List of geodesic polyhedra and Goldberg polyhedra

This is a list of selected

of each other. The geodesic and Goldberg polyhedra are parameterized by integers m and n, with ${\displaystyle m>0}$ and ${\displaystyle n\geq 0}$. T is the triangulation number, which is equal to ${\displaystyle T=m^{2}+mn+n^{2}}$.

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		{3,5} {3,5+}1,0	I		{5,3} {5+,3}1,0 GP5(1,0)	D
2	0	4	I	42	120	80		{3,5+}2,0	uI dcdI		{5+,3}2,0 GP5(2,0)	cD cD
3	0	9	I	92	270	180		{3,5+}3,0	xI ktI		{5+,3}3,0 GP5(3,0)	yD tkD
4	0	16	I	162	480	320		{3,5+}4,0	uuI dccD		{5+,3}4,0 GP5(4,0)	c2D
5	0	25	I	252	750	500		{3,5+}5,0	u5I u5I		{5+,3}5,0 GP5(5,0)	c5D du5I
6	0	36	I	362	1080	720		{3,5+}6,0	uxI dctkdI		{5+,3}6,0 GP5(6,0)	cyD ctkD
7	0	49	I	492	1470	980		{3,5+}7,0	vvI dwrwdI		{5+,3}7,0 GP5(7,0)	wwD wrwD
8	0	64	I	642	1920	1280		{3,5+}8,0	u3I dcccdI		{5+,3}8,0 GP5(8,0)	cccD
9	0	81	I	812	2430	1620		{3,5+}9,0	xxI ktktI		{5+,3}9,0 GP5(9,0)	yyD tktkD
10	0	100	I	1002	3000	2000		{3,5+}10,0	uu5I uu5I		{5+,3}10,0 GP5(10,0)	cc5D
11	0	121	I	1212	3630	2420		{3,5+}11,0	u11I u11I		{5+,3}11,0 GP5(11,0)	c11D
12	0	144	I	1442	4320	2880		{3,5+}12,0	uuxD dcctkD		{5+,3}12,0 GP5(12,0)	ccyD cctkD
13	0	169	I	1692	5070	3380		{3,5+}13,0	u13I u13I		{5+,3}13,0 GP5(13,0)	c13D
14	0	196	I	1962	5880	3920		{3,5+}14,0	uvvI dcwwdI		{5+,3}14,0 GP5(14,0)	cwrwD
15	0	225	I	2252	6750	4500		{3,5+}15,0	u5xI u5ktI		{5+,3}15,0 GP5(15,0)	c5yD c5tkD
16	0	256	I	2562	7680	5120		{3,5+}16,0	dc4dI		{5+,3}16,0 GP5(16,0)	ccccD
1	1	3	II	32	90	60		{3,5+}1,1	nI kD		{5+,3}1,1 GP5(1,1)	yD tI
2	2	12	II	122	360	240		{3,5+}2,2	unI =dctI		{5+,3}2,2 GP5(2,2)	czD cdkD
3	3	27	II	272	810	540		{3,5+}3,3	xnI ktkD		{5+,3}3,3 GP5(3,3)	yzD tkdkD
4	4	48	II	482	1440	960		{3,5+}4,4	u2nI dcctI		{5+,3}4,4 GP5(4,4)	c2zD cctI
5	5	75	II	752	2250	1500		{3,5+}5,5	u5nI		{5+,3}5,5 GP5(5,5)	c5zD
6	6	108	II	1082	3240	2160		{3,5+}6,6	uxnI dctktI		{5+,3}6,6 GP5(6,6)	cyzD ctkdkD
7	7	147	II	1472	4410	2940		{3,5+}7,7	vvnI dwrwtI		{5+,3}7,7 GP5(7,7)	wwzD wrwdkD
8	8	192	II	1922	5760	3840		{3,5+}8,8	u3nI dccckD		{5+,3}8,8 GP5(8,8)	c3zD ccctI
9	9	243	II	2432	7290	4860		{3,5+}9,9	xxnI ktktkD		{5+,3}9,9 GP5(9,9)	yyzD tktktI
12	12	432	II	4322	12960	8640		{3,5+}12,12	uuxnI dccdktkD		{5+,3}12,12 GP5(12,12)	ccyzD cckttI
