Generating 3D Figures with a Given Symmetry Group

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A symmetry of a figure is a transformation, such as a rotation, reflection, inversion, etc., that repositions the figure to be indistinguishable from the original. For example, rotating a circle about its center is a symmetry of the circle.


All the symmetries of a figure form a group called the figure's symmetry group. This Demonstration considers some figures in 3D consisting of finite sets of congruent triangles.


Contributed by: Izidor Hafner (December 2011)
Open content licensed under CC BY-NC-SA



If the figure has only one rotational axis, there are the four possible kinds of symmetries, all cyclic: (there is an axis of rotation and reflection, but there is no mirror plane), (there is a mirror plane, but it is not perpendicular to the axis), (there is a mirror plane that is perpendicular to the axis), and (there is a glide reflection).

If the figure has more than one rotational axis but no more than one -fold axis with , the possibilities are (dihedral symmetries): (no mirror plane), (the mirror plane is not perpendicular to the principal axis), (the mirror plane is perpendicular to the principal axis).

The figure may have more than one 5-fold axis (icosahedral symmetry): (rotations only), (there is a mirror plane).

The figure may have more than one 4-fold axis (octahedral symmetry): (rotations only), (there is a mirror plane).

The figure may have more than one principal 3-fold axis (tetrahedral symmetry): (rotations only), (there is a mirror plane, no inversion), (there is a point of inversion).

This Demonstration is a guessing game to learn about the 14 types of symmetry groups of figures that have a rotational axis.

Not demonstrated are the three symmetry groups with no rotational symmetry: (asymmetric ), (only inversion), and (only one mirror plane).


[1] P. R. Cromwell, Polyhedra, New York: Cambridge University Press, 1999 pp. 289–313.

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