4D Visualization
Projections (3)
Projection Envelopes
The envelope of a projected object is the outer boundary of its image. For example, the following view of the 3D cube has a hexagonal envelope:
The same object may have different envelopes; for example, if a cube is viewed face-on, it has a square envelope:
A cube can never have a triangular or pentagonal envelope. If you know what envelopes an object can have, it narrows down what type of object it can be.
However, one should keep in mind that the envelope of an object really only gives limited information about the object. One should not falsely think that knowing the envelope of an object's projection is sufficient to uniquely identify the object. For example, an octahedron also has a square envelope, when viewed vertex-on:
Hence, if you only knew that an object has a square envelope, that doesn't tell you which object it is. The important thing about a projected image is not the envelope, but its internal structure. It is the internal structure that gives insight into the structure of the object. For example, knowing that a cube has a hexagonal envelope isn't very useful in itself; it is more insightful to see how the 3 projected faces of the cube are laid out inside that hexagon.
Look at the projection of the 3D cube again. What part of the image do you automatically focus on? Your attention naturally falls on the central region of the image, where 3 of the cube's edges meet at the corner that's facing you. In fact, your attention so spontaneously centers itself on this central region, that you are usually unaware that this view of the cube has a hexagonal envelope!
But if you were a 2D creature, your viewpoint would be quite different. You would see this image edge-on, and it would appear as a series of line segments. Upon further investigation, you would find that these line segments form a hexagonal shape. This is what your attention would center on. Being told that this image represents a cube, it would be tempting to identify the hexagonal envelope with the cube. However, the real point of interest lies inside the envelope.
All this may seem obvious, but it is very important to keep in mind when we start examining projections of complex 4D objects into 3D. These projections often have fascinating envelopes, such as rhombic dodecahedra, cuboctahedra, and other interesting polyhedra. It is tempting to unconsciously identify these polyhedral envelopes with the 4D object itself, because we are used to identifying 3D objects by the shape of their surfaces. However, most of the information about the structure of the 4D object lies inside the envelope.
A Projection of the 4D Hypercube
Consider the vertex-first projection of the 4D hypercube. Its envelope is a rhombic dodecahedron, a polyhedron bounded by 12 rhombuses. It is tempting to only regard this rhombic dodecahedral envelope, which is interesting in its own right:
However, if this is all we focus on, we would not know where the hypercube's cells are located in the image. In fact, we would not even see the hypercube vertex that the 4D viewer is looking at! The hypercube vertex in fact projects to the center of this dodecahedron, not to any of its external vertices. It is the central vertex highlighted in yellow below:
Knowing this internal structure of the projection also helps us locate the 4 cubical cells of the hypercube that are currently visible:
These cells appear to be distorted cubes, but they are actually perfectly regular cubes. They only appear distorted because they are foreshortened by perspective projection.
When a 4D being looks at the hypercube, its attention falls primarily on this layout of cubical cells in the image, not on the envelope itself. The envelope is only peripheral; the inside of the image is what is of interest. When examining projections of 4D objects, we should always focus on its internal structure rather than be distracted by its envelope.
Note that in the above views of the hypercube, only 4 of its 8 cells are visible. The reason for this is that the other 4 cells are behind these four, and are therefore obscured. We shall explain this in detail in the next chapter.
