The Stellations of the Rhombic Triacontahedron

This page is under construction in the sense that the 3D models still need to be fine tuned.

Contents

• Introduction
• Models of the Stellations
• Notes
• References
• Links
• Last Updated

Introduction

This page presents interactive 3D models of the stellations of the rhombic triacontahedron . The 3D models have predefined camera points from which the different kind of symmetry axes are recognised. In the article The Stellations of the Rhombic Triacontahedron and beyond P. Messer presents a way to derive all fully supported stellations of the rhombic triacontahedron according to a set of rules. All models on this page are based on that set of rules and the presented methods were used to generate all models.

To be able to view and investigate the 3D models you need to have working OpenGL drivers and JavaScript enabled.

Models of the Stellations

The following table contains the list of fully supported reflexible stellations.

Name	Regions	3D model	Name	Regions	3D model	Name	Regions	3D model
A	1a	No. 1	B	2ab	No. 2	C	2a, 3ac	No. 3
CD	3abc	No. 4	D	2b, 3b	No. 5	DE	3ab, 4ad	No. 6
DG	3ab, 4a, 5a	No. 7	E	2a, 3a, 4ad	No. 8	EF	4abcd	No. 9
EI	4acd, 5c	No. 10	F	3c, 4bc	No. 11	FG	4abc, 5a	No. 12
G	2a, 3a, 4a, 5a	No. 13	GH	4b, 5abd	No. 14	GHI	5abcd	No. 15
GI	4ac, 5ac	No. 16	GK	5ad, 6bcd	No. 17	GM	5ad, 6bd, 7be	No. 18
GMN	5ad, 6d, 7bcef	No. 19	GN	5ad, 6cd, 7cf	No. 20	GQ	5ad, 6d, 7ef, 8cf	No. 21
GU	5ad, 6d, 7ef, 8f, 9g	No. 22	H	4bd, 5bd	No. 23	HI	4d, 5bcd	No. 24
I	3c, 4c, 5c	No. 25	IJ	5bc, 6a	No. 26	J	4b, 5b, 6a	No. 27
JK	6abcd	No. 28	JM	6abd, 7be	No. 29	JMN	6ad, 7bcef	No. 30
JN	6acd, 7cf	No. 31	JQ	6ad, 7ef, 8cf	No. 32	JU	6ad, 7ef, 8f, 9g	No. 33
K	4d, 5d, 6bcd	No. 34	L	6bc, 7ad	No. 35	LM	6b, 7abde	No. 36
LMN	7abcdef	No. 37	LN	6c, 7acdf	No. 38	LQ	7adef, 8cf	No. 39
LU	7adef, 8f, 9g	No. 40	M	4d, 5d, 6bd, 7be	No. 41	MN	4d, 5d, 6d, 7bcef	No. 42
MO	7bcde, 8ad	No. 43	MR	7bcde, 8d, 9ae	No. 44	N	4d, 5d, 6cd, 7cf	No. 45
NP	7abcf, 8be	No. 46	NT	7abcf, 8b, 9c	No. 47	O	6c, 7cd, 8ad	No. 48
OP	7bc, 8abde	No. 49	OPQ	8abcdef	No. 50	OPU	8abdef, 9g	No. 51
OQ	7de, 8acdf	No. 52	OQT	8abcdf, 9c	No. 53	OT	7bc, 8abd, 9c	No. 54
OTU	8abdf, 9cg	No. 55	OU	7de, 8adf, 9g	No. 56	P	6b, 7ab, 8be	No. 57
PQ	7af, 8bcef	No. 58	PQR	8bcdef, 9ae	No. 59	PR	7bc, 8bde, 9ae	No. 60
PRU	8bdef, 9aeg	No. 61	PU	7af, 8bef, 9g	No. 62	Q	4d, 5d, 6d, 7ef, 8cf	No. 63
QR	7de, 8cdf, 9ae	No. 64	QRT	8bcdf, 9ace	No. 65	QT	7af, 8bcf, 9c	No. 66
R	6c, 7cd, 8d, 9ae	No. 67	RS	8ce, 9abdef	No. 68	RST	8c, 9abcdef	No. 69
RSTU	9abcdefg	No. 70	RSU	8e, 9abdefg	No. 71	RT	7bc, 8bd, 9ace	No. 72
RTU	8bdf, 9aceg	No. 73	RTW	10bg, 9abcef	No. 74	RU	7de, 8df, 9aeg	No. 75
RUX	10cd, 9abdeg	No. 76	RW	10bg, 8e, 9abef	No. 77	RWX	10bcdg, 9abe	No. 78
RX	10cd, 8c, 9abde	No. 79	R2A	10bc, 11cf, 9abe	No. 80	S	8ace, 9bdf	No. 81
ST	8ac, 9bcdf	No. 82	STU	8a, 9bcdfg	No. 83	SU	8ae, 9bdfg	No. 84
T	6b, 7ab, 8b, 9c	No. 85	TU	7af, 8bf, 9cg	No. 86	TUV	10aef, 9acdfg	No. 87
TV	10aef, 8c, 9acdf	No. 88	TVW	10abefg, 9acf	No. 89	TW	10bg, 8a, 9bcf	No. 90
TZ	10afg, 11be, 9acf	No. 91	U	4d, 5d, 6d, 7ef, 8f, 9g	No. 92	UV	10aef, 8e, 9adfg	No. 93
UVX	10acdef, 9adg	No. 94	UX	10cd, 8a, 9bdg	No. 95	UY	10ade, 11ad, 9adg	No. 96
V	10aef, 8ce, 9adf	No. 97	VW	10abefg, 8e, 9af	No. 98	VWX	10abcdefg, 9a	No. 99
VX	10acdef, 8c, 9ad	No. 100	V2A	10abcef, 11cf, 9a	No. 101	W	10bg, 8ae, 9bf	No. 102
WX	10bcdg, 8a, 9b	No. 103	WY	10abdeg, 11ad, 9a	No. 104	X	10cd, 8ac, 9bd	No. 105
XZ	10acdfg, 11be, 9a	No. 106	Y	10ade, 11ad, 8c, 9ad	No. 107	YZ	10adg, 11abde, 9a	No. 108
YZ2A	10a, 11abcdef, 9a	No. 109	Y2A	10abe, 11acdf, 9a	No. 110	Z	10afg, 11be, 8e, 9af	No. 111
Z2A	10acf, 11bcef, 9a	No. 112	2A	10bc, 11cf, 8a, 9b	No. 113	2B	10a, 11abf, 12abc, 9a	No. 114

