What can you do with the symmetry group of an equilateral triangle? I saw this episode of How It's Made a couple of weeks ago. In it, they focused on how they make highlighters. At about the 3 minute mark of this clip, they show how they assemble 3-in-1 highlighters. At about the 4:30 minute mark, they show an application of Abstract Algebra. In particular, an application of the symmetry group of an equilateral triangle, D3.
Recall, the symmetry group of the equilateral triangle contains six elements: The identity transformation, a 120-degree rotation, a 240-degree rotation (they are both either clockwise or counter-clockwise), and three reflections, one in the perpendicular bisector of each side. The result of the composition of any two of these transformations is equivalent to one of the others.
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