Group structure When considered as the set of unit quaternions, inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, can be regarded as a real Lie group. It is a nonabelian, compact Lie group of dimension 3. When thought of as a Lie group is often denoted or . It turns out that the only spheres that admit a Lie group structure are , thought of as the set of unit complex numbers, and , the set of unit quaternions (The degenerate case which consists of the real numbers 1 and −1 is also a Lie group, albeit a 0-dimensional one). One might think that , the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give one important property: parallelizability. It turns out that the only spheres that are parallelizable are , , and . By using a matrix representation of the quaternions, , one obtains a matrix representation of . One convenient choice is given by the Pauli matrices: This map gives an injective algebra homomorphism from to the set of 2 × 2 complex matrices. It has the property that the absolute value of a quaternion is equal to the square root of the determinant of the matrix image of . The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group . Thus, as a Lie group is isomorphic to . Using our Hopf coordinates we can then write any element of in the form Another way to state this result is if we express the matrix representation of an element of as a exponential of a linear combination of the Pauli matrices. It is seen that an arbitrary element can be written as The condition that the determinant of is +1 implies that the coefficients are constrained to lie on a 3-sphere. In literature In Edwin Abbott Abbott's Flatland, published in 1884, and in Sphereland, a 1965 sequel to Flatland by Dionys Burger, the 3-sphere is referred to as an oversphere, and a 4-sphere is referred to as a hypersphere. Writing in the American Journal of Physics, Mark A. Peterson describes three different ways of