The following table contains the list of fully supported enantiomorphous stellations.

Name	Regions	3D model	Name	Regions	3D model
E, (F)r	(4ad), (4bc)r, (3ab)l	No. 115	(F)r	(3c), (4bc)r, (3ab)l	No. 116
G, (F)r	(4a, 5a), (4bc)r, (3ab)l	No. 117	G, (K)r	(5ad), (6bcd)r, (5bc)l	No. 118
G, (Q)r	(5ad, 6d, 7ef), (8cf)r, (7bc)l	No. 119	J, (K)r	(6a), (6bcd)r, (5bc)l	No. 120
J, (Q)r	(6ad, 7ef), (8cf)r, (7bc)l	No. 121	(K)r	(4d, 5d), (6bcd)r, (5bc)l	No. 122
L, (Q)r	(7adef), (8cf)r, (7bc)l	No. 123	N, (P)r	(7abcf), (8be)r, (7de)l	No. 124
O, (P)r	(7bc, 8ad), (8be)r, (7de)l	No. 125	OP, (Q)r	(8abde), (8cf)r, (7bc)l	No. 126
OQ, (P)r	(8acdf), (8be)r, (7de)l	No. 127	O, (PQ)r	(8ad), (8bcef)r, (7bcde)l	No. 128
O, (Q)l, (P)r	(8ad), (7bc, 8be)r, (7de, 8cf)l	No. 129	OU, (P)r	(8adf, 9g), (8be)r, (7de)l	No. 130
O, (Q)r	(7de, 8ad), (8cf)r, (7bc)l	No. 131	OT, (Q)r	(8abd, 9c), (8cf)r, (7bc)l	No. 132
(P)r	(6b, 7ab), (8be)r, (7de)l	No. 133	P, (Q)r	(7af, 8be), (8cf)r, (7bc)l	No. 134
Q, (P)r	(7af, 8cf), (8be)r, (7de)l	No. 135	(PQ)r	(7af), (8bcef)r, (7bcde)l	No. 136
(Q)l, (P)r	(7af), (7bc, 8be)r, (7de, 8cf)l	No. 137	PR, (Q)r	(8bde, 9ae), (8cf)r, (7bc)l	No. 138
QR, (P)r	(8cdf, 9ae), (8be)r, (7de)l	No. 139	R, (PQ)r	(8d, 9ae), (8bcef)r, (7bcde)l	No. 140
R, (Q)l, (P)r	(8d, 9ae), (7bc, 8be)r, (7de, 8cf)l	No. 141	R, (P)r	(7bc, 8d, 9ae), (8be)r, (7de)l	No. 142
RU, (P)r	(8df, 9aeg), (8be)r, (7de)l	No. 143	U, (P)r	(7af, 8f, 9g), (8be)r, (7de)l	No. 144
(Q)r	(4d, 5d, 6d, 7ef), (8cf)r, (7bc)l	No. 145	R, (Q)r	(7de, 8d, 9ae), (8cf)r, (7bc)l	No. 146
RT, (Q)r	(8bd, 9ace), (8cf)r, (7bc)l	No. 147	T, (Q)r	(7af, 8b, 9c), (8cf)r, (7bc)l	No. 148
R, (S)r	(8ce, 9ae), (9bdf)r, (8bdf)l	No. 149	R, (PS)r	(8c, 9ae), (8e, 9bdf)r, (7de, 8df)l	No. 150
R, (QS)r	(8e, 9ae), (8c, 9bdf)r, (7bc, 8bd)l	No. 151	R, (PQS)r	(9ae), (8ce, 9bdf)r, (7bcde, 8d)l	No. 152
RT, (S)r	(8c, 9ace), (9bdf)r, (8bdf)l	No. 153	RT, (QS)r	(9ace), (8c, 9bdf)r, (7bc, 8bd)l	No. 154
