قائمة مجسمات جونسون

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. In 1966, Norman Johnson published a list which included all 92 solids, and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller proved in 1969 that Johnson's list was complete.

Other polyhedra can be constructed that have only approximately regular planar polygon faces, and are informally called near-miss Johnson solids; there can be no definitive count of them.

The various sections that follow have tables listing all 92 Johnson solids, and values for some of their most important properties. Each table allows sorting by column so that numerical values, or the names of the solids, can be sorted in order.

الرؤوس والحواف والأوجه والتناظر

Jn	Solid name	Net	Image	V	E	F	F3	F4	F5	F6	F8	F10	Symmetry group	Order
1	Square pyramid			5	8	5	4	1					C4v, [4], (*44)	8
2	Pentagonal pyramid			6	10	6	5		1				C5v, [5], (*55)	10
3	Triangular cupola			9	15	8	4	3		1			C3v, [3], (*33)	6
4	Square cupola			12	20	10	4	5			1		C4v, [4], (*44)	8
5	Pentagonal cupola			15	25	12	5	5	1			1	C5v, [5], (*55)	10
6	Pentagonal rotunda			20	35	17	10		6			1	C5v, [5], (*55)	10
7	Elongated triangular pyramid			7	12	7	4	3					C3v, [3], (*33)	6
8	Elongated square pyramid			9	16	9	4	5					C4v, [4], (*44)	8
9	Elongated pentagonal pyramid			11	20	11	5	5	1				C5v, [5], (*55)	10
10	Gyroelongated square pyramid			9	20	13	12	1					C4v, [4], (*44)	8
11	Gyroelongated pentagonal pyramid			11	25	16	15		1				C5v, [5], (*55)	10
12	Triangular bipyramid			5	9	6	6						D3h, [3,2], (*223)	12
13	Pentagonal bipyramid			7	15	10	10						D5h, [5,2], (*225)	20
14	Elongated triangular bipyramid			8	15	9	6	3					D3h, [3,2], (*223)	12
15	Elongated square bipyramid			10	20	12	8	4					D4h, [4,2], (*224)	16
16	Elongated pentagonal bipyramid			12	25	15	10	5					D5h, [5,2], (*225)	20
17	Gyroelongated square bipyramid			10	24	16	16						D4d, [2+,8], (2*4)	16
18	Elongated triangular cupola			15	27	14	4	9		1			C3v, [3], (*33)	6
19	Elongated square cupola			20	36	18	4	13			1		C4v, [4], (*44)	8
20	Elongated pentagonal cupola			25	45	22	5	15	1			1	C5v, [5], (*55)	10
21	Elongated pentagonal rotunda			30	55	27	10	10	6			1	C5v, [5], (*55)	10
22	Gyroelongated triangular cupola			15	33	20	16	3		1			C3v, [3], (*33)	6
23	Gyroelongated square cupola			20	44	26	20	5			1		C4v, [4], (*44)	8
24	Gyroelongated pentagonal cupola			25	55	32	25	5	1			1	C5v, [5], (*55)	10
25	Gyroelongated pentagonal rotunda			30	65	37	30		6			1	C5v, [5], (*55)	10
26	Gyrobifastigium			8	14	8	4	4					D2d, [2+,4], (2*2)	8
27	Triangular orthobicupola			12	24	14	8	6					D3h, [3,2], (*223)	12
28	Square orthobicupola			16	32	18	8	10					D4h, [4,2], (*224)	16
29	Square gyrobicupola			16	32	18	8	10					D4d, [2+,8], (2*4)	16
30	Pentagonal orthobicupola			20	40	22	10	10	2				D5h, [5,2], (*225)	20
31	Pentagonal gyrobicupola			20	40	22	10	10	2				D5d, [2+,10], (2*5)	20
32	Pentagonal orthocupolarotunda			25	50	27	15	5	7				C5v, [5], (*55)	10
33	Pentagonal gyrocupolarotunda			25	50	27	15	5	7				C5v, [5], (*55)	10
34	Pentagonal orthobirotunda			30	60	32	20		12				D5h, [5,2], (*225)	20
35	Elongated triangular orthobicupola			18	36	20	8	12					D3h, [3,2], (*223)	12
36	Elongated triangular gyrobicupola			18	36	20	8	12					D3d, [2+,6], (2*3)	12
37	Elongated square gyrobicupola			24	48	26	8	18					D4d, [2+,8], (2*4)	16
38	Elongated pentagonal orthobicupola			30	60	32	10	20	2				D5h, [5,2], (*225)	20
39	Elongated pentagonal gyrobicupola			30	60	32	10	20	2				D5d, [2+,10], (2*5)	20
40	Elongated pentagonal orthocupolarotunda			35	70	37	15	15	7				C5v, [5], (*55)	10
41	Elongated pentagonal gyrocupolarotunda			35	70	37	15	15	7				C5v, [5], (*55)	10
42	Elongated pentagonal orthobirotunda			40	80	42	20	10	12				D5h, [5,2], (*225)	20
43	Elongated pentagonal gyrobirotunda			40	80	42	20	10	12				D5d, [2+,10], (2*5)	20
44	Gyroelongated triangular bicupola			18	42	26	20	6					D3, [3,2]+,(223)	6
45	Gyroelongated square bicupola			24	56	34	24	10					D4, [4,2]+, (224)	8
46	Gyroelongated pentagonal bicupola			30	70	42	30	10	2				D5, [5,2]+, (225)	10
47	Gyroelongated pentagonal cupolarotunda			35	80	47	35	5	7				C5, [5]+, (55)	5
48	Gyroelongated pentagonal birotunda			40	90	52	40		12				D5, [5,2]+, (225)	10
49	Augmented triangular prism			7	13	8	6	2					C2v, [2], (*22)	4
50	Biaugmented triangular prism			8	17	11	10	1					C2v, [2], (*22)	4
51	Triaugmented triangular prism			9	21	14	14						D3h, [3,2], (*223)	12
52	Augmented pentagonal prism			11	19	10	4	4	2				C2v, [2], (*22)	4
53	Biaugmented pentagonal prism			12	23	13	8	3	2				C2v, [2], (*22)	4
54	Augmented hexagonal prism			13	22	11	4	5		2			C2v, [2], (*22)	4
55	Parabiaugmented hexagonal prism			14	26	14	8	4		2			D2h, [2,2], (*222)	8
56	Metabiaugmented hexagonal prism			14	26	14	8	4		2			C2v, [2], (*22)	4
57	Triaugmented hexagonal prism			15	30	17	12	3		2			D3h, [3,2], (*223)	12
58	Augmented dodecahedron			21	35	16	5		11				C5v, [5], (*55)	10
59	Parabiaugmented dodecahedron			22	40	20	10		10				D5d, [2+,10], (2*5)	20
60	Metabiaugmented dodecahedron			22	40	20	10		10				C2v, [2], (*22)	4
61	Triaugmented dodecahedron			23	45	24	15		9				C3v, [3], (*33)	6
62	Metabidiminished icosahedron			10	20	12	10		2				C2v, [2], (*22)	4
63	Tridiminished icosahedron			9	15	8	5		3				C3v, [3], (*33)	6
64	Augmented tridiminished icosahedron			10	18	10	7		3				C3v, [3], (*33)	6
65	Augmented truncated tetrahedron			15	27	14	8	3		3			C3v, [3], (*33)	6
66	Augmented truncated cube			28	48	22	12	5			5		C4v, [4], (*44)	8
67	Biaugmented truncated cube			32	60	30	16	10			4		D4h, [4,2], (*224)	16
68	Augmented truncated dodecahedron			65	105	42	25	5	1			11	C5v, [5], (*55)	10
69	Parabiaugmented truncated dodecahedron			70	120	52	30	10	2			10	D5d, [2+,10], (2*5)	20
70	Metabiaugmented truncated dodecahedron			70	120	52	30	10	2			10	C2v, [2], (*22)	4
71	Triaugmented truncated dodecahedron			75	135	62	35	15	3			9	C3v, [3], (*33)	6
72	Gyrate rhombicosidodecahedron			60	120	62	20	30	12				C5v, [5], (*55)	10
73	Parabigyrate rhombicosidodecahedron			60	120	62	20	30	12				D5d, [2+,10], (2*5)	20
74	Metabigyrate rhombicosidodecahedron			60	120	62	20	30	12				C2v, [2], (*22)	4
75	Trigyrate rhombicosidodecahedron			60	120	62	20	30	12				C3v, [3], (*33)	6
76	Diminished rhombicosidodecahedron			55	105	52	15	25	11			1	C5v, [5], (*55)	10
77	Paragyrate diminished rhombicosidodecahedron			55	105	52	15	25	11			1	C5v, [5], (*55)	10
78	Metagyrate diminished rhombicosidodecahedron			55	105	52	15	25	11			1	Cs, [ ], (*11)	2
79	Bigyrate diminished rhombicosidodecahedron			55	105	52	15	25	11			1	Cs, [ ], (*11)	2
80	Parabidiminished rhombicosidodecahedron			50	90	42	10	20	10			2	D5d, [2+,10], (2*5)	20
81	Metabidiminished rhombicosidodecahedron			50	90	42	10	20	10			2	C2v, [2], (*22)	4
82	Gyrate bidiminished rhombicosidodecahedron			50	90	42	10	20	10			2	Cs, [ ], (*11)	2
83	Tridiminished rhombicosidodecahedron			45	75	32	5	15	9			3	C3v, [3], (*33)	6
84	Snub disphenoid			8	18	12	12						D2d, [2+,4], (2*2)	8
85	Snub square antiprism			16	40	26	24	2					D4d, [2+,8], (2*4)	16
86	Sphenocorona			10	22	14	12	2					C2v, [2], (*22)	4
87	Augmented sphenocorona			11	26	17	16	1					Cs, [ ], (*11)	2
88	Sphenomegacorona			12	28	18	16	2					C2v, [2], (*22)	4
89	Hebesphenomegacorona			14	33	21	18	3					C2v, [2], (*22)	4
90	Disphenocingulum			16	38	24	20	4					D2d, [2+,4], (2*2)	8
91	Bilunabirotunda			14	26	14	8	2	4				D2h, [2,2], (*222)	8
92	Triangular hebesphenorotunda			18	36	20	13	3	3	1			C3v, [3], (*33)	6

