List of geodesic polyhedra and Goldberg polyhedra

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.

This is a list of selected geodesic polyhedra and Goldberg polyhedra, two infinite classes of polyhedra. Geodesic polyhedra and Goldberg polyhedra are duals of each other. The geodesic and Goldberg polyhedra are parameterized by integers m and n, with $m>0$ and $n\ge 0$. T is the triangulation number, which is equal to $T=m^2+mn+n^2$.

Icosahedral

m	n	T	Class	Vertices (geodesic) Faces (Goldberg)	Edges	Faces (geodesic) Vertices (Goldberg)	Face triangle	Geodesic			Goldberg
								Symbols	Conway	Image	Symbols	Conway	Image
1	0	1	I	12	30	20	File:Subdivided triangle 01 00.svg	{3,5} {3,5+}1,0	I	File:Icosahedron.svg	{5,3} {5+,3}1,0 GP5(1,0)	D	File:Dodecahedron.svg
2	0	4	I	42	120	80	File:Subdivided triangle 02 00.svg	{3,5+}2,0	uI dcdI	File:Pentakis icosidodecahedron.png	{5+,3}2,0 GP5(2,0)	cD cD	File:Truncated rhombic triacontahedron.png
3	0	9	I	92	270	180	File:Subdivided triangle 03 00.svg	{3,5+}3,0	xI ktI	File:Conway polyhedron K6k5tI.png	{5+,3}3,0 GP5(3,0)	yD tkD	File:Conway polyhedron Dk6k5tI.png
4	0	16	I	162	480	320	File:Subdivided triangle 04 00.svg	{3,5+}4,0	uuI dccD	File:Conway polyhedron k6k5at5daD.png	{5+,3}4,0 GP5(4,0)	c2D	File:Conway polyhedron dk6k5at5daD.png
5	0	25	I	252	750	500	File:Subdivided triangle 05 00.svg	{3,5+}5,0	u5I u5I	File:Icosahedron subdivision5.png	{5+,3}5,0 GP5(5,0)	c5D du5I	File:Goldberg polyhedron 5 0.png
6	0	36	I	362	1080	720	File:Subdivided triangle 06 00.svg	{3,5+}6,0	uxI dctkdI	File:Conway polyhedron kdkt5daD.png	{5+,3}6,0 GP5(6,0)	cyD ctkD	File:Conway polyhedron tkt5daD.png
7	0	49	I	492	1470	980	File:Subdivided triangle 07 00.svg	{3,5+}7,0	vvI dwrwdI	File:Conway dwrwD.png	{5+,3}7,0 GP5(7,0)	wwD wrwD	File:Goldberg polyhedron 7 0.png
8	0	64	I	642	1920	1280	File:Subdivided triangle 08 00.svg	{3,5+}8,0	u3I dcccdI	File:Conway dcccD.png	{5+,3}8,0 GP5(8,0)	cccD	File:Conway polyhedron dk6k5adk6k5at5daD.png
9	0	81	I	812	2430	1620	File:Subdivided triangle 09 00.svg	{3,5+}9,0	xxI ktktI	File:Conway ktktI.png	{5+,3}9,0 GP5(9,0)	yyD tktkD	File:Conway tdtdtkD.png
10	0	100	I	1002	3000	2000	File:Subdivided triangle 10 00.svg	{3,5+}10,0	uu5I uu5I	File:10-subdivided icosahedron.png	{5+,3}10,0 GP5(10,0)	cc5D	File:Goldberg polyhedron 10 0.png
11	0	121	I	1212	3630	2420	File:Subdivided triangle 11 00.svg	{3,5+}11,0	u11I u11I	File:11-subdivided icosahedron.png	{5+,3}11,0 GP5(11,0)	c11D
12	0	144	I	1442	4320	2880	File:Subdivided triangle 12 00.svg	{3,5+}12,0	uuxD dcctkD	File:Conway dcctkD.png	{5+,3}12,0 GP5(12,0)	ccyD cctkD	File:Conway polyhedron dk5k6adk5k6adktI.png
13	0	169	I	1692	5070	3380	File:Subdivided triangle 13 00.svg	{3,5+}13,0	u13I u13I	File:13-subdivided icosahedron.png	{5+,3}13,0 GP5(13,0)	c13D
14	0	196	I	1962	5880	3920	File:Subdivided triangle 14 00.svg	{3,5+}14,0	uvvI dcwwdI	File:Conway dcwrwdI.png	{5+,3}14,0 GP5(14,0)	cwrwD	File:Conway cwrwdI.png
