Talk:Cantellated 5-cell
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Projection terminology?
editNo good place to put talk, so I'll pick here for all of the uniform polychorons.
There's some varied interpretations of the polychoron projections. All of them that show cells shrinking towards the center must qualify as Stereographic projection to reduce from 4D to 3D, even if 3D is viewed as a Perspective projection down to 2D.
I think the Schlegel diagram, is the best description, named after the first person to use this approach to view 4D figures in 3d. I'd change this everywhere if I had time, but since I don't, perhaps better anyway to get some consensus. Tom Ruen 22:12, 3 January 2007 (UTC)
Thoughts?
- It looks like User:Rocchini made changes from Stereographic projection to Perspective projection on the assumption that the curved edge versions are spherical projections. I see BOTH linear and curved edges can be seen as Stereographic projection, depending on whether just the vertices, or edges and vertices are projected. BOTH are lowering the figure from 4D to 3D.
- I'm not sure how to handle this descriptively. The curved ones could be considered polyhedral tilings of the hypersphere, rather than polytopes. All the uniform polyhedrons are shown now as linear edges, so it is somewhat inconsistent to draw "hyperspherical tilings" as polychora, even if both are interesting.
- Since images are far from complete, I won't be choosy, but I'd say primary images in the articles ought to be linear versions if both types exist. Someday if we get more, I'm happy to have 3 types of images: (1) Orthogonal projections (drop 2 dimensions), (2) Linear Schlegel diagrams (3) Arcing Schlegel diagrams. Tom Ruen 22:22, 3 January 2007 (UTC)
IMNSHO, all projections that show cells shrinking towards the center should be categorized as stereographic, because basically the viewpoint lies on the surface of the polytope. A "proper" perspective projection should have a viewpoint that properly lies outside the polytope, and will in general have the nearest cell (to the 4D viewpoint) surrounded by other cells in the 3D projection (the cells expand "outward" before shrinking inwards to the antipodal cell). Except, of course, for polytopes with acute dihedral (dichoral?) angles, such as the pentatope or the tetracube. The analogy that I like to use is that of projecting a 3D object into 2D: the most familiar way of doing a perspective projection is place the target object at some distance away from the viewpoint ("camera") so that it most resembles what we see with our eyes; placing the viewpoint on the surface of the object produces an image quite foreign to what our eyes see, even if it is more appealing mathematically.—Tetracube 18:11, 5 January 2007 (UTC)
- I didn't follow all that, but I think we're in agreement. It is funny there's two "perspectives" involved - one is a special (polar) perspective that maps 4D to 3D, and once the cells are in 3D, a 3D perspective can again map to 2D. I agree best to keep the second perspective OUTSIDE the model, at least for a primary view of each model. What about just calling all of these as Schlegel diagrams? It would be good to get similar projection images of all the convex uniform polyhedra as well. I can do that, just haven't gotten around to it.
- OH - I do like User:Rocchini's use of partially coloring cells. I never thought of that and it is a very good effect for showing the cell elements. Tom Ruen 23:15, 5 January 2007 (UTC)