formulasearchengine - User contributions [en]

Information cascade

2014-02-24T01:43:57Z

178.16.15.145: /* Quantitative description */

Emilia Shryock is my name but you can contact me anything you like. My day job is a meter reader. One of the issues she enjoys most is to do aerobics and now she is trying to make cash with it. California is where I've always been living and I adore each day residing here. Feel free to visit my web page - [http://boomerangfoundation.org/%5Brealname%5D/project/use-efficient-natural-remedies-liver-illnesses over the counter std test]

Faraday effect

2014-01-31T06:53:31Z

178.16.0.56:

In [[mathematics]], a '''quadratic integral''' is an [[integral]] of the form

:<math>\int \frac{dx}{a+bx+cx^2}. </math>

It can be evaluated by [[completing the square]] in the [[denominator]].

:<math>\int \frac{dx}{a+bx+cx^2} = \frac{1}{c} \int \frac{dx}{\left( x+ \frac{b}{2c} \right)^2 + \left( \frac{a}{c} - \frac{b^2}{4c^2} \right)}. </math>

==Positive-discriminant case==

Assume that the [[discriminant]] ''q'' = ''b''2 − 4''ac'' is positive. In that case, define ''u'' and ''A'' by

:<math>u = x + \frac{b}{2c} </math>,

and

:<math> -A^2 = \frac{a}{c} - \frac{b^2}{4c^2} = \frac{1}{4c^2} \left( 4ac - b^2 \right). </math>

The quadratic integral can now be written as

:<math> \int \frac{dx}{a+bx+cx^2} = \frac1c \int \frac{du}{u^2-A^2} = \frac1c \int \frac{du}{(u+A)(u-A)}. </math>

The [[partial fraction decomposition]]

:<math> \frac{1}{(u+A)(u-A)} = \frac{1}{2A} \left( \frac{1}{u-A} - \frac{1}{u+A} \right) </math>

allows us to evaluate the integral:

:<math> \frac1c \int \frac{du}{(u+A)(u-A)} = \frac{1}{2Ac} \ln \left( \frac{u - A}{u + A} \right) + \text{constant}. </math>

The final result for the original integral, under the assumption that ''q'' > 0, is

:<math> \int \frac{dx}{a+bx+cx^2} = \frac{1}{ \sqrt{q}} \ln \left( \frac{2cx + b - \sqrt{q}}{2cx+b+ \sqrt{q}} \right) + \text{constant, where } q = b^2 - 4ac. </math>

==Negative-discriminant case==

:''This (hastily written) section may need attention.''

In case the [[discriminant]] ''q'' = ''b''2 − 4''ac'' is negative, the second term in the denominator in

:<math>\int \frac{dx}{a+bx+cx^2} = \frac{1}{c} \int \frac{dx}{\left( x+ \frac{b}{2c} \right)^2 + \left( \frac{a}{c} - \frac{b^2}{4c^2} \right)}. </math>

is positive. Then the integral becomes

:<math>
\begin{align}
& {} \qquad \frac{1}{c} \int \frac{ du} {u^2 + A^2} \\[9pt]
& = \frac{1}{cA} \int \frac{du/A}{(u/A)^2 + 1 } \\[9pt]
& = \frac{1}{cA} \int \frac{dw}{w^2 + 1} \\[9pt]
& = \frac{1}{cA} \arctan(w) + \mathrm{constant} \\[9pt]
& = \frac{1}{cA} \arctan\left(\frac{u}{A}\right) + \text{constant} \\[9pt]
& = \frac{1}{c\sqrt{\frac{a}{c} - \frac{b^2}{4c^2}}} \arctan
\left(\frac{x + \frac{b}{2c}}{\sqrt{\frac{a}{c} - \frac{b^2}{4c^2}}}\right) + \text{constant} \\[9pt]
& = \frac{2}{\sqrt{4ac - b^2\, }}
\arctan\left(\frac{2cx + b}{\sqrt{4ac - b^2}}\right) + \text{constant}.
\end{align}
</math>

==References==
*Weisstein, Eric W. "[http://mathworld.wolfram.com/QuadraticIntegral.html Quadratic Integral]." From ''MathWorld''--A Wolfram Web Resource, wherein the following is referenced:
*Gradshteyn, I. S. and Ryzhik, I. M. ''Tables of Integrals, Series, and Products,'' 6th ed. San Diego, CA: Academic Press, 2000.

[[Category:Integral calculus]]

Magnetic circuit

2014-01-29T11:41:08Z

178.16.0.56: /* Circuit laws */

{{Probability distribution |
name =Type-2 Gumbel|
type =density|
pdf_image =|
cdf_image =|
parameters =<math>a\!</math> ([[real number|real]]) <math>b\!</math> shape (real)|
support =|
pdf =<math> a b x^{-a-1} e^{-b x^{-a}}\!</math>|
cdf =<math> e^{-b x^{-a}}\!</math>|
mean =|
median =|
mode =|
variance =|
skewness =|
kurtosis =|
entropy =|
mgf =|
char =|
}}
In [[probability theory]], the '''Type-2 Gumbel [[probability density function]]''' is

:<math>f(x|a,b) = a b x^{-a-1} e^{-b x^{-a}}\,</math>

for

:<math>0 < x < \infty</math>.

This implies that it is similar to the [[Weibull distribution]]s, substituting <math>b=\lambda^{-k}</math> and <math>a=-k</math>. Note however that a positive ''k'' (as in the Weibull distribution) would yield a negative ''a'', which is not allowed here as it would yield a negative probability density.

For <math>0<a\le 1</math> the [[mean]] is infinite. For <math>0<a\le 2</math> the [[variance]] is infinite.

The [[cumulative distribution function]] is

:<math>F(x|a,b) = e^{-b x^{-a}}\,</math>

The moments <math> E[X^k] \,</math> exist for <math>k < a\,</math>

The special case b = 1 yields the [[Fréchet distribution]]

----
Based on [http://www.gnu.org/software/gsl/manual/html_node/The-Type_002d2-Gumbel-Distribution.html The GNU Scientific Library], used under GFDL.

==See also==
* [[Extreme value theory]]
* [[Gumbel distribution]]
* [[Type-1 Gumbel distribution]]

{{ProbDistributions|continuous-semi-infinite}}

[[Category:Continuous distributions]]
[[Category:Probability distributions]]

Magnetomotive force

2014-01-29T09:23:45Z

178.16.0.56:

{{Distinguish|Designer drug}}
{{Repetition|date=November 2013}}
'''Drug design''', sometimes referred to as '''rational drug design''' or more simply [[rational design]], is the [[invention|inventive]] process of finding new [[medications]] based on the knowledge of a [[biological target]].<ref name="isbn0-415-...">{{cite book | author = Madsen, Ulf; Krogsgaard-Larsen, Povl; Liljefors, Tommy | authorlink = | editor = | others = | title = Textbook of Drug Design and Discovery | edition = | language = | publisher = Taylor & Francis | location = Washington, DC | year = 2002 | origyear = | pages = | quote = | isbn = 0-415-28288-8 | oclc = | doi = | url = | accessdate = }}</ref> The drug is most commonly an [[organic compound|organic]] [[small molecule]] that activates or inhibits the function of a [[biomolecule]] such as a [[protein]], which in turn results in a [[therapeutic effect|therapeutic]] benefit to the [[patient]]. In the most basic sense, drug design involves the design of small molecules that are complementary in [[shape]] and [[electric charge|charge]] to the biomolecular target with which they interact and therefore will bind to it. Drug design frequently but not necessarily relies on [[molecular modelling|computer modeling]] techniques.<ref name="isbn012178245x">{{cite book | author = Cohen, N. Claude | authorlink = | editor = | others = | title = Guidebook on Molecular Modeling in Drug Design | edition = | language = | publisher = Academic Press | location = Boston | year = 1996 | origyear = | pages = | quote = | isbn = 0-12-178245-X | oclc = | doi = | url = | accessdate = }}</ref> This type of modeling is often referred to as '''computer-aided drug design'''. Finally, drug design that relies on the knowledge of the three-dimensional structure of the biomolecular target is known as '''structure-based drug design'''.

The phrase "drug design" is to some extent a [[misnomer]]. What is really meant by drug design is [[ligand (biochemistry)|ligand]] design (i.e., design of a small molecule that will bind tightly to its target).<ref name="pmid8739258">{{cite journal | author = Tollenaere JP | title = The role of structure-based ligand design and molecular modelling in drug discovery | journal = Pharm World Sci | volume = 18 | issue = 2 | pages = 56–62 |date=April 1996 | pmid = 8739258 | doi = 10.1007/BF00579706 }}</ref> Although modeling techniques for prediction of binding affinity are reasonably successful, there are many other properties, such as [[bioavailability]], [[biological half-life|metabolic half-life]], lack of [[adverse drug reaction|side effects]], etc., that first must be optimized before a ligand can become a safe and efficacious drug. These other characteristics are often difficult to optimize using rational drug design techniques.

==Background==
Typically a drug target is a key [[molecule]] involved in a particular [[metabolic pathway|metabolic]] or [[signal transduction|signaling]] pathway that is specific to a disease condition or [[pathology]] or to the [[infectivity]] or survival of a [[microorganism|microbial]] [[pathogen]]. Some approaches attempt to inhibit the functioning of the pathway in the diseased state by causing a key molecule to stop functioning. Drugs may be designed that bind to the active region and inhibit this key molecule. Another approach may be to enhance the normal pathway by promoting specific molecules in the normal pathways that may have been affected in the diseased state. In addition, these drugs should also be designed so as not to affect any other important "off-target" molecules or [[antitarget]]s that may be similar in appearance to the target molecule, since drug interactions with off-target molecules may lead to undesirable [[adverse effect|side effect]]s. [[Sequence homology]] is often used to identify such risks.

