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Table 2 Number of dimensions in the different components of shape space under object symmetry with rotation only or with rotation and reflection, for landmark data in two and three dimensions

From: Beyond bilateral symmetry: geometric morphometric methods for any type of symmetry

  2D 3D
Symmetry under rotation only:
Symmetric 2k - 2 3k + c - 3
Asymmetric 2k(o - 1) + 2c - 2 3k(o - 1) + 2c - 4
Symmetry under rotation and reflection:
Completely symmetric 2p + b + m - 1
= k - 1
3p + 2b + 2m + c - 2
Reflection symmetry only 2p(o - 1) + b(o - 1) + m(o - 1) + c - 1
= k(o - 1) + c - 1
If o is even:
3p ( o - 1 ) +b ( 3 2 o - 1 ) +m ( 3 2 o - 2 ) +c-2
If o is odd:
3 p o - 1 + 3 b o - 1 2 + 3 m o - 1 2 + c - 2
Rotational symmetry only 2p + b + m - 1
= k - 1
3p + b + m - 1
Completely asymmetric 2p(o - 1) + b(o - 1) + m(o - 1) + c - 1
= k(o - 1) + c - 1
If o is even:
3p ( o - 1 ) +b ( 3 2 o - 2 ) +m ( 3 2 o - 1 ) +c-2
If o is odd:
3p ( o - 1 ) +3b o - 1 2 +3m o - 1 2 +c-2
  1. Notation: For rotational symmetry of order o, the complete landmark configuration can be subdivided into o different sectors (Figure 4). Each sector contains k landmarks. In addition, there are c landmarks on the centre or axis of rotation (for 2D data, c is 0 or 1; for 3D data, c is 0 or greater). The sample consists of n individuals (specimens), and each specimen has been digitized r times.