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Legend:

• J_n – Johnson solid number
• Net – Flattened (unfolded) image
• V – Number of vertices
• E – Number of edges
• F – Number of faces (total)
• F₃–F₁₀ – Number of faces by side counts

The square pyramid J₁ has the fewest vertices (5), the fewest edges (8), and the fewest faces (5).

The triaugmented truncated dodecahedron J₇₁ has the most vertices (75) and the most edges (135). It also has the highest number of faces (62), along with the gyrate rhombicosidodecahedron J₇₂, the parabigyrate rhombicosidodecahedron J₇₃, the metabigyrate rhombicosidodecahedron J₇₄, and the trigyrate rhombicosidodecahedron J₇₅.

المساحة السطحية

Since all faces of Johnson solids are regular polygons with 3, 4, 5, 6, 8, or 10 sides, and since all these polygons have the same edge length a, the surface area of a Johnson solid can be calculated as

$$A={\sum }_{n=3,4,5,6,8,10}{F}_n{A}_n$$

where the F_n are the polygonal face counts in the previous table and

$$A_n=(\frac{n}{4}\cot \frac{\pi }{n})a^2$$

is the area of a regular polygon with n sides of length a. In terms of radicals, one has

$$A_3=\frac{1}{4}\sqrt{3}{a}^2$$

$$A_4=a^2$$

$$A_5=\frac{1}{4}\sqrt{5(5+2\sqrt{5})}{a}^2$$

$$A_6=\frac{3}{2}\sqrt{3}{a}^2$$

$$A_8=2(1+\sqrt{2})a^2$$

$$A_{10}=\frac{5}{2}\sqrt{5+2\sqrt{5}}{a}^2,$$

resulting in the following table of surface areas.