Most commonly, drugs are [[organic compound|organic]] [[small molecule]]s produced through chemical synthesis, but biopolymer-based drugs (also known as [[biologic medical product|biologics]]) produced through biological processes are becoming increasingly more common. In addition, [[mRNA]]-based [[gene silencing]] technologies may have therapeutic applications.

==Types==
[[File:Flow charts of two strategies of structure based drug design.jpg|thumb|500 px|Flow charts of two strategies of structure-based drug design]]There are two major types of drug design. The first is referred to as '''ligand-based drug design''' and the second, '''structure-based drug design'''.

===Ligand-based===
Ligand-based drug design (or '''indirect drug design''') relies on knowledge of other molecules that bind to the biological target of interest. These other molecules may be used to derive a [[pharmacophore]] model that defines the minimum necessary structural characteristics a molecule must possess in order to bind to the target.<ref name="isbn0-9636817-6-1">{{cite book | author = Guner, Osman F. | authorlink = | editor = | others = | title = Pharmacophore Perception, Development, and use in Drug Design | edition = | language = | publisher = International University Line | location = La Jolla, Calif | year = 2000 | origyear = | pages = | quote = | isbn = 0-9636817-6-1 | oclc = | doi = | url = | accessdate = }}</ref> In other words, a model of the biological target may be built based on the knowledge of what binds to it, and this model in turn may be used to design new molecular entities that interact with the target. Alternatively, a [[quantitative structure-activity relationship]] (QSAR), in which a correlation between calculated properties of molecules and their experimentally determined biological activity, may be derived. These QSAR relationships in turn may be used to predict the activity of new analogs.

===Structure-based===
Structure-based drug design (or '''direct drug design''') relies on knowledge of the [[tertiary structure|three dimensional structure]] of the biological target obtained through methods such as [[X-ray_crystallography#Protein_crystallography|x-ray crystallography]] or [[Protein nuclear magnetic resonance spectroscopy|NMR spectroscopy]].<ref name="isbn1-4020-4406-2">{{cite book | author = Leach, Andrew R.; Harren Jhoti | authorlink = | editor = | others = | title = Structure-based Drug Discovery | edition = | language = | publisher = Springer | location = Berlin | year = 2007 | origyear = | pages = | quote = | isbn = 1-4020-4406-2 | oclc = | doi = | url = | accessdate = }}</ref> If an experimental structure of a target is not available, it may be possible to create a [[homology modeling|homology model]] of the target based on the experimental structure of a related protein. Using the structure of the biological target, candidate drugs that are predicted to bind with high [[dissociation constant|affinity]] and [[Ligand_(biochemistry)#Selective_and_non-selective|selectivity]] to the target may be designed using interactive graphics and the intuition of a [[medicinal chemistry|medicinal chemist]]. Alternatively various automated computational procedures may be used to suggest new drug candidates.

As '''experimental methods''' such as X-ray crystallography and NMR develop, the amount of information concerning 3D structures of biomolecular targets has increased dramatically. In parallel, information about the structural dynamics and electronic properties about ligands has also increased. This has encouraged the rapid development of the structure-based drug design. Current methods for structure-based drug design can be divided roughly into two categories. The first category is about “finding” ligands for a given receptor, which is usually referred as database searching. In this case, a large number of potential ligand molecules are screened to find those fitting the binding pocket of the receptor. This method is usually referred as ligand-based drug design. The key advantage of database searching is that it saves synthetic effort to obtain new lead compounds. Another category of structure-based drug design methods is about “building” ligands, which is usually referred as receptor-based drug design. In this case, ligand molecules are built up within the constraints of the binding pocket by assembling small pieces in a stepwise manner. These pieces can be either individual atoms or molecular fragments. The key advantage of such a method is that novel structures, not contained in any database, can be suggested.<ref name="ligbuilder">{{cite journal | author = Wang R,Gao Y,Lai L | title = LigBuilder: A Multi-Purpose Program for Structure-Based Drug Design | journal = Journal of Molecular Modeling | year=2000 | volume=6 | issue = 7–8 | pages=498–516 | doi = 10.1007/s0089400060498}}</ref><ref name="CBDDreview">{{cite journal | author = Schneider G, Fechner U | title = Computer-based de novo design of drug-like molecules | journal = Nat Rev Drug Discov | volume = 4 | issue = 8 | pages = 649–63 |date=August 2005 | pmid = 16056391 | doi = 10.1038/nrd1799 | url = }}</ref><ref name="pmid15031495">{{cite journal | author = Jorgensen WL | title = The many roles of computation in drug discovery | journal = Science | volume = 303 | issue = 5665 | pages = 1813–8 |date=March 2004 | pmid = 15031495 | doi = 10.1126/science.1096361 | url = |bibcode = 2004Sci...303.1813J }}</ref>

====Active site identification====
Active site identification is the first step in this program. It analyzes the protein to find the binding pocket, derives key interaction sites within the binding pocket, and then prepares the necessary data for Ligand fragment link. The basic inputs for this step are the 3D structure of the protein and a pre-docked ligand in PDB format, as well as their atomic properties. Both ligand and protein atoms need to be classified and their atomic properties should be defined, basically, into four atomic types:
*'''hydrophobic atom''': All carbons in hydrocarbon chains or in aromatic groups.
*'''H-bond donor''': Oxygen and nitrogen atoms bonded to hydrogen atom(s).
*'''H-bond acceptor''': Oxygen and [[Orbital_hybridisation#sp2_hybrids|sp2]] or [[Orbital_hybridisation#sp_hybrids|sp hybridized]] nitrogen atoms with lone electron pair(s).
*'''Polar atom''': Oxygen and nitrogen atoms that are neither H-bond donor nor H-bond acceptor, sulfur, phosphorus, halogen, metal, and carbon atoms bonded to hetero-atom(s).

The space inside the ligand binding region would be studied with virtual probe atoms of the four types above so the chemical environment of all spots in the ligand binding region can be known. Hence we are clear what kind of chemical fragments can be put into their corresponding spots in the ligand binding region of the receptor.

====Ligand fragment link====
[[File:Flow chart for structure based drug design.jpg|thumb|500 px|Flow chart for structure-based drug design]]
When we want to plant “seeds” into different regions defined by the previous section, we need a fragments database to choose fragments from. The term “fragment” is used here to describe the building blocks used in the construction process. The rationale of this algorithm lies in the fact that organic structures can be decomposed into basic chemical fragments. Although the diversity of organic structures is infinite, the number of basic fragments is rather limited.

Before the first fragment, i.e. the seed, is put into the binding pocket, and other fragments can be added one by one, it is useful to identify potential problems. First, the possibility for the fragment combinations is huge. A small perturbation of the previous fragment conformation would cause great difference in the following construction process. At the same time, in order to find the lowest binding energy on the [[Potential energy surface]] (PES) between planted fragments and receptor pocket, the scoring function calculation would be done for every step of conformation change of the fragments derived from every type of possible fragments combination. Since this requires a large amount of computation, using different tricks may use less computing power and let the program work more efficiently. When a ligand is inserted into the pocket site of a receptor, groups on the ligand that bind tightly with the receptor should have the highest priority in finding their lowest-energy conformation. This allows us to put several seeds into the program at the same time and optimize the conformation of those seeds that form significant interactions with the receptor, and then connect those seeds into a continuous ligand in a manner that make the rest of the ligand have the lowest energy. The pre-placed seeds ensure high binding affinity and their optimal conformation determines the manner in which the ligand will be built, thus determining the overall structure of the final ligand. This strategy efficiently reduces the calculation burden for fragment construction. On the other hand, it reduces the possibility of the combination of fragments, which reduces the number of possible ligands that can be derived from the program. The two strategies above are widely used in most structure-based drug design programs. They are described as “'''Grow'''” and “'''Link'''”. The two strategies are always combined in order to make the construction result more reliable.<ref name='ligbuilder'/><ref name='CBDDreview'/><ref name="pmid7922037">{{cite journal | author = Verlinde CL, Hol WG | title = Structure-based drug design: progress, results and challenges | journal = Structure | volume = 2 | issue = 7 | pages = 577–87 |date=July 1994 | pmid = 7922037 | doi = 10.1016/S0969-2126(00)00060-5 | url = }}</ref>

====Scoring method====
{{Main|Scoring functions for docking}}
Structure-based drug design attempts to use the structure of proteins as a basis for designing new ligands by applying accepted principles of molecular recognition. The basic assumption underlying structure-based drug design is that a good ligand molecule should bind tightly to its target. Thus, one of the most important principles for designing or obtaining potential new ligands is to predict the binding affinity of a certain ligand to its target and use it as a criterion for selection.


One early method was developed by Böhm<ref name="pmid7964925">{{cite journal | author = Böhm HJ | title = The development of a simple empirical scoring function to estimate the binding constant for a protein-ligand complex of known three-dimensional structure | journal = J. Comput. Aided Mol. Des. | volume = 8 | issue = 3 | pages = 243–56 |date=June 1994 | pmid = 7964925 | doi = 10.1007/BF00126743 | url = |bibcode = 1994JCAMD...8..243B }}</ref> to develop a general-purposed empirical scoring function in order to describe the binding energy. The following “Master Equation” was derived:

<math>\begin{array}{lll}\Delta G_{\text{bind}} = -RT \ln K_{\text{d}}\\[1.3ex]
K_{\text{d}} = \dfrac{[\text{Receptor}][\text{Acceptor}]}{[\text{Complex}]}\\[1.3ex]

\Delta G_{\text{bind}} = \Delta G_{\text{desolvation}} + \Delta G_{\text{motion}} + \Delta G_{\text{configuration}} + \Delta G_{\text{interaction}}\end{array}</math>

where:
* desolvation – [[enthalpy|enthalpic]] penalty for removing the ligand from solvent
* motion – [[entropy|entropic]] penalty for reducing the degrees of freedom when a ligand binds to its receptor
* configuration – conformational strain energy required to put the ligand in its "active" conformation
* interaction – enthalpic gain for "resolvating" the ligand with its receptor

The basic idea is that the overall binding free energy can be decomposed into independent components that are known to be important for the binding process. Each component reflects a certain kind of free energy alteration during the binding process between a ligand and its target receptor. The Master Equation is the linear combination of these components. According to Gibbs free energy equation, the relation between dissociation equilibrium constant, Kd, and the components of free energy was built.

