Tip of the Ice Cube

he film An Inconvenient Truth had quite a lot to say about polar ice, drawing attention to the distinction between the frozen ocean afloat in the Arctic and the massive ice sheets covering 98% of the land area in Antarctica.  Both are melting, but the latter has the greater potential for raising the sea level.  A simple parlor puzzle can confirm that...
Fill a glass with ice cubes and enough water to bring the level to the brim and ask, "Will the table stay dry?"  The floating ice will actually be seen above the brim.
• As ice melts, though, it actually shrinks, so the level of water in the glass does not change, averting a spill.  Think Arctic ocean.
• On the other hand, if walls of the glass are allowed to provide mechnical support for the ice, the water level will rise as shown in this clip.  Think Antarctica's impact on low-lying lands around the world.
Oliver Wood. Ice Cube, Pencil Sketch

metaphor can be too perfect.  Case in point: "Only the tip of the iceberg."  It is difficult to invent a more succinct alternative to it which expresses the concept of the issue is more difficult than it appears.  So that's what we say without thinking -- certainly without thinking about...

That moonless night of 14 April 1912 when the mighty RMS Titanic with 2,223 souls on board collided 37 seconds after the sighting of a particular iceberg and the fateful annoucement by lookouts, "Iceberg right ahead!"
There can be no doubt that what makes "Only the tip of the iceberg" work so well as a metaphor is the universal knowledge of a singular property of frozen water: It floats on water.
Everybody also knows that water is that most abundant substance on the planet, which covers nearly 71% of the surface and constitutes up to 78% of our bodies.
No wonder that water has been appropriated as a scientific standard.  For example, in characterizing density, scientists use the expression specific gravity (SG), which is the ratio of the mass of a given body's volume to the mass of a reference body having an equal volume.  The reference body is -- well, water.  Accordingly, it is kind of ironic to study the SG of water itself, as shown in this little graph...

...in which solvers are invited to make the following observations about the SG of water:

• Starting at a rather hot 40o C (104o F), we see that the SG of liquid water is less than 1.00  becoming less dense with higher temperatures beyond the right edge, as one would expect.
• Removing heat from water results in cooler temperatures toward freezing at 0o C (32o F), where the SG of water has increased to unity (1.00), then something remarkable occurs...
• At a constant temperature of 0o C (32o F), removing heat causes the SG of water to decrease steeply to 0.90, which is the SG of the solid phase of water, called ice.
• Moving farther to the left on the graph, the ice is cooled toward -30o C (-22o F) and the SG increases, as one would expect for any solid getting denser when cooled.
ew substances expand like water when they freeze, specifically certain crystal-forming compounds and the following chemical elements: antimony, bismuth, gallium, germanium, silicon, and plutonium.  One wonders: To what extent does life as we know it here on the planet earth depend on this particular property of H2O?
Our Tip of the Ice Cube puzzle is offered here in the form of a simple thought experiment that pertains to the theoretical ability of a lookout on board a vessel to see the tip of the iceberg far enough away to avoid collision with it
 Given a perfect ice cube, what is the height of its tip  above the surface of the water on which it is floating?