Problem of the day | 01-01-2025 | Abstract Algebra - Symmetries of equilateral triangle

Sachchidanand Prasad

01-01-2025

Problem: With pictures and words, describe each symmetry in $D_3$ (the set of symmetries of an equilateral triangle).

abstract-algebra group-theory

Solution: The set of symmetries of an equilateral triangle, contains six elements: three rotations and three reflections. These symmetries can be described as follows:

Rotation of $0^\circ$ ($R_0$): The triangle remains unchanged. This is the "do nothing" operation.
Rotation by $120^\circ$ ($R_{120}$): Each vertex of the triangle moves to the position of the next vertex in a counter-clockwise direction.
Rotation by $240^\circ$ ($R_{240}$): Each vertex of the triangle moves to the position of the next vertex in a counter-clockwise direction (equivalent to a $120^\circ$ clockwise rotation).
Reflection across a vertical axis ($V$): The triangle is flipped about the vertical line passing through one vertex and the midpoint of the opposite side.
Reflection across a diagonal axis ($D_1$): The triangle is flipped about an axis passing through one vertex and the midpoint of an opposite side.
Reflection across the other diagonal axis ($D_2$): The triangle is flipped about an axis passing through the other vertex and the midpoint of an opposite side.