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near-miss Johnson solid

{{Short description|Convex polyhedron whose faces are almost regular polygons}}

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.{{citation |last1=Kaplan |first1=Craig S. |last2=Hart |first2=George W. |author2-link=George W. Hart |contribution=Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons |title=Bridges: Mathematical Connections in Art, Music and Science |year=2001 |url=https://cs.uwaterloo.ca/~csk/publications/Papers/kaplan_hart_2001.pdf}}. The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.

Some near-misses with high symmetry are also symmetrohedra with some truly regular polygon faces.

Some near-misses are also zonohedra.

Examples

class="wikitable sortable"

! Name
Conway name!! Image!!Vertex
configurations!! V!! E!! F!! F3!! F4!! F5!! F6!! F8!! F10!! F12!! Symmetry

align=center

| Associahedron
[https://levskaya.github.io/polyhedronisme/?recipe=C100A1t4dP3 t4dP3]

80px2 (5.5.5)
12 (4.5.5)		1421936Dih3
order 12
align=center

| Truncated triakis tetrahedron
[https://levskaya.github.io/polyhedronisme/?recipe=C100A1t6kT t6kT]

|80px

|4 (5.5.5)
24 (5.5.6)

| 28

| 42

| 16

|  

|  

| 12

| 4

|  

|  

|  

| Td, [3,3]
order 24

align=center

|Pentahexagonal pyritoheptacontatetrahedron

|80px

|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

align=center

| Chamfered cube
[https://levskaya.github.io/polyhedronisme/?recipe=C100A1cC cC]

|80px

| 24 (4.6.6)
8 (6.6.6)

| 32

| 48

| 18

|  

| 6

|  

| 12

|  

|  

|  

| Oh, [4,3]
order 48

align=center

| --

| 80px

|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

align=center

| --

| 80px

|6 (5.5.5)
9 (3.5.3.5)
12 (3.3.5.5)

| 27

| 51

| 26

| 14

|  

| 12

|  

|  

|  

|  

| D3h, [3,2]
order 12

align=center

| Tetrated dodecahedron

|80px

|4 (5.5.5)
12 (3.5.3.5)
12 (3.3.5.5)

| 28

| 54

| 28

| 16

|  

| 12

|  

|  

|  

|  

| Td, [3,3]
order 24

align=center

| Chamfered dodecahedron
[https://levskaya.github.io/polyhedronisme/?recipe=C100A1cD cD]

|80px

| 60 (5.6.6)
20 (6.6.6)

| 80

| 120

| 42

|  

|  

| 12

| 30

|  

|  

|  

| Ih, [5,3]
order 120

align=center

| Rectified truncated icosahedron
[https://levskaya.github.io/polyhedronisme/?recipe=C400A1atI atI]

| 80px

| 60 (3.5.3.6)
30 (3.6.3.6)

| 90

| 180

| 92

| 60

|  

| 12

| 20

|  

|  

|  

| Ih, [5,3]
order 120

align=center

| Truncated truncated icosahedron
[https://levskaya.github.io/polyhedronisme/?recipe=C1000ttI ttI]

| 80px

| 120 (3.10.12)
60 (3.12.12)

| 180

| 270

| 92

| 60

|  

|  

|  

|  

| 12

| 20

| Ih, [5,3]
order 120

align=center

| Expanded truncated icosahedron
[https://levskaya.github.io/polyhedronisme/?recipe=C1000aatI etI]

| 80px

| 60 (3.4.5.4)
120 (3.4.6.4)

| 180

| 360

| 182

| 60

| 90

| 12

| 20

|  

|  

|  

| Ih, [5,3]
order 120

align=center

| Snub rectified truncated icosahedron
[https://levskaya.github.io/polyhedronisme/?recipe=C1000stI stI]

| 80px

| 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

{{See also|Deltahedron#Non-strictly convex cases}}

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

File:Digonal orthobicupola.svg|Rhombic prism

File:Tet-oct-wedge.png|Wedge

File:Gyroelongated triangular bipyramid.png|Trigonal trapezohedron

File:Augmented octahedron.png|Gyroelongated trigonal pyramid

File:Triangulated monorectified tetrahedron.png|Triangulated monorectified tetrahedron

File:TetOct2 solid2.png|Elongated octahedron

File:Triangulated tetrahedron.png|Tetratetrahedron, triangulated tetrahedron

File:Augmented triangular cupola.png|Augmented triangular cupola

File:Triangulated truncated triangular bipyramid.png|Triangulated truncated triangular bipyramid

File:Double diminished icosahedron.png|Edge-contracted icosahedron

File:Triangulated hexagonal prism.png|Hexagonal prism

File:Augmented hexagonal antiprism flat.png|Hexagonal antiprism,
Gyroelongated hexagonal pyramid

File:Augmented triangular cupula.png|Triangular cupola

File:Triangulated truncated tetrahedron.png|Truncated tetrahedron

File:Triangulated truncated octahedron.png|Truncated octahedron

4.4.4.4

File:Partial cubic honeycomb.png|Square icositetrahedron
(Cube)

3.4.6.4:

File:Hexagonal cupola flat.svg|Hexagonal cupola
(Degenerate)

See also

References

{{Reflist}}

External links

  • [https://polytope.miraheze.org/wiki/Near-miss_Johnson_solid Near-miss Johnson solid], Polytope Wiki (74)
  • [https://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/jsmn.htm Johnson Solid Near Misses], [https://www.orchidpalms.com/polyhedra/ Polyhedra by Jim McNeil] (31)
  • [https://cs.uwaterloo.ca/~csk/other/nearmisses/ Near Misses], Craig S. Kaplan (5)

{{Near-miss Johnson solids navigator}}

Category:Polyhedra