giving a total of eight. The slices can now be rearranged to give the cube as shown at the bottom.
The right hand column shows the individual pieces.
The top row shows the heptagonal prism. Then in each row some of the pieces are rearranged until in the bottom row we have the cube.
The top row shows the octagonal prism. Then in each row some of the pieces are rearranged until in the bottom row we have the cube.
Unfortunately the shaded piece is very thin in the middle, so there is room for improvement here.
The dissection is translational.
The individual pieces are shown to the right.
Starting with the truncated octahedron at the top left, four straight cuts are made. After each cut the pieces are rearranged and shown in the next diagram. After four cuts we arrive at the cube shown at the bottom right.
The individual pieces are shown to the right.
The dissection is translational and hingeable.
Starting with the cube, this dissection can be performed with just five straight cuts. We start with the cube at the top left. After the first cut the two halves are rearranged to get the rhombohedron shown at the top right. A second cut is made and the pieces rearranged to get a second rhombohedron shwon at the upper middle. Similairly a third cut is made and the pieces rearranged to get a third rhombohedron shwon at the lower middle. Finally two more cuts are made to separate the rhombohedron into the two tetrahedra and the octahedron, shown at the bottom.
The dissection is translational.
We start by dividing the rhombic triacontahedron into 5 pieces. If it has a volume of 1 then the five pieces have volumes of 0.4, 0.2, 0.2, 0.1 and 0.1.
The four small pieces with volumes of 0.2, 0.2, 0.1 and 0.1 are rearranged into a single block.
The first cut adds 4 pieces. These pieces are rearranged. The second cut adds a further 5 pieces. Again rearrange the pieces into a clean parallel sided block. A final two parallel cuts are made adding a single piece and 5 more pieces. The pieces are now rearranged to give a final block of dimensions 0.6×1×1. The pieces required are 4+4+5+(1+5) for a total of 19 pieces.
Here are these 19 pieces shown separately and grouped for the original 4 pieces.
The central block must now be dissected. We start by making 3 parallel cuts making a total of 4 pieces. These are rearranged. Now another cut is made adding 3 more pieces. Again rearrange the pieces making a clean parallel sided block. A final cut is made adding a further 3 pieces. The pieces are now rearranged to give a final block of dimensions 0.4×1×1. The pieces required are 4+3+3 for a total of 10 pieces.
The two blocks of 0.4×1×1 and 0.6×1×1 can now be combined to make a cube with a total of 29 pieces
Here are these 10 pieces shown separately.