formulasearchengine - User contributions [en]

Mean value theorem

2014-12-12T11:45:25Z

130.60.188.209: /* Second mean value theorem for integration */ add a new reference and remove claim unverified since 2007

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Convex conjugate

2014-12-09T13:52:01Z

130.60.188.209: /* Table of selected convex conjugates */ \star -> *

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Monotone class theorem

2014-08-06T13:24:41Z

130.60.188.148: /* Monotone class theorem for functions */

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2014-05-15T16:02:58Z

130.60.6.54: minor edit for style

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Hahn decomposition theorem

2014-01-21T13:12:51Z

130.60.188.209: /* External links */

'''Floquet theory''' is a branch of the theory of [[ordinary differential equations]] relating to the class of solutions to [[linear differential equation]]s of the form

:<math>\dot{x} = A(t) x,\,</math>

with <math>\displaystyle A(t)</math> a [[Piecewise#Continuity|piecewise continuous]] periodic function with period <math>T</math>.

The main theorem of Floquet theory, '''Floquet's theorem''', due to {{harvs|txt|authorlink=Gaston Floquet|first=Gaston |last=Floquet|year=1883}}, gives a [[canonical form]] for each [[Fundamental solution|fundamental matrix solution]] of this common [[linear system]]. It gives a [[Change of coordinates|coordinate change]] <math>\displaystyle y=Q^{-1}(t)x</math> with <math>\displaystyle Q(t+2T)=Q(t)</math> that transforms the periodic system to a traditional linear system with constant, real [[coefficients]].

In [[solid-state physics]], the analogous result (generalized to three dimensions) is known as [[Bloch wave|Bloch's theorem]].

Note that the solutions of the linear differential equation form a vector space. A matrix <math>\phi\,(t)</math> is called a fundamental matrix solution if all columns are linearly independent solutions. A matrix <math>\Phi(t)</math> is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists <math>t_0</math> such that <math>\Phi(t_0)</math> is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using <math>\Phi(t)=\phi\,(t){\phi\,}^{-1}(t_0)</math>. The solution of the linear differential equation with the initial condition <math>x(0)=x_0</math> is <math>x(t)=\phi\,(t){\phi\,}^{-1}(0)x_0</math> where <math>\phi \,(t)</math> is any fundamental matrix solution.

== Floquet's theorem == 
Let <math>\dot{x}= A(t) x</math> be a linear first order differential equation,
where <math>x(t)</math> is a column vector of length <math>n</math> and <math>A(t)</math> an <math>n \times n</math> periodic matrix with period <math>T</math> (that is <math>A(t + T) = A(t)</math> for all real values of <math>t</math>). Let <math>\phi\, (t) </math> be a fundamental matrix solution of this differential equation. Then, for all <math>t \in \mathbb{R}</math>,

:<math> \phi(t+T)=\phi(t) \phi^{-1}(0) \phi (T).\ </math>

Here

:<math>\phi^{-1}(0) \phi (T)\ </math>

is known as the [[monodromy matrix]].
In addition, for each matrix <math>B</math> (possibly complex) such that

:<math>e^{TB}=\phi^{-1}(0) \phi (T),\ </math>

there is a periodic (period <math>T</math>) matrix function <math>t \mapsto P(t)</math> such that

:<math>\phi (t) = P(t)e^{tB}\text{ for all }t \in \mathbb{R}.\ </math>

Also, there is a ''real'' matrix <math>R</math> and a ''real'' periodic (period-<math>2T</math>) matrix function <math>t \mapsto Q(t)</math> such that

:<math>\phi (t) = Q(t)e^{tR}\text{ for all }t \in \mathbb{R}.\ </math>

In the above <math>B</math>, <math>P</math>, <math>Q</math> and <math>R</math> are <math>n \times n</math> matrices.

== Consequences and applications ==
This mapping <math>\phi \,(t) = Q(t)e^{tR}</math> gives rise to a time-dependent change of coordinates (<math>y = Q^{-1}(t) x</math>), under which our original system becomes a linear system with real constant coefficients <math>\dot{y} = R y</math>. Since <math>Q(t)</math> is continuous and periodic it must be bounded. Thus the stability of the zero solution for <math>y(t)</math> and <math>x(t)</math> is determined by the eigenvalues of <math>R</math>.

The representation <math>\phi \, (t) = P(t)e^{tB}</math> is called a ''Floquet normal form'' for the fundamental matrix <math>\phi \, (t)</math>.