RTU, (S)r	(9aceg), (9bdf)r, (8bdf)l	No. 155	RU, (S)r	(8e, 9aeg), (9bdf)r, (8bdf)l	No. 156
RU, (PS)r	(9aeg), (8e, 9bdf)r, (7de, 8df)l	No. 157	RU, (X)r	(9abdeg), (10cd)r, (9cf)l	No. 158
RU, (SX)r	(9aeg), (10cd, 9bd)r, (8bdf, 9c)l	No. 159	RW, (X)r	(10bg, 9abe), (10cd)r, (9cf)l	No. 160
R, (X)r	(8c, 9abde), (10cd)r, (9cf)l	No. 161	R, (SX)r	(8c, 9ae), (10cd, 9bd)r, (8bdf, 9c)l	No. 162
R, (QSX)r	(9ae), (10cd, 8c, 9bd)r, (7bc, 8bd, 9c)l	No. 163	(S)r	(8ace), (9bdf)r, (8bdf)l	No. 164
(PS)r	(8ac), (8e, 9bdf)r, (7de, 8df)l	No. 165	(QS)r	(8ae), (8c, 9bdf)r, (7bc, 8bd)l	No. 166
(PQS)r	(8a), (8ce, 9bdf)r, (7bcde, 8d)l	No. 167	T, (S)r	(8ac, 9c), (9bdf)r, (8bdf)l	No. 168
T, (QS)r	(8a, 9c), (8c, 9bdf)r, (7bc, 8bd)l	No. 169	TU, (S)r	(8a, 9cg), (9bdf)r, (8bdf)l	No. 170
U, (S)r	(8ae, 9g), (9bdf)r, (8bdf)l	No. 171	U, (PS)r	(8a, 9g), (8e, 9bdf)r, (7de, 8df)l	No. 172
TU, (V)r	(9acdfg), (10aef)r, (9be)l	No. 173	TU, (SV)r	(9acg), (10aef, 9df)r, (8bdf, 9e)l	No. 174
T, (V)r	(8c, 9acdf), (10aef)r, (9be)l	No. 175	T, (SV)r	(8c, 9ac), (10aef, 9df)r, (8bdf, 9e)l	No. 176
T, (QSV)r	(9ac), (10aef, 8c, 9df)r, (7bc, 8bd, 9e)l	No. 177	TW, (V)r	(10bg, 9acf), (10aef)r, (9be)l	No. 178
T, (Z)r	(10afg, 9acf), (11be)r, (10be)l	No. 179	T, (VZ)r	(10g, 9acf), (10af, 11be)r, (10b, 9be)l	No. 180
U, (V)r	(8e, 9adfg), (10aef)r, (9be)l	No. 181	U, (SV)r	(8e, 9ag), (10aef, 9df)r, (8bdf, 9e)l	No. 182
U, (PSV)r	(9ag), (10aef, 8e, 9df)r, (7de, 8df, 9e)l	No. 183	UV, (X)r	(10aef, 9adg), (10cd)r, (9cf)l	No. 184
UX, (V)r	(10cd, 9adg), (10aef)r, (9be)l	No. 185	U, (VX)r	(9adg), (10acdef)r, (9bcef)l	No. 186
U, (X)l, (V)r	(9adg), (10aef, 9cf)r, (10cd, 9be)l	No. 187	U, (SVX)r	(9ag), (10acdef, 9d)r, (8bdf, 9ce)l	No. 188
U, (X)r	(8a, 9bdg), (10cd)r, (9cf)l	No. 189	U, (SX)r	(8a, 9g), (10cd, 9bd)r, (8bdf, 9c)l	No. 190
(V)r	(8ce, 9adf), (10aef)r, (9be)l	No. 191	(SV)r	(8ce, 9a), (10aef, 9df)r, (8bdf, 9e)l	No. 192
(PSV)r	(8c, 9a), (10aef, 8e, 9df)r, (7de, 8df, 9e)l	No. 193	(QSV)r	(8e, 9a), (10aef, 8c, 9df)r, (7bc, 8bd, 9e)l	No. 194
