[Physicist]

If we were flatlanders, living in a 2D plane, and a 3D rod were perpendicular to our plane, we'd see it only as a 2D circular cross-section of the rod.

The rod isn't perpendicular to the plane (although it could be). The extra dimension is perpendicular to the plane.

If the rod were in motion,

Full stop. The "rod" only represents the trajectory of the particle over time. The particle is pointlike and massless. The particle moves through the space as a massless object. The shadow of the particle on the plane moves on the plane as if it had mass.

[PatrickHenry]

Thanks for the clarification. From my squinty-eyed perspective, it sounds too much like Plato's parable of the cave. I'm betting on the Higgs. (But what do I know?)

[spunkets]

"The particle moves through the space as a massless object. The shadow of the particle on the plane moves on the plane as if it had mass."

I think it's important to note that, "moving as a massless particle" means the particles are all moving at the same speed(c) in the 3d space of your example. In the projection to a lower d space, the particle speeds vary and are all <= to the fixed 3d speed. It's the appearance of the different velocities in the projection that gives rise to the concept of mass.