Global InformationQuaternion information
| ↓ × → | 1 | i | j | k |
|---|---|---|---|---|
| 1 | 1 | i | j | k |
| i | i | −1 | k | −j |
| j | j | −k | −1 | i |
| k | k | j | −i | −1 |
| Left column shows the left factor, top row shows the right factor. Also, and for , . | ||||
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843[1][2] and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space.[3][4] Quaternions are generally represented in the form
where the coefficients a, b, c, d are real numbers, and 1, i, j, k are the basis vectors or basis elements.[5]
Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, magnetic resonance imaging[6] and crystallographic texture analysis.[7] They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
In modern terms, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case of a Clifford algebra, classified as It was the first noncommutative division algebra to be discovered.
According to the Frobenius theorem, the algebra is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra.[8]
The unit quaternions give a group structure on the 3-sphere S3 isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). The positive and negative basis vectors form the eight-element quaternion group.
- In blue:
- 1 ⋅ i = i (1/i plane)
- i ⋅ j = k (i/k plane)
- In red:
- 1 ⋅ j = j (1/j plane)
- j ⋅ i = −k (j/k plane)
- ^ "On Quaternions; or on a new System of Imaginaries in Algebra". Letter to John T. Graves. 17 October 1843.
- ^ Rozenfelʹd, Boris Abramovich (1988). The history of non-euclidean geometry: Evolution of the concept of a geometric space. Springer. p. 385. ISBN 9780387964584.
- ^ Hamilton. Hodges and Smith. 1853. p. 60.
quaternion quotient lines tridimensional space time
- ^ Hardy 1881. Ginn, Heath, & co. 1881. p. 32. ISBN 9781429701860.
- ^ Curtis, Morton L. (1984), Matrix Groups (2nd ed.), New York: Springer-Verlag, p. 10, ISBN 978-0-387-96074-6
- ^ Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
- ^ Kunze, Karsten; Schaeben, Helmut (November 2004). "The Bingham distribution of quaternions and its spherical radon transform in texture analysis". Mathematical Geology. 36 (8): 917–943. Bibcode:2004MatGe..36..917K. doi:10.1023/B:MATG.0000048799.56445.59. S2CID 55009081.
- ^ Smith, Frank (Tony). "Why not sedenion?". Retrieved 8 June 2018.