space of spinors via its representations. It is made up of real linear combinations of 1 and σ1σ2. As a real algebra, Cℓ02,0() is isomorphic to the field of complex numbers . As a result, it admits a conjugation operation (analogous to complex conjugation), sometimes called the reverse of a Clifford element, defined by which, by the Clifford relations, can be written The action of an even Clifford element on vectors, regarded as 1-graded elements of Cℓ2,0(), is determined by mapping a general vector to the vector where is the conjugate of , and the product is Clifford multiplication. In this situation, a spinor is an ordinary complex number. The action of on a spinor is given by ordinary complex multiplication: An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors: On the other hand, in comparison with its action on spinors , the action of on ordinary vectors appears as the square of its action on spinors. Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to , so that the corresponding action on spinors is via . In general, because of logarithmic branching, it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued. In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by abuse of language, the two are often conflated. One may then talk about "the action of a spinor on a vector". In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to computer graphics) they make sense. Examples The even-graded element corresponds to a vector rotation of 90° from σ1 around towards σ2, which can be checked by confirming that It corresponds to a spinor rotation of only 45°, however: Similarly the even-graded element corresponds to a vector rotation of 180°: but a spinor rotation of only 90°: Continuing on further, the even-graded element corresponds to a vector rotation of 360°: but a spinor