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{{Short description|Convex polyhedron whose faces are almost regular polygons}}
⚫
In [[geometry]], a '''near-miss Johnson solid''' is a strictly [[convex set|convex]] [[polyhedron]] in which the faces are close to being [[regular polygon]]s but in which some or all of the faces are not precisely regular. They generalize the [[Johnson solid]]s, polyhedra in which all faces are regular, and "can often be physically constructed without noticing the discrepancy" between their regular and irregular faces.<ref>{{citation|contribution=Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons|title=Bridges: Mathematical Connections in Art, Music and Science|year=2001|url=http://www.cgl.uwaterloo.ca/~csk/papers/kaplan_hart_bridges2001.pdf|first1=Craig S.|last1=Kaplan|first2=George W.|last2=Hart|author2-link=George W. Hart}}.</ref> 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 [[symmetrohedron]] with some perfect regular polygon faces.
{{merge to|Johnson solid|discuss=Talk:Near-miss Johnson solid#Merge Near-miss Johnson solid into Johnson solid|date=June 2024}}


⚫
In [[geometry]], a '''near-miss Johnson solid''' is a strictly [[convex set|convex]] [[polyhedron]] whose [[face (geometry)|faces]] are close to being [[regular polygon]]s 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.<ref>{{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}}.</ref> The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.
⚫
==Examples==


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

Some near-misses are also [[zonohedron|zonohedra]].

