A rotor is an object in the geometric algebra (also called Clifford algebra) of a vector space that represents a rotation about the origin. The term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre). Hestenes defined a rotor to be any element of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies , where is the "reverse" of —that is, the product of the same vectors, but in reverse order. Definition In mathematics, a rotor in the geometric algebra of a vector space V is the same thing as an element of the spin group Spin(V). We define this group below. Let V be a vector space equipped with a positive definite quadratic form q, and let Cl(V) be the geometric algebra associated to V. The algebra Cl(V) is the quotient of the tensor algebra of V by the relations for all . (The tensor product in Cl(V) is what is called the geometric product in geometric algebra and in this article is denoted by .) The Z-grading on the tensor algebra of V descends to a Z/2Z-grading on Cl(V), which we denote by Here, Cleven(V) is generated by even-degree blades and Clodd(V) is generated by odd-degree blades. There is a unique antiautomorphism of Cl(V) which restricts to the identity on V: this is called the transpose, and the transpose of any multivector a is denoted by . On a blade (i.e., a simple tensor), it simply reverses the order of the factors. The spin group Spin(V) is defined to be the subgroup of Cleven(V) consisting of multivectors R such that That is, it consists of multivectors that can be written as a product of an even number of unit vectors. Action as rotation on the vector space Reflections along a vector in geometric algebra may be represented as (minus) sandwiching a multivector M between a non-null vector v perpendicular to the hyperplane of reflection and that vector's inverse v−1: and are of even grade. Under a rotation generated by the rotor R, a general multivector M will transform double-sidedly as This action gives a surjective homomorphism presenting Spin(V) as a double cover of SO(V). (See Spin group for more details.) Restricted alternative formulation For a Euclidean space, it may be convenient to consider an