In Polyhedra and Plane Tilings
Each platonic solid and its dual have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.)
The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
- The rectified tetrahedron, whose dual is the tetrahedron, is the tetratetrahedron, better known as the octahedron.
- The rectified octahedron, whose dual is the cube, is the cuboctahedron.
- The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron.
- A rectified square tiling is a square tiling.
- A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling.
Examples
| Family | Parent | Rectification | Dual |
|---|---|---|---|
Tetrahedron |
Tetratetrahedron |
Tetrahedron |
|
Cube |
Cuboctahedron |
Octahedron |
|
Dodecahedron |
Icosidodecahedron |
Icosahedron |
|
Hexagonal tiling |
Trihexagonal tiling |
Triangular tiling |
|
Order-3 heptagonal tiling |
Triheptagonal tiling |
Order-7 triangular tiling |
|
Square tiling |
Square tiling |
Square tiling |
|
Order-4 pentagonal tiling |
tetrapentagonal tiling |
Order-5 square tiling |
Read more about this topic: Rectification (geometry)
Famous quotes containing the word plane:
“At the moment when a man openly makes known his difference of opinion from a well-known party leader, the whole world thinks that he must be angry with the latter. Sometimes, however, he is just on the point of ceasing to be angry with him. He ventures to put himself on the same plane as his opponent, and is free from the tortures of suppressed envy.”
—Friedrich Nietzsche (1844–1900)