Various computational methods are used to estimate each of the components of the master equation. For example, the change in polar surface area upon ligand binding can be used to estimate the desolvation energy. The number of rotatable bonds frozen upon ligand binding is proportional to the motion term. The configurational or strain energy can be estimated using [[molecular mechanics]] calculations. Finally the interaction energy can be estimated using methods such as the change in non polar surface, statistically derived [[potential of mean force|potentials of mean force]], the number of hydrogen bonds formed, etc. In practice, the components of the master equation are fit to experimental data using multiple linear regression. This can be done with a diverse training set including many types of ligands and receptors to produce a less accurate but more general "global" model or a more restricted set of ligands and receptors to produce a more accurate but less general "local" model.<ref name="pmid10623530">{{cite journal | author = Gohlke H, Hendlich M, Klebe G | title = Knowledge-based scoring function to predict protein-ligand interactions | journal = J. Mol. Biol. | volume = 295 | issue = 2 | pages = 337–56 |date=January 2000 | pmid = 10623530 | doi = 10.1006/jmbi.1999.3371 | url = }}</ref><ref name="pmid11858637">{{cite journal | author = Clark RD, Strizhev A, Leonard JM, Blake JF, Matthew JB | title = Consensus scoring for ligand/protein interactions | journal = J. Mol. Graph. Model. | volume = 20 | issue = 4 | pages = 281–95 |date=January 2002 | pmid = 11858637 | doi = 10.1016/S1093-3263(01)00125-5 | url = }}</ref><ref name="pmid12197663">{{cite journal | author = Wang R, Lai L, Wang S | title = Further development and validation of empirical scoring functions for structure-based binding affinity prediction | journal = J. Comput. Aided Mol. Des. | volume = 16 | issue = 1 | pages = 11–26 |date=January 2002 | pmid = 12197663 | doi = 10.1023/A:1016357811882 | url = |bibcode = 2002JCAMD..16...11W }}</ref>

==Rational drug discovery==
In contrast to traditional methods of [[drug discovery]], which rely on [[trial-and-error]] testing of chemical substances on [[cell culture|cultured cell]]s or [[animal]]s, and matching the apparent effects to treatments, rational drug design begins with a hypothesis that modulation of a specific biological target may have therapeutic value. In order for a biomolecule to be selected as a drug target, two essential pieces of information are required. The first is evidence that modulation of the target will have therapeutic value. This knowledge may come from, for example, disease linkage studies that show an association between mutations in the biological target and certain disease states. The second is that the target is "drugable". This means that it is capable of binding to a small molecule and that its activity can be modulated by the small molecule.

Once a suitable target has been identified, the target is normally [[molecular cloning|cloned]] and [[protein expression|expressed]]. The expressed target is then used to establish a [[Drug_discovery#Screening_and_design|screening assay]]. In addition, the three-dimensional structure of the target may be determined.

The search for small molecules that bind to the target is begun by screening libraries of potential drug compounds. This may be done by using the screening assay (a "wet screen"). In addition, if the structure of the target is available, a [[virtual screening|virtual screen]] may be performed of candidate drugs. Ideally the candidate drug compounds should be "[[druglikeness|drug-like]]", that is they should possess properties that are predicted to lead to [[oral bioavailability]], adequate chemical and metabolic stability, and minimal toxic effects. Several methods are available to estimate druglikeness such as [[Lipinski's Rule of Five]] and a range of scoring methods such as [[Lipophilic efficiency]]. Several methods for predicting drug metabolism have been proposed in the scientific literature, and a recent example is SPORCalc.<ref name="pmid19157988">{{cite journal | author = Smith J, Stein V | title = SPORCalc: A development of a database analysis that provides putative metabolic enzyme reactions for ligand-based drug design | journal = Computational Biology and Chemistry | volume = 33 | issue = 2 | pages = 149–59 |date=April 2009 | pmid = 19157988 | doi = 10.1016/j.compbiolchem.2008.11.002 | url = }}</ref> Due to the complexity of the drug design process, two terms of interest are still [[Serendipity#Pharmacology|serendipity]] and [[bounded rationality]]. Those challenges are caused by the large [[chemical space]] describing potential new drugs without [[adverse effect|side-effect]]s.

==Computer-aided drug design {{anchor|Computer-assisted drug design}}==
Computer-aided drug design uses [[computational chemistry]] to discover, enhance, or study [[drugs]] and related biologically active [[molecule]]s. The most fundamental goal is to predict whether a given molecule will bind to a target and if so how strongly. [[Molecular mechanics]] or [[molecular dynamics]] are most often used to predict the conformation of the [[small molecule]] and to model conformational changes in the biological target that may occur when the small molecule binds to it. [[Semi-empirical quantum chemistry method|Semi-empirical]], [[ab initio quantum chemistry methods]], or [[density functional theory]] are often used to provide optimized parameters for the molecular mechanics calculations and also provide an estimate of the electronic properties (electrostatic potential, [[polarizability]], etc.) of the drug candidate that will influence binding affinity.

Molecular mechanics methods may also be used to provide semi-quantitative prediction of the binding affinity. Also, knowledge-based [[scoring functions for docking|scoring function]] may be used to provide binding affinity estimates. These methods use [[linear regression]], [[machine learning]], [[neural net]]s or other statistical techniques to derive predictive binding affinity equations by fitting experimental affinities to computationally derived interaction energies between the small molecule and the target.<ref name="pmid17554857">{{cite journal | author = Rajamani R, Good AC | title = Ranking poses in structure-based lead discovery and optimization: current trends in scoring function development | journal = Curr Opin Drug Discov Devel | volume = 10 | issue = 3 | pages = 308–15 |date=May 2007 | pmid = 17554857 | doi = | url = }}</ref><ref name="pmid19128212">{{cite journal | author = de Azevedo WF, Dias R | title = Computational methods for calculation of ligand-binding affinity | journal = Curr Drug Targets | volume = 9 | issue = 12 | pages = 1031–9 |date=December 2008 | pmid = 19128212 | doi = 10.2174/138945008786949405| url = }}</ref>

Ideally the computational method should be able to predict affinity before a compound is synthesized and hence in theory only one compound needs to be synthesized. The reality however is that present computational methods are imperfect and provide at best only qualitatively accurate estimates of affinity. Therefore in practice it still takes several iterations of design, synthesis, and testing before an optimal molecule is discovered. On the other hand, computational methods have accelerated discovery by reducing the number of iterations required and in addition have often provided more novel small molecule structures.

Drug design with the help of computers may be used at any of the following stages of drug discovery:
# hit identification using [[virtual screening]] (structure- or ligand-based design)
# [[drug discovery hit to lead|hit-to-lead]] optimization of affinity and selectivity (structure-based design, [[Quantitative structure-activity relationship|QSAR]], etc.)
# [[drug development|lead optimization]] optimization of other pharmaceutical properties while maintaining affinity
[[File:wiki Clustering.png|thumb| 400px |alt=Flowchart of a common Clustering Analysis for Structure-Based Drug Design|Flowchart of a Usual Clustering Analysis for Structure-Based Drug Design]]
In order to overcome the insufficient prediction of binding affinity calculated by recent scoring functions, the protein-ligand interaction and compound 3D structure information are used to analysis. For structure-based drug design, several post-screening analysis focusing on protein-ligand interaction has been developed for improving enrichment and effectively mining potential candidates:
* Consensus scoring<ref name="pmid18831053">{{cite journal | author = Liang S, Meroueh SO, Wang G, Qiu C, Zhou Y | title = Consensus scoring for enriching near-native structures from protein-protein docking decoys | journal = Proteins | volume = 75 | issue = 2 | pages = 397–403 |date=May 2009 | pmid = 18831053 | pmc = 2656599 | doi = 10.1002/prot.22252}}</ref><ref name="pmid16426072">{{cite journal | author = Oda A, Tsuchida K, Takakura T, Yamaotsu N, Hirono S | title = Comparison of consensus scoring strategies for evaluating computational models of protein-ligand complexes | journal = J Chem Inf Model | volume = 46 | issue = 1 | pages = 380–91 | year = 2006 | pmid = 16426072 | doi = 10.1021/ci050283k}}</ref>
** Selecting candidates by voting of multiple scoring functions
** May lose the relationship between protein-ligand structural information and scoring criterion
* Geometric analysis
** Comparing protein-ligand interactions by visually inspecting individual structures
** Becoming intractable when the number of complexes to be analyzed increasing
* Cluster analysis<ref name="pmid14711306">{{cite journal | author = Deng Z, Chuaqui C, Singh J | title = Structural interaction fingerprint (SIFt): a novel method for analyzing three-dimensional protein-ligand binding interactions | journal = J. Med. Chem. | volume = 47 | issue = 2 | pages = 337–44 |date=January 2004 | pmid = 14711306 | doi = 10.1021/jm030331x | url = }}</ref><ref name="pmid16426058">{{cite journal | author = Amari S, Aizawa M, Zhang J, Fukuzawa K, Mochizuki Y, Iwasawa Y, Nakata K, Chuman H, Nakano T | title = VISCANA: visualized cluster analysis of protein-ligand interaction based on the ab initio fragment molecular orbital method for virtual ligand screening | journal = J Chem Inf Model | volume = 46 | issue = 1 | pages = 221–30 | year = 2006 | pmid = 16426058 | doi = 10.1021/ci050262q}}</ref>
** Represent and cluster candidates according to protein-ligand 3D information
** Needs meaningful representation of protein-ligand interactions.

