Point Group - Two Dimensions

Two Dimensions

Point groups in two dimensions, sometimes called rosette groups.

They come in two infinite families:

  1. Cyclic groups Cn of n-fold rotation groups
  2. Dihedral groups Dn of n-fold rotation and reflection groups

Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups.

Group Intl Orbifold Coxeter Order Description
Cn n nn + n Cyclic: n-fold rotations. Abstract group Zn, the group of integers under addition modulo n.
Dn nm *nn 2n Dihedral: cyclic with reflections. Abstract group Dihn, the dihedral group.

The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an isomorphism, mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order.

Reflective Rotational Related polygons
Group Coxeter group Coxeter diagram Order Subgroup Coxeter Order
D1 A1 2 C1 + 1 Digon
D2 A12 4 C2 + 2 Rectangle
D3 A2 6 C3 + 3 Equilateral triangle
D4 BC2 8 C4 + 4 Square
D5 H2 10 C5 + 5 Regular pentagon
D6 G2 12 C6 + 6 Regular hexagon
Dn I2(n) 2n Cn + n Regular polygon
D2×2 A12×2 ] = = 8
D3×2 A2×2 ] = = 12
D4×2 BC2×2 ] = = 16
D5×2 H2×2 ] = = 20
D6×2 G2×2 ] = = 24
Dn×2 I2(n)×2 ] = = 4n

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