14	14	588	II	5882	17640	11760		{3,5+}14,14	uvvnI dcwwkD		{5+,3}14,14 GP5(14,14)	cwwzD cwrwtI
16	16	768	II	7682	23040	15360		{3,5+}16,16	uuuunI dcccctI		{5+,3}16,16 GP5(16,16)	cccczD cccctI
2	1	7	III	72	210	140		{3,5+}2,1	vI dwD		{5+,3}2,1 GP5(2,1)	wD
3	1	13	III	132	390	260		{3,5+}3,1	v3,1I		{5+,3}3,1 GP5(3,1)	w3,1D
3	2	19	III	192	570	380		{3,5+}3,2	v3I		{5+,3}3,2 GP5(3,2)	w3D
4	1	21	III	212	630	420		{3,5+}4,1	dwtI		{5+,3}4,1 GP5(4,1)	wkI
4	2	28	III	282	840	560		{3,5+}4,2	vnI dwtI		{5+,3}4,2 GP5(4,2)	wdkD
4	3	37	III	372	1110	740		{3,5+}4,3	v4I		{5+,3}4,3 GP5(4,3)	w4D
5	1	31	III	312	930	620		{3,5+}5,1	u5,1I		{5+,3}5,1 GP5(5,1)	w5,1D
5	2	39	III	392	1170	780		{3,5+}5,2	u5,2I		{5+,3}5,2 GP5(5,2)	w5,2D
5	3	49	III	492	1470	980		{3,5+}5,3	vvI dwwD		{5+,3}5,3 GP5(5,3)	wwD
6	2	52	III	522	1560	1040		{3,5+}6,2	v3,1uI		{5+,3}6,2 GP5(6,2)	w3,1cD
6	3	63	III	632	1890	1260		{3,5+}6,3	vxI dwdktI		{5+,3}6,3 GP5(6,3)	wyD wtkD
8	2	84	III	842	2520	1680		{3,5+}8,2	vunI dwctI		{5+,3}8,2 GP5(8,2)	wczD wcdkD
8	4	112	III	1122	3360	2240		{3,5+}8,4	vuuI dwccD		{5+,3}8,4 GP5(8,4)	wccD
11	2	147	III	1472	4410	2940		{3,5+}11,2	vvnI dwwtI		{5+,3}11,2 GP5(11,2)	wwzD
12	3	189	III	1892	5670	3780		{3,5+}12,3	vxnI dwtktktI		{5+,3}12,3 GP5(12,3)	wyzD wtktI
10	6	196	III	1962	5880	3920		{3,5+}10,6	vvuI dwwcD		{5+,3}10,6 GP5(10,6)	wwcD
12	6	252	III	2522	7560	5040		{3,5+}12,6	vxuI dwctkD		{5+,3}12,6 GP5(12,6)	cywD wctkD
16	4	336	III	3362	10080	6720		{3,5+}16,4	vuunI dwcctI		{5+,3}16,4 GP5(16,4)	wcczD wcctI
14	7	343	III	3432	10290	6860		{3,5+}14,7	vvvI dwrwwD		{5+,3}14,7 GP5(14,7)	wwwD wrwwD
15	9	441	III	4412	13230	8820		{3,5+}15,9	vvxI dwwtkD		{5+,3}15,9 GP5(15,9)	wwxD wwtkD
16	8	448	III	4482	13440	8960		{3,5+}16,8	vuuuI dwcccD		{5+,3}16,8 GP5(16,8)	wcccD
18	1	343	III	3432	10290	6860		{3,5+}18,1	vvvI dwwwD		{5+,3}18,1 GP5(18,1)	wwwD
18	9	567	III	5672	17010	11340		{3,5+}18,9	vxxI dwtktkD		{5+,3}18,9 GP5(18,9)	wyyD wtktkD
20	12	784	III	7842	23520	15680		{3,5+}20,12	vvuuI dwwccD		{5+,3}20,12 GP5(20,12)	wwccD
20	17	1029	III	10292	30870	20580		{3,5+}20,17	vvvnI dwwwtI		{5+,3}20,17 GP5(20,17)	wwwzD wwwdkD
28	7	1029	III	10292	30870	20580		{3,5+}28,7	vvvnI dwrwwdkD		{5+,3}28,7 GP5(28,7)	wwwzD wrwwdkD

