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Element of an algebra using quaternions and split-complex numbers
In mathematics, a split-biquaternion is a hypercomplex number of the form q = w + x i + y j + z k , {\displaystyle q=w+x\mathrm {i} +y\mathrm {j} +z\mathrm
Split-biquaternion
Quaternions with complex number coefficients
types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions
Biquaternion
a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions
Biquaternion_algebra
Vector space equipped with a bilinear product
integers. A classical example of an algebra over its center is the split-biquaternion algebra, which is isomorphic to H × H, the direct product of two quaternion
Algebra_over_a_field
Reals with an extra square root of +1 adjoined
Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions. He called its elements "motors"
Split-complex_number
British mathematician and philosopher (1845–1879)
from split-complex numbers D or from the dual numbers N. In terms of tensor products, H ⊗ D {\displaystyle H\otimes D} produces split-biquaternions, while
William_Kingdon_Clifford
Element of a unital algebra over the field of real numbers
century, number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature
Hypercomplex_number
Eight-dimensional algebra over the real numbers
quaternions as "Study biquaternions" or "Clifford biquaternions". The latter eponym has also been used to refer to split-biquaternions. Read the article by
Dual_quaternion
Four-dimensional number system
short descriptions of redirect targets Biquaternion – Quaternions with complex number coefficients Biquaternion functions – Functions of complex quaternions
Quaternion
Four-dimensional associative algebra over the reals
Split-quaternions (Rosenfeld 1988) Antiquaternions (Rosenfeld 1988) Pseudoquaternions (Yaglom 1968 Rosenfeld 1988) Pauli matrices Split-biquaternions
Split-quaternion
Branch of mathematics
1848 and coquaternions in 1849. William Kingdon Clifford introduced split-biquaternions in 1873. In addition Cayley introduced group algebras over the real
Abstract_algebra
Branch of algebra
quaternions and biquaternions; James Cockle presented tessarines and coquaternions; and William Kingdon Clifford was an enthusiast of split-biquaternions, which
Ring_theory
Mathematical functions of split-complex numbers
Sketch of Biquaternions" (1873). He used split-complex numbers for scalars in his split-biquaternions. Motor variable is used here in place of split-complex
Motor_variable
Application of Clifford algebra
also most "intuitive" for humans Elliptic Pin(4, 0, 0) Cl4,0,0(R) 1 Split-biquaternions; Spin(4, 0, 0); double cover of 4D rotations Also known as "spherical
Plane-based_geometric_algebra
Functions of complex quaternions
complex plane can be extended to functions of complex quaternions (biquaternions). This is simple when the function can be expressed as a power series
Biquaternion_functions
Algebra based on a vector space with a quadratic form
to the algebra of split-quaternions. Cl0,3(R) is an 8-dimensional algebra isomorphic to the direct sum H ⊕ H, the split-biquaternions. Cl3,0(R) ≅ Cl1,2(R)
Clifford_algebra
Vector on which a quadratic form is zero
In the Cayley–Dickson construction, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the complex
Null_vector
Origin and evolution of the symbols used to write equations and formulas
Kingdon Clifford published his Elements of Dynamic. Clifford developed split-biquaternions (e.g. q = w + x i + y j + z k {\displaystyle q=w+xi+yj+zk} ) which
History of mathematical notation
History_of_mathematical_notation
Generalization of quaternions to other fields
to isomorphism. When F = C {\displaystyle F=\mathbb {C} } , then the biquaternions form the quaternion algebra over F. Quaternion algebra here means something
Quaternion_algebra
Hamiltonian formulation of general relativity
