Why flipping a triangle twice is the same as turning it
Pick up a cardboard equilateral triangle, flip it over along one line of symmetry, then flip it along another. It lands exactly as if you had simply rotated it by 120 degrees. That small trick sits at the heart of dihedral groups, the mathematics of every way a regular polygon can be moved and still look unchanged.
A regular polygon with n sides has n rotations that leave it looking the same, counting the do-nothing rotation, and n mirror flips. Together that makes 2n symmetries. For a triangle there are six: turns of 0, 120 and 240 degrees, plus three flips across lines running from each corner to the middle of the opposite side. A square has eight. Doing one symmetry after another always gives another symmetry, which is exactly what mathematicians need to call the collection a group.
The flip trick generalises neatly: reflecting across one axis and then another produces a rotation through twice the angle between the two axes. Order matters, though. For most polygons, doing move A then move B gives a different result from B then A, so these groups are non-commutative. Only the two smallest cases, D1 and D2, escape this, and D2 turns out to be the same structure as the Klein four-group.
Placing the polygon at the centre of a coordinate grid turns each symmetry into a small table of numbers, a matrix, so combining moves becomes matrix multiplication. The square's eight symmetries can be written as eight such two-by-two matrices, a simple example of what is called a group representation.
The name comes from Greek roots meaning two faces, because a flat polygon has a front and a back and a flip swaps them. Confusingly, geometers call the symmetry group of the n-sided polygon Dn, while algebraists call the same thing D2n after its number of elements. Dihedral groups are among the simplest finite groups, and they turn up in geometry and chemistry as well as pure algebra.
Source: Dihedral group
