The Stellations of the Rhombic Triacontahedron

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



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 a X3D player. See note1. (Click here if you want to watch the models with a VRML player instead.)

Models of the Stellations

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

Name Regions X3D model Name Regions X3D model Name Regions X3D model
A 1a Model B 2ab Model C 2a, 3ac Model
CD 3abc Model D 2b, 3b Model DE 3ab, 4ad Model
DG 3ab, 4a, 5a Model E 2a, 3a, 4ad Model EF 4abcd Model
EI 4acd, 5c Model F 3c, 4bc Model FG 4abc, 5a Model
G 2a, 3a, 4a, 5a Model GH 4b, 5abd Model GHI 5abcd Model
GI 4ac, 5ac Model GK 5ad, 6bcd Model GM 5ad, 6bd, 7be Model
GMN 5ad, 6d, 7bcef Model GN 5ad, 6cd, 7cf Model GQ 5ad, 6d, 7ef, 8cf Model
GU 5ad, 6d, 7ef, 8f, 9g Model H 4bd, 5bd Model HI 4d, 5bcd Model
I 3c, 4c, 5c Model IJ 5bc, 6a Model J 4b, 5b, 6a Model
JK 6abcd Model JM 6abd, 7be Model JMN 6ad, 7bcef Model
JN 6acd, 7cf Model JQ 6ad, 7ef, 8cf Model JU 6ad, 7ef, 8f, 9g Model
K 4d, 5d, 6bcd Model L 6bc, 7ad Model LM 6b, 7abde Model
LMN 7abcdef Model LN 6c, 7acdf Model LQ 7adef, 8cf Model
LU 7adef, 8f, 9g Model M 4d, 5d, 6bd, 7be Model MN 4d, 5d, 6d, 7bcef Model
MO 7bcde, 8ad Model MR 7bcde, 8d, 9ae Model N 4d, 5d, 6cd, 7cf Model
NP 7abcf, 8be Model NT 7abcf, 8b, 9c Model O 6c, 7cd, 8ad Model
OP 7bc, 8abde Model OPQ 8abcdef Model OPU 8abdef, 9g Model
OQ 7de, 8acdf Model OQT 8abcdf, 9c Model OT 7bc, 8abd, 9c Model
OTU 8abdf, 9cg Model OU 7de, 8adf, 9g Model P 6b, 7ab, 8be Model
PQ 7af, 8bcef Model PQR 8bcdef, 9ae Model PR 7bc, 8bde, 9ae Model
PRU 8bdef, 9aeg Model PU 7af, 8bef, 9g Model Q 4d, 5d, 6d, 7ef, 8cf Model
QR 7de, 8cdf, 9ae Model QRT 8bcdf, 9ace Model QT 7af, 8bcf, 9c Model
R 6c, 7cd, 8d, 9ae Model RS 8ce, 9abdef Model RST 8c, 9abcdef Model
RSTU 9abcdefg Model RSU 8e, 9abdefg Model RT 7bc, 8bd, 9ace Model
RTU 8bdf, 9aceg Model RTW 10bg, 9abcef Model RU 7de, 8df, 9aeg Model
RUX 10cd, 9abdeg Model RW 10bg, 8e, 9abef Model RWX 10bcdg, 9abe Model
RX 10cd, 8c, 9abde Model R2A 10bc, 11cf, 9abe Model S 8ace, 9bdf Model
ST 8ac, 9bcdf Model STU 8a, 9bcdfg Model SU 8ae, 9bdfg Model
T 6b, 7ab, 8b, 9c Model TU 7af, 8bf, 9cg Model TUV 10aef, 9acdfg Model
TV 10aef, 8c, 9acdf Model TVW 10abefg, 9acf Model TW 10bg, 8a, 9bcf Model
TZ 10afg, 11be, 9acf Model U 4d, 5d, 6d, 7ef, 8f, 9g Model UV 10aef, 8e, 9adfg Model
UVX 10acdef, 9adg Model UX 10cd, 8a, 9bdg Model UY 10ade, 11ad, 9adg Model
V 10aef, 8ce, 9adf Model VW 10abefg, 8e, 9af Model VWX 10abcdefg, 9a Model
VX 10acdef, 8c, 9ad Model V2A 10abcef, 11cf, 9a Model W 10bg, 8ae, 9bf Model
WX 10bcdg, 8a, 9b Model WY 10abdeg, 11ad, 9a Model X 10cd, 8ac, 9bd Model
XZ 10acdfg, 11be, 9a Model Y 10ade, 11ad, 8c, 9ad Model YZ 10adg, 11abde, 9a Model
YZ2A 10a, 11abcdef, 9a Model Y2A 10abe, 11acdf, 9a Model Z 10afg, 11be, 8e, 9af Model
Z2A 10acf, 11bcef, 9a Model 2A 10bc, 11cf, 8a, 9b Model 2B 10a, 11abf, 12abc, 9a Model

