Versor, the Glossary
In mathematics, a versor is a quaternion of norm one (a unit quaternion).[1]
Table of Contents
106 relations: A K Peters, Additive inverse, Adolf Hurwitz, Alexander Macfarlane, Alfred Robb, American Journal of Mathematics, American Journal of Physics, Angle, Antipodal point, Arthur Sherburne Hardy, Arthur Stafford Hathaway, Axis–angle representation, Baker–Campbell–Hausdorff formula, Bicomplex number, Binary icosahedral group, Binary octahedral group, Binary tetrahedral group, Biquaternion, Bitruncation, Bloch sphere, Cambridge Philosophical Society, Cayley transform, Circle group, Cis (mathematics), Clifford parallel, Cross product, Curve, Doublet state, E. C. George Sudarshan, Elliptic geometry, Encyclopedia of Mathematics, Euclidean plane, Euclidean vector, Exponential map (Lie theory), Fiber bundle, Frame of reference, Geometric algebra, Georges Lemaître, Great circle, Group (mathematics), Group action, Harold Scott MacDonald Coxeter, Homersham Cox (mathematician), Hopf fibration, Hurwitz quaternion, Hyperbolic quaternion, Icosian, Indefinite orthogonal group, Inner automorphism, Internet Archive, ... Expand index (56 more) »
- Quaternions
- Rotation in three dimensions
- Spherical trigonometry
- William Rowan Hamilton
A K Peters
A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science.
Additive inverse
In mathematics, the additive inverse of a number (sometimes called the opposite of) is the number that, when added to, yields zero.
See Versor and Additive inverse
Adolf Hurwitz
Adolf Hurwitz (26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
Alexander Macfarlane
Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician.
See Versor and Alexander Macfarlane
Alfred Robb
Alfred Arthur Robb FRS (18 January 1873 in Belfast – 14 December 1936 in Castlereagh) was a Northern Irish physicist.
American Journal of Mathematics
The American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press.
See Versor and American Journal of Mathematics
American Journal of Physics
The American Journal of Physics is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics.
See Versor and American Journal of Physics
Angle
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.
See Versor and Angle
Antipodal point
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center.
See Versor and Antipodal point
Arthur Sherburne Hardy
Arthur Sherburne Hardy (August 13, 1847 – March 14, 1930) was an American engineer, educator, editor, diplomat, novelist, and poet.
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Arthur Stafford Hathaway
Arthur Stafford Hathaway (1855 — 1934) was an American mathematician.
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Axis–angle representation
In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction (geometry) of an axis of rotation, and an angle of rotation describing the magnitude and sense (e.g., clockwise) of the rotation about the axis. Versor and axis–angle representation are rotation in three dimensions.
See Versor and Axis–angle representation
Baker–Campbell–Hausdorff formula
In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation e^X e^Y.
See Versor and Baker–Campbell–Hausdorff formula
Bicomplex number
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^*.
See Versor and Bicomplex number
Binary icosahedral group
In mathematics, the binary icosahedral group 2I or Coxeter&Moser: Generators and Relations for discrete groups:: Rl.
See Versor and Binary icosahedral group
Binary octahedral group
In mathematics, the binary octahedral group, name as 2O or Coxeter&Moser: Generators and Relations for discrete groups:: Rl.
See Versor and Binary octahedral group
Binary tetrahedral group
In mathematics, the binary tetrahedral group, denoted 2T or, Coxeter&Moser: Generators and Relations for discrete groups:: Rl.
See Versor and Binary tetrahedral group
Biquaternion
In abstract algebra, the biquaternions are the numbers, where, and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. Versor and biquaternion are quaternions and William Rowan Hamilton.
Bitruncation
In geometry, a bitruncation is an operation on regular polytopes.
Bloch sphere
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.
Cambridge Philosophical Society
The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge.
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Cayley transform
In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things.
See Versor and Cayley transform
Circle group
In mathematics, the circle group, denoted by \mathbb T or, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers \mathbb T.
Cis (mathematics)
is a mathematical notation defined by, where is the cosine function, is the imaginary unit and is the sine function.
See Versor and Cis (mathematics)
Clifford parallel
In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. Versor and Clifford parallel are quaternions.
See Versor and Clifford parallel
Cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times.
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
See Versor and Curve
Doublet state
In quantum mechanics, a doublet is a composite quantum state of a system with an effective spin of 1/2, such that there are two allowed values of the spin component, −1/2 and +1/2.
E. C. George Sudarshan
Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian American theoretical physicist and a professor at the University of Texas.
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Elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold.
See Versor and Elliptic geometry
Encyclopedia of Mathematics
The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.
See Versor and Encyclopedia of Mathematics
Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted \textbf^2 or \mathbb^2.
See Versor and Euclidean plane
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction.
See Versor and Euclidean vector
Exponential map (Lie theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.
See Versor and Exponential map (Lie theory)
Fiber bundle
In mathematics, and particularly topology, a fiber bundle (''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure.
Frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points―geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (signaled by conventional markers).
See Versor and Frame of reference
Geometric algebra
In mathematics, a geometric algebra (also known as a Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors.
See Versor and Geometric algebra
Georges Lemaître
Georges Henri Joseph Édouard Lemaître (17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, mathematician, astronomer, and professor of physics at the Catholic University of Louvain.
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Great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Versor and great circle are spherical trigonometry.
Group (mathematics)
In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
See Versor and Group (mathematics)
Group action
In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane.
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.
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Homersham Cox (mathematician)
Homersham Cox (1857–1918) was an English mathematician.
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Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere.
Hurwitz quaternion
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). Versor and Hurwitz quaternion are quaternions.
See Versor and Hurwitz quaternion
Hyperbolic quaternion
In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form where the squares of i, j, and k are +1 and distinct elements of multiply with the anti-commutative property.
See Versor and Hyperbolic quaternion
Icosian
In mathematics, the icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. Versor and icosian are quaternions.
Indefinite orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature, where.
See Versor and Indefinite orthogonal group
Inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element.
See Versor and Inner automorphism
Internet Archive
The Internet Archive is an American nonprofit digital library founded in 1996 by Brewster Kahle.
See Versor and Internet Archive
Intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously.
Irving Stringham
Washington Irving Stringham (December 10, 1847 – October 5, 1909) was an American mathematician born in Yorkshire, New York.
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James Cockle
Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.
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Journal of Mathematical Physics
The Journal of Mathematical Physics is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics.
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Journal of Physics A
The Journal of Physics A: Mathematical and Theoretical is a peer-reviewed scientific journal published by IOP Publishing, the publishing branch of the Institute of Physics.
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Lie algebra
In mathematics, a Lie algebra (pronounced) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity.
Lie group
In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
Lie theory
In mathematics, the mathematician Sophus Lie initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory.
Line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints.
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena.
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former.
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Mathematical Reviews
Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.
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Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
Mathematics Magazine
Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America.
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Narasimhaiengar Mukunda
Narasimhaiengar Mukunda (born 25 January 1939, New Delhi, India) is an Indian theoretical physicist.
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Null vector
In mathematics, given a vector space X with an associated quadratic form q, written, a null vector or isotropic vector is a non-zero element x of X for which.
One-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is injective then \varphi(\mathbb), the image, will be a subgroup of G that is isomorphic to \mathbb as an additive group.
See Versor and One-parameter group
Orthogonal group
In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.
See Versor and Orthogonal group
Perpendicular
In geometry, two geometric objects are perpendicular if their intersection forms right angles (angles that are 90 degrees or π/2 radians wide) at the point of intersection called a foot.
Philosophical Magazine
The Philosophical Magazine is one of the oldest scientific journals published in English.
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Project Gutenberg
Project Gutenberg (PG) is a volunteer effort to digitize and archive cultural works, as well as to "encourage the creation and distribution of eBooks." It was founded in 1971 by American writer Michael S. Hart and is the oldest digital library.
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Quaternion
In mathematics, the quaternion number system extends the complex numbers. Versor and quaternion are quaternions and William Rowan Hamilton.
Quaternion group
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 \ of the quaternions under multiplication. Versor and quaternion group are quaternions.
See Versor and Quaternion group
Quaternions and spatial rotation
Unit quaternions, known as ''versors'', provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Versor and quaternions and spatial rotation are quaternions and rotation in three dimensions.
See Versor and Quaternions and spatial rotation
Radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics.
Rajiah Simon
Rajiah Simon is a professor of physics at the Institute of Mathematical Sciences, Chennai, India.
Rapidity
Rapidity is a measure for relativistic velocity.
Right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn.
Rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn.
See Versor and Rotational symmetry
Rotations in 4-dimensional Euclidean space
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). Versor and rotations in 4-dimensional Euclidean space are quaternions.
See Versor and Rotations in 4-dimensional Euclidean space
Science (journal)
Science, also widely referred to as Science Magazine, is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals.
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Sophus Lie
Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.
Special linear group
In mathematics, the special linear group of degree n over a commutative ring R is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.
See Versor and Special linear group
Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time.
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Special unitary group
In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.
See Versor and Special unitary group
Spherical geometry
A sphere with a spherical triangle on it. Spherical geometry or spherics is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres.
See Versor and Spherical geometry
Split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2.
See Versor and Split-complex number
Split-quaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. Versor and split-quaternion are quaternions.
See Versor and Split-quaternion
Subgroup
In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.
Three-dimensional space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point.
See Versor and Three-dimensional space
Topological group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.
See Versor and Topological group
Trinity College Dublin
Trinity College Dublin (Coláiste na Tríonóide), officially The College of the Holy and Undivided Trinity of Queen Elizabeth near Dublin, is the sole constituent college of the University of Dublin, Ireland.
