Table 10.1.4.3

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000520

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 10.1, pp. 800-801

Table 10.1.4.3

Th. Hahn^a ^* and H. Klapper^a ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtl.rwth-aachen.de

Table 10.1.4.3 | top | pdf |
The two icosahedral point groups

General , special and limiting face forms and point forms , oriented face- and site-symmetry symbols , and `initial' values of (hkl) and x, y, z (see text).

235	I
60	d	1	Pentagon-hexecontahedron	(hkl)
			Snub pentagon-dodecahedron (= pentagon-dodecahedron + icosahedron + pentagon-hexecontahedron)	x, y, z
			$[\left\{\matrix{\hbox{Trisicosahedron}\hfill\cr Pentagon\hbox{-}dodecahedron\ truncated\ by\ icosahedron\hfill\cr (\hbox{poles between axes 2 and 3})\hfill\cr \cr\hbox{Deltoid\hbox{-}hexecontahedron}\hfill\cr Rhomb\hbox{-}triacontahedron\ \&\ \hfill\cr pentagon\hbox{-}dodecahedron\ \&\ icosahedron\hfill\cr (\hbox{poles between axes 3 and 5})\hfill\cr \cr \hbox{Pentakisdodecahedron}\hfill\cr Icosahedron\ truncated\ by\hfill\cr pentagon\hbox{-}dodecahedron\hfill\cr (\hbox{poles between axes 5 and 2)}\hfill\cr}\right.]$	$[\matrix{(0kl)\hbox{ with } \|l\| \lt 0.382 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \lt 0.382 \|y\|\hfill\cr \cr\cr (0kl)\hbox{ with }0.382 \|k\| \lt \|l\| \lt 1.618 \|k\|\hfill\cr 0,y,z\ with\ 0.382 \|y\| \lt \|z\| \lt 1.618 \|y\|\hfill\cr\cr\cr\cr (0kl)\hbox{ with }\|l\| \gt 1.618 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \gt 1.618 \|y\|\hfill\cr\cr\cr}]$
30	c	2..	$[\!\matrix{\hbox{Rhomb-triacontahedron}\hfill\cr Icosadodecahedron\ (\!= pentagon\hbox{-}\hfill\cr dodecahedron\ \&\ icosahedron)\hfill\cr}]$	$[\!\matrix{(100)\hfill\cr x,0,0\hfill\cr\cr}]$

20	b	.3.	$[\!\matrix{\hbox{Regular icosahedron}\hfill\cr Regular\ pentagon\hbox{-}dodecahedron\hfill\cr}]$	$[\!\matrix{(111)\hfill\cr x,\;x,\;x\hfill\cr}]$
12	a	..5	$[\!\matrix{\hbox{Regular pentagon-dodecahedron}\hfill\cr {Regular\ icosahedron}\hfill\cr}]$	$[\left.\matrix{(01\tau)\hfill\cr 0,y,\tau y\hfill\cr}\right\} \hbox{ with } \tau = {\textstyle{1 \over 2}}(\sqrt{5} + 1) = 1.618]$
Symmetry of special projections
$[\matrix{\hbox{Along } [001]&&&\hbox{Along } [111]&&&\hbox{Along } [1\tau 0]\cr 2mm&&&3m&&&5m}]$
$[\matrix{m\bar{3}\bar{5}\hfill\cr\cr\displaystyle{2 \over m}\bar{3}\bar{5}\hfill\cr}]$	$[I_{h}]$
120	e	l	Hecatonicosahedron or hexaicosahedron	(hkl)
			Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron	x, y, z
60	d	m..	$[\left\{\matrix{\hbox{Trisicosahedron}\hfill\cr Pentagon\hbox{-}dodecahedron\ truncated\ by\ icosahedron\hfill\cr \hbox{(poles between axes 2 and }\overline{3})\hfill\cr \cr \hbox{Deltoid-hexecontahedron}\hfill\cr Rhomb\hbox{-}triacontahedron\ \& \ pentagon\hbox{-}dodecahedron\ \& \hfill\cr icosahedron\hfill\cr \hbox{(poles between axes }\overline{3} \hbox{ and } \overline{5})\hfill\cr \cr \hbox{Pentakisdodecahedron}\hfill\cr Icosahedron\ truncated\ by\ pentagon\hbox{-}dodecahedron\hfill\cr \hbox{(poles between axes }\overline{5} \hbox{ and } 2)\hfill\cr}\right.]$	$[\matrix{(0kl) \hbox{ with } \|l\| \;\lt\; 0.382 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \;\lt\; 0.382 \|y\|\hfill\cr \cr\cr (0kl) \hbox{ with } 0.382 \|k\| \;\lt\; \|l\| \;\lt\; 1.618 \|k\|\hfill\cr 0,y,z\ with\ 0.382 \|y\| \;\lt\; \|z\| \;\lt\; 1.618 \|y\|\hfill\cr \cr\cr\cr (0kl) \hbox{ with } \|l\| \;\gt\; 1.618 \|k\|\hfill\cr 0,y,z\ with\ \|z\| \;\gt\; 1.618 \|y\|\hfill\cr\cr}]$
30	c	2mm..	Rhomb-triacontahedron	(100)
			Icosadodecahedron (= pentagon-dodecahedron $[\&]$ icosahedron)	x, 0, 0
20	b	3m (m3.)	Regular icosahedron	(111)
			Regular pentagon-dodecahedron	x, x, x
12	a	5m (m.5)	$[\!\matrix{\hbox{Regular pentagon-dodecahedron}\hfill\cr Regular\ icosahedron\hfill\cr}]$	$[\left.\matrix{(01\tau)\hfill\cr 0,y,\tau y\hfill\cr}\right\} \hbox{ with }\tau = {1 \over 2}(\sqrt{5} + 1) = 1.618]$
Symmetry of special projections
$[\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[1\tau 0]\cr 2mm&&6mm&&10mm\cr}]$