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Geometry without the parallel postulate
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally
Absolute_geometry
Two geometries based on axioms closely related to those specifying Euclidean geometry
non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the
Non-Euclidean_geometry
On distance between centers of a triangle
Victor; Schacht, Celia (2018), "Euler's inequality in absolute geometry", Journal of Geometry, 109 (Art. 8): 1–11, doi:10.1007/s00022-018-0414-6, S2CID 125459983
Euler's_theorem_in_geometry
Type of non-Euclidean geometry
resulting geometry is absolute geometry. There are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of absolute geometry, including
Hyperbolic_geometry
Study of geometries as axiomatic systems
theorems of absolute geometry hold in hyperbolic geometry as well as in Euclidean geometry. Absolute geometry is inconsistent with elliptic geometry: in elliptic
Foundations_of_geometry
in hyperbolic geometry. Similarly, the real line is the absolute of the Poincaré half-plane model. The extent of Cayley–Klein geometry was summarized
Cayley–Klein_metric
Line intersecting 2 coplanar lines at 2 points
27 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of
Transversal_(geometry)
References Absolute differential calculus An older name of Ricci calculus Absolute geometry Also called neutral geometry, a synthetic geometry similar to
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Theory in number theory
in varieties (absolute, mono-anabelian, and combinatorial versions) and with multiple interactions with number theory, algebraic geometry, and low-dimensional
Anabelian_geometry
Type of metric geometry
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined
Taxicab_geometry
Russian programmer and mathematician (born 1980)
a generalized algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field with one element and an algebraic analogue of
Nikolai_Durov
Optimisation problem in triangle geometry
orthic triangle can be proven in a more general setting, that of absolute geometry and even weaker settings. Set TSP problem, a more general task of
Fagnano's_problem
Distance from zero to a number
then its absolute value is necessarily positive ( | x | = − x > 0 {\displaystyle |x|=-x>0} ). From an analytic geometry point of view, the absolute value
Absolute_value
On sums of distances in triangles
_{i=1}^{n}w_{i}\geq \left(\sec {\frac {\pi }{n}}\right)\sum _{i=1}^{n}r_{i}} In absolute geometry the Erdős–Mordell inequality is equivalent, as proved in Pambuccian
Erdős–Mordell_inequality
Point or an area on Earth's surface or elsewhere
human or social attributes of place identity and sense of place than on geometry. A populated place is called a settlement. A locality, settlement, or populated
Location
Exterior angle of a triangle is greater than either of the remote interior angles
fundamental result in absolute geometry because its proof does not depend upon the parallel postulate. In several high school treatments of geometry, the term "exterior
Exterior_angle_theorem
Topics referred to by the same term
station in the UK Absolute Security, specializes in security and data risk management Absolut Vodka, a brand of Swedish vodka Absolute (geometry), the quadric
Absolute
Mathematical model of the physical space
Euclidean geometry is a model. Absolute geometry Analytic geometry Birkhoff's axioms Cartesian coordinate system Hilbert's axioms Incidence geometry List of
Euclidean_geometry
Non-Euclidean geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel
Elliptic_geometry
Hungarian mathematician (1802–1860)
developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consistent alternative geometry that
János_Bolyai
Branch of mathematics
algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field of one element, and an algebraic analogue of Arakelov's geometry were
Algebraic_geometry
Overview of and topical guide to geometry
Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry
Outline_of_geometry
Concept in projective geometry
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions
Duality_(projective_geometry)
Geometric axiom
equivalent of the first four postulates) is known as an absolute geometry (or sometimes "neutral geometry"). Probably the best-known equivalent of Euclid's
Parallel_postulate
Study of the 3D shapes of molecules
will vibrate faster than at absolute zero. [citation needed] As stated above, rotation hardly influences the molecular geometry. But, as a quantum mechanical
Molecular_geometry
In absolute geometry, the sum of the angles in a triangle is at most 180°
absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained
Saccheri–Legendre_theorem
Geometry without using coordinates
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic
Synthetic_geometry
Form of geometry without distances
measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective
Ordered_geometry
Statistical error measure
F_{Y|X}(a)=0.5.} Least absolute deviations Taxicab geometry Mean absolute percentage error Mean percentage error Symmetric mean absolute percentage error Willmott
Mean_absolute_error
Branch of mathematics
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.