==Examples==
A particular example of rational drug design involves the use of three-dimensional information about biomolecules obtained from such techniques as X-ray crystallography and NMR spectroscopy. Computer-aided drug design in particular becomes much more tractable when there is a high-resolution structure of a target protein bound to a potent ligand. This approach to drug discovery is sometimes referred to as structure-based drug design. The first unequivocal example of the application of [[QSAR|structure-based drug design]] leading to an approved drug is the carbonic anhydrase inhibitor [[dorzolamide]], which was approved in 1995.<ref name="pmid8164249">{{cite journal | author = Greer J, Erickson JW, Baldwin JJ, Varney MD | title = Application of the three-dimensional structures of protein target molecules in structure-based drug design | journal = Journal of Medicinal Chemistry | volume = 37 | issue = 8 | pages = 1035–54 |date=April 1994 | pmid = 8164249 | doi = 10.1021/jm00034a001| url = }}</ref><ref name="isbn3-527-29343-4">{{cite book | author = Hendrik Timmerman; Klaus Gubernator; Hans-Joachim Böhm; Raimund Mannhold; Hugo Kubinyi | authorlink = | editor = | others = | title = Structure-based Ligand Design (Methods and Principles in Medicinal Chemistry) | edition = | language = | publisher = Wiley-VCH | location = Weinheim | year = 1998 | origyear = | pages = | quote = | isbn = 3-527-29343-4 | oclc = | doi = | url = | accessdate = }}</ref>

Another important case study in rational drug design is [[imatinib]], a [[tyrosine kinase]] inhibitor designed specifically for the ''bcr-abl'' fusion protein that is characteristic for [[Philadelphia chromosome]]-positive [[leukemia]]s ([[chronic myelogenous leukemia]] and occasionally [[acute lymphocytic leukemia]]). Imatinib is substantially different from previous drugs for [[cancer]], as most agents of [[chemotherapy]] simply target rapidly dividing cells, not differentiating between cancer cells and other tissues.

Additional examples include:
{{columns-list|2|
* Many of the [[atypical antipsychotic]]s
* [[Cimetidine]], the prototypical [[H2-receptor antagonist|H2-receptor antagonist]] from which the later members of the class were developed
* Selective [[Cyclooxygenase|COX-2]] inhibitor [[NSAID]]s
* [[Dorzolamide]], a [[carbonic anhydrase]] inhibitor used to treat glaucoma
* [[Enfuvirtide]], a peptide HIV entry inhibitor
* [[Nonbenzodiazepines]] like [[zolpidem]] and [[zopiclone]]
* [[Probenecid]]
* [[Selective serotonin reuptake inhibitor|SSRI]]s (selective serotonin reuptake inhibitors), a class of [[antidepressant]]s
* [[Zanamivir]], an [[antiviral drug]]
* [[Isentress]], HIV Integrase inhibitor<ref name="url_AutoDock_Integrase_Inhibitor">{{cite web | url = http://autodock.scripps.edu/news/autodocks-role-in-developing-the-first-clinically-approved-hiv-integrase-inhibitor | title = AutoDock's role in Developing the First Clinically-Approved HIV Integrase Inhibitor | author = | date = 2007-12-17 | work = Press Release | publisher = The Scripps Research Institute }}</ref>
}}

;Case studies
* [[5-HT3 antagonist#5-HT3 antagonists drug design|5-HT3 antagonists]]
* [[Development of nicotinic acetylcholine receptor agonists|Acetylcholine receptor agonists]]
* [[Discovery and development of angiotensin receptor blockers|Angiotensin receptor blockers]]
* [[Bcr-Abl tyrosine kinase inhibitors]]
* [[Cannabinoid receptor antagonist#Drug design|Cannabinoid receptor antagonists]]
* [[Discovery and development of CCR5 receptor antagonists|CCR5 receptor antagonists]]
* [[Discovery and development of cyclooxygenase 2 inhibitors|Cyclooxygenase 2 inhibitors]]
* [[Development of dipeptidyl peptidase-4 inhibitors|Dipeptidyl peptidase-4 inhibitors]]
* [[Discovery and development of HIV protease inhibitors|HIV protease inhibitors]]
* [[NK1 receptor antagonist#Drug discovery and development|NK1 receptor antagonists]]
* [[Discovery and development of non-nucleoside reverse transcriptase inhibitors|Non-nucleoside reverse transcriptase inhibitors]]
* [[Discovery and development of proton pump inhibitors|Proton pump inibitors]]
* [[Discovery and development of triptans|Triptans]]
* [[Discovery and development of TRPV1 antagonists|TRPV1 antagonists]]
* [[Discovery and Development of Renin Inhibitors|Renin inhibitors]]
* [[c-Met inhibitors]]
* [[Discovery and development of phosphodiesterase 5 inhibitors|PDE5 inhibitors]]

==See also==
{{columns-list|2|
* [[Bioinformatics]]
* [[Cheminformatics]]
* [[Drug development]]
* [[Drug discovery]]
* [[List of pharmaceutical companies]]
* [[Medicinal chemistry]]
* [[Molecular Conceptor]]
* [[Molecular design software]]
}}

==References==
{{Reflist|colwidth=35em}}

==External links==
* {{MeshName|Drug+Design}}

{{Medicinal chemistry}}
{{Drug design}}

[[Category:Pharmacology]]
[[Category:Design of experiments]]
[[Category:Clinical research]]
[[Category:Medicinal chemistry]]
[[Category:Drug discovery]]

Magnetic reluctance

2014-01-27T02:57:53Z

178.16.0.56:

In [[geometry]], a '''[[point group]] in three dimensions''' is an [[isometry group]] in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a [[sphere]]. It is a [[subgroup]] of the [[orthogonal group]] O(3), the group of all [[isometry|isometries]] that leave the origin fixed, or correspondingly, the group of [[orthogonal matrix|orthogonal matrices]]. O(3) itself is a subgroup of the [[Euclidean group]] ''E''(3) of all isometries.

[[Symmetry group]]s of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible [[symmetry|symmetries]]. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them.

The symmetry group of an object is sometimes also called '''full symmetry group''', as opposed to its '''rotation group''' or '''proper symmetry group''', the intersection of its full symmetry group and the [[rotation group SO(3)]] of the 3D space itself. The rotation group of an object is equal to its full symmetry group [[if and only if]] the object is [[chirality (mathematics)|chiral]].

The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a [[molecule]] and of [[molecular orbital]]s forming [[covalent bond]]s, and in this context they are also called '''[[List of character tables for chemically important 3D point groups|molecular point groups]]'''.

[[Finite Coxeter group]]s are a special set of ''point groups'' generated purely by a set of reflectional mirrors passing through the same point. A rank ''n'' Coxeter group has ''n'' mirrors and is represented by a [[Coxeter-Dynkin diagram]]. [[Coxeter notation]] offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups.

==Group structure==
SO(3) is a subgroup of [[Euclidean group#Direct and indirect isometries|''E''+(3)]], which consists of [[Euclidean group#Direct and indirect isometries|''direct isometries'']], i.e., isometries preserving [[orientation (mathematics)|orientation]]; it contains those that leave the origin fixed.

O(3) is the [[direct product of groups|direct product]] of SO(3) and the group generated by [[Inversion in a point|inversion]] (denoted by its matrix −''I''):
:O(3) = SO(3) × { ''I'' , −''I'' }

Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. Also there is a 1-to-1 correspondence between all groups of direct isometries ''H'' in O(3) and all groups ''K'' of isometries in O(3) that contain inversion:
:''K'' = ''H'' × { ''I'' , −''I'' }
:''H'' = ''K'' ∩ SO(3)
For instance, if ''H'' is ''C''2, then ''K'' is ''C''2h, or if ''H'' is ''C''3, then ''K'' is ''S''6. (See lower down for the definitions of these groups.)

If a group of direct isometries ''H'' has a subgroup ''L'' of [[Index of a subgroup|index]] 2, then, apart from the corresponding group containing inversion there is also a corresponding group that contains indirect isometries but no inversion:
:''M'' = ''L'' ∪ ( (''H'' \ ''L'') × { − ''I'' } )
where isometry ( ''A'', ''I'' ) is identified with ''A''. An example would be ''C''4 for ''H'' and ''S''4 for ''M''.

Thus ''M'' is obtained from ''H'' by inverting the isometries in ''H'' \ ''L''. This group ''M'' is as abstract group isomorphic with ''H''. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries. This is clarifying when categorizing isometry groups, see below.

In 2D the [[cyclic group]] of ''k''-fold [[rotation]]s ''Ck'' is for every positive integer ''k'' a normal subgroup of O(2,'''R''') and SO(2,'''R'''). Accordingly, in 3D, for every axis the cyclic group of ''k''-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations (''Cn'') is normal both in the group obtained by adding reflections in planes through the axis (''Cnv'') and in the group obtained by adding a reflection plane perpendicular to the axis (''Cnh'').

==3D isometries that leave origin fixed==
The isometries of '''R'''''3'' that leave the origin fixed, forming the group O(''3'','''R'''), can be categorized as follows:
*SO(''3'','''R'''):
**identity
**rotation about an axis through the origin by an angle not equal to 180°
**rotation about an axis through the origin by an angle of 180°
*the same with [[Inversion in a point|inversion]] ('''x''' is mapped to −'''x'''), i.e. respectively:
**inversion
**rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis
**reflection in a plane through the origin

The 4th and 5th in particular, and in a wider sense the 6th also, are called [[improper rotation]]s.

See also the similar [[Euclidean group#Overview of isometries in up to three dimensions|overview including translations]].

==Conjugacy==
When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups ''H''1, ''H''2 of a group ''G'' are [[Conjugacy class#Conjugacy of subgroups and general subsets|''conjugate'']], if there exists ''g'' ∈ ''G'' such that ''H''1 = g−1''H''2''g'' ).

