Near-miss Johnson solid

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A near-miss Johnson solid is a polyhedron that visually appears to qualify as a Johnson solid, but which does not meet all of the requirements to be one. It can have slightly irregular faces, usually close enough to regular that one can make a physical model with regular faces and not notice a problem, by having slightly differing edge lengths and/or internal angles of faces.

Near-miss Johnson solids can be divided into two types:

If all vertices of a near-miss Johnson solid have polyhedral angles measuring less than 2π (if the polygons surrounding those vertices were regular), the polyhedron is considered locally spherical.

Examples of locally spherical near-miss Johnson solids:
Tetrated dodecahedron Order-6 truncated triakis tetrahedron Sesquitruncated octahedron Hexagonally expanded snub dodecahedron

If a near-miss Johnson solid has any vertices that are planar (if the polygons surrounding those vertices were regular), the polyhedron is considered locally Euclidean.

Examples of locally Euclidean near-miss Johnson solids:
Chamfered cube Chamfered dodecahedron Rectified truncated icosahedron Icosahedrally expanded chamfered dodecahedron


The Goldberg polyhedra and their dual geodesic polyhedra may also be thought of as locally Euclidean near-miss Johnson solids since they approach the regular hexagonal tiling and the triangular tiling respectively as the number of faces increases.

Unlike Johnson solids which have at most prismatic or antiprismatic symmetry, these near-misses can have high degrees of symmetry.

List of near-miss Johnson solids[edit | edit source]

Name Image Vertex types Vertices Edges Faces Face types Symmetry Notes Type
Sesquitruncated square pyramid

(Heptagonal-faced polyhedron)

6x 3.7.8


4+4x 3.7.7


4x 3.4.3.7

20 32 14 4+4 triangles

1 square

4 heptagons

1 octagons

B2×I Locally spherical
Sesquitruncated triangular bipyramid

(Octagonal-faced polyhedron)

6+6+6x 3.8.8


12x 3.4.3.8

30 51 23 6+6+2 triangles

3 squares

6 octagons

A2×A1 Locally spherical
Sesquitruncated octahedron

(Enneagonal-faced polyhedron)

24x 3.4.3.9
24x 3.9.9 		48 84 38 24 triangles
6 squares
8 enneagons 		BC3 As seen here Locally spherical
Sesquitruncated rhombic dodecahedron

(Decagonal-faced polyhedron)

24x 3.4.3.10


24+24x 3.10.10

72 120 50 8+24 triangles
6 squares
12 decagons 		BC3 Locally spherical
Sesquitruncated trapezo-rhombic dodecahedron 6+12+12+6+6+6x 3.10.10


12+12x 3.4.3.10

72 120 50 2+6+6+6+12 triangles

6 squares

6+6 decagons

A2×A1 Locally spherical
Sesquitruncated pentagonal bifrustum 20+10+10+10x 3.10.10


20x 3.4.3.10

70 115 47 10+10+10 triangles

5 squares

2+10 decagons

H2×A1 Locally spherical
Sesquitruncated hexagonal bifrustum 12+12+12x 3.10.10


24x 3.4.3.10


24x 3.10.12

84 138 56 12+12+12 triangles

6 squares

12 decagons

2 dodecagons

G2×A1 Locally spherical
Sesquitruncated pentagonal icositetrahedron

(Hendecagonal-faced polyhedron)

24x 3.4.3.11
24+24+24+24+24x 3.11.11 		144 228 86 8+24+24 triangles
6 squares
24 hendecagons 		BC3+ As seen here Locally spherical
Sesquitruncated deltoidal icositetrahedron 24+24+24+24x 3.11.11


48+24x 3.4.3.11

168 288 122 8+24+24+24 triangles

6+12 squares

24 hendecagons

BC3 As seen here Locally spherical
Sesquitruncated gyrodeltoidal icositetrahedron

(Sesquitruncated dual of J37)

16+16+16+8+8+8+8+8+8x 3.11.11


16+16+16+16+8x 3.4.3.11

168 288 122 8+8+8+8+8+8+16+16 triangles

2+8+8 squares

8+16 hendecagons

B2×A1 Locally spherical
Sesquitruncated tetradeltoctapentagonal icositetrahedron

(Sesquitruncated dual of J45)

8+8+8+8+8+8+8+8+8+8+8+8+8+8x 3.11.11


8+8+8+8+8x 3.4.4.11

152 248 98 8+8+8+8+8+8+8+8 triangles

2+8 squares

8+8+8 hendecagons

B2×A1+ Locally spherical
Sesquitruncated order-6 truncated tetrakis hexahedron 24+24+24x 3.11.11


