Assuming tungsten lies on the price/density frontier, and we need to verify the claim that an object is (almost) pure tungsten, we need to measure its density. Mass is easy, volume is trickier. Although the volume of an arbitrary shape can (probably) be easily measured by displacement of liquid, we would prefer a less messy way, a geometric way. What shapes can have their volume easily determined geometrically?

The volume of a tetrahedron can be determined by measuring its edge lengths. Comparing with a straightedge can determine that the edges are straight. Is there a simple way to determine that the faces are flat? Especially need to detect concave.

Cube is more difficult because we need to determine that faces are at right angles.

How easy is to to determine the diameter of a sphere? How easy is it to verify that a shape is a sphere? Rolling won't work: Meissner tetrahedra.

In this modern age, probably laser scanning or structured light.

Tungsten is hard so maybe the vertices of the tetrahedron won't break off. But tungsten is brittle, so maybe they will.

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