Jn	Solid name	A/a2 (approx.)	A/a2 (exact)
1	Square pyramid	2.732050808	$1+\sqrt{3}$
2	Pentagonal pyramid	3.885540910	$\frac{1}{4}(5\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
3	Triangular cupola	7.330127019	$\frac{1}{2}(6+5\sqrt{3})$
4	Square cupola	11.560477932	$7+2\sqrt{2}+\sqrt{3}$
5	Pentagonal cupola	16.579749753	$\frac{1}{4}(20+5\sqrt{3}+10\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
6	Pentagonal rotunda	22.347200265	$\frac{1}{2}(5\sqrt{3}+5\sqrt{5+2\sqrt{5}}+3\sqrt{5(5+2\sqrt{5})})$
7	Elongated triangular pyramid	4.732050808	$3+\sqrt{3}$
8	Elongated square pyramid	6.732050808	$5+\sqrt{3}$
9	Elongated pentagonal pyramid	8.885540910	$\frac{1}{4}(20+5\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
10	Gyroelongated square pyramid	6.196152423	$1+3\sqrt{3}$
11	Gyroelongated pentagonal pyramid	8.215667929	$\frac{1}{4}(15\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
12	Triangular dipyramid	2.598076211	$\frac{3\sqrt{3}}{2}$
13	Pentagonal dipyramid	4.330127019	$\frac{5\sqrt{3}}{2}$
14	Elongated triangular dipyramid	5.598076211	$\frac{3}{2}(2+\sqrt{3})$
15	Elongated square dipyramid	7.464101615	$2(2+\sqrt{3})$
16	Elongated pentagonal dipyramid	9.330127019	$\frac{5}{2}(2+\sqrt{3})$
17	Gyroelongated square dipyramid	6.928203230	$4\sqrt{3}$
18	Elongated triangular cupola	13.330127019	$\frac{1}{2}(18+5\sqrt{3})$
19	Elongated square cupola	19.560477932	$15+2\sqrt{2}+\sqrt{3}$
20	Elongated pentagonal cupola	26.579749753	$\frac{1}{4}(60+5\sqrt{3}+10\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
21	Elongated pentagonal rotunda	32.347200265	$\frac{1}{2}(20+5\sqrt{3}+5\sqrt{5+2\sqrt{5}}+3\sqrt{5(5+2\sqrt{5})})$
22	Gyroelongated triangular cupola	12.526279442	$\frac{1}{2}(6+11\sqrt{3})$
23	Gyroelongated square cupola	18.488681163	$7+2\sqrt{2}+5\sqrt{3}$
24	Gyroelongated pentagonal cupola	25.240003791	$\frac{1}{4}(20+25\sqrt{3}+10\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
25	Gyroelongated pentagonal rotunda	31.007454303	$\frac{1}{2}(15\sqrt{3}+5\sqrt{5+2\sqrt{5}}+3\sqrt{5(5+2\sqrt{5})})$
26	Gyrobifastigium	5.732050808	$4+\sqrt{3}$
27	Triangular orthobicupola	9.464101615	$2(3+\sqrt{3})$
28	Square orthobicupola	13.464101615	$2(5+\sqrt{3})$
29	Square gyrobicupola	13.464101615	$2(5+\sqrt{3})$
30	Pentagonal orthobicupola	17.771081820	$\frac{1}{2}(20+5\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
31	Pentagonal gyrobicupola	17.771081820	$\frac{1}{2}(20+5\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
32	Pentagonal orthocupolarotunda	23.538532333	$\frac{1}{4}(20+15\sqrt{3}+7\sqrt{5(5+2\sqrt{5})})$
33	Pentagonal gyrocupolarotunda	23.538532333	$\frac{1}{4}(20+15\sqrt{3}+7\sqrt{5(5+2\sqrt{5})})$
34	Pentagonal orthobirotunda	29.305982845	$5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
35	Elongated triangular orthobicupola	15.464101615	$2(6+\sqrt{3})$
36	Elongated triangular gyrobicupola	15.464101615	$2(6+\sqrt{3})$
37	Elongated square gyrobicupola	21.464101615	$2(9+\sqrt{3})$
38	Elongated pentagonal orthobicupola	27.771081820	$\frac{1}{2}(40+5\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
39	Elongated pentagonal gyrobicupola	27.771081820	$\frac{1}{2}(40+5\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
40	Elongated pentagonal orthocupolarotunda	33.538532333	$\frac{1}{4}(60+15\sqrt{3}+7\sqrt{5(5+2\sqrt{5})})$
41	Elongated pentagonal gyrocupolarotunda	33.538532333	$\frac{1}{4}(60+15\sqrt{3}+7\sqrt{5(5+2\sqrt{5})})$
42	Elongated pentagonal orthobirotunda	39.305982845	$10+5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
43	Elongated pentagonal gyrobirotunda	39.305982845	$10+5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
44	Gyroelongated triangular bicupola	14.660254038	$6+5\sqrt{3}$
45	Gyroelongated square bicupola	20.392304845	$2(5+3\sqrt{3})$
46	Gyroelongated pentagonal bicupola	26.431335858	$\frac{1}{2}(20+15\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
47	Gyroelongated pentagonal cupolarotunda	32.198786370	$\frac{1}{4}(20+35\sqrt{3}+7\sqrt{5(5+2\sqrt{5})})$
48	Gyroelongated pentagonal birotunda	37.966236883	$10\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
49	Augmented triangular prism	4.598076211	$\frac{1}{2}(4+3\sqrt{3})$
50	Biaugmented triangular prism	5.330127019	$\frac{1}{2}(2+5\sqrt{3})$