48+48x 3.11.12


24x 3.4.3.11

192 300 110 24+24+24 triangles

6 squares

24 hendecagons

8 dodecagons

BC3 Locally spherical
Sesquitruncated cuboctahedron 24x 3.4.3.12


24x 3.4.3.9


48x 3.9.12

96 168 74 48 triangles

12 squares

8 enneagons

6 dodecagons

BC3 locally Euclidean
Truncated order-6 truncated triakis tetrahedron 12+12+12x 3.10.10
24+24x 3.10.12 		84 126 44 4+12+12 triangles
12 decagons
4 dodecagons 		A3 Truncated #22[1]As seen here Locally spherical
Rectified order-6 truncated triakis tetrahedron 12+6x 5.3.5.3


24x 5.3.6.3

42 84 44 4+12+12 triangles
12 pentagons
4 hexagons 		A3 Rectified #22[1] Locally spherical
Expanded order-6 truncated triakis tetrahedron 12x 3.4.5.4


24+24x 4.5.12

60 96 38 4 triangles

6+12 squares

12 pentagons

4 dodecagons

A3 Expanded #22[1] Locally spherical
Tetrated dodecahedron 4x 5.5.5
12x 3.5.3.5

12x 3.3.5.5

28 54 28 4+12 triangles
12 pentagons 		A3 [2] Locally spherical
Chiral tetrated dodecahedron 4x 5.5.5
6x 3.5.3.5

12x 3.3.5.5

12x 3.3.3.3.5

34 72 40 4+12+12 triangles
12 pentagons 		A3+ As seen here, here, and here Locally spherical
Tetrahedrally expanded tetrated dodecahedron 12x 3.5.3.5
12x 3.4.5.4

24x 3.4.4.5

48 96 50 4+4+12 triangles
6+12 squares
12 pentagons 		A3 [3]

Also seen here

Locally spherical
Snub expanded tetrated dodecahedron 12x 3.5.3.5
12+12+12x 3.3.3.3.5 		48 114 68 4+4+12+12+12+12 triangles
12 pentagons 		A3+ [3] Locally spherical
Hexagonally expanded snub dodecahedron

(Pentahexagonal pyritoheptacontatetrahedron)

12x 3.5.6.5
24x 3.3.5.6

24x 3.3.3.3.5

60 132 74 24+24+8 triangles
12 pentagons
6 hexagons 		BC3/2 [2], [3] Locally spherical
Bipentagonal snub dodecahedron 12x 3.3.5.5
12+24x 3.3.3.3.5 		48 114 68 8+24+24 triangles
12 pentagons 		BC3/2 [3] Locally spherical
Order-4 truncated triangular bipyramid

(Three-dimensional associahedron)

2x 5.5.5
12x 4.5.5 		14 21 9 3 squares
6 pentagons 		A2×A1 [2] Locally spherical
Disdyakis order-4 truncated triangular bipyramid 2x 5.5.5
3x 3.3.3.3

6+6x 3.3.5.5

17 33 18 12 triangles
6 pentagons 		A2×A1 As seen here Locally spherical
Partially expanded disdyakis order-4 truncated triangular bipyramid 6x 3.4.5
6x 3.3.4.4

6x 3.3.5.5

12x 3.4.4.5

30 60 32 2+12 triangles
6+6 squares
6 pentagons 		A2×A1 Locally spherical
Elongated disdyakis order-4 truncated triangular bipyramid 2x 5.5.5
6x 3.3.4.4

6x 3.3.5.5

12x 3.4.4.5

26 51 27 12 triangles
3+6 squares
6 pentagons 		A2×A1 As seen here Locally spherical
Expanded disdyakis order-4 truncated triangular bipyramid 6x 3.4.5.4
12+12x 3.4.4.5

12x 3.4.4.4

42 84 44 2+12 triangles
3+3+6+6+6 squares
6 pentagons 		A2×A1 Locally spherical
Triated dodecahedron 6x 5.5.5
3+6x 3.5.3.5

12x 3.3.5.5

27 51 26 2+12 triangles
6+6 pentagons 		A2×A1 [2] Locally spherical
Partially truncated trapezo-rhombic dodecahedron 2x 5.5.5


