Rotation, mirroring and inversion are noncommutative and hence non Abelian transformation operations in set theory. The result depends on the sequence of operations.
The simulation demonstrates this for the operations of rotation and mirroring, applied to a triangle.
When the simulation is started, one sees two identical coordinate systems, with a mirror located at the center x = 0 of each. At its left side there is a triangle with its base initially parallel to the x axis. Its upper edge is marked by the letter A. The triangle is pink in color. At the right side of the mirror one sees the mirror image of the triangle, marked by a letter M, and colored in blue
The slider Base Orientation defines the orientation of the original triangle. One observes the counterclockwise rotation of the mirror image, which is the same in both charts.
The slider Rotation Angle creates a rotation operation on the object in its base orientation.
In the left chart the rotation is operated on the object A, and the mirror image M of the rotated object R is displayed. In the right chart the rotation is operated on the mirrored object M and results in a different R. The sequence of operations is discernable by the coloring of the triangles: green is first, blue is second.
The left chart displays the mirrored-rotated object, the right one the rotated-mirrored object. In general both are different in orientation. The transformation operations are noncommutative.
Reset restores the initial orientation.