such as π = g i j π i j {\displaystyle \pi =g^{ij}\pi _{ij}} . The ADM split denotes the separation of the spacetime metric into three spatial components
ADM_formalism
Setting of relativistic physics in geometric algebra
Lorentz transformation formulas in terms of complex quaternions or biquaternions by making some simple identifications. The quaternion Lorentz transformations
Spacetime_algebra
German mathematician (1862 – 1930)
hypercomplex systems in that study are dual numbers, dual quaternions, and split-biquaternions, all being associative algebras over R. Study's work with dual numbers
Eduard_Study
Branch of mathematical analysis
Banach algebras is called functional analysis. Giovanni Battista Rizza Biquaternion functions Felix Gantmacher (1959) The Theory of Matrices, two volumes
Hypercomplex_analysis
German mathematician and Nazi Party official (1869–1945)
anti-commuting square roots of −1. Vahlen also recounts split-biquaternions and parabolic biquaternions originated by Clifford. But Vahlen cites Eduard Study
Theodor_Vahlen
Unbounded quadric surface
which included presentation of biquaternions. The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from quaternions
Hyperboloid
Type of algebras, possibly non associative
composition algebras over C are C itself, the bicomplex numbers, the biquaternions (isomorphic to the 2×2 complex matrix ring M(2, C)), and the bioctonions
Composition_algebra
Relativistic correction
is the product of the norms, making the biquaternions a composition algebra. The norm of a non-zero biquaternion can be any complex number, including zero
Thomas_precession
Physics concept expressed as E = mc²
an object up on earth does. This energy is equal to the work required to split the particles apart. The mass of the Solar System is slightly less than
Mass–energy_equivalence
Hypercomplex number system
algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matrices over the real numbers, or that studied
Sedenion
Quaternion of norm 1 (unit quaternion)
extend the concept to 4-space. Problems in that algebra led to use of biquaternions after 1900. In a widely seen review, Macfarlane wrote: ... the root
Versor
Compact astronomical body
black holes and other compact objects. In this method, a laser beam is split, sent down two long arms of a tunnel, then reflected at the far end of the
Black_hole
carving remains. Since the algebra of the split-quaternions is isomorphic to M(2, R) and the algebra of biquaternions is isomorphic to M(2, C), there is a
History_of_quaternions
Element in a ring whose some power is 0
numbers that contain nilpotent spaces include split-quaternions (coquaternions), split-octonions, biquaternions C ⊗ H {\displaystyle \mathbb {C} \otimes \mathbb
Nilpotent
Reduction of a ring by one of its ideals
in both the quadratic binomials also results in split-quaternions. The three types of biquaternions can also be written as quotients by use of the free
Quotient_ring
Mutation of quaternions where unit vectors square to +1
features of the older and larger algebra of biquaternions. They both contain subalgebras isomorphic to the split-complex number plane. Furthermore, just as
Hyperbolic_quaternion
Measured time difference as explained by relativity theory
Terrell rotation Spacetime Light cone World line Minkowski diagram Biquaternions Minkowski space General relativity Background Introduction Mathematical
Time_dilation
Speed of electromagnetic waves in vacuum
coherent beam of light (e.g. from a laser), with a known frequency f, is split to follow two paths and then recombined. By adjusting the path length while
Speed_of_light
Theory of interwoven space and time by Albert Einstein
illustrated in Fig. 5-1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing
Special_relativity
Commutative, associative algebra of two complex dimensions
Chang, Ja-Han; Ding, Jian-Jiun (21 June 2004). "Commutative reduced biquaternions and their Fourier transform for signal and image processing" (PDF).