Differential_geometry
Simple curve of Euclidean geometry
mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Annulus: a ring-shaped object, the region bounded
Circle
Convex quadrilateral with at least one pair of parallel sides
In geometry, a trapezoid (/ˈtræpəzɔɪd/) in North American English, or trapezium (/trəˈpiːziəm/) in British English, is a quadrilateral that has at least
Trapezoid
Geometrical term
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line R {\displaystyle R} through a point P {\displaystyle
Limiting_parallel
Theoretical object in mathematics
Oliver (2016), "A blueprinted view on F1‑geometry", in Koen, Thas (ed.), Absolute arithmetic and F1‑geometry, European Mathematical Society Publishing
Field_with_one_element
Mathematical concept
which is the real projective line. Projective geometry also refers to a line at infinity in plane geometry, a plane at infinity in three-dimensional space
Infinity
Modern formulation of Euclid's parallel postulate
axiom in discussions of the parallel postulate. Within the context of absolute geometry the two statements are equivalent, meaning that each can be proved
Playfair's_axiom
Absolute dielectric permittivity of free space
ε0 (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to
Vacuum_permittivity
Figure formed by two rays meeting at a common point
In geometry, an angle is formed by two lines that meet at a point. Each line is called a side of the angle, and the point they share is called the vertex
Angle
Mathematical theorem
In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: given a regular
Theorem_of_absolute_purity
Mathematical term
The steepness, incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined
Slope
Length of a line segment
ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's Elements, distances were not represented as numbers
Euclidean_distance
Product of a number by itself
the real numbers. There are several major uses of the square function in geometry. The name of the square function shows its importance in the definition
Square_(algebra)
Saccheri–Legendre theorem (absolute geometry) Six circles theorem (circles) Steiner–Lehmus theorem (triangle geometry) Symphonic theorem (triangle geometry) Tangent-secant
List_of_theorems
analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
German polymath and scholar (1777–1855)
contributions spanned the branches of number theory, algebra, analysis, geometry, statistics, and probability. Gauss was director of the Göttingen Observatory
Carl_Friedrich_Gauss
37th Johnson solid (26 faces)
Sommerville, D. M. Y. (1905), "Semi-regular networks of the plane in absolute geometry", Transactions of the Royal Society of Edinburgh, 41 (3): 725–747
Elongated_square_gyrobicupola
In mathematics, straight line touching a plane curve without crossing it
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at
Tangent
Axiom in the foundations of geometry
Victor (2019), "The elementary Archimedean axiom in absolute geometry (Paper No. 52)", Journal of Geometry, 110: 1–9, doi:10.1007/s00022-019-0507-x, S2CID 209943756
Aristotle's_axiom
Length in a vector space
following properties, where | s | {\displaystyle |s|} denotes the usual absolute value of a scalar s {\displaystyle s} : Subadditivity / Triangle inequality:
Norm_(mathematics)
Vector relating the initial and the final positions of a moving point
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing
Displacement_(geometry)
Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for taxation, commerce, trade, and in astronomy, to record time and formulate
History_of_mathematics
Mathematical space with a notion of distance
setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean
Metric_space
Branch of applied mathematics
provided multiple examples and ideas in differential geometry (e.g., several notions in symplectic geometry and vector bundles). Within mathematics proper,
Mathematical_physics
Statistical method
In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso, LASSO or L1 regularization) is a regression analysis
Lasso_(statistics)
and geometry – for instance in 1872 and 1876 he wrote summaries of the then current knowledge about non-Euclidean geometry (which he called "absolute geometry")
Johannes_Frischauf
Solid with twenty equal triangular faces
(4): 459–462. MR 1426716. Zbl 0877.51021. Barnes, John (2012). Gems of Geometry (2nd ed.). Springer. doi:10.1007/978-3-642-30964-9. ISBN 978-3-642-30964-9
Regular_icosahedron
Mathematical set with some added structure
coordinates (analytic geometry) was adopted by René Descartes in 1637. At that time, geometric theorems were treated as absolute objective truths knowable
Space_(mathematics)
Euclidean geometry without distance and angles
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance
Affine_geometry
Polyhedron with 12 faces
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with
Dodecahedron
Loss function used in robust regression
551699. PMID 18282924. Hartley, R.; Zisserman, A. (2003). Multiple View Geometry in Computer Vision (2nd ed.). Cambridge University Press. p. 619. ISBN 978-0-521-54051-3
Huber_loss
Causal relationships between points in a manifold
A.A. Robb; The absolute relations of time and space; Cambridge University Press, 1921; (Geometry, Causal Structure) A.A. Robb; Geometry of Time and Space;
Causal_structure
Theoretical foundation of Newtonian mechanics
Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred
Absolute_space_and_time
Curve from a cone intersecting a plane
type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is
Conic_section
Method for specifying point positions
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points
Coordinate_system
German mathematician (1862–1943)
variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations
David_Hilbert
Point at infinity in hyperbolic geometry
together form the Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model
Ideal_point
Arc-like path that the Sun appears to follow across the sky
plane. At solar noon, the zenith angle is at a minimum and is equal to the absolute value of latitude minus solar declination angle. This is the basis by which
