been formulated in terms of the plate trick, tangloids and other examples of orientation entanglement. Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah's statement that is recounted by Dirac's biographer Graham Farmelo: History The most general mathematical form of spinors was discovered by Élie Cartan in 1913. The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics. Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices. The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group. By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors. Spinor spaces were represented as left ideals of a matrix algebra in 1930, by G. Juvet and by Fritz Sauter. More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in . In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras. In 1966/1967, David Hestenes replaced spinor spaces by the even subalgebra Cℓ01,3() of the spacetime algebra Cℓ1,3(). As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals. Examples Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra . This is an algebra built up from an orthonormal basis of mutually orthogonal vectors under addition and multiplication, p of which have norm +1 and q of which have norm −1, with the product rule for the basis vectors Two dimensions The Clifford algebra Cℓ2,0() is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ1 and σ2, and one unit pseudoscalar . From the definitions above, it is evident that , and . The even subalgebra Cℓ02,0(), spanned by even-graded basis elements of Cℓ2,0(), determines the