Folding Iso-Area Rectangular Solids of Various Dimensions

A few months back I was folding Kawasaki cubes and an idea occured to me. What would happen if I folded in the usual way but added more divisions? I fold cubes from a design by Toshikazu Kawasaki in "Origami for the Connoisseur." I like to fold cube #2. I do it a little differently from the way it is diagrammed in the aformentioned book. I fold the squares with half the divisions used in the book because the extra divisions are not really needed for cube #2. The diagrams I have provided use my method. If you find it more difficult, use the method in the book as a guide.

What I found was that by increasing the number of divisions, I could fold things other than cubes. Instructions are included for a 1x1x2 and (less detailed) for a 2x2x1. It is possible to fold any rectangular solid with square ends whose dimensions can be expressed in positive integers. Of course, the paper has limitations, but with two dimensional paper it would be possible.

You will note that the 1x1x2 solid uses 5 divisions each way and the diagonal pleats are one division in width. The 2x2x1 solid uses 7 divisions each way and the pleats are two divisions wide. Here is a table showing a few possibilities. I hope you can see the pattern that emerges.

   solid      divisions     width

   1x1x1          4           1
   1x1x2          5           1
   1x1x3          6           1
   1x1x4          7           1

   2x2x1          7           2
   2x2x2          8           2    (It's the same as 1x1x1 with extra folds)
   2x2x3          9           2

   3x3x1         10           3
   3x3x2         11           3

   4x4x1         13           4
You can see that if 'a' represents the number of units on an edge of the square end that 3a + 1 is the smallest number of divisions that will be sufficient to fold a solid. If you try to fold a solid with say 4 units on the square edge from a square with 12 divisions, you will get a flat coaster (another Kawasaki design). If you fold a 1x1x1 and a 4x4x4 from the same size square you will get identical cubes except for the extra creases. Similarly, if all your dimensions are divisible by a common denominator, you can always fold a solid of the same dimensions by reducing the numbers with division by the common number. Finally, the pleat width is equal to the number of units on a square edge.

I hope you have as much fun with this as I have.

        - - - Tom May - - -