Sun_path
mathematician János Bolyai (1802 – 1860), mathematician who developed absolute geometry László Borbély (born 1954), economist and politician Edmond Bordeaux
List_of_Székelys
Change in the position of an object
Modern physics holds that, as there is no absolute frame of reference, Isaac Newton's concept of absolute motion cannot be determined. Everything in
Motion
Number divisible only by 1 and itself
prime ideals of the ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization
Prime_number
Galois group of the separable closure
In mathematics, particularly in anabelian geometry and p-adic geometry, the absolute Galois group G K {\displaystyle G_{K}} of a field K {\displaystyle
Absolute_Galois_group
Vector representing the position of a point with respect to a fixed origin
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its
Position_(geometry)
Philosophical view that there is no correct way of perceiving the passage of time
relativity. It can be argued that special relativity eliminates the concept of absolute simultaneity and a universal present: according to the relativity of simultaneity
Eternalism (philosophy of time)
Eternalism_(philosophy_of_time)
Theories in mathematical logic
systems of geometry include ordered geometry, absolute geometry, affine geometry, Euclidean geometry, projective geometry, and hyperbolic geometry. For each
List_of_first-order_theories
(1879, p. 356). absolute 1. A fixed choice of something in projective space, used to construct some other geometry from projective geometry. For example
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
German mathematician (1909–1982)
"Eine Begründung der absoluten Geometrie in der Ebene" [Rationale for absolute geometry in the plane]. Mathematische Annalen. 113 (1): 424–451. doi:10.1007/BF01571645
Friedrich_Bachmann
Mathematical notion of infinitesimal difference
various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebraic topology. The term differential is used nonrigorously
Differential_(mathematics)
Geometric axiom
a_i and a_j also intersects a_k. Its role in Friedrich Bachmann's absolute geometry based on line-reflections, in the absence of order or free mobility
Lotschnittaxiom
Isometric automorphisms of a hyperbolic space
hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is
Hyperbolic_motion
Specification of a derivative along a tangent vector of a manifold
context to include a wider range of possible geometries. In the 1940s, practitioners of differential geometry began introducing other notions of covariant
Covariant_derivative
Geometric space with four dimensions
traditional absolute space and time cosmology previously used in a universe of three space dimensions and one time dimension. The geometry of four-dimensional
Four-dimensional_space
Framework of distances and directions
framework. In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather
Space
Number, approximately 3.14
Because it relates to a circle, π is found in formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is found
Pi
Algebra associated to any vector space
product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude
Exterior_algebra
Geometrical concept
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional
Cross_section_(geometry)
Approximation technique in integral calculus
fit this definition. Swokowski, Earl W. (1979). Calculus with Analytic Geometry (Second ed.). Boston, MA: Prindle, Weber & Schmidt. pp. 821–822. ISBN 0-87150-268-2
Riemann_sum
Natural number
Lozano-Robledo, Álvaro (2019). Number Theory and Geometry: An Introduction to Arithmetic Geometry. American Mathematical Society. p. 413. ISBN 978-1-4704-5016-8
1729_(number)
Natural number
{\displaystyle \mathrm {D} _{5}} demihypercubic group. In two-dimensional geometry, the regular 23-sided icositrigon is the first regular polygon that is
23_(number)
American mathematician
American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his translations of works
G._B._Halsted
Notation in organic chemistry for double bonds
the E–Z convention, is the IUPAC's preferred method of describing the absolute stereochemistry of double bonds in organic chemistry. It is an extension
E–Z_notation
Concept that simultaneity depends on choice of reference frame
whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame. This possibility was raised
Relativity_of_simultaneity
Model of hyperbolic geometry
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside
Poincaré_disk_model
Tensor index notation for tensor-based calculations
connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), tensor calculus
Ricci_calculus
Manifold upon which it is possible to perform calculus
study of calculus on differentiable manifolds is known as differential geometry. "Differentiability" of a manifold has been given several meanings, including:
Differentiable_manifold
Philosophical term referring to "making" or "doing"
"practical knowledge", techne can include various fields such as mathematics, geometry, medicine, shoemaking, rhetoric, philosophy, music, and astronomy. One
Techne
Mathematical metric
It is an example of an injective metric. In two dimensions, i.e. plane geometry, if the points a and b have Cartesian coordinates ( x 1 , y 1 ) {\displaystyle
Chebyshev_distance
Mathematical model combining space and time
of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances
Spacetime
Concept in physics and mathematics
acting on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view. In special relativity
Galilean_transformation
Complex numbers with non-negative imaginary part
metric space is the hyperbolic plane. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model. Mathematicians
Upper_half-plane
Mathematical transformation in physics
transformations. Different symmetries form different groups with different geometries. Time independent Hamiltonian systems form a group of time translations
Time-translation_symmetry
ABSOLUTE GEOMETRY
ABSOLUTE GEOMETRY
Boy/Male
Hindu
Resolute
Boy/Male
Indian, Sanskrit
Alone; One; Absolute
Boy/Male
Arabic, Muslim
Absolute; Unlimited
Boy/Male
Indian
Absolute.