51	Triaugmented triangular prism	6.062177826	$\frac{7\sqrt{3}}{2}$
52	Augmented pentagonal prism	9.173005609	$\frac{1}{2}(8+2\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
53	Biaugmented pentagonal prism	9.905056416	$\frac{1}{2}(6+4\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
54	Augmented hexagonal prism	11.928203230	$5+4\sqrt{3}$
55	Parabiaugmented hexagonal prism	12.660254038	$4+5\sqrt{3}$
56	Metabiaugmented hexagonal prism	12.660254038	$4+5\sqrt{3}$
57	Triaugmented hexagonal prism	13.392304845	$3(1+2\sqrt{3})$
58	Augmented dodecahedron	21.090314916	$\frac{1}{4}(5\sqrt{3}+11\sqrt{5(5+2\sqrt{5})})$
59	Parabiaugmented dodecahedron	21.534901025	$\frac{5}{2}(\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
60	Metabiaugmented dodecahedron	21.534901025	$\frac{5}{2}(\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
61	Triaugmented dodecahedron	21.979487134	$\frac{3}{4}(5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})})$
62	Metabidiminished icosahedron	7.771081820	$\frac{1}{2}(5\sqrt{3}+\sqrt{5(5+2\sqrt{5})})$
63	Tridiminished icosahedron	7.326495711	$\frac{1}{4}(5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})})$
64	Augmented tridiminished icosahedron	8.192521115	$\frac{1}{4}(7\sqrt{3}+3\sqrt{5(5+2\sqrt{5})})$
65	Augmented truncated tetrahedron	14.258330249	$\frac{1}{2}(6+13\sqrt{3})$
66	Augmented truncated cube	34.338288046	$15+10\sqrt{2}+3\sqrt{3}$
67	Biaugmented truncated cube	36.241911729	$2(9+4\sqrt{2}+2\sqrt{3})$
68	Augmented truncated dodecahedron	102.182092220	$\frac{1}{4}(20+25\sqrt{3}+110\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
69	Parabiaugmented truncated dodecahedron	103.373424287	$\frac{1}{2}(20+15\sqrt{3}+50\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
70	Metabiaugmented truncated dodecahedron	103.373424287	$\frac{1}{2}(20+15\sqrt{3}+50\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
71	Triaugmented truncated dodecahedron	104.564756354	$\frac{1}{4}(60+35\sqrt{3}+90\sqrt{5+2\sqrt{5}}+3\sqrt{5(5+2\sqrt{5})})$
72	Gyrate rhombicosidodecahedron	59.305982845	$30+5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
73	Parabigyrate rhombicosidodecahedron	59.305982845	$30+5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
74	Metabigyrate rhombicosidodecahedron	59.305982845	$30+5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
75	Trigyrate rhombicosidodecahedron	59.305982845	$30+5\sqrt{3}+3\sqrt{5(5+2\sqrt{5})}$
76	Diminished rhombicosidodecahedron	58.114650778	$\frac{1}{4}(100+15\sqrt{3}+10\sqrt{5+2\sqrt{5}}+11\sqrt{5(5+2\sqrt{5})})$
77	Paragyrate diminished rhombicosidodecahedron	58.114650778	$\frac{1}{4}(100+15\sqrt{3}+10\sqrt{5+2\sqrt{5}}+11\sqrt{5(5+2\sqrt{5})})$
78	Metagyrate diminished rhombicosidodecahedron	58.114650778	$\frac{1}{4}(100+15\sqrt{3}+10\sqrt{5+2\sqrt{5}}+11\sqrt{5(5+2\sqrt{5})})$
79	Bigyrate diminished rhombicosidodecahedron	58.114650778	$\frac{1}{4}(100+15\sqrt{3}+10\sqrt{5+2\sqrt{5}}+11\sqrt{5(5+2\sqrt{5})})$
80	Parabidiminished rhombicosidodecahedron	56.923318711	$\frac{5}{2}(8+\sqrt{3}+2\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
81	Metabidiminished rhombicosidodecahedron	56.923318711	$\frac{5}{2}(8+\sqrt{3}+2\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
82	Gyrate bidiminished rhombicosidodecahedron	56.923318711	$\frac{5}{2}(8+\sqrt{3}+2\sqrt{5+2\sqrt{5}}+\sqrt{5(5+2\sqrt{5})})$
83	Tridiminished rhombicosidodecahedron	55.731986644	$\frac{1}{4}(60+5\sqrt{3}+30\sqrt{5+2\sqrt{5}}+9\sqrt{5(5+2\sqrt{5})})$
84	Snub disphenoid	5.196152423	$3\sqrt{3}$
85	Snub square antiprism	12.392304845	$2(1+3\sqrt{3})$
86	Sphenocorona	7.196152423	$2+3\sqrt{3}$
87	Augmented sphenocorona	7.928203230	$1+4\sqrt{3}$
88	Sphenomegacorona	8.928203230	$2(1+2\sqrt{3})$
89	Hebesphenomegacorona	10.794228634	$\frac{3}{2}(2+3\sqrt{3})$
90	Disphenocingulum	12.660254038	$4+5\sqrt{3}$
91	Bilunabirotunda	12.346011217	$2+2\sqrt{3}+\sqrt{5(5+2\sqrt{5})}$
92	Triangular hebesphenorotunda	16.388673538	$\frac{1}{4}(12+19\sqrt{3}+3\sqrt{5(5+2\sqrt{5})})$

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For a fixed edge length, the triangular dipyramid J₁₂ has the smallest surface area and the triaugmented truncated dodecahedron J₇₁ has the largest, more than 40 times larger.