⚫
==Examples==
{| class="wikitable sortable"
{| class="wikitable sortable"
! Name<BR>[[Conway polyhedron notation|Conway name]]!! Image!![[Vertex configuration|Vertex<BR>configurations]]!! V!! E!! F!! F<sub>3</sub>!! F<sub>4</sub>!! F<sub>5</sub>!! F<sub>6</sub>!! F<sub>8</sub>!! F<sub>10</sub>!! F<sub>12</sub>!! [[List of spherical symmetry groups|Symmetry]]
! Name<br>[[Conway polyhedron notation|Conway name]]!! Image!![[Vertex configuration|Vertex<br>configurations]]!! V!! E!! F!! F<sub>3</sub>!! F<sub>4</sub>!! F<sub>5</sub>!! F<sub>6</sub>!! F<sub>8</sub>!! F<sub>10</sub>!! F<sub>12</sub>!! [[List of spherical symmetry groups|Symmetry]]
|- align=center
|- align=center
| Truncated [[triangular bipyramid]]<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C100A1t4dP3 t4dP3]||[[File:Associahedron.gif|80px]]|| 3 (5.5.5)<BR>12 (4.5.5)|| 14|| 21|| 9|| || 3|| 6|||||||||| Dih<sub>3</sub>
| [[Associahedron]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C100A1t4dP3 t4dP3]||[[File:Associahedron.gif|80px]]|| 2 (5.5.5)<br>12 (4.5.5)|| 14|| 21|| 9|| || 3|| 6|||||||||| Dih<sub>3</sub><br>order 12
|- align=center
|- align=center
| [[Truncated triakis tetrahedron]]<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C100A1t6kT t6kT]
| [[Truncated triakis tetrahedron]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C100A1t6kT t6kT]
|[[Image:Truncated triakis tetrahedron.png|80px]]
|[[File:Truncated triakis tetrahedron.png|80px]]
|4 (5.5.5)<BR>24 (5.5.6)
|4 (5.5.5)<br>24 (5.5.6)
| 28
| 28
| 42
| 42
Line 21: Line 27:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''T''<sub>d</sub>, [3,3]<BR>order 24
| ''T''<sub>d</sub>, [3,3]<br>order 24
|- align=center
|[[Pentahexagonal pyritoheptacontatetrahedron]]
|[[File:Pyritohedral_near-miss_johnson.png|80px]]
|12 (3.5.3.6)<br>24 (3.3.5.6)<br>24 (3.3.3.3.5)
|60
|132
|74
|56
|
|12
|6
|
|
|
| ''T''<sub>h</sub>, [3<sup>+</sup>,4]<br>order 24
|- align=center
|- align=center
| [[Chamfered cube]]<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C100A1cC cC]
| [[Chamfered cube]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C100A1cC cC]
|[[File:Truncated rhombic dodecahedron.png|80px]]
|[[File:Truncated rhombic dodecahedron.png|80px]]
| 24 (4.6.6)<BR>8 (6.6.6)
| 24 (4.6.6)<br>8 (6.6.6)
| 32
| 32
| 48
| 48
Line 36: Line 57:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''O''<sub>h</sub>, [4,3]<BR>order 48
| ''O''<sub>h</sub>, [4,3]<br>order 48
|- align=center
|- align=center
| --
| --
| [[Image:Hexpenttri near-miss Johnson solid.png|80px]]
| [[File:Hexpenttri near-miss Johnson solid.png|80px]]
|12 (5.5.6)<BR>6 (3.5.3.5)<BR>12 (3.3.5.5)
|12 (5.5.6)<br>6 (3.5.3.5)<br>12 (3.3.5.5)
| 30
| 30
| 54
| 54
Line 51: Line 72:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''D''<sub>6h</sub>, [6,2]<BR>order 24
| ''D''<sub>6h</sub>, [6,2]<br>order 24
|- align=center
|- align=center
| --
| --
| [[Image:Dh3 symmetry dodecahedral nearmiss johnson.png|80px]]
| [[File:Dh3 symmetry dodecahedral nearmiss johnson.png|80px]]
|6 (5.5.5)<BR>9 (3.5.3.5)<BR>12 (3.3.5.5)
|6 (5.5.5)<br>9 (3.5.3.5)<br>12 (3.3.5.5)
| 27
| 27
| 51
| 51
Line 66: Line 87:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''D''<sub>3h</sub>, [3,2]<BR>order 12
| ''D''<sub>3h</sub>, [3,2]<br>order 12
|- align=center
|- align=center
| [[Tetrated dodecahedron]]
| [[Tetrated dodecahedron]]
|[[Image:Tetrated Dodecahedron.gif|80px]]
|[[File:Tetrated dodecahedron.svg|80px]]
|4 (5.5.5)<BR>12 (3.5.3.5)<BR>12 (3.3.5.5)
|4 (5.5.5)<br>12 (3.5.3.5)<br>12 (3.3.5.5)
| 28
| 28
| 54
| 54
Line 81: Line 102:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''T''<sub>d</sub>, [3,3]<BR>order 24
| ''T''<sub>d</sub>, [3,3]<br>order 24
|- align=center
|- align=center
| [[Chamfered dodecahedron]]<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C100A1cD cD]
| [[Chamfered dodecahedron]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C100A1cD cD]
|[[File:Truncated rhombic triacontahedron.png|80px]]
|[[File:Truncated rhombic triacontahedron.png|80px]]
| 60 (5.6.6)<BR>20 (6.6.6)
| 60 (5.6.6)<br>20 (6.6.6)
| 80
| 80
| 120
| 120
Line 96: Line 117:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''I''<sub>h</sub>, [5,3]<BR>order 120
| ''I''<sub>h</sub>, [5,3]<br>order 120
|- align=center
|- align=center
| [[Rectified truncated icosahedron]]<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C400A1atI atI]
| [[Rectified truncated icosahedron]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C400A1atI atI]
| [[File:Rectified truncated icosahedron.png|80px]]
| [[File:Rectified truncated icosahedron.png|80px]]
| 60 (3.5.3.6)<BR>30 (3.6.3.6)
| 60 (3.5.3.6)<br>30 (3.6.3.6)
| 90
| 90
| 180
| 180
Line 111: Line 132:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''I''<sub>h</sub>, [5,3]<BR>order 120
| ''I''<sub>h</sub>, [5,3]<br>order 120
|- align=center
|- align=center
| Truncated truncated icosahedron<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C1000ttI ttI]
| [[Truncated truncated icosahedron]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C1000ttI ttI]
| [[File:Truncated truncated icosahedron.png|80px]]
| [[File:Truncated truncated icosahedron.png|80px]]
| 120 (3.10.12)<BR>60 (3.12.12)
| 120 (3.10.12)<br>60 (3.12.12)
| 180
| 180
| 270
| 270
Line 126: Line 147:
| 12
| 12
| 20
| 20
| ''I''<sub>h</sub>, [5,3]<BR>order 120
| ''I''<sub>h</sub>, [5,3]<br>order 120
|- align=center
|- align=center
| Expanded truncated icosahedron<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C1000aatI etI]
| [[Expanded truncated icosahedron]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C1000aatI etI]
| [[File:Expanded truncated icosahedron.png|80px]]
| [[File:Expanded truncated icosahedron.png|80px]]
| 60 (3.4.5.4)<BR>120 (3.4.6.4)
| 60 (3.4.5.4)<br>120 (3.4.6.4)
| 180
| 180
| 360
| 360
Line 141: Line 162:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''I''<sub>h</sub>, [5,3]<BR>order 120
| ''I''<sub>h</sub>, [5,3]<br>order 120
|- align=center
|- align=center
| Snub rectified truncated icosahedron<BR>[http://levskaya.github.io/polyhedronisme/?recipe=C1000stI stI]
| [[Snub rectified truncated icosahedron]]<br>[https://levskaya.github.io/polyhedronisme/?recipe=C1000stI stI]
| [[File:Snub rectified truncated icosahedron.png|80px]]
| [[File:Snub rectified truncated icosahedron.png|80px]]
| 60 (3.3.3.3.5)<BR>120 (3.3.3.3.6)
| 60 (3.3.3.3.5)<br>120 (3.3.3.3.6)
| 180
| 180
| 450
| 450
Line 156: Line 177:
| &nbsp;
| &nbsp;
| &nbsp;
| &nbsp;
| ''I'', [5,3]<sup>+</sup><BR>order 60
| ''I'', [5,3]<sup>+</sup><br>order 60
|}
|}


==Coplanar misses==
==Coplanar misses==
{{See|Deltahedron#Non-strictly convex cases}}
{{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.
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 [[polycube]]s) or [[alternated cubic honeycomb]], ignoring any obscured faces.


