Near-miss Johnson solid

In geometry, a near-miss Johnson solid is a strictly convex polyhedron in which the faces are close to being regular polygons but in which some or all of the faces are not precisely regular. They generalize the Johnson solids, polyhedra in which all faces are regular, and "can often be physically constructed without noticing the discrepancy" between their regular and irregular faces.[1] The precise number of near misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.

Examples

Name
Conway name		 Image verfs V E F F3 F4 F5 F6 F8 F10 F12 Symmetry
Truncated triakis tetrahedron
t6kT		 4 (5.5.5)
24 (5.5.6)		 28 42 16     12 4       Td, [3,3]
order 24
Chamfered cube
cC		 24 (4.6.6)
8 (6.6.6)		 32 48 18   6   12       Oh, [4,3]
order 48
-- 12 (5.5.6)
6 (3.5.3.5)
12 (3.3.5.5)		 30 54 26 12   12 2       D6h, [6,2]
order 24
-- 6 (5.5.5)
9 (3.5.3.5)
12 (3.3.5.5)		 27 51 26 14   12         D3h, [3,2]
order 12
Tetrated dodecahedron 4 (5.5.5)
12 (3.5.3.5)
12 (3.3.5.5)		 28 54 28 16   12         Td, [3,3]
order 24
Chamfered dodecahedron
cD		 60 (5.6.6)
20 (6.6.6)		 80 120 42     12 30       Ih, [5,3]
order 120
Rectified truncated icosahedron
rtI		 60 (3.5.3.6)
30 (3.6.3.6)		 90 180 92 60   12 20       Ih, [5,3]
order 120
Truncated truncated icosahedron
ttI		 120 (3.10.12)
60 (3.12.12)		 180 270 92 60         12 20 Ih, [5,3]
order 120
Expanded truncated icosahedron
etI		 60 (3.4.5.4)
120 (3.4.6.4)		 180 360 182 60 90 12 20       Ih, [5,3]
order 120
Snub rectified 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

Coplanar misses

Some failed Johnson solid candidates have coplanar faces. These polyhedra can be perturbed to become convex with faces that are arbitrarily close to regular polygons. These cases use 4.4.4.4 vertex figures of the square tiling, 3.3.3.3.3.3 vertex figure of the triangular tiling, as well as 60 degree rhombi divided double equilateral triangle faces, or a 60 degree trapezoid as three equilateral triangles.

Examples: 3.3.3.3.3.3

4.4.4.4

3.4.6.4:

See also

References

  1. ↑ Kaplan, Craig S.; Hart, George W. (2001), "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons", Bridges: Mathematical Connections in Art, Music and Science.

External links