For example two 3D objects have the same symmetry type:
*if both have mirror symmetry, but with respect to a different mirror plane
*if both have 3-fold rotational symmetry, but with respect to a different axis.
In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type [[if and only if]] there is a rotation mapping the whole structure of the first symmetry group to that of the second. (In fact there will be more than one such rotation, but not an infinite number as when there is only one mirror or axis.) The conjugacy definition would also allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. (The structure '''''is''''' chiral for 11 pairs of '''''[[space group]]s''''' with a screw axis.)

==Infinite isometry groups==
There are many infinite isometry groups; for example, the "[[cyclic group]]" (meaning that it is generated by one element – not to be confused with a [[torsion group]]) generated by a rotation by an irrational number of turns about an axis. We may create non-cyclical [[abelian group]]s by adding more rotations around the same axis. There are also non-abelian groups generated by rotations around different axes. These are usually (generically) [[free group]]s. They will be infinite unless the rotations are specially chosen.

All the infinite groups mentioned so far are not [[Closed (topology)|closed]] as [[topological group|topological subgroups]] of O(3). We now discuss topologically closed subgroups of O(3).

The whole O(3) is the symmetry group of spherical symmetry; [[SO(3)]] is the corresponding rotation group. The other infinite isometry groups consist of all [[rotation]]s about an axis through the origin, and those with additionally reflection in the planes through the axis, and/or reflection in the plane through the origin, perpendicular to the axis. Those with reflection in the planes through the axis, with or without reflection in the plane through the origin perpendicular to the axis, are the symmetry groups for the two types of cylindrical symmetry. Note that any physical object having infinite rotational symmetry will also have the symmetry of mirror planes through the axis.

See also [[Rotational symmetry#Rotational symmetry with respect to any angle|rotational symmetry with respect to any angle]].

==Finite isometry groups==
Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin. For finite 3D point groups, see also [[list of spherical symmetry groups|spherical symmetry groups]].

Up to conjugacy the set of finite 3D point groups consists of:
*7 infinite series with at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite [[Cylinder (geometry)|cylinder]], or equivalently, those on a finite cylinder. They are sometimes called the axial or prismatic point groups.
*7 point groups with multiple 3-or-more-fold rotation axes; they can also be characterized as point groups with multiple 3-fold rotation axes, because all 7 include these axes; with regard to 3-or-more-fold rotation axes the possible combinations are:
**4 3-fold axes
**4 3-fold axes and 3 4-fold axes
**10 3-fold axes and 6 5-fold axes
A selection of point groups is compatible with discrete [[translational symmetry]]: 27 from the 7 infinite series, and 5 of the 7 others, the 32 so-called crystallographic point groups. See also the [[crystallographic restriction theorem]].

==The seven infinite series of axial groups==

The infinite series of axial or prismatic groups have an index ''n'', which can be any integer; in each series, the ''n''th symmetry group contains ''n''-fold [[rotational symmetry]] about an axis, i.e. symmetry with respect to a rotation by an angle 360°/''n''. ''n''=1 covers the cases of no rotational symmetry at all. There are four series with no other axes of rotational symmetry (see [[cyclic symmetries]]) and three with additional axes of 2-fold symmetry (see [[dihedral symmetry]]). They can be understood as [[point groups in two dimensions]] extended with an axial coordinate and reflections in it. They are related to the [[frieze group]]s;<ref>{{citation | first1=G.L. | last1=Fisher | first2=B. | last2=Mellor | title= Three-dimensional finite point groups and the symmetry of beaded beads | journal=[[Journal of Mathematics and the Arts]] | year=2007 | url=http://myweb.lmu.edu/bmellor/beadedbeads.pdf}}</ref> they can be interpreted as frieze-group patterns repeated ''n'' times around a cylinder.

The following table lists several notations for point groups: [[Hermann–Mauguin notation]], [[Arthur Moritz Schönflies|Schönflies]] notation, [[orbifold notation]], and [[Coxeter notation]]. The latter two are not only conveniently related to its properties, but also to the order of the group, see below. It is a unified notation, also applicable for [[wallpaper group]]s and [[frieze group]]s. The crystallographic groups have ''n'' restricted to 1, 2, 3, 4, and 6; removing crystallographic restriction allows any positive integer.

The series are:
{| class="wikitable"
|-
! colspan=2 | Hermann–Mauguin
! rowspan=2 | Schönflies
! rowspan=2 | Orbifold
! rowspan=2 | Coxeter
! rowspan=2 | Frieze
! rowspan=2 | Order
! rowspan=2 | Abstract group
! rowspan=2 | Comments
|-
! Even ''n'' || Odd ''n''
|-
| colspan=2 | ''n''
| C''n''
| ''nn''
| [n]+
| p1
| ''n''
| Z''n''
| ''n''-fold rotational symmetry
|-
| {{overline|2''n''}}
| {{overline|''n''}}
| S2''n''
| ''n''x
| [2n+,2+]
| p11g
| 2''n''
| Z2''n''
| Not to be confused with the [[symmetric group]]s
|-
| ''n''/m
| {{overline|2''n''}}
| C''n''h
| ''n''*
| [n+,2]
| p11m
| 2''n''
| Z''n'' × Z2
|
|-
| ''n''mm
| ''n''m
| C''n''v
| *''nn''
| [n]
| p1m1
| 2''n''
| Dih''n''
| Pyramidal symmetry; in biology, biradial symmetry
|-
| ''n''22
| ''n''2
| D''n''
| 22''n''
| [n,2]+
| p211
| 2''n''
| Dih''n''
| Dihedral symmetry
|-
| {{overline|2''n''}}2m
| {{overline|''n''}}m
| D''n''d, D''n''v
| 2*''n''
| [2n,2+]
| p2mg
| 4''n''
| Dih2''n''
| Antiprismatic symmetry
|-
| ''n''/mmm
| {{overline|2''n''}}2m
| D''n''h
| *22''n''
| [n,2]
| p2mm
| 4''n''
| Dih''n'' × Z2
| Prismatic symmetry
|}

For odd ''n'' we have Z2''n'' = Z''n'' × Z2 and Dih2''n'' = Dih''n'' × Z2.

The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal).

The simplest nontrivial ones have [[Involution (mathematics)|Involution]]al symmetry (abstract group Z2 ):
*''C''''i'' – '''[[inverse (mathematics)|inversion]] symmetry'''
*''C''2 – '''2-fold [[rotational symmetry]]'''
*''C''''s'' – '''[[reflection symmetry]]''', also called '''[[symmetry (biology)#Bilateral symmetry|bilateral symmetry]]'''.

[[Image:Uniaxial.png|right|thumb|200px|Patterns on a cylindrical band illustrating the case ''n'' = 6 for each of the 7 infinite families of point groups. The symmetry group of each pattern is the indicated group.]]
The second of these is the first of the uniaxial groups ([[cyclic group]]s) ''Cn'' of order ''n'' (also applicable in 2D), which are generated by a single rotation of angle 360°/''n''. In addition to this, one may add a mirror plane perpendicular to the axis, giving the group ''Cnh'' of order 2''n'', or a set of ''n'' mirror planes containing the axis, giving the group ''Cnv'', also of order 2''n''. The latter is the symmetry group for a regular ''n''-sided [[pyramid (geometry)|pyramid]]. A typical object with symmetry group ''C''''n'' or ''D''''n'' is a [[propeller]].

If both horizontal and vertical reflection planes are added, their intersections give ''n'' axes of rotation through 180°, so the group is no longer uniaxial. This new group of order 4''n'' is called ''Dnh''. Its subgroup of rotations is the [[dihedral group]] ''Dn'' of order 2''n'', which still has the 2-fold rotation axes perpendicular to the primary rotation axis, but no mirror planes. Note that in 2D ''Dn'' includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside, but in 3D the two operations are distinguished: the group contains "flipping over", not reflections.

There is one more group in this family, called ''Dnd'' (or ''Dnv''), which has vertical mirror planes containing the main rotation axis, but instead of having a horizontal mirror plane, it has an isometry that combines a reflection in the horizontal plane and a rotation by an angle 180°/''n''. ''Dnh'' is the symmetry group for a regular ''(n+2)''-sided [[Prism (geometry)|prisms]] and also for a regular (2n)-sided [[bipyramid]]. ''Dnd'' is the symmetry group for a regular ''(n+2)''-sided [[antiprism]], and also for a regular ''(2n)''-sided [[trapezohedron]]. ''Dn'' is the symmetry group of a partially rotated prism.

The groups ''D''2 and ''D''2''h'' are noteworthy in that there is no special rotation axis. Rather, there are three perpendicular 2-fold axes. ''D''2 is a subgroup of all the polyhedral symmetries (see below), and ''D''2''h'' is a subgroup of the polyhedral groups Th and Oh. ''D''2 can occur in [[homotetramer]]s such as [[Concanavalin A]], in tetrahedral [[coordination compound]]s with four identical [[chiral ligand]]s, or in a molecule such as tetrakis(chlorofluoromethyl)methane if all the chlorofluoromethyl groups have the same chirality. The elements of ''D''2 are in 1-to-2 correspondence with the rotations given by the [[Unit (ring theory)|unit]] [[Lipschitz quaternion]]s.

The group ''Sn'' is generated by the combination of a reflection in the horizontal plane and a rotation by an angle 360°/n. For ''n'' odd this is equal to the group generated by the two separately, ''Cnh'' of order 2''n'', and therefore the notation ''Sn'' is not needed; however, for ''n'' even it is distinct, and of order ''n''. Like ''Dnd'' it contains a number of [[improper rotation]]s without containing the corresponding rotations.

All symmetry groups in the 7 infinite series are different, except for the following four pairs of mutually equal ones:
*''C1h'' and ''C1v'': group of order 2 with a single reflection (''Cs'' )
*''D''1 and ''C''2: group of order 2 with a single 180° rotation
*''D''1''h'' and ''C''2''v'': group of order 4 with a reflection in a plane and a 180° rotation through a line in that plane
*''D''1''d'' and ''C''2''h'': group of order 4 with a reflection in a plane and a 180° rotation through a line perpendicular to that plane

''S''2 is the group of order 2 with a single inversion (''Ci'' )

"Equal" is meant here as the same up to conjugacy in space. This is stronger than "up to algebraic isomorphism". For example, there are three different groups of order two in the first sense, but there is only one in the second sense. Similarly, e.g. ''S2n'' is algebraically isomorphic with Z''2n''.

