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 

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: If o is odd: 
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: If o is odd: 


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. 

Savriama and Klingenberg BMC Evolutionary Biology 2011 11:280 doi:10.1186/1471214811280 