What is the perfect shape for ice cube trays?

Factual Questions
1

Inspired by this thread, but I want to talk about math theory, not practice. (Ice-cube tray question… )

I once had the perfect ice cube trays, and have never seen them again, but would like to define them.

(Please assume for discussion that “cube” refers to any chunk of ice, not strictly cubical in shape.)

The key to the perfect ice cube tray is that as the ice forms the cubes expand:

  • a worst case tray would have perfectly square slots, and the ice would wedge itself from side to side, locking in solidly.

  • a V-shaped trough or point-down cone would at first appear to produce easily-freed cubes, but this is not the case. As the cubes expand, they simply slide to new positions along the shape, and there is full contact between the cube and the walls.

  • A parabolic slot would seem ideal, since although increasing the size of a parabola leaves it “the same shape”, it really only makes contact with its old shape at the top edges, thus being essentially free. Of course, since it is sealed at all times at the top edge, the gap forming beneath the cube is a vacuum, now forcing the ice to again become hard to budge, although only the edge seal must be broken, not the entire bottom and side walls.

  • I believe that a quadratic curve would have the same effect (one that looks like a barrel half, but with a bulge rather than a flat bottom)

Now, for the shape of these lost perfect trays - They were parabolic from port to starboard, if you will, and quadratic from fore to aft.
The result was that as the ice formed the cube would expand and rise out of the slot, but since the expansion in one direction was greater than the other direction, no seal would form at the rim and the cube would be free, resting on only two points of the rim.

I’m sure this shape was accidental, although I wonder whether some unsung genius had a long overlooked and now expired patent on the perfect shape.

Is this analysis correct? I.e., would it really work? Or have I misremembered the whole thing?

2

So if I understand, did the trays look like a (for lack of a better term) “butt imprint.” Like if someone in the buff were to sit in the sand???

3

Umm… parabolas are a type of quadratic curves. What’s the mathematical distinction you were trying to illustrate here?

4

I think they meant a second degree equation for the parabola and a fourth degree equation for the quadratic. (That confused me too).

5

In that case, quadratic is the incorrect term. What Joey P is describing is a quartic.

6

Ignoring for the moment the negligible effective difference between a parabola (I’m assuming that the poster meant a 3-d shape that appears parabolic in two dimensions) and a sphere, I’m going to have to side with the makers of automatic ice “cube” machines. Most that I’ve seen use a flexible form that grows ice in the shape of a drawfed hemisphere.

(Slice the earth in half through the equator fo you get two hemispheres. Slice it through 30°N lat. and you get a “dwarfed” hemisphere.)

This shape allows the ice to fall almost effortlessly out of the mold. Any other shape would create more static friction between the ice & the mold, therefore more effort required to overcome this and get the ice our of the mold.

Now, what do you mean by “ideal”? If you mean best shape for cooling down drinks, it would have to be a cube (assuming that crushed ice is disallowed for this discussion). Cubes have something like 19% more surface area compared to spheres of equal volume.

7

Yes, I meant “quartic”, not quadratic, and “paraboloid”, not parabolic. I’m not a math person.

There’s a picture of a quartic here

As to the most area, I hadn’t thought of that, so no, it’s not part of the question. (The shape with the highest area/volume that I’ve seen are the novelty ice cubes that are shaped like nudes.)

So, would a “dwarfed hemisphere” be any different from the paraboloid? Wouldn’t it still have the rim seal and partial vacuum?