6+6x 3.3.5.5


3+12x 3.5.3.5

29 57 30 6+12 triangles

6+6 pentagons

A2×A1 Locally spherical
Bisquare icosahedron 1x 3.4.3.4


2x 3.3.3.3.4


2x 3.3.3.4


2+2+2+2x 3.3.3.3.3

13 31 20 2+2+2+2+2+2+2+2+2 triangles

2 squares

K2+×I Locally spherical
Parabiaugmented 5-5-4 acrohedron 2x 3.3.3.3.4


2+2x 3.4.3.5


2x 3.3.3.3.4


2+2+2+2x 3.3.4.5


2x 3.3.3.3.3

18 40 24 2 pentagons

2+2 squares

2+2+2+2+2+2+2+2+2 triangles

K2+×I #26 on the Orchidpalms list[1], and a parabiaugmented #6 Locally spherical
Semi-rectified pentagonal bipyramid 10x 5.5.5


5x 3.5.3.5


10x 3.3.5.5

25 45 22 10 triangles

2+10 pentagons

H2×A1 Locally spherical
Diminished triaugmented truncated octahedron 6x 4.6.6


6x 3.3.4.6


6x 3.6.9


3x 3.3.3.9

21 36 17 6+3 triangles

3 squares

3+1 hexagons

1 enneagon

A2×I Locally spherical
Diminished triaugmented truncated icosahedron 10x 3.5.15


5x 3.3.3.15


10x 3.3.5.6


5+5x 5.6.6

35 60 27 10+5 triangles

5+1 pentagons

5 hexagons

1 pentadecagon

H2×I Locally spherical
Diminished rectified order-6 truncated triakis tetrahedron 6x 3.6.12


6x 3.5.12


6x 3.5.3.6


3+3+6x 3.5.3.5

24 42 20 3+3+3 triangles
6 pentagons
1 hexagon

1 dodecagon

A2×I Locally spherical
Sphenomegalunacorona 1x 3.3.3.3
2x 3.3.3.3.3

2x 3.3.3.3.5

4x 3.3.4.5

4x 4.4.5

13 26 15 2+4+4 triangles
1+2 squares
2 pentagons 		K2×I #1 on the orchidpalms list[1] Locally spherical
Tetrasquare icosioctahedron 2x 3.3.4.4
2+2x 3.3.3.3.3

2+2+4+4x 3.3.3.3.4

 18 44 28 2+2+4+4+4+4+4 triangles
2+2 squares 		K2×I #2 on the orchidpalms list[1] Locally spherical
Bisquare dodecahedron 1x 3.4.4
1x 3.3.4.4

1x 3.3.3.3.3

2x 3.3.3.4

2x 3.3.3.3.4

1+1x 3.3.3.3

9 19 12 1+1+2+2+2+2 triangles
2 squares 		A1×I×I #3 on the orchidpalms list[1] Locally spherical
Trisquare enneadecahedron 1x 4.4.4
3x 3.3.4.4

3x 3.3.3.3.4

3+3x 3.3.3.3.3

13 30 19 1+3+3+3+3+3 triangles
3 squares 		A2×I #4 on the orchidpalms list[1] Locally spherical
Belted bilunarotunda 1x 3.5.3.5
4x 3.4.5

4x 3.4.4.4

4x 3.4.3.5

13 24 13 2+4 triangles
1+2+2 squares
2 pentagons 		K2×I #5 on the orchidpalms list[1] Locally spherical
5-5-4 acrohedron 4x 4.4.5
4x 3.4.3.5

8x 3.3.4.5

16 30 16 8 triangles
4 squares
4 pentagons 		BC2×A1/2 #6 on the orchidpalms list[1] (diminished #23) Locally spherical
Diminished sphenocorona 2x 3.3.4.4
2x 3.3.3.4

2x 3.4.5

1+2x 3.3.3.5

9 17 10 1+2+2+2 triangles
1+1 squares
1 pentagon 		A1×I×I #7 on the orchidpalms list[1] [4] Locally spherical
1x 4.4.5
1x 3.3.4.4

2x 3.3.4.5

2x 3.3.3.3.5

1+1+2+2x 3.3.3.3.4


1+2+2x 3.3.3.3.3

17 40 25 1+2+2+2+2+2+2+2+2+2+2 ​triangles
1+2 squares
1 pentagons 		A1×I×I #8 on the orchidpalms list[1] Locally spherical
2x 3.6.6
4x 3.4.3.6

4x 3.3.4.4

4x 3.4.4.4

4x 3.3.4.6

18 35 19 2+4+4 triangles
1+2+2+2 squares
2 hexagons 		K2×I #9 on the orchidpalms list[1] Locally spherical
Trisquare icosiheptahedron 1x 3.4.3.4
2+4x 3.3.3.3.3

2+2+2+4x 3.3.3.3.4

 17 42 27 2+2+4+4+4+4+4 triangles
2+1 squares 		K2×I #10 on the orchidpalms list[1] Locally spherical
1x 3.3.3.3.5
1x 3.3.3.3.3

