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.


    Virtual Polyhedra, (c) 1996,George W. Hart