In geometry, the Johnson almost polyhedron is a strictly convex polyhedron in which the faces are close to regular polygons , but some or all of them are not quite regular. The concept generalizes Johnson’s polyhedra and “can often be physically constructed without any noticeable difference” between irregular faces and regular ones. [1] The exact number of "almost" Johnson polyhedra depends on the requirements of how accurately the edges approach regular polygons.
Content
Examples
| Title Conway Name | Picture | Vertex configuration | V | E | F | F 3 | F 4 | F 5 | F 6 | F 8 | F 10 | F 12 | Symmetry |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
t6kT | 4 (5.5.5) 24 (5.5.6) | 28 | 42 | sixteen | 12 | four | T d , [3,3] order 24 | ||||||
cC | 24 (4.6.6) 8 (6.6.6) | 32 | 48 | 18 | 6 | 12 | O h , [4.3] order 48 | ||||||
| - | 12 (5.5.6) 6 (3.5.3.5) 12 (3.3.5.5) | thirty | 54 | 26 | 12 | 12 | 2 | D 6h , [6.2] order 24 | |||||
| - | 6 (5.5.5) 9 (3.5.3.5) 12 (3.3.5.5) | 27 | 51 | 26 | 14 | 12 | D 3h , [3,2] order 12 | ||||||
| 4 (5.5.5) 12 (3.5.3.5) 12 (3.3.5.5) | 28 | 54 | 28 | sixteen | 12 | T d , [3,3] order 24 | |||||||
cD | 60 (5.6.6) 20 (6.6.6) | 80 | 120 | 42 | 12 | thirty | I h , [5.3] order 120 | ||||||
rtI | 60 (3.5.3.6) 30 (3.6.3.6) | 90 | 180 | 92 | 60 | 12 | 20 | I h , [5.3] order 120 | |||||
| Truncated Truncated Icosahedron ttI | 120 (3.10.12) 60 (12/3/12) | 180 | 270 | 92 | 60 | 12 | 20 | I h , [5.3] order 120 | |||||
| Extended truncated icosahedron etI | 60 (3.4.5.4) 120 (3.4.6.4) | 180 | 360 | 182 | 60 | 90 | 12 | 20 | I h , [5.3] order 120 | ||||
| Flat Nose Truncated Truncated Icosahedron stI | 60 (3.3.3.3.5) 120 (3.3.3.3.6) | 180 | 450 | 272 | 240 | 12 | 20 | I , [5.3] + order 60 |
Johnson's almost polyhedrons with coplanar faces
Some candidates for Johnson's almost polyhedra have coplanar faces. These polyhedra can be slightly deformed so that the faces are arbitrarily close to regular polygons. These cases use vertex figures 4.4.4.4 of a square mosaic , vertex figures 3.3.3.3.3.3 of a triangular mosaic , and rhombuses with a 60 ° angle divided by two regular triangles, or trapezoid with a 60 ° angle as three regular triangles.
Examples: 3.3.3.3.3.3
Rhombic prism
Wedge
Twisted elongated triangular pyramid
Triangulated one-truncated tetrahedron
Triangulated tetrahedron
Extended triangular dome
Triangulated Truncated Bipyramid
Octahedron
Hexagonal prism
Hexagonal antiprism
Triangular dome
Truncated tetrahedron
Truncated octahedron
4.4.4.4
Square icotetrahedron
( Cube )
3.4.6.4:
Hexagonal dome
(degenerate)
See also
- Regular polyhedron
- Semi-Regular Polyhedron
- Archimedean body
- Prism
- Antiprism
- Johnson's Polyhedron
- Geodesic dome
Notes
- ↑ Craig S. Kaplan, George W. Hart. Bridges: Mathematical Connections in Art, Music and Science. - 2001.