The Johnson solids are a class of geometric solids .
properties Johnson solids are strictly convex polyhedra , which are made up exclusively of regular polygons, but are neither Platonic solids , Archimedean solids , prisms nor antiprisms . What is common to the Catalan bodies is that the corners of a Johnson body are not identical. A specialty among the Johnson solids is the pseudo-rhombic cuboctahedron (J 37 ), the corners of which are locally uniform, but not global.
In 1966, Norman Johnson published a list of 92 such polyhedra; its assumption that it is complete was proven by Wictor Salgaller in 1969 .
list Johnson bodies are often referred to with , where the body number is in the list below . For example, the triangular dome .
J
n
{\ displaystyle J_ {n}}
n
{\ displaystyle n}
J
3
{\ displaystyle J_ {3}}
The following list shows the number of corners, the number of edges, the number of corner faces and the number of all faces of the respective body.
E.
{\ displaystyle E}
K
{\ displaystyle K}
F.
n
{\ displaystyle F_ {n}}
n
{\ displaystyle n}
F.
: =
F.
3
+
F.
4th
+
F.
5
+
...
{\ displaystyle F: = F_ {3} + F_ {4} + F_ {5} + \ ldots}
Pyramids, domes and rotundas
Y n
body
Illustration
Type
E.
K
F.
F 3
F 4
F 5
F 6
F 8
F 10
symmetry
1
Square pyramid
pyramid
5
8th
5
4th
1
0
0
0
0
C 4v
2
Pentagonal pyramid
pyramid
6th
10
6th
5
0
1
0
0
0
C 5v
3
Triangular dome
dome
9
15th
8th
4th
3
0
1
0
0
C 3v
4th
Square dome
dome
12
20th
10
4th
5
0
0
1
0
C 4v
5
Pentagonal dome
dome
15th
25th
12
5
5
1
0
0
1
C 5v
6th
Pentagonal rotunda
rotunda
20th
35
17th
10
0
6th
0
0
1
C 5v
Modified pyramids
Y n
Surname
Illustration
Type
E.
K
F.
F 3
F 4
F 5
symmetry
7th
elongated triangular pyramid
elongated pyramid
7th
12
7th
4th
3
0
C 3v
8th
extended square pyramid (at the same time extended hexahedron or square prism)
elongated pyramid
9
16
9
4th
5
0
C 4v
9
elongated pentagonal pyramid
elongated pyramid
11
20th
11
5
5
1
C 5v
10
twisted elongated square pyramid
twisted elongated pyramid
9
20th
13
12
1
0
C 4v
11
twisted elongated pentagonal pyramid (trimmed icosahedron)
twisted elongated pyramid
11
25th
16
15th
0
1
C 5v
12
Triangular pyramid
Bipyramid
5
9
6th
6th
0
0
D 3h
13
Pentagonal pyramid
Bipyramid
7th
15th
10
10
0
0
D 5h
14th
elongated triangular bipyramid
elongated bipyramid
8th
15th
9
6th
3
0
D 3h
15th
elongated square bipyramid (at the same time double expanded hexahedron or square prism)
elongated bipyramid
10
20th
12
8th
4th
0
D 4h
16
elongated pentagonal bipyramid
elongated bipyramid
12
25th
15th
10
5
0
D 5h
17th
twisted elongated square bipyramid
twisted elongated bipyramid
10
24
16
16
0
0
D 4d
Modified domes and rotundas
Y n
body
Illustration
Type
E.
K
F.