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

Name Regions X3D model Name Regions X3D model
E, (F)r (4ad), (4bc)r, (3ab)l Model (F)r (3c), (4bc)r, (3ab)l Model
G, (F)r (4a, 5a), (4bc)r, (3ab)l Model G, (K)r (5ad), (6bcd)r, (5bc)l Model
G, (Q)r (5ad, 6d, 7ef), (8cf)r, (7bc)l Model J, (K)r (6a), (6bcd)r, (5bc)l Model
J, (Q)r (6ad, 7ef), (8cf)r, (7bc)l Model (K)r (4d, 5d), (6bcd)r, (5bc)l Model
L, (Q)r (7adef), (8cf)r, (7bc)l Model N, (P)r (7abcf), (8be)r, (7de)l Model
O, (P)r (7bc, 8ad), (8be)r, (7de)l Model OP, (Q)r (8abde), (8cf)r, (7bc)l Model
OQ, (P)r (8acdf), (8be)r, (7de)l Model O, (PQ)r (8ad), (8bcef)r, (7bcde)l Model
O, (Q)l, (P)r (8ad), (7bc, 8be)r, (7de, 8cf)l Model OU, (P)r (8adf, 9g), (8be)r, (7de)l Model
O, (Q)r (7de, 8ad), (8cf)r, (7bc)l Model OT, (Q)r (8abd, 9c), (8cf)r, (7bc)l Model
(P)r (6b, 7ab), (8be)r, (7de)l Model P, (Q)r (7af, 8be), (8cf)r, (7bc)l Model
Q, (P)r (7af, 8cf), (8be)r, (7de)l Model (PQ)r (7af), (8bcef)r, (7bcde)l Model
(Q)l, (P)r (7af), (7bc, 8be)r, (7de, 8cf)l Model PR, (Q)r (8bde, 9ae), (8cf)r, (7bc)l Model
QR, (P)r (8cdf, 9ae), (8be)r, (7de)l Model R, (PQ)r (8d, 9ae), (8bcef)r, (7bcde)l Model
R, (Q)l, (P)r (8d, 9ae), (7bc, 8be)r, (7de, 8cf)l Model R, (P)r (7bc, 8d, 9ae), (8be)r, (7de)l Model
RU, (P)r (8df, 9aeg), (8be)r, (7de)l Model U, (P)r (7af, 8f, 9g), (8be)r, (7de)l Model
(Q)r (4d, 5d, 6d, 7ef), (8cf)r, (7bc)l Model R, (Q)r (7de, 8d, 9ae), (8cf)r, (7bc)l Model
RT, (Q)r (8bd, 9ace), (8cf)r, (7bc)l Model T, (Q)r (7af, 8b, 9c), (8cf)r, (7bc)l Model
R, (S)r (8ce, 9ae), (9bdf)r, (8bdf)l Model R, (PS)r (8c, 9ae), (8e, 9bdf)r, (7de, 8df)l Model
R, (QS)r (8e, 9ae), (8c, 9bdf)r, (7bc, 8bd)l Model R, (PQS)r (9ae), (8ce, 9bdf)r, (7bcde, 8d)l Model
RT, (S)r (8c, 9ace), (9bdf)r, (8bdf)l Model RT, (QS)r (9ace), (8c, 9bdf)r, (7bc, 8bd)l Model
RTU, (S)r (9aceg), (9bdf)r, (8bdf)l Model RU, (S)r (8e, 9aeg), (9bdf)r, (8bdf)l Model
RU, (PS)r (9aeg), (8e, 9bdf)r, (7de, 8df)l Model RU, (X)r (9abdeg), (10cd)r, (9cf)l Model
RU, (SX)r (9aeg), (10cd, 9bd)r, (8bdf, 9c)l Model RW, (X)r (10bg, 9abe), (10cd)r, (9cf)l Model
R, (X)r (8c, 9abde), (10cd)r, (9cf)l Model R, (SX)r (8c, 9ae), (10cd, 9bd)r, (8bdf, 9c)l Model
R, (QSX)r (9ae), (10cd, 8c, 9bd)r, (7bc, 8bd, 9c)l Model (S)r (8ace), (9bdf)r, (8bdf)l Model
(PS)r (8ac), (8e, 9bdf)r, (7de, 8df)l Model (QS)r (8ae), (8c, 9bdf)r, (7bc, 8bd)l Model
(PQS)r (8a), (8ce, 9bdf)r, (7bcde, 8d)l Model T, (S)r (8ac, 9c), (9bdf)r, (8bdf)l Model
T, (QS)r (8a, 9c), (8c, 9bdf)r, (7bc, 8bd)l Model TU, (S)r (8a, 9cg), (9bdf)r, (8bdf)l Model
U, (S)r (8ae, 9g), (9bdf)r, (8bdf)l Model U, (PS)r (8a, 9g), (8e, 9bdf)r, (7de, 8df)l Model
TU, (V)r (9acdfg), (10aef)r, (9be)l Model TU, (SV)r (9acg), (10aef, 9df)r, (8bdf, 9e)l Model