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		{3,4} {3,4+}1,0	O		{4,3} {4+,3}1,0 GP4(1,0)	C
2	0	4	I	18	48	32		{3,4+}2,0	dcC dcC		{4+,3}2,0 GP4(2,0)	cC cC
3	0	9	I	38	108	72		{3,4+}3,0	ktO		{4+,3}3,0 GP4(3,0)	tkC
4	0	16	I	66	192	128		{3,4+}4,0	uuO dccC		{4+,3}4,0 GP4(4,0)	ccC
5	0	25	I	102	300	200		{3,4+}5,0	u5O		{4+,3}5,0 GP4(5,0)	c5C
6	0	36	I	146	432	288		{3,4+}6,0	uxO dctkdO		{4+,3}6,0 GP4(6,0)	cyC ctkC
7	0	49	I	198	588	392		{3,4+}7,0	dwrwO		{4+,3}7,0 GP4(7,0)	wrwO
8	0	64	I	258	768	512		{3,4+}8,0	uuuO dcccC		{4+,3}8,0 GP4(8,0)	cccC
9	0	81	I	326	972	648		{3,4+}9,0	xxO ktktO		{4+,3}9,0 GP4(9,0)	yyC tktkC
1	1	3	II	14	36	24		{3,4+}1,1	kC		{4+,3}1,1 GP4(1,1)	tO
2	2	12	II	50	144	96		{3,4+}2,2	ukC dctO		{4+,3}2,2 GP4(2,2)	czC ctO
3	3	27	II	110	324	216		{3,4+}3,3	ktkC		{4+,3}3,3 GP4(3,3)	tktO
4	4	48	II	194	576	384		{3,4+}4,4	uunO dcctO		{4+,3}4,4 GP4(4,4)	cczC cctO
2	1	7	III	30	84	56		{3,4+}2,1	vO dwC		{4+,3}2,1 GP4(2,1)	wC

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		{3,3} {3,3+}1,0	T		{3,3} {3+,3}1,0 GP3(1,0)	T
1	1	3	II	8	18	12		{3,3+}1,1	kT kT		{3+,3}1,1 GP3(1,1)	tT tT
2	0	4	I	10	24	16		{3,3+}2,0	dcT dcT		{3+,3}2,0 GP3(2,0)	cT cT
3	0	9	I	20	54	36		{3,3+}3,0	ktT		{3+,3}3,0 GP3(3,0)	tkT
4	0	16	I	34	96	64		{3,3+}4,0	uuT dccT		{3+,3}4,0 GP3(4,0)	ccT
5	0	25	I	52	150	100		{3,3+}5,0	u5T		{3+,3}5,0 GP3(5,0)	c5T
6	0	36	I	74	216	144		{3,3+}6,0	uxT dctkdT		{3+,3}6,0 GP3(6,0)	cyT ctkT
7	0	49	I	100	294	196		{3,3+}7,0	vrvT dwrwT		{3+,3}7,0 GP3(7,0)	wrwT
8	0	64	I	130	384	256		{3,3+}8,0	u3T dcccdT		{3+,3}8,0 GP3(8,0)	c3T cccT
9	0	81	I	164	486	324		{3,3+}9,0	xxT ktktT		{3+,3}9,0 GP3(9,0)	yyT tktkT
3	3	27	II	56	162	108		{3,3+}3,3	ktkT		{3+,3}3,3 GP3(3,3)	tktT
2	1	7	III	16	42	28		{3,3+}2,1	dwT		{3+,3}2,1 GP5(2,1)	wT

References

• ISBN 978-0-486-40921-4