Symmetries of a Square and an Equilateral Triangle

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In "single symmetry" mode, the Demonstration shows the action of a selected symmetry of a square or an equilateral triangle. In "composition" mode, the effect of two symmetries applied successively is shown. The set of symmetries of each figure forms a group under composition and the group multiplication table is shown in "Cayley table" mode.

Contributed by: Marc Brodie (July 2012)
(Wheeling Jesuit University)
Open content licensed under CC BY-NC-SA



The notation is based on that used in [1]. Rotations are counterclockwise and composition is right-to-left, that is, in , acts first. The reflection over the horizontal axis through the center of a figure is denoted ; stands for a reflection over the central vertical axis. The reflection over the "main" diagonal (upper-left to lower-right) of a square is denoted ; is the reflection over the other diagonal. Similarly, when the figure is set to "triangle," denotes the axis of reflection that cuts through the upper-left edge and lower-right vertex of the triangle, and is the axis of reflection sloping the other way. The elements of the group in the Cayley table are listed with rotations preceding reflections.


[1] J. Gallian, Contemporary Abstract Algebra, 7th ed., Belmont, CA: Brooks/Cole, Cengage Learning, 2010.

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