F 3
F 4
F 5
F 6
F 8
F 10
symmetry
18th
elongated triangular dome
elongated dome
15th
27
14th
4th
9
0
1
0
0
C 3v
19th
elongated square dome (trimmed small rhombic cuboctahedron)
elongated dome
20th
36
18th
4th
13
0
0
1
0
C 4v
20th
elongated pentagonal dome
elongated dome
25th
45
22nd
5
15th
1
0
0
1
C 5v
21st
elongated pentagonal rotunda
extended rotunda
30th
55
27
10
10
6th
0
0
1
C 5v
22nd
twisted elongated triangular dome
twisted elongated dome
15th
33
20th
16
3
0
1
0
0
C 3v
23
twisted elongated square dome
twisted elongated dome
20th
44
26th
20th
5
0
0
1
0
C 4v
24
twisted elongated pentagonal dome
twisted elongated dome
25th
55
32
25th
5
1
0
0
1
C 5v
25th
twisted elongated pentagonal rotunda
twisted elongated rotunda
30th
65
37
30th
0
6th
0
0
1
C 5v
26th
twisted double wedge
Double dome
8th
14th
8th
4th
4th
0
0
0
0
D 2d
27
Triangular double dome (twisted cuboctahedron, disheptahedron )
Double dome
12
24
14th
8th
6th
0
0
0
0
D 3h
28
Square double dome
Double dome
16
32
18th
8th
10
0
0
0
0
D 4h
29
twisted square double dome
Double dome
16
32
18th
8th
10
0
0
0
0
D 4d
30th
Pentagonal double dome
Double dome
20th
40
22nd
10
10
2
0
0
0
D 5h
31
twisted pentagonal double dome
Double dome
20th
40
22nd
10
10
2
0
0
0
D 5d
32
Pentagonal dome rotunda
Dome rotunda
25th
50
27
15th
5
7th
0
0
0
C 5v
33
twisted pentagonal dome rotunda
Dome rotunda
25th
50
27
15th
5
7th
0
0
0
C 5v
34
Pentagonal double rotunda (twisted icosidodecahedron )
Double rotunda
30th
60
32
20th
0
12
0
0
0
D 5h
35
elongated triangular double dome
elongated double dome
18th
36
20th
8th
12
0
0
0
0
D 3h
36
elongated twisted triangular double dome
elongated double dome
18th
36
20th
8th
12
0
0
0
0
D 3d
37
elongated twisted square double dome (twisted small rhombic cuboctahedron)
elongated double dome
24
48
26th
8th
18th
0
0
0
0
D 4d
38
elongated pentagonal double dome
elongated double dome
30th
60
32
10
20th
2
0
0
0
D 5h
39
elongated twisted pentagonal double dome
elongated double dome
30th
60
32
10
20th
2
0
0
0
D 5d
40
elongated pentagonal dome rotunda
extended dome rotunda
35
70
37
15th
15th
7th
0
0
0
C 5v
41
elongated twisted pentagonal dome rotunda
extended dome rotunda
35
70
37
15th
15th
7th
0
0
0
C 5v
42
elongated pentagonal double rotunda
extended double rotunda
40
80
42
20th
10
12
0
0
0
D 5h
43
elongated twisted pentagonal double rotunda
extended double rotunda
40
80
42
20th
10
12
0
0
0
D 5d
44
twisted elongated triangular double dome
twisted elongated double dome
18th
42
26th
20th
6th
0
0
0
0
D 3
45
twisted elongated square double dome
twisted elongated double dome
24
56
34
24
10
0
0
0
0
D 4
46
twisted elongated pentagonal double dome
twisted elongated double dome
30th
70
42
30th
10
2
0
0
0
D 5
47
twisted elongated pentagonal dome rotunda
twisted elongated dome rotunda
35
80
47
35
5
7th
0
0
0
C 5
48
twisted elongated pentagonal double rotunda
twisted elongated double rotunda
40
90
52
40
0
12
0
0
0
D 5
Extended prisms
Y n
body
Illustration
E.
K
F.
F 3
F 4
F 5
F 6
symmetry
49
extended triangular prism
7th
13
8th
6th
2
0
0
C 2v
50
double extended triangular prism
8th
17th
11
10
1
0
0
C 2v
51
triple expanded triangular prism
9
21st
14th
14th
0
0
0
D 3h
52
extended pentagonal prism
11
19th
10
4th
4th
2
0
C 2v
53
double extended pentagonal prism
12
23
13
8th
3
2
0
C 2v
54
extended hexagonal prism
13
22nd
11
4th
5
0
2
C 2v
55
double expanded hexagonal prism (para)
14th
26th
14th
8th
4th
0
2
D 2h
56
double expanded hexagonal prism (meta)
14th
26th
14th
8th
4th
0
2
C 2v
57
triple expanded hexagonal prism
15th
30th
17th
12
3
0
2
D 3h
Modified platonic solids
Y n
body
Illustration
Type
E.
K
F.
F 3
F 4
F 5
symmetry
58
extended dodecahedron
extended dodecahedron
21st
35
16
5
0
11
C 5v
59
double expanded dodecahedron (para)
extended dodecahedron
22nd
40
20th
10
0
10
D 5d
60
double expanded dodecahedron (meta)
extended dodecahedron
22nd
40
20th
10
0
10
C 2v
61
threefold expanded dodecahedron
extended dodecahedron
23
45
24
15th
0
9
C 3v
62
double trimmed icosahedron (meta)
trimmed icosahedron
10
20th
12
10
0
2
C 2v
63
triple trimmed icosahedron
trimmed icosahedron
9
15th
8th
5
0
3
C 3v
64
extended triple-trimmed icosahedron
-
10
18th
10
7th
0
3
C 3v
Modified Archimedean Solids
Y n
body
Illustration
Type
E.