الحجم

The following table lists the volume of each Johnson solid. Here V is the volume (not the number of vertices, as in the first table) and a is the edge length.

The source for this table is the PolyhedronData[..., "Volume"] command in Wolfram Research's Mathematica.

These volumes can be calculated from a set of vertex coordinates; such coordinates are known for all 92 Johnson solids. A conceptually simple approach is to triangulate the surface of the solid (for example, by adding an extra point in the center of each non-triangular face) and choose some interior point as an "origin" so that the interior can be subdivided into irregular tetrahedra. Each tetrahedron has one vertex at the origin inside and three vertices on the surface. The volume of the solid is then the sum of the volumes of these tetrahedra. There is a simple formula for the volume of an irregular tetrahedron.

Jn	اسم المجسم	V/a3 (approx.)	V/a3 (exact)
1	Square pyramid	0.235702260	$\frac{1}{3\sqrt{2}}$
2	Pentagonal pyramid	0.301502832	$\frac{1}{24}(5+\sqrt{5})$
3	Triangular cupola	1.178511302	$\frac{5}{3\sqrt{2}}$
4	Square cupola	1.942809042	$1+\frac{2\sqrt{2}}{3}$
5	Pentagonal cupola	2.324045318	$\frac{1}{6}(5+4\sqrt{5})$
6	Pentagonal rotunda	6.917762968	$\frac{1}{12}(45+17\sqrt{5})$
7	Elongated triangular pyramid	0.550863832	$\frac{1}{12}(\sqrt{2}+3\sqrt{3})$
8	Elongated square pyramid	1.235702260	$\frac{1}{6}(6+\sqrt{2})$
9	Elongated pentagonal pyramid	2.021980233	$\frac{1}{24}(5+\sqrt{5}+6\sqrt{5(5+2\sqrt{5})})$
10	Gyroelongated square pyramid	1.192702242	$\frac{1}{6}(\sqrt{2}+2\sqrt{4+3\sqrt{2}})$
11	Gyroelongated pentagonal pyramid	1.880192158	$\frac{1}{24}(25+9\sqrt{5})$
12	Triangular dipyramid	0.235702260	$\frac{1}{3\sqrt{2}}$
13	Pentagonal dipyramid	0.603005665	$\frac{1}{12}(5+\sqrt{5})$
14	Elongated triangular dipyramid	0.668714962	$\frac{1}{12}(2\sqrt{2}+3\sqrt{3})$
15	Elongated square dipyramid	1.471404521	$\frac{1}{3}(3+\sqrt{2})$
16	Elongated pentagonal dipyramid	2.323483065	$\frac{1}{12}(5+\sqrt{5}+3\sqrt{5(5+2\sqrt{5})})$
17	Gyroelongated square dipyramid	1.428404503	$\frac{1}{3}(\sqrt{2}+\sqrt{4+3\sqrt{2}})$
18	Elongated triangular cupola	3.776587513	$\frac{1}{6}(5\sqrt{2}+9\sqrt{3})$
19	Elongated square cupola	6.771236166	$3+\frac{8\sqrt{2}}{3}$
20	Elongated pentagonal cupola	10.018254161	$\frac{1}{6}(5+4\sqrt{5}+15\sqrt{5+2\sqrt{5}})$
21	Elongated pentagonal rotunda	14.611971811	$\frac{1}{12}(45+17\sqrt{5}+30\sqrt{5+2\sqrt{5}})$
22	Gyroelongated triangular cupola	3.516053091	$\frac{1}{3}\sqrt{\frac{61}{2}+18\sqrt{3}+30\sqrt{1+\sqrt{3}}}$
23	Gyroelongated square cupola	6.210765792	root of polynomial of degree 8
24	Gyroelongated pentagonal cupola	9.073333194	root of polynomial of degree 8
25	Gyroelongated pentagonal rotunda	13.667050844	root of polynomial of degree 8
26	Gyrobifastigium	0.866025404	$\frac{\sqrt{3}}{2}$
27	Triangular orthobicupola	2.357022604	$\frac{5\sqrt{2}}{3}$
28	Square orthobicupola	3.885618083	$2+\frac{4\sqrt{2}}{3}$
29	Square gyrobicupola	3.885618083	$2+\frac{4\sqrt{2}}{3}$
30	Pentagonal orthobicupola	4.648090637	$\frac{1}{3}(5+4\sqrt{5})$
31	Pentagonal gyrobicupola	4.648090637	$\frac{1}{3}(5+4\sqrt{5})$
32	Pentagonal orthocupolarotunda	9.241808286	$\frac{5}{12}(11+5\sqrt{5})$
33	Pentagonal gyrocupolarotunda	9.241808286	$\frac{5}{12}(11+5\sqrt{5})$
34	Pentagonal orthobirotunda	13.835525936	$\frac{1}{6}(45+17\sqrt{5})$
35	Elongated triangular orthobicupola	4.955098815	$\frac{5\sqrt{2}}{3}+\frac{3\sqrt{3}}{2}$
36	Elongated triangular gyrobicupola	4.955098815	$\frac{5\sqrt{2}}{3}+\frac{3\sqrt{3}}{2}$
37	Elongated square gyrobicupola	8.714045208	$4+\frac{10\sqrt{2}}{3}$
38	Elongated pentagonal orthobicupola	12.342299480	$\frac{1}{6}(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}})$
39	Elongated pentagonal gyrobicupola	12.342299480	$\frac{1}{6}(10+8\sqrt{5}+15\sqrt{5+2\sqrt{5}})$
40	Elongated pentagonal orthocupolarotunda	16.936017129	$\frac{5}{12}(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}})$
41	Elongated pentagonal gyrocupolarotunda	16.936017129	$\frac{5}{12}(11+5\sqrt{5}+6\sqrt{5+2\sqrt{5}})$
