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a Kleinian integer is a complex number of the form m + n 1 + − 7 2 {\displaystyle m+n{\frac {1+{\sqrt {-7}}}{2}}} , with m and n rational integers. They
Kleinian_integer
Root of a quadratic polynomial with a unit leading coefficient
{-7}}~\!{\bigr )}\right]} are called Kleinian integers The first two rings mentioned above are rings of integers of cyclotomic fields Q(ζ4) and Q(ζ3)
Quadratic_integer
Discrete group of Möbius transformations
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable
Kleinian_group
Field (mathematics) generated by the square root of an integer
({\sqrt {d}})} where d {\displaystyle d} is a (uniquely defined) square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} . If d > 0 {\displaystyle
Quadratic_field
Klein surface Klein quadric Klein quartic Kleinian group Kleinian integer Kleinian model Kleinian ring Kleinian singularity Klein's icosahedral cubic surface
List of things named after Felix Klein
List_of_things_named_after_Felix_Klein
in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras
Arithmetic hyperbolic 3-manifold
Arithmetic_hyperbolic_3-manifold
Curve whose range contains the unit square
space-filling, or rather sphere-filling, curves in the theory of doubly degenerate Kleinian groups. For example, Cannon & Thurston (2007) showed that the circle at
Space-filling_curve
module with a derivation from the ring to it. Kleinian integer The Kleinian integers are the integers of the imaginary quadratic field of discriminant
Glossary of commutative algebra
Glossary_of_commutative_algebra
Mathematical group
PSL 2 ( C ) {\displaystyle {\text{PSL}}_{2}(\mathbb {C} )} , now termed Kleinian groups. As a subgroup of PSL 2 ( C ) {\displaystyle {\text{PSL}}_{2}(\mathbb
Bianchi_group
Fractal named after mathematician Benoit Mandelbrot
by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations
Mandelbrot_set
S2CID 56322745. Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Discrete subgroup of the real projective special linear group of dimension 2
contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup
Fuchsian_group
Fractal composed of tangent circles
gasket is the limit set of a group of Möbius transformations known as a Kleinian group. For Euclidean symmetry transformations rather than Möbius transformations
Apollonian_gasket
Type of group in group theory
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm
Arithmetic_group
Aspect of group theory in mathematics
is commonly attributed to Felix Klein who used it to study subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space
Ping-pong_lemma
Type of topological group
Kleinian groups are, by definition, discrete subgroups of the isometry group of hyperbolic 3-space. These include quasi-Fuchsian groups. A Kleinian group
Discrete_group
Smallest cardinality of a generating set for a group
word hyperbolic groups. The rank problem is decidable for torsion-free Kleinian groups. The rank problem is open for finitely generated virtually abelian
Rank_of_a_group
C2 and the quotient is called a Picard modular surface. Fuchsian group Kleinian group Langlands, Robert P.; Ramakrishnan, Dinakar, eds. (1992), The zeta
Picard_modular_group
classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups. In geometric group theory the fundamental groups of such bundles
Surface bundle over the circle
Surface_bundle_over_the_circle
Orientation-preserving mapping class group of the torus
{PSL} (2,\mathbb {Z} )} of 2 × 2 {\displaystyle 2\times 2} matrices with integer coefficients and determinant 1 {\displaystyle 1} , such that the matrices
Modular_group
Type of mathematical group
strictly contained in a larger discrete subgroup. These are called maximal Kleinian groups, and it is possible to give a complete classification in a given
Arithmetic_Fuchsian_group
Rational function of the form (az + b)/(cz + d)
surface is a discrete subgroup of the Möbius group (see Fuchsian group and Kleinian group). A particularly important discrete subgroup of the Möbius group
Möbius_transformation
Branch of mathematics that studies the properties of groups