Examples:
Examples:
Line 169: Line 190:
File:Tet-oct-wedge.png|[[Wedge (geometry)|Wedge]]
File:Tet-oct-wedge.png|[[Wedge (geometry)|Wedge]]
File:Gyroelongated triangular bipyramid.png|[[Trigonal trapezohedron]]
File:Gyroelongated triangular bipyramid.png|[[Trigonal trapezohedron]]
File:Augmented_octahedron.png|Gyroelongated trigonal pyramid
File:Augmented octahedron.png|Gyroelongated trigonal pyramid
File:Triangulated monorectified tetrahedron.png|Triangulated monorectified tetrahedron
File:Triangulated monorectified tetrahedron.png|Triangulated monorectified tetrahedron
File:TetOct2_solid2.png|[[Elongated octahedron]]
File:TetOct2 solid2.png|[[Elongated octahedron]]
File:Triangulated_tetrahedron.png|[[Tetratetrahedron]], Triangulated tetrahedron
File:Triangulated tetrahedron.png|[[Tetratetrahedron]], triangulated tetrahedron
File:Augmented triangular cupola.png|Augmented triangular cupola
File:Augmented triangular cupola.png|Augmented triangular cupola
File:Triangulated_truncated_triangular_bipyramid.png|Triangulated truncated triangular bipyramid
File:Triangulated truncated triangular bipyramid.png|Triangulated truncated triangular bipyramid
File:Double diminished icosahedron.png|[[Edge-contracted icosahedron]]
File:Double_diminished_icosahedron.png|[[Octadecahedron]]
File:Triangulated hexagonal prism.png|[[Hexagonal prism]]
File:Triangulated hexagonal prism.png|[[Hexagonal prism]]
File:Augmented hexagonal antiprism flat.png|[[Hexagonal antiprism]],<BR>[[Gyroelongated hexagonal pyramid]]
File:Augmented hexagonal antiprism flat.png|[[Hexagonal antiprism]],<br>[[Gyroelongated hexagonal pyramid]]
File:Augmented triangular cupula.png|[[Triangular cupola]]
File:Augmented triangular cupula.png|[[Triangular cupola]]
File:Triangulated truncated tetrahedron.png|[[Truncated tetrahedron]]
File:Triangulated truncated tetrahedron.png|[[Truncated tetrahedron]]
Line 184: Line 205:
4.4.4.4
4.4.4.4
<gallery>
<gallery>
File:Partial cubic honeycomb.png|[[Square icositetrahedron]]<BR>([[Cube]])
File:Partial cubic honeycomb.png|[[Square icositetrahedron]]<br>([[Cube]])
</gallery>
</gallery>
3.4.6.4:
3.4.6.4:
<gallery>
<gallery>
File:Hexagonal cupola flat.png|Hexagonal cupola<BR>(Degenerate)
File:Hexagonal cupola flat.png|Hexagonal cupola<br>(Degenerate)
</gallery>
</gallery>


==See also==
==See also==
⚫
*[[Geodesic polyhedron]]
*[[Goldberg polyhedron]]
⚫
*[[Johnson solid]]
*[[Platonic solid]]
*[[Platonic solid]]
*[[Semiregular polyhedron]]
*[[Semiregular polyhedron]]
Line 197: Line 221:
**[[Prism (geometry)|Prism]]
**[[Prism (geometry)|Prism]]
**[[Antiprism]]
**[[Antiprism]]
⚫
*[[Johnson solids]]
*[[Geodesic sphere]]
⚫
*[[Goldberg polyhedron]]


==References==
==References==
{{reflist}}
{{Reflist}}


==External links==
==External links==
*[https://polytope.miraheze.org/wiki/Near-miss_Johnson_solid Near-miss Johnson solid], Polytope Wiki (74)
⚫
*[http://www.cgl.uwaterloo.ca/~csk/projects/nearmisses Near Misses]
*[http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/jsmn.htm 24 Johnson Solid Near Misses]
*[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}}
{{Near-miss Johnson solids navigator}}

Latest revision as of 07:21, 10 June 2024

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.[1] 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

[edit]
Name
Conway name		 Image Vertex
configurations		 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 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

[edit]
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

4.4.4.4

3.4.6.4:

  • Hexagonal cupola (Degenerate)
    Hexagonal cupola
    (Degenerate)

See also

[edit]

References

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