42	Elongated pentagonal orthobirotunda	21.529734779	$\frac{1}{6}(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}})$
43	Elongated pentagonal gyrobirotunda	21.529734779	$\frac{1}{6}(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}})$
44	Gyroelongated triangular bicupola	4.694564393	$\frac{1}{3}\sqrt{68+18\sqrt{3}+60\sqrt{1+\sqrt{3}}}$
45	Gyroelongated square bicupola	8.153574834	root of polynomial of degree 8
46	Gyroelongated pentagonal bicupola	11.397378512	root of polynomial of degree 8
47	Gyroelongated pentagonal cupolarotunda	15.991096162	root of polynomial of degree 8
48	Gyroelongated pentagonal birotunda	20.584813812	root of polynomial of degree 8
49	Augmented triangular prism	0.668714962	$\frac{1}{12}(2\sqrt{2}+3\sqrt{3})$
50	Biaugmented triangular prism	0.904417223	$\sqrt{\frac{59}{144}+\frac{1}{\sqrt{6}}}$
51	Triaugmented triangular prism	1.140119483	$\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{4}$
52	Augmented pentagonal prism	1.956179661	$\frac{1}{12}\sqrt{233+90\sqrt{5}+12\sqrt{50+20\sqrt{5}}}$
53	Biaugmented pentagonal prism	2.191881921	$\frac{1}{12}\sqrt{257+90\sqrt{5}+24\sqrt{50+20\sqrt{5}}}$
54	Augmented hexagonal prism	2.833778472	$\frac{1}{6}(\sqrt{2}+9\sqrt{3})$
55	Parabiaugmented hexagonal prism	3.069480732	$\frac{1}{6}(2\sqrt{2}+9\sqrt{3})$
56	Metabiaugmented hexagonal prism	3.069480732	$\frac{1}{6}(2\sqrt{2}+9\sqrt{3})$
57	Triaugmented hexagonal prism	3.305182993	$\frac{1}{\sqrt{2}}+\frac{3\sqrt{3}}{2}$
58	Augmented dodecahedron	7.964621793	$\frac{1}{24}(95+43\sqrt{5})$
59	Parabiaugmented dodecahedron	8.266124625	$\frac{1}{6}(25+11\sqrt{5})$
60	Metabiaugmented dodecahedron	8.266124625	$\frac{1}{6}(25+11\sqrt{5})$
61	Triaugmented dodecahedron	8.567627458	$\frac{5}{8}(7+3\sqrt{5})$
62	Metabidiminished icosahedron	1.578689326	$\frac{1}{6}(5+2\sqrt{5})$
63	Tridiminished icosahedron	1.277186493	$\frac{5}{8}+\frac{7\sqrt{5}}{24}$
64	Augmented tridiminished icosahedron	1.395037624	$\frac{1}{24}(15+2\sqrt{2}+7\sqrt{5})$
65	Augmented truncated tetrahedron	3.889087297	$\frac{11}{2\sqrt{2}}$
66	Augmented truncated cube	15.542472333	$8+\frac{16\sqrt{2}}{3}$
67	Biaugmented truncated cube	17.485281374	$9+6\sqrt{2}$
68	Augmented truncated dodecahedron	87.363709878	$\frac{505}{12}+\frac{81\sqrt{5}}{4}$
69	Parabiaugmented truncated dodecahedron	89.687755196	$\frac{1}{12}(515+251\sqrt{5})$
70	Metabiaugmented truncated dodecahedron	89.687755196	$\frac{1}{12}(515+251\sqrt{5})$
71	Triaugmented truncated dodecahedron	92.011800514	$\frac{7}{12}(75+37\sqrt{5})$
72	Gyrate rhombicosidodecahedron	41.615323782	$20+\frac{29\sqrt{5}}{3}$
73	Parabigyrate rhombicosidodecahedron	41.615323782	$20+\frac{29\sqrt{5}}{3}$
74	Metabigyrate rhombicosidodecahedron	41.615323782	$20+\frac{29\sqrt{5}}{3}$
75	Trigyrate rhombicosidodecahedron	41.615323782	$20+\frac{29\sqrt{5}}{3}$
76	Diminished rhombicosidodecahedron	39.291278464	$\frac{115}{6}+9\sqrt{5}$
77	Paragyrate diminished rhombicosidodecahedron	39.291278464	$\frac{115}{6}+9\sqrt{5}$
78	Metagyrate diminished rhombicosidodecahedron	39.291278464	$\frac{115}{6}+9\sqrt{5}$
79	Bigyrate diminished rhombicosidodecahedron	39.291278464	$\frac{115}{6}+9\sqrt{5}$
80	Parabidiminished rhombicosidodecahedron	36.967233146	$\frac{5}{3}(11+5\sqrt{5})$
81	Metabidiminished rhombicosidodecahedron	36.967233146	$\frac{5}{3}(11+5\sqrt{5})$
82	Gyrate bidiminished rhombicosidodecahedron	36.967233146	$\frac{5}{3}(11+5\sqrt{5})$
83	Tridiminished rhombicosidodecahedron	34.643187827	$\frac{35}{2}+\frac{23\sqrt{5}}{3}$
84	Snub disphenoid	0.859493646	root of polynomial of degree 6
85	Snub square antiprism	3.601222010	root of polynomial of degree 12
86	Sphenocorona	1.515351640	$\frac{1}{2}\sqrt{1+3\sqrt{\frac{3}{2}}+\sqrt{13+3\sqrt{6}}}$
87	Augmented sphenocorona	1.751053900	root of polynomial of degree 16
88	Sphenomegacorona	1.948108229	root of polynomial of degree 32
89	Hebesphenomegacorona	2.912910415	root of polynomial of degree 20
90	Disphenocingulum	3.777645342	root of polynomial of degree 24
91	Bilunabirotunda	3.093717650	$\frac{1}{12}(17+9\sqrt{5})$
92	Triangular hebesphenorotunda	5.108745974	$\frac{5}{2}+\frac{7\sqrt{5}}{6}$

