In geometry, a near-miss Johnson solid is a convex polyhedron whose facesBy definition, each face is the intersection of the convex polyhedron with a different bounding plane, so no two faces are coplanar — any two adjacent faces form an angle less than 180 degrees. If instead a convex polyhedron is presented by giving a collection of polygons that a priori may be coplanar (e.g., by subdividing a face), one could write "strictly convex polyhedron" here to indicate the condition that no two of the polygons are coplanar, that no two meet in a 180-degree angle. This notion of "strictly convex" for polyhedra is not the same as the standard notion used for general convex sets: no convex polyhedra are strictly convex in the latter sense; see p. 263 of A. G. Khovanskii, Geometry of generalized virtual polyhedra, J. Math. Sciences 269 (2023), 256–269. 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. Because "close to" is not precisely defined, the notion of near-miss Johnson solid is not precisely defined either.
Some near-misses with high symmetry are also symmetrohedra with some truly regular polygon faces.
Some near-misses are also zonohedra.
Examples NameConway name ImageVertexconfigurations V E F F3 F4 F5 F6 F8 F10 F12 Symmetry Associahedron t4dP3 2 (5.5.5)12 (4.5.5) 14 21 9 3 6 Dih3order 12 Truncated triakis tetrahedront6kT4 (5.5.5)24 (5.5.6) 28 42 16 12 4 Td, [3,3]order 24Pentahexagonal pyritoheptacontatetrahedron12 (3.5.3.6)24 (3.3.5.6)24 (3.3.3.3.5)601327456126 Th, [3+,4]order 24 Chamfered cubecC 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 dodecahedron4 (5.5.5)12 (3.5.3.5)12 (3.3.5.5) 28 54 28 16 12 Td, [3,3]order 24 Chamfered dodecahedroncD 60 (5.6.6)20 (6.6.6) 80 120 42 12 30 Ih, [5,3]order 120 Rectified truncated icosahedronatI 60 (3.5.3.6)30 (3.6.3.6) 90 180 92 60 12 20 Ih, [5,3]order 120 Truncated truncated icosahedronttI 120 (3.10.12)60 (3.12.12) 180 270 92 60 12 20 Ih, [5,3]order 120 Expanded truncated icosahedronetI 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 icosahedronstI 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. It is possible to take an infinite amount of distinct coplanar misses from sections of the cubic honeycomb (alternatively convex polycubes) or alternated cubic honeycomb, ignoring any obscured faces.
Examples: 3.3.3.3.3.3
4.4.4.4
3.4.6.4:
See also Geodesic polyhedron Goldberg polyhedron Johnson solid Platonic solid Semiregular polyhedron Archimedean solid Prism Antiprism
References
External links Near-miss Johnson solid, Polytope Wiki (74) Johnson Solid Near Misses, Polyhedra by Jim McNeil (31) Near Misses, Craig S. Kaplan (5)