> It's like a diagonal of a cube 1x1x1 has length sqrt(3), but if you apply orthogonal projection onto R^2, its image will be a diagonal of a square and it will have length sqrt(2). Shorter distance -> shorter time to travel.

This example doesn't make sense to me. In that analogy, wouldn't anything on that diagonal appear to move more slowly in 2D than the same thing moving along the diagonal of a face? The cube diagonal would make it move farther than it does in 2D space.

I remember seeing a simulator in my optics class that combined multiple wavelengths of light. The interference pattern moved faster than the speed of light, but that was fine because information wasn't moving faster. That was just the result of adding them together.

But when you move the laser emitter in your hand you're controlling the speed in that 2d space, not in 3d. You don't ever affect the position of photons in the Z dimension. So you’re not constrained by speed in 3d which would later be slowed down after being projected. So you move your laser emitter along the diagonal of a face with velocity v. And the perceived light which would get projected onto a plane needs to match the position of the emitter on the face. Which creates the illusion that light travelled along the 3d longer diagonal faster than at v (in order to match the projection which describes how you/camera sensor see the light). But in reality the light never travelled along this longer diagonal. It’s only an illusion. And it is this illusion that we’re measuring the speed of. Photons on this diagonal arrived straight from the emitter, i.e. each of them appeared in only one point of the diagonal throughout its entire history. In other words, the photon at the beginning of the perceived movement is a different photon than at the end. They travelled along different paths. And when some photons were at the diagonal, some others were on their way there.