15	0	225	I	2252	6750	4500	File:Subdivided triangle 15 00.svg	{3,5+}15,0	u5xI u5ktI	File:15-subdivided icosahedron.png	{5+,3}15,0 GP5(15,0)	c5yD c5tkD	File:Goldberg polyhedron 15 0.png
16	0	256	I	2562	7680	5120	File:Subdivided triangle 16 00.svg	{3,5+}16,0	dc4dI	File:Conway dccccD.png	{5+,3}16,0 GP5(16,0)	ccccD	File:Conway polyhedron dk6k5adk6k5adk6k5at5daD.png
1	1	3	II	32	90	60	File:Subdivided triangle 01 01.svg	{3,5+}1,1	nI kD	File:Conway polyhedron kD.png	{5+,3}1,1 GP5(1,1)	yD tI	File:Uniform polyhedron-53-t12.png
2	2	12	II	122	360	240	File:Subdivided triangle 02 02.svg	{3,5+}2,2	unI =dctI	File:Conway polyhedron kt5daD.png	{5+,3}2,2 GP5(2,2)	czD cdkD	File:Conway polyhedron dkt5daD.png
3	3	27	II	272	810	540	File:Subdivided triangle 03 03.svg	{3,5+}3,3	xnI ktkD	File:Conway polyhedron kdktI.png	{5+,3}3,3 GP5(3,3)	yzD tkdkD	File:Conway polyhedron dkdktI.png
4	4	48	II	482	1440	960	File:Subdivided triangle 04 04.svg	{3,5+}4,4	u2nI dcctI	File:Conway polyhedron k5k6akdk5aD.png	{5+,3}4,4 GP5(4,4)	c2zD cctI	File:Conway polyhedron dadkt5daD.png
5	5	75	II	752	2250	1500	File:Subdivided triangle 05 05.svg	{3,5+}5,5	u5nI	File:Conway u5zI.png	{5+,3}5,5 GP5(5,5)	c5zD	File:Conway du5zI.png
6	6	108	II	1082	3240	2160	File:Subdivided triangle 06 06.svg	{3,5+}6,6	uxnI dctktI	File:Conway polyhedron kdkt5daD.png	{5+,3}6,6 GP5(6,6)	cyzD ctkdkD	File:Conway cyzD.png
7	7	147	II	1472	4410	2940	File:Subdivided triangle 07 07.svg	{3,5+}7,7	vvnI dwrwtI	File:Conway dwrwtI.png	{5+,3}7,7 GP5(7,7)	wwzD wrwdkD	File:Conway wrwdkD.png
8	8	192	II	1922	5760	3840	File:Subdivided triangle 08 08.svg	{3,5+}8,8	u3nI dccckD	File:Conway dccctI.png	{5+,3}8,8 GP5(8,8)	c3zD ccctI	File:Conway cccdkD.png
9	9	243	II	2432	7290	4860	File:Subdivided triangle 09 09.svg	{3,5+}9,9	xxnI ktktkD	File:Conway ktktkdI.png	{5+,3}9,9 GP5(9,9)	yyzD tktktI	File:Conway tktkdkD.png
12	12	432	II	4322	12960	8640	File:Subdivided triangle 12 12.svg	{3,5+}12,12	uuxnI dccdktkD	File:Conway dccdktkdI.png	{5+,3}12,12 GP5(12,12)	ccyzD cckttI	File:CctkdkD.png
14	14	588	II	5882	17640	11760	File:Subdivided triangle 14 14.svg	{3,5+}14,14	uvvnI dcwwkD	File:Conway dcwrwtI.png	{5+,3}14,14 GP5(14,14)	cwwzD cwrwtI	File:Conway cwwdkD.png
16	16	768	II	7682	23040	15360	File:Subdivided triangle 16 16.svg	{3,5+}16,16	uuuunI dcccctI	File:Conway dcccctI.png	{5+,3}16,16 GP5(16,16)	cccczD cccctI	File:Conway ccccdkD.png
2	1	7	III	72	210	140	File:Subdivided triangle 02 01.svg	{3,5+}2,1	vI dwD	File:Conway polyhedron K5sI.png	{5+,3}2,1 GP5(2,1)	wD	File:Conway polyhedron Dk5sI.png
3	1	13	III	132	390	260	File:Subdivided triangle 03 01.svg	{3,5+}3,1	v3,1I	File:Conway polyhedron u5I.png	{5+,3}3,1 GP5(3,1)	w3,1D	File:Goldberg polyhedron 3 1.png
3	2	19	III	192	570	380	File:Subdivided triangle 03 02.svg	{3,5+}3,2	v3I	File:Geodesic polyhedron 3 2.png	{5+,3}3,2 GP5(3,2)	w3D	File:Goldberg polyhedron 3 2.png
4	1	21	III	212	630	420	File:Subdivided triangle 04 01.svg	{3,5+}4,1	dwtI	File:Conway polyhedron K5k6st.png	{5+,3}4,1 GP5(4,1)	wkI	File:Conway polyhedron Dk5k6st.png
