In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form q = a + b i + c...

In mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form q = exp ⁡ ( a r ) = cos ⁡ a + r sin ⁡ a , r 2 = − 1...

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton...

form Hyperbolic quaternions, a non-associative algebra, precursor to Minkowski space Hyperbolic rotation, a synonym for squeeze mapping Hyperbolic sector...

In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {...

Similar to Homersham Cox (1882/83), Macfarlane uses the hyperbolic versor as the hyperbolic quaternion corresponding to the versor of Hamilton. The presentation...

generates a complex plane with the real line. For split-quaternions there are hyperboloids of hyperbolic and imaginary units that generate split-complex or...

Thomas Little Heath Hilbert's problems History of topos theory Hyperbolic quaternion Indian mathematics Islamic mathematics Italian school of algebraic...

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a...

theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They...

in the Poincaré ball model, another standard model of hyperbolic geometry. Hyperbolic quaternion Hyperspace Introduction to the mathematics of general...

was charting this submanifold through his Algebra of Physics and hyperbolic quaternions, though Macfarlane did not use cosmological language as Minkowski...

"Circular and hyperbolic quaternions, octonions, and sedenions", Appl. Math. Comput. 28:47–72. K. Carmody, (1997) "Circular and hyperbolic quaternions, octonions...

of the nineteenth century, including Quaternions Tessarines Coquaternions Biquaternions Hyperbolic quaternions This disambiguation page lists articles...

variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions...

The Quaternion Society was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied Systems...

precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group...

cubic extension of Q. The (2,3,7) hyperbolic triangle group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative...

Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number. As Hawkins explains...

other than GF(2) (see previous section), and the sedenions. The hyperbolic quaternion algebra over R, which was an experimental algebra before the adoption...

stores hyperbolic manifolds. The construction of arithmetic Kleinian groups from quaternion algebras gives rise to particularly interesting hyperbolic manifolds...

is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations...

of hyperbolic functions". Resonance. 11 (8): 81–85. doi:10.1007/BF02855783. S2CID 119753430. Brand, Louis (October 1942). "The roots of a quaternion"....

particularly regular behaviour among Fuchsian groups and hyperbolic surfaces. A quaternion algebra over a field F {\displaystyle F} is a four-dimensional...

(although, strictly speaking, it is a pseudovector). Matrices, versors (quaternions), and other algebraic things: see the section Linear and Multilinear...

algebra Examples Cayley–Dickson construction Octonions Sedenions Hyperbolic quaternions Virasoro algebra Algebraic structure Universal algebra Variety (universal...

as in the quaternion group forms a nonassociative loop of order 8. See hyperbolic quaternions for its application. (The hyperbolic quaternions themselves...

in the polar view of 4-space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the three-sphere...

term itself.: v  At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond...

then be either cross-sectioned or projected into a 3D structure. The quaternion (4-dimensional) Mandelbrot set is simply a solid of revolution of the...