Bicomplex_number
Theory of gravitation as curved spacetime
universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example
General_relativity
Belgian scientist and Catholic priest (1894–1966)
Terrell rotation Spacetime Light cone World line Minkowski diagram Biquaternions Minkowski space General relativity Background Introduction Mathematical
Georges_Lemaître
Lie group of Lorentz transformations
representation of the action of the Lorentz group on Minkowski space uses biquaternions, which form a composition algebra. The isometry property of Lorentz
Lorentz_group
German mathematician (1885–1955)
This equation describes massless fermions. A normal Dirac fermion could be split into two Weyl fermions or formed from two Weyl fermions. Neutrinos were
Hermann_Weyl
Concept in mathematical group theory
traditional since the work of Ludwik Silberstein in 1914 to use the ring of biquaternions to represent the Lorentz group. For the spacetime conformal group, it
Conformal_group
Latin quasi quasar, quasiparticle quatern- four each Latin quaternī biquaternion, quatern, quaternary, quaternate, quaternion, quaternity, quire quati-
List of Greek and Latin roots in English/P–Z
List_of_Greek_and_Latin_roots_in_English/P–Z
Projective construction in ring theory
that ring. Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation
Projective_line_over_a_ring
Foundational issues principle of relativity speed of light faster-than-light biquaternion conjugate diameters four-vector four-acceleration four-force four-gradient
List of mathematical topics in relativity
List_of_mathematical_topics_in_relativity
Mathematical transformation in physics
Terrell rotation Spacetime Light cone World line Minkowski diagram Biquaternions Minkowski space General relativity Background Introduction Mathematical
Time-translation_symmetry
Equation used in relativistic physics
velocity to the speed of light in vacuum. Hyperbolic law of cosines Biquaternion Relative velocity These formulae follow from inverting αv for v2 and
Velocity-addition_formula
Non-abelian group of order eight
\mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} } is the algebra of biquaternions. The two-dimensional irreducible complex representation described above
Quaternion_group
Algebra of 4D spacetime
"algebra of physical space" (APS) originally stems from the use of the biquaternions via its definition as the real Clifford or geometric algebra Cl3,0(R)
Algebra_of_physical_space
Smallest stable circular orbit of a particle
{GM}{c^{2}}}=2\,R_{S}} . If the particle is also spinning there is a further split in ISCO radius depending on whether the spin is aligned with or against
Innermost stable circular orbit
Innermost_stable_circular_orbit
Family of linear transformations
change, while those coordinates perpendicular do not. With this in mind, split the spatial position vector r as measured in F, and r′ as measured in F′
Lorentz_transformation
Sum of directed areas in exterior algebra
the latter in his Lectures on Quaternions (1853) as he introduced biquaternions, which have bivectors for their vector parts. It was not until English
Bivector
Theoretical physics phenomenon
means there are two inequivalent transformations. Each of these can be split into a boost then rotation, or a rotation then boost, doubling the number
Wigner_rotation
Irish mathematician and physicist (1805–1865)
Hamilton introduced, as a method of analysis, both quaternions and biquaternions, the extension to eight dimensions by the introduction of complex number
William_Rowan_Hamilton
American physicist (1932–2023)
Deser he published a Hamiltonian formulation of the Einstein equation that split Einstein's unified spacetime back into separated space and time. This set
Charles_W._Misner
techniques Biophysics Biot number Biot–Savart law Biplane Bipolaron (physics) Biquaternion Birch's law Birch–Murnaghan equation of state Bird flight Birefringence
Index_of_physics_articles_(B)
Contraction of length in the direction of propagation in Minkowski space
Terrell rotation Spacetime Light cone World line Minkowski diagram Biquaternions Minkowski space General relativity Background Introduction Mathematical
Length_contraction
Development of linear transformations forming the Lorentz group
transformation, transformation of the Sine-Gordon equation, Biquaternion algebra, split-complex numbers, Clifford algebra, and others. Learning materials
History of Lorentz transformations
History_of_Lorentz_transformations
German physicist
became an adjunct professor at LMU Munich. In March 1991, the institute split into the Max Planck Institute for Physics and the Max Planck Institute for
Jürgen_Ehlers
SPLIT BIQUATERNION
SPLIT BIQUATERNION
Boy/Male
Tamil
Inside viewer, Spilt second
Surname or Lastname
English and French
English and French : metonymic occupational name for a turnspit, i.e. a servant who turned the spit, from Old French haste ‘(roasting) spit’.A bearer of the name Haste from Paris is documented in Montreal in 1662.
Boy/Male
Tamil
Inside viewer, Spilt second
Boy/Male
English
From the split meadow.
Surname or Lastname
English
English : habitational name from a place in Lancashire, near Rishton, recorded in 1246 as Kunteclive, from Old English cunte ‘cunt’ + clif ‘slope’, i.e. ‘slope with a slit or crack in it’.