4	2	28	III	282	840	560	File:Subdivided triangle 04 02.svg	{3,5+}4,2	vnI dwtI	File:Conway polyhedron dcwdI.png	{5+,3}4,2 GP5(4,2)	wdkD	File:Conway polyhedron dk6k5adk5sD.png
4	3	37	III	372	1110	740	File:Subdivided triangle 04 03.svg	{3,5+}4,3	v4I		{5+,3}4,3 GP5(4,3)	w4D	File:Goldberg polyhedron 4 3.png
5	1	31	III	312	930	620	File:Subdivided triangle 05 01.svg	{3,5+}5,1	u5,1I		{5+,3}5,1 GP5(5,1)	w5,1D	File:Goldberg polyhedron 5 1.png
5	2	39	III	392	1170	780	File:Subdivided triangle 05 02.svg	{3,5+}5,2	u5,2I	File:Geodesic polyhedron 5 2.png	{5+,3}5,2 GP5(5,2)	w5,2D	File:Goldberg polyhedron 5 2.png
5	3	49	III	492	1470	980	File:Subdivided triangle 05 03.svg	{3,5+}5,3	vvI dwwD	File:Conway dwwD.png	{5+,3}5,3 GP5(5,3)	wwD	File:Goldberg polyhedron 5 3.png
6	2	52	III	522	1560	1040	File:Subdivided triangle 06 02.svg	{3,5+}6,2	v3,1uI	File:Geodesic polyhedron 6 2.png	{5+,3}6,2 GP5(6,2)	w3,1cD	File:Goldberg polyhedron 6 2.png
6	3	63	III	632	1890	1260	File:Subdivided triangle 06 03.svg	{3,5+}6,3	vxI dwdktI	File:Conway polyhedron dtkwdI.png	{5+,3}6,3 GP5(6,3)	wyD wtkD	File:Conway polyhedron tkwD.png
8	2	84	III	842	2520	1680	File:Subdivided triangle 08 02.svg	{3,5+}8,2	vunI dwctI		{5+,3}8,2 GP5(8,2)	wczD wcdkD	File:Conway dkcwD.png
8	4	112	III	1122	3360	2240	File:Subdivided triangle 08 04.svg	{3,5+}8,4	vuuI dwccD		{5+,3}8,4 GP5(8,4)	wccD	File:Conway ccwD.png
11	2	147	III	1472	4410	2940	File:Subdivided triangle 11 02.svg	{3,5+}11,2	vvnI dwwtI		{5+,3}11,2 GP5(11,2)	wwzD	File:Conway dkwwD.png
12	3	189	III	1892	5670	3780	File:Subdivided triangle 12 03.svg	{3,5+}12,3	vxnI dwtktktI		{5+,3}12,3 GP5(12,3)	wyzD wtktI	File:Conway wtkdkD.png
10	6	196	III	1962	5880	3920	File:Subdivided triangle 10 06.svg	{3,5+}10,6	vvuI dwwcD		{5+,3}10,6 GP5(10,6)	wwcD	File:Conway cwwD.png
12	6	252	III	2522	7560	5040	File:Subdivided triangle 12 06.svg	{3,5+}12,6	vxuI dwctkD		{5+,3}12,6 GP5(12,6)	cywD wctkD	File:Conway ctkwD.png
16	4	336	III	3362	10080	6720	File:Subdivided triangle 16 04.svg	{3,5+}16,4	vuunI dwcctI		{5+,3}16,4 GP5(16,4)	wcczD wcctI	File:Conway ccwdkD.png
14	7	343	III	3432	10290	6860	File:Subdivided triangle 14 07.svg	{3,5+}14,7	vvvI dwrwwD		{5+,3}14,7 GP5(14,7)	wwwD wrwwD	File:Conway wrwwD.png
15	9	441	III	4412	13230	8820	File:Subdivided triangle 15 09.svg	{3,5+}15,9	vvxI dwwtkD		{5+,3}15,9 GP5(15,9)	wwxD wwtkD	File:Conway dkwdkwD.png
16	8	448	III	4482	13440	8960	File:Subdivided triangle 16 08.svg	{3,5+}16,8	vuuuI dwcccD		{5+,3}16,8 GP5(16,8)	wcccD	File:Conway cccwD.png
18	1	343	III	3432	10290	6860		{3,5+}18,1	vvvI dwwwD		{5+,3}18,1 GP5(18,1)	wwwD	File:Conway wwwD.png
18	9	567	III	5672	17010	11340		{3,5+}18,9	vxxI dwtktkD		{5+,3}18,9 GP5(18,9)	wyyD wtktkD	File:Conway tktkwD.png
20	12	784	III	7842	23520	15680		{3,5+}20,12	vvuuI dwwccD		{5+,3}20,12 GP5(20,12)	wwccD	File:Conway cwcwD.png
20	17	1029	III	10292	30870	20580		{3,5+}20,17	vvvnI dwwwtI		{5+,3}20,17 GP5(20,17)	wwwzD wwwdkD	File:Conway wwwdkD.png
28	7	1029	III	10292	30870	20580		{3,5+}28,7	vvvnI dwrwwdkD		{5+,3}28,7 GP5(28,7)	wwwzD wrwwdkD	File:Conway wrwwdkD.png