2x 3.4.5

2x 3.3.3.4

2x 3.3.4.4

2x 3.4.4.4

2x 3.4.3.5

12 24 14 1+2+2+2+2 triangles
1+1+2 squares
1 pentagon 		A1×I×I #11 on the orchidpalms list[1] Locally spherical
3x 3.4.3.4
6x 3.4.3.6

3+6x 3.4.5.4

6x 4.5.6

 24 45 23 3+3+3 triangles
3+6 squares
3 pentagons
2 hexagon 		A2×I #12 on the orchidpalms list[1] Locally spherical
Semi-rectified hexagonal bipyramid 12x 5.5.6
6x 3.5.3.5

12x 3.3.5.5

30 54 26 12 triangles
12 pentagons
2 hexagons 		G2×A1 #13 on the orchidpalms list[1] Locally spherical
Semi-rectified order-6 truncated hexagonal antitegum 12x 3.4.3.6
12x 3.4.4.4 		24 48 26 12 triangles
12 squares
2 hexagons 		I2(12)×A1/2 #14 on the orchidpalms list[1] Locally spherical
Bidiminished expanded semi-rectified hexagonal bipyramid 6x 3.5.3.5
24x 4.5.12

24x 3.4.4.5

54 96 44 12 triangles
6+12 squares
12 pentagons
2 dodecagons 		G2×A1 #15 on the orchidpalms list[1] Locally spherical
Tetrasquare icosioctahedron 2x 3.4.3.4
4x 3.3.3.3.3

4+4+4x 3.3.3.3.4

 18 44 28 4+4+4+4+4+4 triangles
4 squares 		K2+×I #16 on the orchidpalms list[1] Locally spherical
2x 3.5.5
2x 3.3.3.3.5

4x 3.4.3.5

2+4x 3.3.4.4

14 28 16 2+4+4 triangles
2+2 squares
2 pentagons 		K2×I #17 on the orchidpalms list[1] Locally spherical
Trisquare tridecahedron 1x 3.3.3
1x 3.4.4

1x 3.3.4.4

1x 3.3.3.3.3

2x 3.4.3.4

2x 3.3.3.4

2x 3.3.3.3.4

10 21 13 1+1+2+2+2+2 triangles
1+2 squares 		A1×I×I #18 on the orchidpalms list[1] Locally spherical
1x 3.5.5
1x 3.3.3.3

1x 3.3.5.5

1x 3.3.3.3.3

2x 3.4.3.5

2x 3.3.3.3.4

2+2x 3.3.4.4

2+2x 3.3.3.3.5

 16 35 21 1+1+2+2+2+2+2+2+2 triangles
1+1+1 squares
2 pentagons 		A1×I×I #19 on the orchidpalms list[1] Locally spherical
1x 3.3.3.3
2x 3.5.5

2x 3.3.4.4

4x 3.4.3.5

2+2+2x 3.3.5.5

 15 29 16 2+4+4 triangles
2 squares
4 pentagons 		K2×I #20 on the orchidpalms list[1] (contracted #29) Locally spherical
Tetrasquare pentadecahedron 2x 3.3.4.4
1+2x 3.4.3.4

1+1+2x 3.3.3.3.3

2+2+2x 3.3.3.3.4

 15 35 22 1+1+1+1+2+2+2+2+2+2+2 triangles
1+1+2 squares 		A1×I×I #21 on the orchidpalms list[1] (augmented #27) Locally spherical
Order-6 truncated triakis tetrahedron 4x 5.5.5
12+12x 5.5.6 		28 42 16 12 pentagons
4 hexagons 		A3 #22 on the orchidpalms list[1][2] Locally spherical
Pentakis 5-5-4 acrohedron 4+4+8x 3.3.3.3.4
4x 3.3.3.3.3 		20 50 32 4+8+8+8 triangles
4 squares 		BC2×A1/2 #23 on the orchidpalms list[1] (augmented #6) Locally spherical
1x 3.3.3.3.5
2+2x 3.3.4.5

1+1+1+2x 3.3.3.3.3

2+2+2+2x 3.3.3.3.4

 18 39 23 1+1+1+2+2+2+2+2+2+2+2 triangles
1+2 squares
1 pentagon 		A1×I×I #24 on the orchidpalms list[1] Locally spherical
Quasigyrodisphenocingulum 4x 3.3.4.4
4x 3.3.3.3.3

8x 3.3.3.3.4

16 38 24 4+4+4+8 triangles
4 squares 		K2+×I #25 on the orchidpalms list[1] Locally spherical
Diminished tetrasquare pentadecahedron 2x 3.3.4.4
2x 3.3.4.5

