Reference Polyhedra

The geometries of 92 reference polyhedra, ranging from 2 to 60 vertices, are internally defined in Cosmochlore. This list was derived entirely from the SHAPE 2.1 list of reference polyhedra and has been discussed in numerous articles by Álvarez, Alemany, Avnir, LLunell, Cirera et al. ¹ Select them with `cshm -s/—sh` by their index within each vertex count, or add your own with user-defined shapes.

Shape Indices

The index of each shape differs from SHAPE’s 2.1 by 1: indices here start at 0. (The original numbering starting from one is probably due to Fortran arrays being silly.)

Vertices	Index	Label	Shape	Symmetry
2	0	L-2	Linear	$D_{\infty h}$
2	1	vT-2	Divacant tetrahedron (V-shape, 109.47º)	$C_{2v}$
2	2	vOC-2	Tetravacant octahedron (L-shape, 90º)	$C_{2v}$
3	0	TP-3	Trigonal planar	$D_{3h}$
3	1	vT-3	Pyramid‡ (vacant tetrahedron)	$C_{3v}$
3	2	fac-vOC-3	Trivacant octahedron	$C_{3v}$
3	3	mer-vOC-3	Trivacant octahedron (T-shape)	$C_{2v}$
4	0	SP-4	Square	$D_{4h}$
4	1	T-4	Tetrahedron	$T_d$
4	2	SS-4	Seesaw or sawhorse‡ (cis-divacant octahedron)	$C_{2v}$
4	3	vTBPY-4	Axially vacant trigonal bipyramid	$C_{3v}$
5	0	PP-5	Pentagon	$D_{5h}$
5	1	vOC-5	Vacant octahedron‡ (Johnson square pyramid, J1)	$C_{4v}$
5	2	TBPY-5	Trigonal bipyramid	$D_{3h}$
5	3	SPY-5	Square pyramid§	$C_{4v}$
5	4	JTBPY-5	Johnson trigonal bipyramid (J12)	$D_{3h}$
6	0	HP-6	Hexagon	$D_{6h}$
6	1	PPY-6	Pentagonal pyramid	$C_{5v}$
6	2	OC-6	Octahedron	$O_h$
6	3	TPR-6	Trigonal prism	$D_{3h}$
6	4	JPPY-6	Johnson pentagonal pyramid (J2)	$C_{5v}$
7	0	HP-7	Heptagon	$D_{7h}$
7	1	HPY-7	Hexagonal pyramid	$C_{6v}$
7	2	PBPY-7	Pentagonal bipyramid	$D_{5h}$
7	3	COC-7	Capped octahedron*	$C_{3v}$
7	4	CTPR-7	Capped trigonal prism*	$C_{2v}$
7	5	JPBPY-7	Johnson pentagonal bipyramid (J13)	$D_{5h}$
7	6	JETPY-7	Elongated triangular pyramid (J7)	$C_{3v}$
8	0	OP-8	Octagon	$D_{8h}$
8	1	HPY-8	Heptagonal pyramid	$C_{7v}$
8	2	HBPY-8	Hexagonal bipyramid	$D_{6h}$
8	3	CU-8	Cube	$O_h$
8	4	SAPR-8	Square antiprism	$D_{4d}$
8	5	TDD-8	Triangular dodecahedron	$D_{2d}$
8	6	JGBF-8	Johnson-Gyrobifastigium (J26)	$D_{2d}$
8	7	JETBPY-8	Johnson-Elongated triangular bipyramid (J14)	$D_{3h}$
8	8	JBTPR-8	Johnson-Biaugmented trigonal prism (J50)	$C_{2v}$
8	9	BTPR-8	Biaugmented trigonal prism	$C_{2v}$
8	10	JSD-8	Snub disphenoid (J84)	$D_{2d}$
8	11	TT-8	Triakis tetrahedron	$T_d$
8	12	ETBPY-8	Elongated trigonal bipyramid (see 8)	$D_{3h}$
9	0	EP-9	Enneagon	$D_{9h}$
9	1	OPY-9	Octagonal pyramid	$C_{8v}$
9	2	HBPY-9	Heptagonal bipyramid	$D_{7h}$
9	3	JTC-9	Triangular cupola (J3) = trivacant cuboctahedron	$C_{3v}$
9	4	JCCU-9	Capped cube (Elongated square pyramid, J8)	$C_{4v}$
9	5	CCU-9	Capped cube	$C_{4v}$
9	6	JCSAPR-9	Capped sq. antiprism (Gyroelongated square pyramid J10)	$C_{4v}$