T, (V)r (8c, 9acdf), (10aef)r, (9be)l Model T, (SV)r (8c, 9ac), (10aef, 9df)r, (8bdf, 9e)l Model
T, (QSV)r (9ac), (10aef, 8c, 9df)r, (7bc, 8bd, 9e)l Model TW, (V)r (10bg, 9acf), (10aef)r, (9be)l Model
T, (Z)r (10afg, 9acf), (11be)r, (10be)l Model T, (VZ)r (10g, 9acf), (10af, 11be)r, (10b, 9be)l Model
U, (V)r (8e, 9adfg), (10aef)r, (9be)l Model U, (SV)r (8e, 9ag), (10aef, 9df)r, (8bdf, 9e)l Model
U, (PSV)r (9ag), (10aef, 8e, 9df)r, (7de, 8df, 9e)l Model UV, (X)r (10aef, 9adg), (10cd)r, (9cf)l Model
UX, (V)r (10cd, 9adg), (10aef)r, (9be)l Model U, (VX)r (9adg), (10acdef)r, (9bcef)l Model
U, (X)l, (V)r (9adg), (10aef, 9cf)r, (10cd, 9be)l Model U, (SVX)r (9ag), (10acdef, 9d)r, (8bdf, 9ce)l Model
U, (X)r (8a, 9bdg), (10cd)r, (9cf)l Model U, (SX)r (8a, 9g), (10cd, 9bd)r, (8bdf, 9c)l Model
(V)r (8ce, 9adf), (10aef)r, (9be)l Model (SV)r (8ce, 9a), (10aef, 9df)r, (8bdf, 9e)l Model
(PSV)r (8c, 9a), (10aef, 8e, 9df)r, (7de, 8df, 9e)l Model (QSV)r (8e, 9a), (10aef, 8c, 9df)r, (7bc, 8bd, 9e)l Model
(PQSV)r (9a), (10aef, 8ce, 9df)r, (7bcde, 8d, 9e)l Model W, (V)r (10bg, 8e, 9af), (10aef)r, (9be)l Model
VW, (X)r (10abefg, 9a), (10cd)r, (9cf)l Model WX, (V)r (10bcdg, 9a), (10aef)r, (9be)l Model
W, (VX)r (10bg, 9a), (10acdef)r, (9bcef)l Model W, (X)l, (V)r (10bg, 9a), (10aef, 9cf)r, (10cd, 9be)l Model
V, (X)r (10aef, 8c, 9ad), (10cd)r, (9cf)l Model X, (V)r (10cd, 8c, 9ad), (10aef)r, (9be)l Model
(VX)r (8c, 9ad), (10acdef)r, (9bcef)l Model (X)l, (V)r (8c, 9ad), (10aef, 9cf)r, (10cd, 9be)l Model
(SVX)r (8c, 9a), (10acdef, 9d)r, (8bdf, 9ce)l Model (QSVX)r (9a), (10acdef, 8c, 9d)r, (7bc, 8bd, 9ce)l Model
2A, (V)r (10bc, 11cf, 9a), (10aef)r, (9be)l Model W, (X)r (10bg, 8a, 9b), (10cd)r, (9cf)l Model
(X)r (8ac, 9bd), (10cd)r, (9cf)l Model (SX)r (8ac), (10cd, 9bd)r, (8bdf, 9c)l Model
(QSX)r (8a), (10cd, 8c, 9bd)r, (7bc, 8bd, 9c)l Model X, (Z)r (10acdfg, 9a), (11be)r, (10be)l Model
Z, (X)r (10afg, 11be, 9a), (10cd)r, (9cf)l Model (XZ)r (10afg, 9a), (10cd, 11be)r, (10be, 9cf)l Model
(Z)l, (X)r (10afg, 9a), (10bcde)r, (11be, 9cf)l Model X, (VZ)r (10cdg, 9a), (10af, 11be)r, (10b, 9be)l Model
(VXZ)r (10g, 9a), (10acdf, 11be)r, (10b, 9bcef)l Model (VZ)l, (X)r (10g, 9a), (10bcd, 9be)r, (10af, 11be, 9cf)l Model
Y, (Z)r (10adg, 11ad, 9a), (11be)r, (10be)l Model Y2A, (Z)r (10a, 11acdf, 9a), (11be)r, (10be)l Model
(Z)r (10afg, 8e, 9af), (11be)r, (10be)l Model (VZ)r (10g, 8e, 9af), (10af, 11be)r, (10b, 9be)l Model
2A, (Z)r (10acf, 11cf, 9a), (11be)r, (10be)l Model 2A, (VZ)r (10c, 11cf, 9a), (10af, 11be)r, (10b, 9be)l Model
(2B)r (10a, 11abf, 9a), (12abc)r, (11cde)l Model (Z2B)r (10a, 11af, 9a), (11b, 12abc)r, (10be, 11cd)l Model


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


Last Updated

2006-02-18, 23:24