1+2x 3.4.3.4

1+2x 3.3.3.5

2+2x 3.3.3.3.4

 14 30 18 1+1+1+2+2+2+2+2 triangles
1+1+2 squares
1 pentagon 		A1×I×I #27 on the orchidpalms list[1] (diminished #21) Locally spherical
Second trisquare enneadecahedron 1x 3.3.3.3
1+1x 3.3.4.4

2x 3.3.3.4

1+1x 3.3.3.3.3

2+2+2x 3.3.3.3.4

13 30 19 1+1+2+2+2+2+2+2+2 triangles
1+2 squares 		A1×I×I #28 on the orchidpalms list[1] Locally spherical
2x 3.3.4.4
2x 3.5.5

2x 3.3.5.5

4x 3.4.4.4

4x 3.4.3.5

4+4x 3.4.4.5

 22 43 23 2+4+4 triangles
1+2+2+2+2 squares
4 pentagons 		K2×I #29 on the orchidpalms list[1] (elongated #20) Locally spherical
Gyrate partially expanded disdyakis order-4 truncated triangular bipyramid 6x 3.4.5.4
12x 3.3.4.5

12x 3.4.4.5

30 60 32 2+12 triangles
6+6 squares
6 pentagons 		G2×A1/2 #30 on the orchidpalms list[1] Locally spherical
Diminished disphenocingulum 2x 3.3.4.4
2x 3.3.3.3.3

4x 3.3.3.3.4

2+4x 3.3.3.6

14 31 19 2+2+4+4+4 triangles
2 squares
1 hexagon 		K2×I #31 on the orchidpalms list[1] Locally spherical
11E 2x 4.4.4

6x 3.3.4.4
3x 3.4.3.4

11 21 12 6 squares

6 triangles

A2×A1 As seen here[5] Locally spherical
14E 2x 4.4.4

6x 3.3.4.4
6x 3.3.3.3.4

14 30 18 6 squares

12 triangles

G2×A1/2 As seen here[5] Locally spherical
Chamfered cube 8x 6.6.6
24x 4.6.6 		32 48 18 6 squares
12 hexagons 		BC3 locally Euclidean
Sesquitruncated icosidodecahedron 60x 3.4.3.15


60x 3.4.3.9


120x 3.9.14

240 420 182 120 triangles

30 squares

20 enneagons

12 pentadecagons

H3 Locally

Hyperbolic

Chamfered dodecahedron 20x 6.6.6
60x 5.6.6 		80 120 42 12 pentagons
30 hexagons 		H3 locally Euclidean
Rectified truncated icosahedron 30x 3.6.3.6
60x 3.5.3.6 		90 180 92 60 triangles
12 pentagons
20 hexagons 		H3 [2]

Also mentioned here

locally Euclidean
Truncated truncated icosahedron 60x 3.12.12
120x 3.10.12 		180 270 92 60 triangles
12 decagons
20 dodecagons 		H3 [2] locally Euclidean
Rectified chamfered dodecahedron

(Rectified truncated rhombic triacontahedron)

60x 3.5.3.6
60x 3.6.3.6 		120 240 122 20+60 triangles
12 pentagons
30 hexagons 		H3 As seen here and here locally Euclidean
Icosahedrally expanded chamfered dodecahedron

(Dodecadiminished expanded truncated icosahedron)

60x 3.4.6.4
120x 3.4.6 		180 300 122 20 triangles
60 squares
30 hexagons
12 decagons 		H3 Mentioned here and here locally Euclidean
Expanded truncated icosahedron 60x 3.4.5.4
120x 3.4.6.4 		180 360 182 60 triangles
30+60 squares
12 pentagons
20 hexagons 		H3 [2]

Also mentioned here

locally Euclidean
Snub rectified truncated icosahedron 60x 3.3.3.3.5
60+60x 3.3.3.3.6 		180 450 272 60+60+120 triangles
12 pentagons
20 hexagons 		H3+ [2] locally Euclidean

External links[edit | edit source]

  • Orchidpalms. "Near Miss Polyhedra."
  • Robertlovespi. "Tag: Johnson solid."
  • Ed Baumann. "Modified Johnson Near Misses"

References[edit | edit source]

  1. ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 Johnson Solid Near Misses
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Near-miss Johnson solid
  3. ↑ 3.0 3.1 3.2 3.3 Near Misses based on dodecahedra
  4. ↑ Wedges
  5. ↑ 5.0 5.1 Miscellaneous Polyhedra
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