The groups may be constructed as follows:
* C''n''. Generated by an element also called C''n'', which corresponds to a rotation by angle 2π/''n'' around the axis. Its elements are E (the identity), C''n'', C''n''2, ..., C''n''''n''−1, corresponding to rotation angles 0, 2π/''n'', 4π/''n'', ..., 2(''n'' − 1)π/''n''.
* S2''n''. Generated by element C2''n''σh, where σh is a reflection in the direction of the axis. Its elements are the elements of C''n'' with C''2n''σh, C2''n''3σh, ..., C2''n''2''n''−1σh added.
* C''n''h. Generated by element C''n'' and reflection σh. Its elements are the elements of group C''n'', with elements σh, C''n''σh, C''n''2σh, ..., C''n''''n''−1σh added.
* C''n''v. Generated by element C''n'' and reflection σv in a direction in the plane perpendicular to the axis. Its elements are the elements of group C''n'', with elements σv, C''n''σv, C''n''2σv, ..., C''n''''n''−1σv added.
* D''n''. Generated by element C''n'' and 180° rotation U = σhσv around a direction in the plane perpendicular to the axis. Its elements are the elements of group C''n'', with elements U, C''n''U, C''n''2U, ..., C''n''''n'' − 1U added.
* D''n''d. Generated by elements C2''n''σh and σv. Its elements are the elements of group C''n'' and the additional elements of S2''n'' and C''n''v, with elements C2''n''σhσv, C2''n''3σhσv, ..., C2''n''2''n'' − 1σhσv added.
* D''n''h. Generated by elements C''n'', σh, and σv. Its elements are the elements of group C''n'' and the additional elements of C''n''h, C''n''v, and D''n''.

Taking ''n'' to ∞ yields groups with continuous axial rotations:

{| class="wikitable"
|-
! H–M
! Schönflies
! Orbifold
! Coxeter
! Limit of
! Abstract group
|-
| ∞
| C∞
| ∞∞
| [∞]+
| C''n''
| SO(2)
|-
| {{overbar|∞}}, ∞/m
| C∞h
| ∞*
| [2,∞+]
| C''n''h, S2''n''
| SO(2) × Z2
|-
| ∞m
| C∞v
| *∞∞
| [∞]
| C''n''v
| O(2)
|-
| ∞2
| D∞
| 22∞
| [2,∞]+
| D''n''
| O(2)
|-
| {{overbar|∞}}m, ∞/mm
| D∞h
| *22∞
| [2,∞]
| D''n''h, D''n''d
| O(2) × Z2
|}

==The seven remaining point groups==
The remaining point groups are said to be of very high or [[polyhedron|polyhedral]] symmetry because they have more than one rotation axis of order greater than 2. Here, Cn denotes an axis of rotation through 360°/n and Sn denotes an axis of improper rotation through the same. In parentheses are the [[orbifold notation]], [[Coxeter notation]], the full [[Hermann–Mauguin notation]], and the abbreviated one if different. The groups are:

*'''T''' (332, [3,3]+, 23) of order 12 – '''chiral [[tetrahedral symmetry]]'''. There are four C3 axes, each through two vertices of a [[cube]] (body diagonals) or one of a regular [[tetrahedron]], and three C2 axes, through the centers of the cube's faces, or the midpoints of the tetrahedron's edges. This group is [[isomorphic]] to ''A''4, the [[alternating group]] on 4 elements, and is the rotation group for a regular tetrahedron. It is a [[normal subgroup]] of Td, Th, and the octahedral symmetries. The elements of the group correspond 1-to-2 to the rotations given by the 24 [[Unit (ring theory)|unit]] [[Hurwitz quaternion]]s (the "[[binary tetrahedral group]]").

*'''Td''' (*332, [3,3] {{overline|4}}3m) of order 24 – '''full [[tetrahedral symmetry]]'''. This group has the same rotation axes as T, but with six mirror planes, each containing two edges of the cube or one edge of the tetrahedron, a single C2 axis and two C3 axes. The C2 axes are now actually S4 axes. This group is the symmetry group for a regular [[tetrahedron]]. Td is isomorphic to ''S''4, the [[symmetric group]] on 4 letters, because there is a 1-to-1 correspondence between the elements of Td and the 24 permutations of the four 3-fold axes. An object of ''C''3v symmetry under one of the 3-fold axes gives rise under the action of Td to an [[Group action#orbstab|orbit]] consisting of four such objects, and Td corresponds to the set of permutations of these four objects. Td is a normal subgroup of Oh. See also [[Tetrahedron#The isometries of the regular tetrahedron|the isometries of the regular tetrahedron]].

*'''Th''' (3*2, [3+,4], 2/m{{overline|3}}, m{{overline|3}}) of order 24 – '''[[tetrahedral symmetry|pyritohedral symmetry]]'''.[[Image:Volleyball seams diagram.png|thumb|The seams of a [[volleyball (ball)|volleyball]] have Th symmetry.]] This group has the same rotation axes as ''T'', with mirror planes parallel to the cube faces. The C3 axes become S6 axes, and there is inversion symmetry. Th is isomorphic to ''A''4 × ''C''2 (since T and Ci are both normal subgroups), and not to the [[symmetric group]] S4. It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a [[pyritohedron]], which is similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup (but not a normal subgroup) of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes. It is a normal subgroup of Oh.

*'''O''' (432, [4,3]+, 432) of order 24 – '''chiral [[octahedral symmetry]]'''. This group is like T, but the C2 axes are now C4 axes, and additionally there are 6 C2 axes, through the midpoints of the edges of the cube. This group is also isomorphic to ''S''4 because its elements are in 1-to-1 correspondence to the 24 permutations of the 3-fold axes, as with T. An object of ''D''3 symmetry under one of the 3-fold axes gives rise under the action of O to an [[Group action#orbstab|orbit]] consisting of four such objects, and O corresponds to the set of permutations of these four objects. It is the rotation group of the [[Cube (geometry)|cube]] and [[octahedron]]. Representing rotations with [[quaternion]]s, O is made up of the 24 [[Unit (ring theory)|unit]] [[Hurwitz quaternion]]s and the 24 [[Lipschitz quaternion]]s of squared norm 2 normalized by dividing by <math>\sqrt 2</math>. As before, this is a 1-to-2 correspondence.

*'''Oh''' (*432, [4,3], 4/m{{overline|3}}2/m, m{{overline|3}}m) of order 48 – '''full octahedral symmetry'''. This group has the same rotation axes as ''O'', but with mirror planes, comprising both the mirror planes of ''Td'' and ''Th''. This group is isomorphic to ''S''4 × ''C''2 (because both O and ''Ci'' are normal subgroups), and is the symmetry group of the [[Cube (geometry)|cube]] and [[octahedron]]. See also [[octahedral symmetry#The isometries of the cube|the isometries of the cube]].

*'''I''' (532, [5,3]+, 532) of order 60 – '''chiral [[icosahedral symmetry]]'''; the rotation group of the [[icosahedron]] and the [[dodecahedron]]. It is a [[normal subgroup]] of [[index of a subgroup|index]] 2 in the full group of symmetries '''Ih'''. The group contains 10 versions of ''D3'' and 6 versions of ''D5'' (rotational symmetries like prisms and antiprisms). It also contains five versions of Th (see [[Compound of five tetrahedra]]). The group '''I''' is [[isomorphic]] to ''A''5, the [[alternating group]] on 5 letters, since its elements correspond 1-to-1 with even permutations of the five Th symmetries (or the five tetrahedra just mentioned).

*'''Ih''' (*532, [5,3], {{overline|5}}{{overline|3}}2/m, {{overline|5}}{{overline|3}}m) of order 120 – '''full icosahedral symmetry'''; the symmetry group of the icosahedron and the dodecahedron. The group '''Ih''' is isomorphic to ''A''5 × ''C''2 because I and ''C''i are both normal subgroups. The group contains 10 versions of ''D3d'', 6 versions of ''D5d'' (symmetries like antiprisms), and 5 versions of Th.

The continuous groups related to these groups are:
* '''K''' or SO(3), all possible rotations.
* '''Kh''' or O(3), all possible rotations and reflections.
As noted above for infinite rotation groups, any physical object having K symmetry will also have Kh symmetry.

==Relation between orbifold notation and order==
The order of each group is 2 divided by the [[orbifold]] [[Euler characteristic]]; the latter is 2 minus the sum of the feature values, assigned as follows:
*''n'' without or before * counts as (''n''−1)/''n''
*''n'' after * counts as (''n''−1)/(2''n'')
* * and x count as 1
This can also be applied for [[wallpaper group]]s and [[frieze group]]s: for them, the sum of the feature values is 2, giving an infinite order; see [[2D crystallographic group#Why there are exactly seventeen groups|orbifold Euler characteristic for wallpaper groups]]

==Rotation groups==
The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups ''C''n (the rotation group of a regular [[pyramid (geometry)|pyramid]]), the dihedral groups ''D''n (the rotation group of a regular [[prism (geometry)|prism]], or regular [[bipyramid]]), and the rotation groups ''T'', ''O'' and ''I'' of a regular [[tetrahedron]], [[octahedron]]/[[cube]] and [[icosahedron]]/[[dodecahedron]].

In particular, the dihedral groups ''D''3, ''D''4 etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore it is also called a ''[[dihedron]]'' (Greek: solid with two faces), which explains the name ''dihedral group''.

*An object with symmetry group ''Cn'', ''Cnh'', ''Cnv'' or ''S2n'' has rotation group ''Cn.
*An object with symmetry group ''Dn'', ''Dnh'', or ''Dnd'' has rotation group ''Dn.
*An object with one of the other seven symmetry groups has as rotation group the corresponding one without subscript: ''T'', ''O'' or ''I''.