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For a fixed edge length, the square pyramid J₁ and the triangular dipyramid J₁₂ have the smallest volume and the triaugmented truncated dodecahedron J₇₁ has the largest, more than 390 times larger.

Thirteen of the 92 Johnson solids have volumes for which V/a³ is not a number expressible using radicals. These values are the greatest real root of the following polynomials.

Jn	Polynomial
23	6561 x8 − 52488 x7 + 113724 x6 − 9720 x5 − 1616922 x4 + 396360 x3 + 1537020 x2 − 178632 x − 3391
24	1679616 x8 − 11197440 x7 + 27060480 x6 + 35769600 x5 − 4456749600 x4 − 10714248000 x3 + 3828402000 x2 + 13859430000 x + 5340175625
25	1679616 x8 − 50388480 x7 + 603262080 x6 − 3520972800 x5 + 5215460400 x4 + 4128624000 x3 − 8894943000 x2 + 3881385000 x − 424924375
45	6561 x8 − 104976 x7 + 594864 x6 − 1384128 x5 − 552096 x4 + 1569024 x3 + 246528 x2 − 119808 x + 4352
46	6561 x8 − 87480 x7 + 313470 x6 + 753300 x5 − 22424850 x4 − 84591000 x3 − 85909500 x2 + 8715000 x + 35547500
47	1679616 x8 − 61585920 x7 + 851472000 x6 − 5108832000 x5 + 4745790000 x4 + 21346200000 x3 − 29019375000 x2 − 4576875000 x − 405859375
48	6561 x8 − 393660 x7 + 9316620 x6 − 108207900 x5 + 601832025 x4 − 1417189500 x3 + 965841750 x2 + 597667500 x − 668786875
84	5832 x6 − 1377 x4 − 2160 x2 − 4
85	531441 x12 − 85726026 x8 − 48347280 x6 + 11588832 x4 + 4759488 x2 − 892448
87	45137758519296 x16 − 110336743047168 x14 − 191069246324736 x12 + 209269081571328 x10 + 364547659290624 x8 − 58793017190400 x6 + 3306865979520 x4 − 1275399855936 x2 + 1439671249
88	521578814501447328359509917696 x32 − 985204427391622731345740955648 x30 − 16645447351681991898880656015360 x28 + 79710816694053483249372512649216 x26 − 152195045391070538203422101864448 x24 + 156280253448056209478031589244928 x22 − 96188116617075838858708654227456 x20 + 30636368373570166303441645731840 x18 + 5828527077458909552923002273792 x16 − 8060049780765551057159394951168 x14 + 1018074792115156107372011716608 x12 + 35220131544370794950945931264 x10 + 327511698517355918956755959808 x8 − 116978732884218191486738706432 x6 + 10231563774949176791703149568 x4 − 366323949299263261553952192 x2 + 3071435678740442112675625
89	47330370277129322496 x20 − 722445512980071186432 x18 + 3596480447590271287296 x16 − 8432333285523990773760 x14 + 8973584611317745975296 x12 − 3065290664181478981632 x10 + 366229890219212144640 x8 − 8337259437908852736 x6 − 22211277300912896 x4 + 132615435213216 x2 + 2693461945329
90	1213025622610333925376 x24 + 54451372392730545094656 x22 − 796837093078664749252608 x20 − 4133410366404688544268288 x18 + 20902529024429842816303104 x16 − 133907540390420673677230080 x14 + 246234688242991598853881856 x12 − 63327534106871321714442240 x10 + 14389309497459555704164608 x8 + 48042947402464500749392128 x6 − 5891096640600351061013664 x4 − 3212114716816853362953264 x2 + 479556973248657693884401

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Inradius, midradius, and circumradius

The following table lists the radius R_i of the insphere, the radius R_m of the midsphere, and the radius R_c of the circumsphere, each divided by the edge length a, when these spheres exist.

A polyhedron does not necessarily have an insphere, or a midsphere, or a circumsphere. For example, it does not have a circumsphere unless all its vertices lie on some sphere. The Johnson solids, having less symmetry than, say, the Platonic solids, lack many of these spheres. Only J₁ and J₂ possess all three of these spheres.

The source for this table is the PolyhedronData[..., "Inradius"], PolyhedronData[..., "Midradius"], and PolyhedronData[..., "Circumradius"] commands in Wolfram Research's Mathematica. The output has been simplified to a consistent form in terms of radicals.

