## Friday, December 14, 2012

### Curious Cuboids

A few more diversions from Insomnia's frozen shore

Recently I was thinking about rectangular parallelepipeds, also known as cuboids. A cuboid is a simple box shape. Think of a shoebox, a cube, or the black monolith in the film 2001: A Space Odyssey.

A cuboid has 12 edges with lengths x, x, x, x, y, y, y, y, z, z, z, z. By analogy to triangles, I call a cuboid scalene if x, y, and z are all distinct, isosceles otherwise. I further classify isosceles cuboids as being rod-shaped (x = y ≤ z), tile-shaped (x = ≥ z), or both (x = y = z; a cube).

 Isosceles cuboid, rod-shaped
 Isosceles cuboid, tile-shaped

To find the volume, multiply length times height times width. To find the surface area, compute the area of each of the six faces, then add them all up. For a linear measure, let's define the edge sum: length plus width plus height.

Below are some hopefully amusing questions about all of this. Also, I made a grainy animation that relates to all this. (It will make more sense after having thought about the questions below.)

Some of these questions can be settled with visualization, and some by simple arithmetic; others are less simple. One day I'll post some further observations on these matters...in the meantime, enjoy! And feel free to put any answers or thoughts in the comments.

Volume and Surface Area. The (6, 6, 6) cuboid is notable in that its volume is 6 × 6 × 6 = 216, and its surface area is (6 × 6) × 6 = 216.

1) Can you find another example of a cuboid with volume and surface area numerically equal when expressed in the same system of units? Is your example scalene, rod-shaped, or tile-shaped? Can you find an example of each?

Surface Area and Edge Sum. The (½, ½, ½) cuboid is notable in that its surface area is (½ × ½) × 6 = 3/2, and its edge sum is ½ + ½ + ½ =  3/2.

2) Can you find another example of a cuboid with surface area and edge sum numerically equal when expressed in the same system of units? Is your example scalene, rod-shaped, or tile-shaped? Can you find an example of each?

Edge Sum and Volume. The (Sqrt[3], Sqrt[3], Sqrt[3]) cuboid is notable in that its edge sum is 3Sqrt[3], and its volume is 3Sqrt[3].

3) Can you find another example of a cuboid with edge sum and volume numerically equal when expressed in the same system of units? Is your example scalene, rod-shaped, or tile-shaped? Can you find an example of each?

In each case above, two measures were equal.

4) Are there any cuboids for which all three measures are numerically equal when expressed in the same system of units?

A few more existential questions...

5) Is there a cuboid with a volume of 1 cubic inch and a surface area of a million square inches?

6) How about a cuboid with a volume of 1 cubic inch and a surface area of a millionth of a square inch?

7) Is there a cuboid with a volume of a million cubic inches and an edge sum of 1 inch?

8) Is there a cuboid with volume 80 cubic inches, surface area 122 square inches, and edge sum 16 inches?

Finally, some optimization problems. In each case, the desired maximizing/minimizing cuboids exist and have whole-number edge lengths.

9) Of all cuboids with surface area 48 square inches and edge sum 9 inches, which has greatest volume? Which has least volume?

10) Of all cuboids with volume 72 cubic inches and edge sum 14 inches, which has greatest surface area? Which has least surface area?

11) Of all cuboids with volume 162 cubic inches and surface area 234 inches, which has greatest edge sum? Which has least edge sum?