Various meanings of the terms
In
mathematics
,
orthogonality
is the generalization of the geometric notion of
perpendicularity
.
Orthogonality
is also used with various meanings that are often weakly related or not related at all with the mathematical meanings.
Etymology
[
edit
]
The word comes from the
Ancient Greek
?ρθ??
(
orthos
), meaning "upright",
[1]
and
γων?α
(
g?nia
), meaning "angle".
[2]
The Ancient Greek
?ρθογ?νιον
(
orthog?nion
) 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
[
edit
]
Physics
[
edit
]
| This section
needs expansion
. You can help by
adding to it
.
(
September 2022
)
|
Optics
[
edit
]
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
[
edit
]
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
[
edit
]
In
geometry
, the relation of
hyperbolic orthogonality
between two lines separated by the asymptotes of a
hyperbola
is a concept used in
special relativity
to define simultaneous events. Two events will be simultaneous when they are on a line hyperbolically orthogonal to a particular time line. This dependence on a certain time line is determined by velocity, and is the basis for the
relativity of simultaneity
.
Quantum mechanics
[
edit
]
In
quantum mechanics
, a sufficient (but not necessary) condition that two
eigenstates
of a
Hermitian operator
,
and
, are orthogonal is that they correspond to different eigenvalues. This means, in
Dirac notation
, that
if
and
correspond to different eigenvalues. This follows from the fact that
Schrodinger's equation
is a
Sturm?Liouville
equation (in Schrodinger's formulation) or that observables are given by Hermitian operators (in Heisenberg's formulation).
[
citation needed
]
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
[
edit
]
Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results.
[6]
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.
[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
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.
Orthogonal instruction set
[
edit
]
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)
[8]
and is designed such that instructions can use any
register
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
uniquely encodes all combinations of registers and addressing modes.
[9]
Telecommunications
[
edit
]
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").
Orthogonal frequency-division multiplexing
[
edit
]
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
;
WiMAX
;
ITU-T
G.hn
,
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
.
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
[
edit
]
When performing statistical analysis,
independent variables
that affect a particular
dependent variable
are said to be orthogonal if they are uncorrelated,
[10]
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
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
[
edit
]
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
[
edit
]
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
[
edit
]
In
organic synthesis
,
orthogonal protection
is a strategy allowing the deprotection of
functional groups
independently of each other.
Bioorthogonal chemistry
[
edit
]
The term
bioorthogonal chemistry
refers to any
chemical reaction
that can occur inside of
living systems
without interfering with native biochemical processes.
[12]
[13]
[14]
The term was coined by
Carolyn R. Bertozzi
in 2003.
[15]
[16]
Since its introduction, the concept of the bioorthogonal reaction has enabled the study of biomolecules such as
glycans
,
proteins
,
[17]
and
lipids
[18]
in real time in living systems without cellular toxicity. A number of
chemical ligation
strategies have been developed that fulfill the requirements of bioorthogonality, including the
1,3-dipolar cycloaddition
between
azides
and
cyclooctynes
(also termed
copper-free click chemistry
),
[19]
between
nitrones
and cyclooctynes,
[20]
oxime
/
hydrazone
formation from
aldehydes
and
ketones
,
[21]
the
tetrazine
ligation,
[22]
the
isocyanide
-based click reaction,
[23]
and most recently, the
quadricyclane
ligation.
[24]
Supramolecular chemistry
[
edit
]
In
supramolecular chemistry
the notion of orthogonality refers to the possibility of two or more supramolecular, often
non-covalent
, interactions being compatible; reversibly forming without interference from the other.
Analytical chemistry
[
edit
]
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
[
edit
]
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
[
edit
]
In
neuroscience
, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.
Philosophy
[
edit
]
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
[
edit
]
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
[
edit
]
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
[
edit
]
Look up
orthogonal
in Wiktionary, the free dictionary.
References
[
edit
]
- ^
Liddell and Scott,
A Greek?English Lexicon
s.v.
?ρθ??
- ^
Liddell and Scott,
A Greek?English Lexicon
s.v.
γων?α
- ^
Liddell and Scott,
A Greek?English Lexicon
s.v.
?ρθογ?νιον
- ^
"orthogonal".
Oxford English Dictionary
(3rd ed.).
Oxford University Press
. September 2004.
- ^
J.A. Wheeler; C. Misner; K.S. Thorne (1973).
Gravitation
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ISBN
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- ^
Michael L. Scott,
Programming Language Pragmatics
, p. 228.
- ^
1968, Adriaan van Wijngaarden et al., Revised Report on the Algorithmic Language ALGOL 68, section 0.1.2, Orthogonal design
- ^
Null, Linda & Lobur, Julia (2006).
The essentials of computer organization and architecture
(2nd ed.). Jones & Bartlett Learning. p. 257.
ISBN
978-0-7637-3769-6
.
- ^
Linda Null (2010).
The Essentials of Computer Organization and Architecture
(PDF)
. Jones & Bartlett Publishers. pp. 287?288.
ISBN
978-1449600068
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Archived
(PDF)
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- ^
Athanasios Papoulis; S. Unnikrishna Pillai (2002).
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ISBN
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.
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.
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.
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.
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.
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.
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.
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- ^
"chessvariants.org chess glossary"
.
- ^
For an illustration, see
YouTube
.