Jn	Ri/a (approx.)	Ri/a (exact)	Rm/a (approx.)	Rm/a (exact)	Rc/a (approx.)	Rc/a (exact)
1	0.258819045	$\frac{\sqrt{2-\sqrt{3}}}{2}$	0.500000000	$\frac{1}{2}$	0.707106781	$\frac{1}{\sqrt{2}}$
2	0.232788309	$\frac{1}{\sqrt{25-7\sqrt{5}+\sqrt{30(5-\sqrt{5})}}}$	0.809016994	$\frac{1}{4}(1+\sqrt{5})$	0.951056516	$\frac{1}{2}\sqrt{\frac{1}{2}(5+\sqrt{5})}$
3	-	-	0.866025404	$\frac{\sqrt{3}}{2}$	1.000000000	$1$
4	-	-	1.306562965	$\sqrt{1+\frac{1}{\sqrt{2}}}$	1.398966326	$\sqrt{\frac{5}{4}+\frac{1}{\sqrt{2}}}$
5	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
6	-	-	1.538841769	$\frac{1}{2}\sqrt{5+2\sqrt{5}}$	1.618033989	$\frac{1}{2}(1+\sqrt{5})$
7	-	-	-	-	-	-
8	-	-	-	-	-	-
9	-	-	-	-	-	-
10	-	-	-	-	-	-
11	-	-	0.809016994	$\frac{1}{4}(1+\sqrt{5})$	0.951056516	$\frac{1}{2}\sqrt{\frac{1}{2}(5+\sqrt{5})}$
12	0.272165527	$\frac{\sqrt{\frac{2}{3}}}{3}$	-	-	-	-
13	0.417774579	$\sqrt{\frac{1}{30}(3+\sqrt{5})}$	-	-	-	-
14	-	-	-	-	-	-
15	-	-	-	-	-	-
16	-	-	-	-	-	-
17	-	-	-	-	-	-
18	-	-	-	-	-	-
19	-	-	1.306562965	$\sqrt{1+\frac{1}{\sqrt{2}}}$	1.398966326	$\sqrt{\frac{5}{4}+\frac{1}{\sqrt{2}}}$
20	-	-	-	-	-	-
21	-	-	-	-	-	-
22	-	-	-	-	-	-
23	-	-	-	-	-	-
24	-	-	-	-	-	-
25	-	-	-	-	-	-
26	-	-	-	-	-	-
27	-	-	0.866025404	$\frac{\sqrt{3}}{2}$	1.000000000	$1$
28	-	-	-	-	-	-
29	-	-	-	-	-	-
30	-	-	-	-	-	-
31	-	-	-	-	-	-
32	-	-	-	-	-	-
33	-	-	-	-	-	-
34	-	-	1.538841769	$\frac{1}{2}\sqrt{5+2\sqrt{5}}$	1.618033989	$\frac{1}{2}(1+\sqrt{5})$
35	-	-	-	-	-	-
36	-	-	-	-	-	-
37	-	-	1.306562965	$\sqrt{1+\frac{1}{\sqrt{2}}}$	1.398966326	$\sqrt{\frac{5}{4}+\frac{1}{\sqrt{2}}}$
38	-	-	-	-	-	-
39	-	-	-	-	-	-
40	-	-	-	-	-	-
41	-	-	-	-	-	-
42	-	-	-	-	-	-
43	-	-	-	-	-	-
44	-	-	-	-	-	-
45	-	-	-	-	-	-
46	-	-	-	-	-	-
47	-	-	-	-	-	-
48	-	-	-	-	-	-
49	-	-	-	-	-	-
50	-	-	-	-	-	-
51	-	-	-	-	-	-
52	-	-	-	-	-	-
53	-	-	-	-	-	-
54	-	-	-	-	-	-
55	-	-	-	-	-	-
56	-	-	-	-	-	-
57	-	-	-	-	-	-
58	-	-	-	-	-	-
59	-	-	-	-	-	-
60	-	-	-	-	-	-
61	-	-	-	-	-	-
62	-	-	0.809016994	$\frac{1}{4}(1+\sqrt{5})$	0.951056516	$\frac{1}{2}\sqrt{\frac{1}{2}(5+\sqrt{5})}$
63	-	-	0.809016994	$\frac{1}{4}(1+\sqrt{5})$	0.951056516	$\frac{1}{2}\sqrt{\frac{1}{2}(5+\sqrt{5})}$
64	-	-	-	-	-	-
65	-	-	-	-	-	-
66	-	-	-	-	-	-
67	-	-	-	-	-	-
68	-	-	-	-	-	-
69	-	-	-	-	-	-
70	-	-	-	-	-	-
71	-	-	-	-	-	-
72	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
73	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
74	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
75	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
76	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
77	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
78	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
79	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
80	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
81	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
82	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
83	-	-	2.176250899	$\sqrt{\frac{5}{2}+\sqrt{5}}$	2.232950509	$\frac{1}{2}\sqrt{11+4\sqrt{5}}$
84	-	-	-	-	-	-
85	-	-	-	-	-	-
86	-	-	-	-	-	-
87	-	-	-	-	-	-
88	-	-	-	-	-	-
89	-	-	-	-	-	-
90	-	-	-	-	-	-
91	-	-	-	-	-	-
92	-	-	-	-	-	-

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References

• Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
• Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.

وصلات خارجية

• Sylvain Gagnon, "Convex polyhedra with regular faces", Structural Topology, No. 6, 1982, 83-95.
• Johnson Solids by George W. Hart.
• Images of all 92 solids, categorized, on one page
• Eric W. Weisstein, Johnson Solid at MathWorld.
• VRML models of Johnson Solids by Jim McNeill
• VRML models of Johnson Solids by Vladimir Bulatov