Projections
The vertex-first parallel projection of the 24-cell into 3-dimensional space has a rhombic dodecahedral envelope. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.
The cell-first parallel projection of the 24-cell into 3-dimensional space has a cuboctahedral envelope. Two of the octahedral cells, the nearest and farther from the viewer along the W-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.
The edge-first parallel projection has an elongated hexagonal dipyramidal envelope, and the face-first parallel projection has a nonuniform hexagonal bi-antiprismic envelope.
The vertex-first perspective projection of the 24-cell into 3-dimensional space has a tetrakis hexahedral envelope. The layout of cells in this image is similar to the image under parallel projection.
The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.
| Cell-first perspective projection | ||
|---|---|---|
In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled. |
In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent). |
Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta. Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the "equator" of the 24-cell, and bridge the two sets of cells. |
| Coxeter plane | F4 | |
|---|---|---|
| Graph | ||
| Dihedral symmetry | ||
| Coxeter plane | B3 / A2 (a) | B3 / A2 (b) |
| Graph | ||
| Dihedral symmetry | ||
| Coxeter plane | B4 | B2 / A2 |
| Graph | ||
| Dihedral symmetry | ||
Stereographic projection |
||
Animated cross-section of 24-cell |
A 3D projection of a 24-cell performing a simple rotation. |
|
A stereoscopic 3D projection of an icositetrachoron (24-cell). |
||
|
Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell |
||
Read more about this topic: 24-cell
Famous quotes containing the word projections:
“Western man represents himself, on the political or psychological stage, in a spectacular world-theater. Our personality is innately cinematic, light-charged projections flickering on the screen of Western consciousness.”
—Camille Paglia (b. 1947)
“Predictions of the future are never anything but projections of present automatic processes and procedures, that is, of occurrences that are likely to come to pass if men do not act and if nothing unexpected happens; every action, for better or worse, and every accident necessarily destroys the whole pattern in whose frame the prediction moves and where it finds its evidence.”
—Hannah Arendt (1906–1975)