K
F.
F 3
F 4
F 5
F 6
F 8
F 10
symmetry
65
extended truncated tetrahedron
extended truncated tetrahedron
15th
27
14th
8th
3
0
3
0
0
C 3v
66
extended truncated hexahedron
extended truncated hexahedron
28
48
22nd
12
5
0
0
5
0
C 4v
67
double expanded truncated hexahedron
extended truncated hexahedron
32
60
30th
16
10
0
0
4th
0
D 4h
68
extended truncated dodecahedron
extended truncated dodecahedron
65
105
42
25th
5
1
0
0
11
C 5v
69
doubly extended truncated dodecahedron (para)
extended truncated dodecahedron
70
120
52
30th
10
2
0
0
10
D 5d
70
doubly expanded truncated dodecahedron (meta)
extended truncated dodecahedron
70
120
52
30th
10
2
0
0
10
C 2v
71
triple expanded truncated dodecahedron
extended truncated dodecahedron
75
135
62
35
15th
3
0
0
9
C 3v
72
twisted rhombicosidodecahedron
twisted small rhombicosidodecahedron
60
120
62
20th
30th
12
0
0
0
C 5v
73
double twisted small rhombicosidodecahedron (para)
twisted small rhombicosidodecahedron
60
120
62
20th
30th
12
0
0
0
D 5d
74
double twisted small rhombicosidodecahedron (meta)
twisted small rhombicosidodecahedron
60
120
62
20th
30th
12
0
0
0
C 2v
75
small three-fold rhombicosidodecahedron
twisted small rhombicosidodecahedron
60
120
62
20th
30th
12
0
0
0
C 3v
76
trimmed small rhombicosidodecahedron
trimmed small rhombicosidodecahedron
55
105
52
15th
25th
11
0
0
1
C 5v
77
twisted trimmed small rhombicosidodecahedron (para)
twisted trimmed small rhombicosidodecahedron
55
105
52
15th
25th
11
0
0
1
C 5v
78
twisted trimmed small rhombicosidodecahedron (meta)
twisted trimmed small rhombicosidodecahedron
55
105
52
15th
25th
11
0
0
1
C s
79
double twisted trimmed small rhombicosidodecahedron
twisted trimmed small rhombicosidodecahedron
55
105
52
15th
25th
11
0
0
1
C s
80
double trimmed small rhombicosidodecahedron (para)
trimmed small rhombicosidodecahedron
50
90
42
10
20th
10
0
0
2
D 5d
81
double trimmed small rhombicosidodecahedron (meta)
trimmed small rhombicosidodecahedron
50
90
42
10
20th
10
0
0
2
C 2v
82
twisted double trimmed small rhombicosidodecahedron
twisted trimmed small rhombicosidodecahedron
50
90
42
10
20th
10
0
0
2
C 2v
83
threefold trimmed small rhombicosidodecahedron
trimmed small rhombicosidodecahedron
45
75
32
5
15th
9
0
0
3
C 3v
Rest
Y n
body
Illustration
E.
K
F.
F 3
F 4
F 5
F 6
symmetry
84
Trigondodecahedron
8th
18th
12
12
0
0
0
D 2d
85
beveled square anti-prism
16
40
26th
24
2
0
0
D 4d
86
Sphenocorona
10
22nd
14th
12
2
0
0
C 2v
87
enlarged sphenocorona
11
26th
17th
16
1
0
0
C s
88
Sphenomegacorona
12
28
18th
16
2
0
0
C 2v
89
Hebesphenomegacorona
14th
33
21st
18th
3
0
0
C 2v
90
Disphenocingulum
16
38
24
20th
4th
0
0
D 2d
91
Biluna double rotunda
14th
26th
14th
8th
2
4th
0
D 2h
92
Triangular lever phenorotunda
18th
36
20th
13
3
3
1
C 3v
Web links Individual evidence
^ Norman W. Johnson: Convex Solids with Regular Faces. In: Canadian Journal of Mathematics. Volume 18, 1966, ISSN 0008-414X
↑ Viktor A. Zalgaller: Convex Polyhedra with Regular Faces (= Seminars in Mathematics. Vol. 2, ISSN 0080-8873
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