Girl/Female
Gujarati, Hindu, Indian, Kannada, Sanskrit
Absolute; Aloneness
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Telugu
Absolute
Boy/Male
Hindu, Indian, Kannada, Marathi, Telugu, Traditional
Absolute Brahma
Boy/Male
Tamil
Keval Kumar | கேவலகà¯à®®à®¾à®°
Absolute
Keval Kumar | கேவலகà¯à®®à®¾à®°
Boy/Male
Tamil
Parabrahmana | பரபà¯à®°à®¹à¯à®®à®¨à®¾
The supreme absolute truth
Parabrahmana | பரபà¯à®°à®¹à¯à®®à®¨à®¾
Boy/Male
Tamil
Kevalin | கேவாலீந
Seeker of the absolute
Kevalin | கேவாலீந
Girl/Female
Indian, Sanskrit
Alone; One; Absolute
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Telugu
Absolute
Boy/Male
Tamil
Chidakash | சிதாகாஷ
Absolute Brahma
Chidakash | சிதாகாஷ
Boy/Male
Indian, Sanskrit
Alone; One; Absolute
Boy/Male
Tamil
Chidaakaash | சிதாகாஷ
Absolute Brahma
Chidaakaash | சிதாகாஷ
Boy/Male
Indian, Sanskrit
Absolute; Aloneness
Boy/Male
Hindu
Absolute
Boy/Male
Hindu
The supreme absolute truth
Boy/Male
Indian, Sanskrit
Seeker of the Absolute
Boy/Male
Gujarati, Hindu, Indian, Malayalam, Marathi, Sanskrit
Absolute Brahma
ABSOLUTE GEOMETRY
ABSOLUTE GEOMETRY
Boy/Male
Biblical
Avenging; or establishing; or resurrection; of the Lord.
Girl/Female
Arabic, Gujarati, Indian, Kannada, Muslim, Parsi
Delicate Body; Worthy of Praise
Boy/Male
Arabic, Muslim
Sina was the Father of Abu Ali Ib-e-sina
Female
English
Variant spelling of English Teal, TEALE means "blue-green" or "teal duck."
Girl/Female
Indian, Sikh
Lion; King or Queen
Girl/Female
Indian
Knowledge
Boy/Male
Hindu
Husband of lotus Sun
Girl/Female
Indian
Colour
Girl/Female
Swedish Greek
Divine fame.
Boy/Male
Indian, Sanskrit
Has Plenty
ABSOLUTE GEOMETRY
ABSOLUTE GEOMETRY
ABSOLUTE GEOMETRY
ABSOLUTE GEOMETRY
ABSOLUTE GEOMETRY
n.
Doctrine of absolute decrees.
a.
Pure; unmixed; as, absolute alcohol.
n.
One who is resolute; hence, a desperado.
n.
Absolute municipal self-government.
a.
No longer in use; gone into disuse; disused; neglected; as, an obsolete word; an obsolete statute; -- applied chiefly to words, writings, or observances.
v. t.
To absolve; as, to solute sin.
a.
Loosed from any limitation or condition; uncontrolled; unrestricted; unconditional; as, absolute authority, monarchy, sovereignty, an absolute promise or command; absolute power; an absolute monarch.
a.
Soluble; as, a solute salt.
v. i.
To become obsolete; to go out of use.
a.
Complete in itself; perfect; consummate; faultless; as, absolute perfection; absolute beauty.
a.
Unqualified; absolute; entire; sheer.
a.
Absolute.
v. t.
To set free, or release, as from some obligation, debt, or responsibility, or from the consequences of guilt or such ties as it would be sin or guilt to violate; to pronounce free; as, to absolve a subject from his allegiance; to absolve an offender, which amounts to an acquittal and remission of his punishment.
a.
Not immediately dependent on the other parts of the sentence in government; as, the case absolute. See Ablative absolute, under Ablative.
a.
Loose; free; liberal; as, a solute interpretation.
n.
Absolute renunciation or repudiation.
adv.
In an absolute, independent, or unconditional manner; wholly; positively.
v. t. & i.
Resolving, or explaining; as, the Resolute Doctor Durand.
a.
Viewed apart from modifying influences or without comparison with other objects; actual; real; -- opposed to relative and comparative; as, absolute motion; absolute time or space.
a.
Not adhering; loose; -- opposed to adnate; as, a solute stipule.