Ref:

1. https://en.wikipedia.org/wiki/Squaring_the_square,

2. Cubing the cube - as 'perfectly' as possible

3. Perfect squaring of rectangles

We try to add a bit to ref 2. In this post, by 'cuboid', we mean only rectangular cuboids - hexahedra with all faces rectangles and adjacent faces meeting only at right angles. A special case of such cuboids is when a pair of opposite faces are squares - the square rectangular cuboid.

Definition: a perfect cuboiding of a cuboid C is a partition of C into a finite number of cuboids that are similar to one another (equal length: width: height ratios) but are mutually non-congruent. Requiring that the small cuboids be similar to C as well is a potentially interesting subcase. Obviously, the pieces needing to be cubes (perfect cubing) is a very restricted case of cuboiding.

Note: As proved in ref 1, a cube has no perfect cubing. The same argument also implies that no perfect cubing can be done for any general cuboid.

• Does a cube allow a perfect cuboiding - a partition into finitely many mutually similar and mutually non-congruent rectangular cuboids? Special case: Does a cube have a perfect cuboiding into square rectangular cuboids?

• And what can one say about perfect cuboidings of general rectangular cuboids/square rectangular cuboids?

We try to add a bit to ref 2 listed below. In this post, by 'cuboid', we mean only rectangular cuboids - hexahedra with all faces rectangles and adjacent faces meeting only at right angles. A special case of such cuboids is when a pair of opposite faces are squares - the square rectangular cuboid.

Definition: a perfect cuboiding of a cuboid C is a partition of C into a finite number of cuboids that are similar to one another (equal length: width: height ratios) but are mutually non-congruent. Requiring that the small cuboids be similar to C as well is a potentially interesting subcase. Obviously, perfect cubing, wherein the pieces need to be cubes, is a very restricted case of cuboiding.

• Does a cube allow perfect cuboiding - partition into finitely many mutually similar and mutually non-congruent rectangular cuboids? Special case: Does a cube have a perfect cuboiding into square rectangular cuboids?

• And what can one say about perfect cuboidings of general rectangular cuboids/square rectangular cuboids?

Note: As proved in ref 1, a cube has no perfect cubing. The same argument also implies that no perfect cubing can be done for any cuboid.

Ref:

1. https://en.wikipedia.org/wiki/Squaring_the_square,

2. Cubing the cube - as 'perfectly' as possible

3. Perfect squaring of rectangles