Near-miss Johnson solid

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Template:Short description

In geometry, a near-miss Johnson solid is a convex polyhedron whose faces[1] 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.[2] 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

Name
Conway name		 Image Vertex
configurations		 V E F F3 F4 F5 F6 F8 F10 F12 Symmetry
Associahedron K5
t4dP3		 2 (5.5.5)
12 (4.5.5)		 14 21 9 3 6 Dih3
order 12
Truncated triakis tetrahedron
t6kT 		4 (5.5.5)
24 (5.5.6) 		28 42 16     12 4       Td, [3,3]
order 24
Pentahexagonal pyritoheptacontatetrahedron 12 (3.5.3.6)
24 (3.3.5.6)
24 (3.3.3.3.5) 		60 132 74 56 12 6 Th, [3+,4]
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
atI 		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

Template:See also 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:

  • Hexagonal cupola (Degenerate)
    Hexagonal cupola
    (Degenerate)

See also

References

Template:Reflist

Template:Near-miss Johnson solids navigator

  1. ↑ By 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.
  2. ↑ Template:Citation.
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