Exercise: Look through the entire
set of Johnson solids and determine which of them have more than one
axis
of symmetry.
Answer: By my count, there are 35 with the axes of symmetry
of a prism, listed here: (and none with any other polyhedral symmetry)
-
triangular dipyramid (J12)
-
pentagonal dipyramid (J13)
-
elongated triangular
dipyramid (J14)
-
elongated square dipyramid
(J15)
-
elongated pentagonal
dipyramid (J16)
-
gyroelongated square
dipyramid (J17)
-
gyrobifastigium (J26)
-
triangular orthobicupola
(J27)
-
square orthobicupola (J28)
-
square gyrobicupola (J29)
-
pentagonal orthobicupola
(J30)
-
pentagonal gyrobicupola
(J31)
-
pentagonal orthobirotunda
(J34)
-
elongated triangular
orthobicupola (J35)
-
elongated triangular
gyrobicupola (J36)
-
elongated square
gyrobicupola (J37)
-
elongated pentagonal
orthobicupola (J38)
-
elongated pentagonal
gyrobicupola (J39)
-
elongated
pentagonal orthobirotunda (J42)
-
elongated pentagonal
gyrobirotunda (J43)
-
gyroelongated
triangular bicupola (J44)
-
gyroelongated square
bicupola (J45)
-
gyroelongated
pentagonal bicupola (J46)
-
gyroelongated
pentagonal birotunda (J48)
-
triaugmented triangular
prism (J51)
-
parabiaugmented
hexagonal prism (J55)
-
triaugmented hexagonal
prism (J57)
-
parabiaugmented dodecahedron
(J59)
-
biaugmented truncated
cube (J67)
-
parabiaugmented
truncated dodecahedron (J69)
-
parabigyrate
rhombicosidodecahedron (J73)
-
parabidiminished
rhombicosidodecahedron (J80)
-
snub disphenoid (J84)
-
snub square antiprism (J85)
-
disphenocingulum (J90)
-
bilunabirotunda (J91)
Further: The last one on this list (J91) has the symmetry of
a 2-gonal prism: three orthogonal 2-fold axes and three planes of symmetry
each containing two axes. The one before it (J90) has the symmetry of a
2-gonal antiprism: three 2-fold axes and three planes of symmetry of which
two pass halfway between axes. Find another with 2-gonal anti-prism symmetry.
Answer: this one.