The rotation group of an object is equal to its full symmetry group [[if and only if]] the object is [[chirality (mathematics)|chiral]]. In other words, the chiral objects are those with their symmetry group in the list of rotation groups.

Given in [[Schönflies notation]], [[Coxeter notation]], ([[orbifold notation]]), the rotation subgroups are:
{| class=wikitable
!Reflectional
!Reflection/rotational
!Improper rotation
!Rotation
|- align=center
| ''Cnv'', [n], (*nn)
| ''Cnh'', [n+,2], (n*)
| ''S2n'', [2n+,2+], (nx)
| ''Cn'', [n]+, (nn)
|- align=center
| ''Dnh'', [2,n], (*n22)
| ''Dnd'', [2+,2n], (2*n)
|
| ''Dn'', [2,n]+, (n22)
|- align=center
| ''Td'', [3,3], (*332)
| ''Th'', [3+,4], (3*2)
|
| ''T'', [3,3]+, (332)
|- align=center
| ''Oh'', [4,3], (*432)
|
|
| ''O'', [4,3]+, (432)
|- align=center
| ''Ih'', [5,3], (*532)
|
|
| ''I'', [5,3]+, (532)
|-
|}

==Correspondence between rotation groups and other groups==
The following groups contain [[Inversion in a point|inversion]]:
*''C''''nh'' and ''D''''nh'' for even ''n''
*''S''2''n'' and ''D''''nd'' for odd ''n'' (''S2'' = ''Ci'' is the group generated by inversion; ''D''''1d'' = ''C''''2h'')
*''Th'', ''Oh'', and ''Ih''

As explained above, there is a 1-to-1 correspondence between these groups and all rotation groups:
*''C''''nh'' for even ''n'' and ''S''2''n'' for odd ''n'' correspond to ''C''''n''
*''D''''nh'' for even ''n'' and ''D''''nd'' for odd ''n'' correspond to ''D''''n''
*''Th'', ''Oh'', and ''Ih'' correspond to ''T'', ''O'', and ''I'', respectively.

The other groups contain indirect isometries, but not inversion:
*''C''''nv''
*''C''''nh'' and ''D''''nh'' for odd ''n''
*''S''2''n'' and ''D''''nd'' for even ''n''
*''Td''

They all correspond to a rotation group ''H'' and a subgroup ''L'' of index 2 in the sense that they are obtained from ''H'' by inverting the isometries in ''H'' \ ''L'', as explained above:
*''C''''n'' is subgroup of ''D''''n'' of index 2, giving ''C''''nv''
*''C''''n'' is subgroup of ''C''''2n'' of index 2, giving ''C''''nh'' for odd ''n'' and ''S''2''n'' for even ''n''
*''D''''n'' is subgroup of ''D''''2n'' of index 2, giving ''D''''nh'' for odd ''n'' and ''D''''nd'' for even ''n''
*T is subgroup of ''O'' of index 2, giving ''Td''

==Maximal symmetries==
There are two discrete point groups with the property that no discrete point group has it as proper subgroup: ''Oh'' and ''Ih''. Their largest common subgroup is ''Th''. The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively. Alternatively the two groups are generated by adding for each a reflection plane to ''Th''.

There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: ''Oh'' and ''D6h''. Their maximal common subgroups, depending on orientation, are ''D3d'' and ''D2h''.

==The groups arranged by abstract group type==
Below the groups explained above are arranged by abstract group type.

The smallest abstract groups that are ''not'' any symmetry group in 3D, are the [[quaternion group]] (of order 8), Z3 × Z3 (of order 9), the [[dicyclic group]] Dic3 (of order 12), and 10 of the 14 groups of order 16.

The column "# of order 2 elements" in the following tables shows the total number of isometry subgroups of types ''C2'', ''Ci'', ''Cs''. This total number is one of the characteristics helping to distinguish the various abstract group types, while their isometry type helps to distinguish the various isometry groups of the same abstract group.

Within the possibilities of isometry groups in 3D, there are infinitely many abstract group types with 0, 1 and 3 elements of order 2, there are two with 2''n'' + 1 elements of order 2, and there are three with 2''n'' + 3 elements of order 2 (for each ''n'' ≥ 2 ). There is never a positive even number of elements of order 2.

{{anchor|Cyclic 3D symmetry groups}}

===Symmetry groups in 3D that are cyclic as abstract group===
The [[symmetry group]] for ''n''-fold rotational [[symmetry]] is ''Cn''; its abstract group type is [[cyclic group]] Zn, which is also denoted by ''Cn''. However, there are two more infinite series of symmetry groups with this abstract group type:
*For even order 2''n'' there is the group [[Improper rotation|''S2n'']] (Schoenflies notation) generated by a rotation by an angle 180°/n about an axis, combined with a reflection in the plane perpendicular to the axis. For ''S2'' the notation ''Ci'' is used; it is generated by inversion.
*For any order 2''n'' where ''n'' is odd, we have ''Cnh''; it has an ''n''-fold rotation axis, and a perpendicular plane of reflection. It is generated by a rotation by an angle 360°/''n'' about the axis, combined with the reflection. For ''C''1''h'' the notation ''Cs'' is used; it is generated by reflection in a plane.

Thus we have, with bolding of the 10 cyclic crystallographic point groups, for which the [[crystallographic restriction theorem|crystallographic restriction]] applies:
{| class=wikitable
|-
!Order !!Isometry groups !! Abstract group !! # of order 2 elements
|- align=center
|1 || '''''C1''''' || Z1 || 0
|- align=center
|2 || '''''C2''''', '''''Ci''''', '''''Cs'''''|| Z2 || 1
|- align=center
|3 || '''''C3''''' || Z3 || 0
|- align=center
|4 || '''''C4''''', '''''S4''''' || Z4 || 1
|- align=center
|5 || ''C5'' || Z5 || 0
|- align=center
|6 || '''''C6''''', '''''S6''''', '''''C3h''''' || Z6 = Z3 × Z2 || 1
|- align=center
|7 || ''C7'' || Z7 || 0
|- align=center
|8 || ''C8'', ''S8'' || Z8 || 1
|- align=center
|9 || ''C9'' || Z9 || 0
|- align=center
|10 || ''C10'', ''S10'', ''C5h'' || Z10 = Z5 × Z2 || 1
|}
etc.

===Symmetry groups in 3D that are dihedral as abstract group===
In 2D [[dihedral group]] ''Dn'' includes reflections, which can also be viewed as flipping over flat objects without distinction of front- and backside.

However, in 3D the two operations are distinguished: the symmetry group denoted by ''Dn'' contains ''n'' 2-fold axes perpendicular to the ''n''-fold axis, not reflections. ''Dn'' is the [[rotation group SO(3)|rotation group]] of the ''n''-sided [[prism (geometry)|prism]] with regular base, and ''n''-sided [[bipyramid]] with regular base, and also of a regular, ''n''-sided [[antiprism]] and of a regular, ''n''-sided [[trapezohedron]]. The group is also the full symmetry group of such objects after making them [[chirality (mathematics)|chiral]] by e.g. an identical chiral marking on every face, or some modification in the shape.

The abstract group type is [[dihedral group]] Dih''n'', which is also denoted by ''Dn''. However, there are three more infinite series of symmetry groups with this abstract group type:

*''Cnv'' of order 2''n'', the symmetry group of a regular ''n''-sided [[Pyramid (geometry)|pyramid]]
*''Dnd'' of order 4''n'', the symmetry group of a regular ''n''-sided [[antiprism]]
*''Dnh'' of order 4''n'' for odd ''n''. For ''n'' = 1 we get ''D''2, already covered above, so ''n'' ≥ 3.

Note the following property:
:Dih''4n+2'' <math>\cong</math> Dih''2n+1'' × Z2

Thus we have, with bolding of the 12 crystallographic point groups, and writing ''D1d'' as the equivalent ''C2h'':
{| class=wikitable
|-
!Order !!Isometry groups !! Abstract group !! # of order 2 elements
|- align=center
|4 || '''''D2''''', '''''C2v''''', '''''C2h''''' || Dih2 = Z2 × Z2 || 3
|- align=center
|6 || '''''D3''''', '''''C3v''''' || Dih3 || 3
|- align=center
|8 || '''''D4''''', '''''C4v''''', '''''D2d''''' || Dih4 || 5
|- align=center
|10 || ''D''5, ''C''5''v'' || Dih5 || 5
|- align=center
|12 || '''''D6''''', '''''C6v''''', '''''D3d''''', '''''D3h''''' || Dih6 = Dih3 × Z2 || 7
|- align=center
|14 || ''D''7, ''C''7''v'' || Dih7 || 7
|- align=center
|16 || ''D''8, ''C''8''v'', ''D''4''d'' || Dih8 || 9
|- align=center
|18 || ''D''9, ''C''9''v'' || Dih9 || 9
|- align=center
|20 || ''D''10, ''C''10''v'', ''D''5''h'', ''D''5''d'' || Dih10 = ''D''5 × Z2 || 11
|}
etc.

===Other===
''C2n,h'' of order 4''n'' is of abstract group type Z2''n'' × Z2. For ''n'' = 1 we get Dih2, already covered above, so ''n'' ≥ 2.

Thus we have, with bolding of the 2 cyclic crystallographic point groups:
{| class=wikitable
|-
!Order !!Isometry group !! Abstract group !! # of order 2 elements !! [[Cycle diagram]]
|- align=center
|8 || '''''C4h''''' || Z4 × Z2 || 3 || [[Image:GroupDiagramMiniC2C4.png]]
|- align=center
|12 || '''''C6h''''' ||Z6 × Z2 = Z3 × Z2 × Z2 = Z3 × Dih2|| 3 || [[Image:GroupDiagramMiniC2C6.png]]
|- align=center
|16 || ''C8h'' || Z8 × Z2 || 3 || [[Image:GroupDiagramMiniC2C8.png]]
|- align=center
|20 || ''C10h'' || Z10 × Z2 = Z5 × Z2 × Z2 || 3 ||  
|}
etc.