(PQSV)r	(9a), (10aef, 8ce, 9df)r, (7bcde, 8d, 9e)l	No. 195	W, (V)r	(10bg, 8e, 9af), (10aef)r, (9be)l	No. 196
VW, (X)r	(10abefg, 9a), (10cd)r, (9cf)l	No. 197	WX, (V)r	(10bcdg, 9a), (10aef)r, (9be)l	No. 198
W, (VX)r	(10bg, 9a), (10acdef)r, (9bcef)l	No. 199	W, (X)l, (V)r	(10bg, 9a), (10aef, 9cf)r, (10cd, 9be)l	No. 200
V, (X)r	(10aef, 8c, 9ad), (10cd)r, (9cf)l	No. 201	X, (V)r	(10cd, 8c, 9ad), (10aef)r, (9be)l	No. 202
(VX)r	(8c, 9ad), (10acdef)r, (9bcef)l	No. 203	(X)l, (V)r	(8c, 9ad), (10aef, 9cf)r, (10cd, 9be)l	No. 204
(SVX)r	(8c, 9a), (10acdef, 9d)r, (8bdf, 9ce)l	No. 205	(QSVX)r	(9a), (10acdef, 8c, 9d)r, (7bc, 8bd, 9ce)l	No. 206
2A, (V)r	(10bc, 11cf, 9a), (10aef)r, (9be)l	No. 207	W, (X)r	(10bg, 8a, 9b), (10cd)r, (9cf)l	No. 208
(X)r	(8ac, 9bd), (10cd)r, (9cf)l	No. 209	(SX)r	(8ac), (10cd, 9bd)r, (8bdf, 9c)l	No. 210
(QSX)r	(8a), (10cd, 8c, 9bd)r, (7bc, 8bd, 9c)l	No. 211	X, (Z)r	(10acdfg, 9a), (11be)r, (10be)l	No. 212
Z, (X)r	(10afg, 11be, 9a), (10cd)r, (9cf)l	No. 213	(XZ)r	(10afg, 9a), (10cd, 11be)r, (10be, 9cf)l	No. 214
(Z)l, (X)r	(10afg, 9a), (10bcde)r, (11be, 9cf)l	No. 215	X, (VZ)r	(10cdg, 9a), (10af, 11be)r, (10b, 9be)l	No. 216
(VXZ)r	(10g, 9a), (10acdf, 11be)r, (10b, 9bcef)l	No. 217	(VZ)l, (X)r	(10g, 9a), (10bcd, 9be)r, (10af, 11be, 9cf)l	No. 218
Y, (Z)r	(10adg, 11ad, 9a), (11be)r, (10be)l	No. 219	Y2A, (Z)r	(10a, 11acdf, 9a), (11be)r, (10be)l	No. 220
(Z)r	(10afg, 8e, 9af), (11be)r, (10be)l	No. 221	(VZ)r	(10g, 8e, 9af), (10af, 11be)r, (10b, 9be)l	No. 222
2A, (Z)r	(10acf, 11cf, 9a), (11be)r, (10be)l	No. 223	2A, (VZ)r	(10c, 11cf, 9a), (10af, 11be)r, (10b, 9be)l	No. 224
(2B)r	(10a, 11abf, 9a), (12abc)r, (11cde)l	No. 225	(Z2B)r	(10a, 11af, 9a), (11b, 12abc)r, (10be, 11cd)l	No. 226

References

Messer, Peter W., The Stellations of the Rhombic Triacontahedron and beyond, Structural Topology 21 (p. 25-46, 1995)

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Last Updated

2024-04-16