(Assuming we’re talking about the regular platonic shapes. I’ll accept no “look at this dagger I made with X sides!”.

I have my own answer and reasoning but I’m curious what others think).

Okay so I did some checking and even my assumption was wrong.

When you slice the points off of the shapes (at same distance from a vertex), the octahedron has less surface area, less volume, is less flat. However you want to measure it the octahedron point is more pointy. The cube's point is simply more blunt.

HOWEVER, the cube has more points (vertices) and is therefore more pointy.