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Truncated 24-cells
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
See Versor and Truncated 24-cells
Turn (angle)
The turn (symbol tr or pla) is a unit of plane angle measurement that is the angular measure subtended by a complete circle at its center.
Unit (ring theory)
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring.
See Versor and Unit (ring theory)
Unit hyperbola
In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation x^2 - y^2.
Unit sphere
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space.
Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.
United States National Library of Medicine
The United States National Library of Medicine (NLM), operated by the United States federal government, is the world's largest medical library.
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Wiley (publisher)
John Wiley & Sons, Inc., commonly known as Wiley, is an American multinational publishing company that focuses on academic publishing and instructional materials.
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William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was a British mathematician and philosopher.
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William Rowan Hamilton
Sir William Rowan Hamilton (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist.
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24-cell
In four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
3-sphere
In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. Versor and 3-sphere are quaternions.
3D rotation group
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. Versor and 3D rotation group are rotation in three dimensions.
See Versor and 3D rotation group
See also
Quaternions
- 3-sphere
- Applications of dual quaternions to 2D geometry
- Biquaternion
- Biquaternion algebra
- Bivector (complex)
- Clifford parallel
- Conversion between quaternions and Euler angles
- Dicyclic group
- Dual quaternion
- Frobenius theorem (real division algebras)
- Hurwitz quaternion
- Hurwitz quaternion order
- Hyperkähler manifold
- Icosian
- Linked field
- Quaternion
- Quaternion algebra
- Quaternion group
- Quaternionic analysis
- Quaternionic manifold
- Quaternionic polytope
- Quaternionic projective space
- Quaternionic structure
- Quaternionic vector space
- Quaternions and spatial rotation
- Rotations in 4-dimensional Euclidean space
- Slerp
- Split-quaternion
- Versor
Rotation in three dimensions
- 3D rotation group
- Axes conventions
- Axis–angle representation
- Chaotic rotation
- Charts on SO(3)
- Chasles' theorem (kinematics)
- Clebsch–Gordan coefficients
- Conversion between quaternions and Euler angles
- Davenport chained rotations
- Euler angles
- Euler's equations (rigid body dynamics)
- Euler's rotation theorem
- Euler–Rodrigues formula
- Gimbal lock
- Givens rotation
- Nutation
- Orientation (geometry)
- Plane of rotation
- Plate trick
- Precession
- Quaternion estimator algorithm
- Quaternions and spatial rotation
- Representation theory of SU(2)
- Rodrigues' rotation formula
- Rotation formalisms in three dimensions
- Slerp
- Spacecraft attitude control
- Spin quantum number
- Spin-1/2
- Spinor
- Total angular momentum quantum number
- Triad method
- Versor
Spherical trigonometry
- Great circle
- Great-circle distance
- Half-side formula
- Haversine formula
- Lénárt sphere
- Legendre's theorem on spherical triangles
- Lexell's theorem
- Pentagramma mirificum
- Schwarz triangle
- Solution of triangles
- Spherical law of cosines
- Spherical trigonometry
- Triangle group
- Trigonometry of a tetrahedron
- Versor
William Rowan Hamilton
- Biquaternion
- Bivector (complex)
- Cayley–Hamilton theorem
- Classical Hamiltonian quaternions
- Hamilton Walk
- Hamilton's principle
- Hamilton–Jacobi equation
- Hamilton–Jacobi–Bellman equation
- Hamiltonian (quantum mechanics)
- Hamiltonian mechanics
- Hamiltonian path
- Hamiltonian vector field
- Icosian calculus
- Icosian game
- List of things named after William Rowan Hamilton
- Nabla symbol
- Quaternion
- Quaternions
- Robert Perceval Graves
- Versor
- William Edwin Hamilton
- William Rowan Hamilton
References
[1] https://en.wikipedia.org/wiki/Versor
Also known as Hyperbolic versor, Unit quaternion, Unit quaternions, Versors.
, Intersection, Irving Stringham, James Cockle, John Horton Conway, Journal of Mathematical Physics, Journal of Physics A, Lie algebra, Lie group, Lie theory, Line segment, Lorentz group, Lorentz transformation, Mathematical Reviews, Mathematics, Mathematics Magazine, Narasimhaiengar Mukunda, Null vector, One-parameter group, Orthogonal group, Perpendicular, Philosophical Magazine, Project Gutenberg, Quaternion, Quaternion group, Quaternions and spatial rotation, Radian, Rajiah Simon, Rapidity, Right angle, Rotational symmetry, Rotations in 4-dimensional Euclidean space, Science (journal), Sophus Lie, Special linear group, Special relativity, Special unitary group, Spherical geometry, Split-complex number, Split-quaternion, Subgroup, Three-dimensional space, Topological group, Trinity College Dublin, Truncated 24-cells, Turn (angle), Unit (ring theory), Unit hyperbola, Unit sphere, Unit vector, United States National Library of Medicine, Wiley (publisher), William Kingdon Clifford, William Rowan Hamilton, 24-cell, 3-sphere, 3D rotation group.