# Group Theory

The subgroup from abstract algebra is symmetry theory. Symmetry theory is the study of symmetry of an assortment of elements. These elements can then be interpreted through specific rules.  Symmetry groups are geometric entities such as a specific line, plane or a point. A symmetry operation has the ability to be carried out. Such operation of symmetry is a spatial transform that after the transform of the elements is carried out, every point is coincident with an equivalent point of the original element. The operations of symmetry transpose into a similar configuration to the original in which it is indistinguishable from. However, one must consider elements in a three dimensions. The symmetry group of interest to Vassar Applied Optics Laboratory is those composed of the sets of symmetry operations carried out on diffraction images on C. elegans . Most cases some C. elegans have a high degree of symmetry whereas other C. elegans posses very few. In order to take full advantage of the mathematical formalisms of symmetry theory, a strict and vigorous process for determination of symmetry was constructed. In order to establish these rules; a C.elegan must have certain symmetry features.

Symmetry elements are

1.)  Rotation axis, Cn: The n indicates the angle in which the rotation of the object takes place. ( the Angle is 360/n so 360/2=180) 2.) Mirror plane: σh, σv, σd: n, v  d:This indicate  the orientation of plane with respect to any rotation axes. 3.) Inversion center,this operation occurs through a single point i located at the center of the object. 4.) Improper rotation axis, Sn:This involves a combination of rotation and reflection. 5.) Identity : E is always a symmetry element as doing nothing to an object always leaves it looking just the same as it originally did.