Girl/Female
Hindu, Indian, Telugu
Motherly Love; Energetic Sprit
Boy/Male
Muslim
Split, Cleavage
Boy/Male
Muslim
Strong, Solid, Firm, Sharp
Boy/Male
Arabic, Muslim, Sindhi
Split
Boy/Male
Arabic, Muslim
Strong; Solid; Firm; Sharp
Surname or Lastname
English
English : habitational name from any of the numerous places so called, which split more or less evenly into two groups with different etymologies. One set (with examples in Berkshire, Dorset, Gloucestershire, Hampshire, Herefordshire, Somerset, and Wiltshire) is named from the Old English weak dative hēan (originally used after a preposition and article) of hēah ‘high’ + Old English tūn ‘enclosure’, ‘settlement’. The other (with examples in Cambridgeshire, Dorset, Gloucestershire, Herefordshire, Northamptonshire, Shropshire, Somerset, Suffolk, and Wiltshire) has Old English hīwan ‘household’, ‘monastery’. Compare Hine as the first element.
Boy/Male
Gujarati, Hindu, Indian
One who Lives Life Long; Gains Victory Within Splits
Girl/Female
Hindu, Indian
Momentary; Split Second
Girl/Female
American, Christian, Hebrew, Indian
Narrow Split of Land
Boy/Male
Muslim/Islamic
Split Cleavage
Surname or Lastname
English
English : from Middle English clevere ‘one who cleaves’ (a derivative of Old English clēofan ‘to split’), hence an occupational name for someone who split wood into planks using a wedge rather than a saw, or possibly for a butcher.English : topographic name from Middle English cleve ‘bank’, ‘slope’ (from the dative of Old English clif) + the suffix -er, denoting an inhabitant.Americanized spelling of German Kliewer or Klüver (see Kluver).
Boy/Male
American, British, English
From the Split Meadow
Boy/Male
Hindu
Inside viewer, Spilt second
Boy/Male
Hindu
Inside viewer, Spilt second
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Momentary; Lord Rama's Ancestor; Spilt-second; Lord Vishnu
SPLIT BIQUATERNION
SPLIT BIQUATERNION
Boy/Male
Arthurian Legend
A knight.
Girl/Female
Swedish
Pure.
Boy/Male
Tamil
Boy/Male
Hindu, Indian, Marathi
Lord Shiva
Girl/Female
American, Australian, British, Chinese, Christian, Danish, Dutch, English, French, German, Hebrew, Japanese
Lily
Girl/Female
Indian
German origin and means noble, Kind
Girl/Female
Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Loving; Dashing; Friendly
Girl/Female
Hindu
The best in number & quality, Most Happy or prosperous
Girl/Female
Hindu, Indian
Senseless
Female
English
 Italian form of Latin Viatrix, BEATRICE means "voyager (through life)."
SPLIT BIQUATERNION
SPLIT BIQUATERNION
SPLIT BIQUATERNION
SPLIT BIQUATERNION
SPLIT BIQUATERNION
a.
Divided; split; partly divided or split.
imp. & p. p.
of Split
imp. & p. p.
of Spit
v. i.
To part asunder; to be rent; to burst; as, vessels split by the freezing of water in them.
v. t.
A piece split off; a splinter.
v. i.
To attend to a spit; to use a spit.
n.
To cut lengthwise; to cut into long pieces or strips; as, to slit iron bars into nail rods; to slit leather into straps.
n.
A piece that is split off, or made thin, by splitting; a splinter; a fragment.
v. t.
A disease affecting the splint bones, as a callosity or hard excrescence.
n.
A long cut; a narrow opening; as, a slit in the ear.
imp. & p. p.
of Slit
n.
the substitution of more than one share of a corporation's stock for one share. The market price of the stock usually drops in proportion to the increase in outstanding shares of stock. The split may be in any ratio, as a two-for-one split; a three-for-two split.
v. t.
To split into splints, or thin, slender pieces; to splinter; to shiver.
v. t.
To divide or separate into components; -- often used with up; as, to split up sugar into alcohol and carbonic acid.
v. t.
One of the small plates of metal used in making splint armor. See Splint armor, below.
v. t.
Splint, or splent, coal. See Splent coal, under Splent.
n.
To thrust a spit through; to fix upon a spit; hence, to thrust through or impale; as, to spit a loin of veal.
v. t.
To divide lengthwise; to separate from end to end, esp. by force; to divide in the direction of the grain layers; to rive; to cleave; as, to split a piece of timber or a board; to split a gem; to split a sheepskin.
v. t.
To fasten or confine with splints, as a broken limb. See Splint, n., 2.
v. t.
A splint bone.