Octahedral

m	n	T	Class	Vertices (geodesic) Faces (Goldberg)	Edges	Faces (geodesic) Vertices (Goldberg)	Face triangle	Geodesic			Goldberg
								Symbols	Conway	Image	Symbols	Conway	Image
1	0	1	I	6	12	8	File:Subdivided triangle 01 00.svg	{3,4} {3,4+}1,0	O	File:Octahedron.svg	{4,3} {4+,3}1,0 GP4(1,0)	C	File:Hexahedron.svg
2	0	4	I	18	48	32	File:Subdivided triangle 02 00.svg	{3,4+}2,0	dcC dcC	File:Tetrakis cuboctahedron.png	{4+,3}2,0 GP4(2,0)	cC cC	File:Truncated rhombic dodecahedron2.png
3	0	9	I	38	108	72	File:Subdivided triangle 03 00.svg	{3,4+}3,0	ktO	File:Octahedral geodesic polyhedron 03 00.svg	{4+,3}3,0 GP4(3,0)	tkC	File:Octahedral goldberg polyhedron 03 00.svg
4	0	16	I	66	192	128	File:Subdivided triangle 04 00.svg	{3,4+}4,0	uuO dccC	File:Octahedral geodesic polyhedron 04 00.svg	{4+,3}4,0 GP4(4,0)	ccC	File:Octahedral goldberg polyhedron 04 00.svg
5	0	25	I	102	300	200	File:Subdivided triangle 05 00.svg	{3,4+}5,0	u5O	File:Octahedral geodesic polyhedron 05 00.svg	{4+,3}5,0 GP4(5,0)	c5C	File:Octahedral goldberg polyhedron 05 00.svg
6	0	36	I	146	432	288	File:Subdivided triangle 06 00.svg	{3,4+}6,0	uxO dctkdO	File:Octahedral geodesic polyhedron 06 00.svg	{4+,3}6,0 GP4(6,0)	cyC ctkC	File:Octahedral goldberg polyhedron 06 00.svg
7	0	49	I	198	588	392	File:Subdivided triangle 07 00.svg	{3,4+}7,0	dwrwO	File:Octahedral geodesic polyhedron 07 00.svg	{4+,3}7,0 GP4(7,0)	wrwO	File:Octahedral goldberg polyhedron 07 00.svg
8	0	64	I	258	768	512	File:Subdivided triangle 08 00.svg	{3,4+}8,0	uuuO dcccC	File:Octahedral geodesic polyhedron 08 00.svg	{4+,3}8,0 GP4(8,0)	cccC	File:Octahedral goldberg polyhedron 08 00.svg
9	0	81	I	326	972	648	File:Subdivided triangle 09 00.svg	{3,4+}9,0	xxO ktktO	File:Octahedral geodesic polyhedron 09 00.svg	{4+,3}9,0 GP4(9,0)	yyC tktkC	File:Octahedral goldberg polyhedron 09 00.svg
1	1	3	II	14	36	24	File:Subdivided triangle 01 01.svg	{3,4+}1,1	kC	File:Tetrakishexahedron.jpg	{4+,3}1,1 GP4(1,1)	tO	File:Truncated octahedron.png
2	2	12	II	50	144	96	File:Subdivided triangle 02 02.svg	{3,4+}2,2	ukC dctO		{4+,3}2,2 GP4(2,2)	czC ctO	File:Conway polyhedron dkt4daC.png
3	3	27	II	110	324	216	File:Subdivided triangle 03 03.svg	{3,4+}3,3	ktkC		{4+,3}3,3 GP4(3,3)	tktO	File:Conway polyhedron tktO.png
4	4	48	II	194	576	384	File:Subdivided triangle 04 04.svg	{3,4+}4,4	uunO dcctO		{4+,3}4,4 GP4(4,4)	cczC cctO	File:Conway polyhedron dk6k4adk6k4adkC.png
2	1	7	III	30	84	56	File:Subdivided triangle 02 01.svg	{3,4+}2,1	vO dwC	File:Conway polyhedron dwC.png	{4+,3}2,1 GP4(2,1)	wC	File:Conway polyhedron wC.png

