Near-miss Johnson solid


In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons. Some high symmetry near-misses are also symmetrohedra with some perfect regular polygon faces.

Examples

Name
Conway name		ImageVertex
configurations		VEFF3F4F5F6F8F10F12Symmetry
Truncated triangular bipyramid
2
12 		1421936Dih3
order 12
Truncated triakis tetrahedron
4
24 		284216 124 Td,
order 24
Pentahexagonal pyritoheptacontatetrahedron601327456126Th,
order 24
Chamfered cube
24
8 		324818 6 12 Oh,
order 48
--12
6
12 		30542612 122 D6h,
order 24
--6
9
12 		27512614 12 D3h,
order 12
Tetrated dodecahedron4
12
12 		28542816 12 Td,
order 24
Chamfered dodecahedron
60
20 		8012042 1230 Ih,
order 120
Rectified truncated icosahedron
60
30 		901809260 1220 Ih,
order 120
Truncated truncated icosahedron
120
60 		1802709260 1220Ih,
order 120
Expanded truncated icosahedron
60
120 		18036018260901220 Ih,
order 120
Snub rectified truncated icosahedron
60
120 		180450272240 1220 I, +
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: