Orthogonality

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In

.

Orthogonality is also used with various meanings that are often weakly related or not related at all with the mathematical meanings.

Etymology

The word comes from the Ancient Greek ὀρθός (orthós), meaning "upright",^[1] and γωνία (gōnía), meaning "angle".^[2]

The Ancient Greek ὀρθογώνιον (orthogṓnion) and Classical Latin orthogonium originally denoted a rectangle.^[3] 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.^[4]

Mathematics

Physics

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Optics

In optics, 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.

Special relativity

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.

Hyperbolic orthogonality

Quantum mechanics

In

, ${\displaystyle \psi _{m}}$ and ${\displaystyle \psi _{n}}$, are orthogonal is that they correspond to different eigenvalues. This means, in , that ${\displaystyle \langle \psi _{m}|\psi _{n}\rangle =0}$ if ${\displaystyle \psi _{m}}$ and ${\displaystyle \psi _{n}}$ 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).^{[citation needed]}

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 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 of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours." Archived 2009-01-31 at the Wayback Machine

Computer science

Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results.

:

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.^[7]

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

, 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.

Orthogonal instruction set

An

Telecommunications

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In

(TDMA), where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").

Orthogonal frequency-division multiplexing

Another scheme is

.

In OFDM, the subcarrier frequencies are chosen^[how?] 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,

(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

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 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.^[11]

Organic synthesis

In

independently of each other.

Bioorthogonal chemistry

Supramolecular chemistry

In

, interactions being compatible; reversibly forming without interference from the other.

Analytical chemistry

In

.

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.

Philosophy

In philosophy, two topics, authors, or pieces of writing are said to be "orthogonal" to each other when they do not substantively cover what could be considered potentially overlapping or competing claims. Thus, texts in philosophy can either support and complement one another, they can offer competing explanations or systems, or they can be orthogonal to each other in cases where the scope, content, and purpose of the pieces of writing are entirely unrelated.

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".^[25] 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.^[26] 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
• Up tack

References