Tetrahedral

m	n	T	Class	Vertices (geodesic) Faces (Goldberg)	Edges	Faces (geodesic) Vertices (Goldberg)	Face triangle	Geodesic			Goldberg
								Symbols	Conway	Image	Symbols	Conway	Image
1	0	1	I	4	6	4	File:Subdivided triangle 01 00.svg	{3,3} {3,3+}1,0	T	File:Tetrahedron.svg	{3,3} {3+,3}1,0 GP3(1,0)	T	File:Tetrahedron.svg
1	1	3	II	8	18	12	File:Subdivided triangle 01 01.svg	{3,3+}1,1	kT kT	File:Triakistetrahedron.jpg	{3+,3}1,1 GP3(1,1)	tT tT	File:Truncated tetrahedron.png
2	0	4	I	10	24	16	File:Subdivided triangle 02 00.svg	{3,3+}2,0	dcT dcT	File:Dual chamfered tetrahedron.png	{3+,3}2,0 GP3(2,0)	cT cT	File:Alternate truncated cube.png
3	0	9	I	20	54	36	File:Subdivided triangle 03 00.svg	{3,3+}3,0	ktT	File:Tetrahedral geodesic polyhedron 03 00.svg	{3+,3}3,0 GP3(3,0)	tkT	File:Tetrahedral Goldberg polyhedron 03 00.svg
4	0	16	I	34	96	64	File:Subdivided triangle 04 00.svg	{3,3+}4,0	uuT dccT	File:Tetrahedral geodesic polyhedron 04 00.svg	{3+,3}4,0 GP3(4,0)	ccT	File:Tetrahedral Goldberg polyhedron 04 00.svg
5	0	25	I	52	150	100	File:Subdivided triangle 05 00.svg	{3,3+}5,0	u5T	File:Tetrahedral geodesic polyhedron 05 00.svg	{3+,3}5,0 GP3(5,0)	c5T	File:Tetrahedral Goldberg polyhedron 05 00.svg
6	0	36	I	74	216	144	File:Subdivided triangle 06 00.svg	{3,3+}6,0	uxT dctkdT	File:Tetrahedral geodesic polyhedron 06 00.svg	{3+,3}6,0 GP3(6,0)	cyT ctkT	File:Tetrahedral Goldberg polyhedron 06 00.svg
7	0	49	I	100	294	196	File:Subdivided triangle 07 00.svg	{3,3+}7,0	vrvT dwrwT	File:Tetrahedral geodesic polyhedron 07 00.svg	{3+,3}7,0 GP3(7,0)	wrwT	File:Tetrahedral Goldberg polyhedron 07 00.svg
8	0	64	I	130	384	256	File:Subdivided triangle 08 00.svg	{3,3+}8,0	u3T dcccdT	File:Tetrahedral geodesic polyhedron 08 00.svg	{3+,3}8,0 GP3(8,0)	c3T cccT	File:Tetrahedral Goldberg polyhedron 08 00.svg
9	0	81	I	164	486	324	File:Subdivided triangle 09 00.svg	{3,3+}9,0	xxT ktktT	File:Tetrahedral geodesic polyhedron 09 00.svg	{3+,3}9,0 GP3(9,0)	yyT tktkT	File:Tetrahedral Goldberg polyhedron 09 00.svg
3	3	27	II	56	162	108	File:Subdivided triangle 03 03.svg	{3,3+}3,3	ktkT		{3+,3}3,3 GP3(3,3)	tktT	File:Conway polyhedron tktT.png
2	1	7	III	16	42	28	File:Subdivided triangle 02 01.svg	{3,3+}2,1	dwT	File:Conway polyhedron dwT.png	{3+,3}2,1 GP5(2,1)	wT	File:Conway polyhedron wT.png

References

• Wenninger, Magnus (1979), Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, MR 0552023, archived from the original on July 4, 2008 Reprinted by Dover 1999 ISBN 978-0-486-40921-4