''Dnh'' of order 4''n'' is of abstract group type Dihn × Z2. For odd ''n'' this is already covered above, so we have here ''D''2''n''h of order 8''n'', which is of abstract group type Dih2''n'' × Z2 (''n''≥1).

Thus we have, with bolding of the 3 dihedral crystallographic point groups:
{| class=wikitable
|-
!Order !!Isometry group !! Abstract group !! # of order 2 elements !! [[Cycle diagram]]
|- align=center
|8 || '''''D2h''''' || Dih2 × Z2 || 7 ||[[Image:GroupDiagramMiniC2x3.png]]
|- align=center
|16 || '''''D4h''''' || Dih4 × Z2 || 11 ||[[Image:GroupDiagramMiniC2D8.png]]
|- align=center
|24 || '''''D6h''''' || Dih6 × Z2|| 15 ||  
|- align=center
|32 || ''D8h'' || Dih8 × Z2 || 19 || 
|}
etc.

The remaining seven are, with bolding of the 5 crystallographic point groups (see also above):

*order 12: of type ''A''4 ([[alternating group]]): '''T'''
*order 24:
**of type ''S''4 ([[symmetric group]], not to be confused with the symmetry group with this notation): '''Td''', '''O'''
**of type ''A''4 × Z2: '''Th''' .
*order 48, of type ''S''4 × Z2: '''Oh'''
*order 60, of type ''A''5: ''I''
*order 120, of type ''A''5 × Z2: ''Ih''
See also [[icosahedral symmetry]].

==Impossible discrete symmetries==
Since the overview is exhaustive, it also shows implicitly what is ''not'' possible as discrete symmetry group. For example:
*a ''C''6 axis in one direction and a ''C''3 in another
*a ''C''5 axis in one direction and a ''C''4 in another
*a ''C''3 axis in one direction and another ''C''3 axis in a perpendicular direction
etc.

==Fundamental domain==
The [[fundamental domain]] of a point group is a [[conic solid]]. An object with a given symmetry in a given orientation is characterized by the fundamental domain. If the object is a surface it is characterized by a surface in the fundamental domain continuing to its radial bordal faces or surface. If the copies of the surface do not fit, radial faces or surfaces can be added. They fit anyway if the fundamental domain is bounded by reflection planes.

For a polyhedron this surface in the fundamental domain can be part of an arbitrary plane. For example, in the [[disdyakis triacontahedron]] one full face is a fundamental domain. Adjusting the orientation of the plane gives various possibilities of combining two or more adjacent faces to one, giving various other polyhedra with the same symmetry. The polyhedron is convex if the surface fits to its copies and the radial line perpendicular to the plane is in the fundamental domain.

Also the surface in the fundamental domain may be composed of multiple faces.

==Binary polyhedral groups==
The map Spin(3) → SO(3) is the double cover of the rotation group by the [[spin group]] in 3 dimensions. (This is the only connected cover of SO(3), since Spin(3) is simply connected.)
By the [[lattice theorem]], there is a [[Galois connection]] between subgroups of Spin(3) and subgroups of SO(3) (rotational point groups): the image of a subgroup of Spin(3) is a rotational point group, and the preimage of a point group is a subgroup of Spin(3).

The preimage of a finite point group is called a '''binary polyhedral group''', represented as <l,n,m>, and is called by the same name as its point group, with the prefix '''binary''', with double the order of the related [[polyhedral group]] (l,m,n). For instance, the preimage of the [[icosahedral group]] (2,3,5) is the [[binary icosahedral group]], <2,3,5>.

The binary polyhedral groups are:
* <math>A_n</math>: [[binary cyclic group]] of an (''n'' + 1)-gon
* <math>D_n</math>: [[binary dihedral group]] of an ''n''-gon, <2,2,n>
* <math>E_6</math>: [[binary tetrahedral group]], <2,3,3>
* <math>E_7</math>: [[binary octahedral group]], <2,3,4>
* <math>E_8</math>: [[binary icosahedral group]], <2,3,5>
These are classified by the [[ADE classification]], and the quotient of '''C'''2 by the action of a binary polyhedral group is a [[Du Val singularity]].<ref>[http://enriques.mathematik.uni-mainz.de/burban/singul.pdf Du Val Singularities, by Igor Burban]</ref>

For point groups that reverse orientation, the situation is more complicated, as there are two [[pin group]]s, so there are two possible binary groups corresponding to a given point group.

Note that this is a covering of ''groups,'' not a covering of ''spaces'' – the sphere is [[simply connected]], and thus has no [[covering space]]s. There is thus no notion of a "binary polyhedron" that covers a 3-dimensional polyhedron. Binary polyhedral groups are discrete subgroups of a Spin group, and under a representation of the spin group act on a vector space, and may stabilize a polyhedron in this representation – under the map Spin(3) → SO(3) they act on the same polyhedron that the underlying (non-binary) group acts on, while under [[spin representation]]s or other representations they may stabilize other polyhedra.

This is in contrast to [[projective polyhedra]] – the sphere does cover [[projective space]] (and also [[lens space]]s), and thus a tessellation of projective space or lens space yields a distinct notion of polyhedron.

==See also==
<div style="-moz-column-count:2; column-count:2;">
*[[List of spherical symmetry groups]]
*[[List of character tables for chemically important 3D point groups]]
*[[Point groups in two dimensions]]
*[[Symmetry]]
*[[Euclidean plane isometry]]
*[[Group action]]
*[[Point group]]
*[[Crystal system]]
*[[Space group]]
*[[List of small groups]]
*[[Molecular symmetry]]
</div>

==Footnotes==
{{reflist}}

==References==
{{Refimprove|date=May 2010}}
{{refbegin}}
* {{Citation | authorlink = Harold Scott MacDonald Coxeter | last = Coxeter | first = H. S. M. | title = Regular Complex Polytopes | publisher = Cambridge University Press | year = 1974 | chapter = 7 The Binary Polyhedral Groups | pages = [http://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA73 73–82] }}.
*{{cite book | author=Coxeter, H. S. M. and Moser, W. O. J. | title=Generators and Relations for Discrete Groups, 4th edition | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9}} 6.5 The binary polyhedral groups, p. 68
*{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Huson | first2=Daniel H. | title= The Orbifold Notation for Two-Dimensional Groups | publisher=Springer Netherlands | doi=10.1023/A:1015851621002 | year=2002 | journal=Structural Chemistry | volume=13 | issue=3 | pages=247–257}}
{{refend}}

==External links==
*[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Graphic overview of the 32 crystallographic point groups] – form the first parts (apart from skipping ''n''=5) of the 7 infinite series and 5 of the 7 separate 3D point groups
*[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/CC_All.html Overview of properties of point groups]
*[http://homepage.mac.com/dmccooey/polyhedra/Simplest.html Simplest Canonical Polyhedra of Each Symmetry Type] (uses Java)
* [http://www.stanford.edu/~yishuwei/crystal.pdf] Point Groups and Crystal Systems, by Yi-Shu Wei, pp. 4–6
* [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node45.html The Geometry Center: 10.1 Formulas for Symmetries in Cartesian Coordinates (three dimensions)]

{{DEFAULTSORT:Point Groups In Three Dimensions}}
[[Category:Euclidean symmetries]]
[[Category:Group theory]]

Additive identity

2013-11-24T22:55:24Z

178.16.9.65: /* The additive identity annihilates ring elements */

{{About|bivariate processes|arrival processes to queues|Markovian arrival process}}
In [[applied probability]], a '''Markov additive process''' ('''MAP''') is a bivariate [[Markov process]] where the future states depends only on one of the variables.<ref name="magiera" />

==Definition==

===Finite or countable state space for ''J''(''t'')===

The process {(''X''(''t''),''J''(''t'')) : ''t'' ≥ 0} is a Markov additive process with continuous time parameter ''t'' if<ref name="magiera">{{cite doi|10.1007/978-1-4612-2234-7_12}}</ref>

# {(''X''(''t''),''J''(''t'')) : ''t'' ≥ 0} is a [[Markov process]]
# the conditional distribution of (''X''(''t'' + ''s'') − ''X''(''t''),''J''(''s'' + ''t'')) given (''X''(''s''),''J''(''s'')) depends only on ''J''(''s'').

The state space of the process is '''R''' × ''S'' where ''X''(''t'') takes real values and ''J''(''t'') takes values in some countable set ''S''.

===General state space for ''J''(''t'')===

For the case where ''J''(''t'') takes a more general state space the evolution of ''X''(''t'') is governed by ''J''(''t'') in the sense that for any ''f'' and ''g'' we require<ref>{{cite doi|10.1007/0-387-21525-5_11}}</ref>

::<math>\mathbb E[f(X_{t+s}-X_t)g(J_{t+s})|\mathcal F_t] = \mathbb E_{J_t,0}[f(X_s)g(J_s)]</math>.

==Example==

A [[fluid queue]] is a Markov additive process where ''J''(''t'') is a [[continuous-time Markov chain]].

==Applications==

Çinlar uses the unique structure of the MAP to prove that, given a [[gamma process]] with a shape parameter that is a function of [[Brownian motion]], the resulting lifetime is distributed according to the [[Weibull distribution]].

Kharoufeh presents a compact transform expression for the failure distribution for wear processes of a component degrading according to a Markovian environment inducing state-dependent continuous linear wear by using the properties of a MAP and assuming the wear process to be temporally homogeneous and that the environmental process has a finite [[state space]].

==Notes==
{{Reflist}}

{{Stochastic processes}}
{{probability-stub}}

[[Category:Stochastic processes]]

Circle packing in a square

2012-06-22T15:04:56Z

178.16.148.1:

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