are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. Symmetries are not restricted to geometrical objects
Group_theory
is 9 or 10. Thurston, William P. (1982). "Three Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry" (PDF). American Mathematical Society. 6
Hyperbolic_Dehn_surgery
Mathematical space
infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem. It also implies the Ahlfors measure
3-manifold
Branch of topology
hyperbolic isometries acting freely and properly discontinuously. See also Kleinian model. Its thick-thin decomposition has a thin part consisting of tubular
Low-dimensional_topology
On tangency patterns of circles
different way to study Kleinian groups, discrete subgroups of the symmetries of three-dimensional hyperbolic space. A Kleinian group is geometrically
Circle_packing_theorem
Transformations induced by a mathematical group
xxvii+467, ISBN 978-0-8176-4912-8, Zbl 1180.57001 Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag
Group_action
Generalized manifold
also gave a 3-dimensional version of this result for Kleinian groups: in this case the Kleinian group Γ is generated by hyperbolic reflections and the
Orbifold
Branch of mathematics
transformations which were normally regarded as giving the fundamental Kleinian geometry on projective space, they concerned themselves also with the higher-degree
Algebraic_geometry
Mathematical classification
van Hoboken, Joris (2002), Platonic solids, binary polyhedral groups, Kleinian singularities and Lie algebras of type A,D,E (PDF), Master's Thesis, University
ADE_classification
Lie group of Lorentz transformations
the Riemann sphere—so the homogeneous space SO+(1, 3) / Sim(2) is the Kleinian geometry that represents conformal geometry on the sphere S2. The (identity
Lorentz_group
Caroline Series (born 1951), English specialist in hyperbolic geometry, Kleinian groups and dynamical systems Lily Serna (born 1986), Israeli-Australian
List_of_women_in_mathematics
Awarded every year by the American Mathematical Society
1090/S0002-9904-1974-13432-4. Bers, Lipman (1972). "Uniformization, moduli, and Kleinian groups". Bulletin of the London Mathematical Society. 4 (3): 257–300. doi:10
Leroy_P._Steele_Prize
American mathematician
3, pp. 263–282 Floyd, William J., Group completions and limit sets of Kleinian groups. Inventiones Mathematicae, vol. 57 (1980), no. 3, pp. 205–218 Karlsson
William_Floyd_(mathematician)
Spherical triangle that can be used to tile a sphere
1016/s0001-8708(71)80003-8 Maskit, Bernard (1988). "Poincaré's theorem". Kleinian groups. Grundlehren der mathematischen Wissenschaften. Vol. 287. Springer-Verlag
Schwarz_triangle
KLEINIAN INTEGER
KLEINIAN INTEGER
Surname or Lastname
English
English : occupational name for a person who worked in a clay pit or one who prepared clay for use in brick making. See Clay.Americanized form of German and Jewish Kleimann (see Kleiman).
Boy/Male
Anglo Saxon
Reward.
Boy/Male
Welsh
Anvil.
Girl/Female
Latin
A Lemnian woman.
KLEINIAN INTEGER
KLEINIAN INTEGER
Surname or Lastname
English
English : from Middle English colhope, col(l)hop ‘fried eggs and ham or bacon’, which Reaney believes to have been applied as a metonymic occupational name for the keeper of a cook house.
Girl/Female
Muslim/Islamic
Opening
Boy/Male
Indian, Punjabi, Sikh
Lion King
Girl/Female
British, English
Noble
Girl/Female
Hindu, Indian, Sanskrit, Tamil
Heaven
Boy/Male
British, English, German, Jamaican
Son of Robert
Boy/Male
American, Australian, British, Christian, Danish, English, German, Norse, Teutonic
Ruler of All; Sacred Ruler; Noble Leader; Ever Ruler
Boy/Male
Indian, Punjabi, Sikh
Liberation through Company
Girl/Female
Arabic
Servant; Lady-in-waiting
Boy/Male
Hindu
KLEINIAN INTEGER
KLEINIAN INTEGER
KLEINIAN INTEGER
KLEINIAN INTEGER
KLEINIAN INTEGER
n.
Lemnian earth.
n.
An Asiatic species of Cacalia (C. Kleinia), used medicinally in India.
a.
Of pertaining to the Plotinists or their doctrines.
a.
Of or pertaining to the isle of Lemnos.
n.
A complete entity; a whole number, in contradistinction to a fraction or a mixed number.
n.
That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided.
a.
Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.