Projection (ConceptTopic, 3)

From Hi.gher. Space

Projection is the removal of one or more dimensions of information from the object. There are two main types of projections: parallel projection, and perspective projection.

Parallel projection

Parallel projection is to completely throw away one dimension from the object. For example, we can take each point (w,x,y,z) from the object, and simply throw out the w-coordinate, leaving us with (x,y,z), a point in 3D space. By doing this to the whole object, we end up with a 3D image of the original object.

This can be thought of as ignoring the axis of "gravity" from a fourth-dimensional object. When you remove this dimension, you have a three-dimensional object. Since you removed the axis of gravity, you ignore gravity since 4D gravity wouldn't be acting on any remaining axis. Then realmspace becomes a surface for objects of the fourth dimension to rest on. Thus, you can imagine roads coming out of your floor and going towards the ceiling, intersecting with roads that travel in midair from each wall to the opposite wall. For a tetronian, roads arranged in this manner could be laid flat on their ground.

Perspective projection

Instead of completely throwing away one dimension from the object, we can instead scale the other 3 dimensions with it, so that in the resulting image, we have some "hint" as to what the value of the 4th coordinate is. For example, for each point (w,x,y,z) in a 4D object, we project it to (x/w, y/w, z/w).

This is similar to how our own eyes see 3D objects: our retina is really only 2D, and when we look at a square from an angle, it makes a trapezoidal shape on our retina. Our brains infer from the unequal edges that the shape must be a square bent into the 3rd dimension. Similarly, when we project a cube residing at an angle in 4D space into 3D, we may end up with a frustum-like shape. A 4D being would infer from the unequal faces that this frustum shape is really a 3D cube bent into 4D space.