9	7	CSAPR-9	Capped square antiprism	$C_{4v}$
9	8	JTCTPR-9	Tricapped trigonal prism (J51)	$D_{3h}$
9	9	TCTPR-9	Tricapped trigonal prism	$D_{3h}$
9	10	JTDIC-9	Tridiminished icosahedron (J63)	$C_{3v}$
9	11	HH-9	Hula-hoop	$C_{2v}$
9	12	MFF-9	Muffin	$C_s$
10	0	DP-10	Decagon	$D_{10h}$
10	1	EPY-10	Enneagonal pyramid	$C_{9v}$
10	2	OBPY-10	Octagonal bipyramid	$D_{8h}$
10	3	PPR-10	Pentagonal prism	$D_{5h}$
10	4	PAPR-10	Pentagonal antiprism	$D_{5d}$
10	5	JBCCU-10	Bicapped cube (Elongated square bipyramid J15)	$D_{4h}$
10	6	JBCSAPR-10	Bicapped square antiprism (Gyroelongated square bipyramid J17)	$D_{4d}$
10	7	JMBIC-10	Metabidiminished icosahedron (J62)	$C_{2v}$
10	8	JATDI-10	Augmented tridiminished icosahedron (J64)	$C_{3v}$
10	9	JSPC-10	Sphenocorona (J87)	$C_{2v}$
10	10	SDD-10	Staggered dodecahedron (2:6:2)#	$D_2$
10	11	TD-10	Tetradecahedron (2:6:2)	$C_{2v}$
10	12	HD-10	Hexadecahedron (2:6:2, or 1:4:4:1)	$D_{4h}$
11	0	HP-11	Hendecagon	$D_{11h}$
11	1	DPY-11	Decagonal pyramid	$C_{10v}$
11	2	EBPY-11	Enneagonal bipyramid	$D_{9h}$
11	3	JCPPR-11	Capped pent. prism (Elongated pentagonal pyramid J9)	$C_{5v}$
11	4	JCPAPR-11	Capped pent. antiprism (Gyroelongated pentagonal pyramid J11)	$C_{5v}$
11	5	JAPPR-11	Augmented pentagonal prism (J52)	$C_{2v}$
11	6	JASPC-11	Augmented sphenocorona (J87)	$C_s$
12	0	DP-12	Dodecagon	$D_{12h}$
12	1	HPY-12	Hendecagonal pyramid	$C_{11v}$
12	2	DBPY-12	Decagonal bipyramid	$D_{10h}$
12	3	HPR-12	Hexagonal prism	$D_{6h}$
12	4	HAPR-12	Hexagonal antiprism	$D_{6d}$
12	5	TT-12	Truncated tetrahedron	$T_d$
12	6	COC-12	Cuboctahedron	$O_h$
12	7	ACOC-12	Anticuboctahedron (Triangular orthobicupola J27)	$D_{3h}$
12	8	IC-12	Icosahedron	$I_h$
12	9	JSC-12	Square cupola (J4)	$C_{4v}$
12	10	JEPBPY-12	Elongated pentagonal bipyramid (J16)	$D_{6h}$
12	11	JBAPPR-12	Biaugmented pentagonal prism (J53)	$C_{2v}$
12	12	JSPMC-12	Sphenomegacorona (J88)	$C_s$
20	0	DD-20	Dodecahedron†	$I_h$
24	0	TCU-24	Truncated cube	$O_h$
24	1	TOC-24	Truncated octahedron	$O_h$
48	0	TCOC-48	Truncated cuboctahedron	$O_h$
60	0	TIC-60	Truncated icosahedron (fullerene)	$I_h$

* Non-regular polyhedron; see the references for the definition of the reference shape.

‡ A regular polyhedron with one or two vertices removed.

§ Spherical distribution of vertices with the centre of mass at the origin (apical-basal bond angles of 104.45º).

† For polyhedra with more than 12 vertices, calculation times may be impractical.

# Chiral polyhedron. cshm’s algorithm does not distinguish the two enantiomers of a chiral shape, so the resulting shape measures may refer to the enantiomer of the polyhedron rather than to the polyhedron itself.

1.

Alvarez, S. et al. Shape maps and polyhedral interconversion paths in transition metal chemistry. Coordination Chemistry Reviews 249, 1693–1708 (2005).