International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 770-771

Table 3.2.3.3 

H. Klapper,a Th. Hahna and M. I. Aroyoc

Table 3.2.3.3| top | pdf |
The two icosahedral point groups

Each point group is specified by its Hermann–Mauguin and Schoenflies symbol. For each point group, the stereographic projections show (on the left) the general position and (on the right) the symmetry elements. The list of the Wyckoff positions includes: Columns 1 to 3: multiplicity, Wyckoff letter, oriented site-symmetry symbol; Under the left stereographic projection: face forms (in roman type) and point forms (in italics), corresponding to the values for the Miller indices and coordinates listed in the last column; only `initial' Miller indices and coordinates are given (see text).

235 I [Scheme scheme58]
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.. Rhomb-triacontahedron (100)
      Icosadodecahedron (= pentagon-dodecahedron [\&] icosahedron) x, 0, 0
20 b .3. Regular icosahedron (111)
      Regular pentagon-dodecahedron x, x, x
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]
1 o 235 Point in origin 0, 0, 0
      Symmetry of special projections
      [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[1\tau 0]\cr 2mm&&3m&&5m\cr}]
[\matrix{m\bar{3}\bar{5}\hfill\cr\cr\displaystyle{2 \over m}\bar{3}\bar{5}\hfill\cr}] [I_{h}] [Scheme scheme59]
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\hfill\cr icosahedron\ \hbox{(poles between axes 2 and }\overline{3})\hfill\cr \cr \hbox{Deltoid-hexecontahedron}\hfill\cr Rhomb\hbox{-}triacontahedron\ \& \ pentagon\hbox{-}\hfill\cr dodecahedron\ \& \ 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{-}\hfill\cr dodecahedron\ \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]
1 o [2/m\,\overline{3}\,\overline{5}] Point in origin 0, 0, 0
      Symmetry of special projections
      [\matrix{\hbox{Along }[001]&&\hbox{Along }[111]&&\hbox{Along }[1\tau 0]\cr 2mm&&6mm&&10mm\cr}]