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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.


Etymology

The word comes from the
Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...
('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle.


Mathematics


Physics

* In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simultaneous events, also determined by the rapidity. The theory features relativity of simultaneity. * In quantum mechanics, a sufficient (but not necessary) condition that two eigenstates of a
Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to i ...
, \psi_m and \psi_n , are orthogonal is that they correspond to different eigenvalues. This means, in Dirac notation, that \langle \psi_m , \psi_n \rangle = 0 if \psi_m and \psi_n correspond to different eigenvalues. This follows from the fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation).


Art

In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines". The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and
Burgoyne Diller Burgoyne A. Diller (January 13, 1906 – January 30, 1965) was an American abstract painter. Many of his best-known works are characterized by orthogonal geometric forms that reflect his strong interest in the De Stijl movement and the work of ...
are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the
web site A website (also written as a web site) is a collection of web pages and related content that is identified by a common domain name and published on at least one web server. Examples of notable websites are Google, Facebook, Amazon, and W ...
of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours."


Computer science

Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results. This usage was introduced by Van Wijngaarden in the design of Algol 68:
The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities.
Orthogonality is a system design property which guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through the separation of concerns and encapsulation, and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them. An instruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task) and is designed such that instructions can use any
register Register or registration may refer to: Arts entertainment, and media Music * Register (music), the relative "height" or range of a note, melody, part, instrument, etc. * ''Register'', a 2017 album by Travis Miller * Registration (organ), th ...
in any addressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An
orthogonal instruction set In computer engineering, an orthogonal instruction set is an instruction set architecture where all instruction types can use all addressing modes. It is "orthogonal" in the sense that the instruction type and the addressing mode vary independentl ...
uniquely encodes all combinations of registers and addressing modes.


Telecommunications

In telecommunications, multiple access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions. One such scheme is time-division multiple access (TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots"). Another scheme is orthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a, g, and n) versions of 802.11
Wi-Fi Wi-Fi () is a family of wireless network protocols, based on the IEEE 802.11 family of standards, which are commonly used for local area networking of devices and Internet access, allowing nearby digital devices to exchange data by radio waves ...
; WiMAX;
ITU-T The ITU Telecommunication Standardization Sector (ITU-T) is one of the three sectors (divisions or units) of the International Telecommunication Union (ITU). It is responsible for coordinating standards for telecommunications and Information Co ...
G.hn G.hn is a specification for home networking with data rates up to 2 Gbit/s and operation over four types of legacy wires: telephone wiring, coaxial cables, power lines and plastic optical fiber. A single G.hn semiconductor device is able to n ...
, DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of
ADSL Asymmetric digital subscriber line (ADSL) is a type of digital subscriber line (DSL) technology, a data communications technology that enables faster data transmission over copper telephone lines than a conventional voiceband modem can provide. ...
. In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required.


Statistics, econometrics, and economics

When performing statistical analysis,
independent variables Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
that affect a particular dependent variable are said to be orthogonal if they are uncorrelated, since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with
simple regression In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the ''x'' a ...
or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions). One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.


Taxonomy

In taxonomy, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.


Chemistry and biochemistry

In synthetic organic chemistry orthogonal protection is a strategy allowing the deprotection of
functional group In organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the r ...
s independently of each other. In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of the other pair. For example, DNA has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively.
Bioorthogonal chemistry The term bioorthogonal chemistry refers to any chemical reaction that can occur inside of living systems without interfering with native biochemical processes. The term was coined by Carolyn R. Bertozzi in 2003. Since its introduction, the concept ...
refers to chemical reactions occurring inside living systems without reacting with naturally present cellular components. In supramolecular chemistry the notion of orthogonality refers to the possibility of two or more supramolecular, often
non-covalent In chemistry, a non-covalent interaction differs from a covalent bond in that it does not involve the sharing of electrons, but rather involves more dispersed variations of electromagnetic interactions between molecules or within a molecule. Th ...
, interactions being compatible; reversibly forming without interference from the other. In analytical chemistry, analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing the reliability of the measurement. Orthogonal testing thus can be viewed as "cross-checking" of results, and the "cross" notion corresponds to the etymologic origin of ''orthogonality''. Orthogonal testing is often required as a part of a new drug application.


System reliability

In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure.


Neuroscience

In neuroscience, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.


Gaming

In board games such as chess which feature a grid of squares, 'orthogonal' is used to mean "in the same row/'rank' or column/'file'". This is the counterpart to squares which are "diagonally adjacent". In the ancient Chinese board game Go a player can capture the stones of an opponent by occupying all orthogonally-adjacent points.


Other examples

Stereo vinyl records encode both the left and right stereo channels in a single groove. The V-shaped groove in the vinyl has walls that are 90 degrees to each other, with variations in each wall separately encoding one of the two analogue channels that make up the stereo signal. The cartridge senses the motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side.For an illustration, se
YouTube
A pure horizontal motion corresponds to a mono signal, equivalent to a stereo signal in which both channels carry identical (in-phase) signals.


See also

* Orthogonal ligand-protein pair


References

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