The [[eigenvalue]]s of <math>e^{TB}</math> are called the [[characteristic multiplier]]s of the system. They are also the eigenvalues of the (linear) Poincaré maps <math>x(t) \to x(t+T)</math>. A '''Floquet exponent''' (sometimes called a characteristic exponent), is a complex <math>\mu</math> such that <math>e^{\mu T}</math> is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since <math>e^{(\mu + \frac{2 \pi i k}{T})T}=e^{\mu T}</math>, where <math>k</math> is an integer. The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, [[Lyapunov stability|Lyapunov stable]] if the Lyapunov exponents are nonpositive and unstable otherwise.

* Floquet theory is very important for the study of [[dynamical systems]].
* Floquet theory shows stability in [[Hill differential equation]] (introduced by [[George William Hill]]) approximating the motion of the [[moon]] as a [[harmonic oscillator]] in a periodic [[gravitational field]].
* [[Bond softening]] and [[bond hardening]] in intense laser fields can be described in terms of solutions obtained from the Floquet theorem.

== Floquet's theorem applied to Mathieu equation==

Mathieu's equation is related to the wave equation for the elliptic cylinder.

Given <math>a \in \mathbb{R}, q \in \mathbb{C}</math>, the [[Mathieu equation]] is given by

: <math>\frac {d^2 y} {dw^2} +(a-2q \cos 2w )y=0.</math>

The Mathieu equation is a linear second-order differential equation with periodic coefficients.

One of the most powerful results of Mathieu's functions is the Floquet's Theorem [1, 2].
It states that solutions of Mathieu equation for any pair (''a'', ''q'') can be expressed in the form

: <math>y(w)=F_{\nu}(w)=e^{iw \nu} P(w) \,</math>

or

: <math>y(w)=F_{\nu}(-w)=e^{-iw \nu} P(-w) \,</math>

where <math> \nu</math> is a constant depending on ''a'' and ''q'' and ''P''(.) is <math> \pi </math>-periodic in ''w''.

The constant <math> \nu</math> is called the ''characteristic exponent''.

If <math> \nu</math> is an integer, then <math>F_{\nu}(w)</math> and <math>F_{\nu}(-w)</math> are linear dependent solutions. Furthermore,

: <math>y(w+k \pi) =e^{i \nu k \pi}y(w)\text{ or }y(w+k \pi) =e^{-i \nu k \pi}y(w), \,</math>

for the solution <math>F_{\nu}(w)</math> or <math>F_{\nu}(-w)</math>, respectively.

We assume that the pair (''a'', ''q'') is such that <math>| \cosh (i \nu \pi) | <1</math> so that the solution <math> y(w)</math> is bounded on the real axis. General solution of Mathieu's equation (<math>q \in \mathbb{R}</math>, <math> \nu</math> non-integer) is the form

: <math>y(w) =c_1 e^{i w \nu}P(w)+ c_2e^{-i w \nu}P(-w), \,</math>

where <math>c_1</math> and <math>c_2</math> are arbitrary constants.

All bounded solutions −those of fractional as well as integral order− are described by an infinite series of [[harmonic oscillation]]s whose amplitudes decrease with increasing frequency.

Another very important property of Mathieu's functions is the orthogonality [3]:

If <math>a( \nu +2p,q)</math> and <math>a( \nu +2s,q)</math> are simple roots of

: <math> \cos(\pi\nu) - y(\pi = 0) = 0, \, </math>

then:

: <math>\int_0^\pi F_{\nu+2p} (w) F_{\nu+2s}(-w) \, dw = 0,\qquad p \ne s,</math>

i.e.,

: <math>\langle F_{\nu +2p} (w),F_{\nu +2s} (w)\rangle = 0, \qquad p \ne s,</math>

where <·,·> denotes an [[inner product]] defined from 0 to ''π''.

== References ==

*C. Chicone. ''Ordinary Differential Equations with Applications.'' Springer-Verlag, New York 1999.
* {{cite book|last=Ekeland|first=Ivar|authorlink=Ivar Ekeland|chapter=One|title=Convexity methods in Hamiltonian mechanics|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]|volume=19|publisher=Springer-Verlag|location=Berlin|year=1990|pages=x+247|isbn=3-540-50613-6|mr=1051888|ref=harv}}
* {{citation|first=Gaston|last= Floquet|title=Sur les équations différentielles linéaires à coefficients périodiques|journal=Annales de l'École Normale Supérieure|volume=12|pages= 47–88 |year=1883|url= http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1883_2_12_/ASENS_1883_2_12__47_0/ASENS_1883_2_12__47_0.pdf}}
* {{Citation
| surname = Krasnosel'skii
| given = M.A.
|authorlink=Mark Krasnosel'skii | title = The Operator of Translation along the Trajectories of Differential Equations
| publisher=[[American Mathematical Society]]
| place = [[Providence, Rhode Island|Providence]]
| year=1968}}, Translation of Mathematical Monographs, 19, 294p.
*W. Magnus, S. Winkler. ''Hill's Equation'', Dover-Phoenix Editions, ISBN 0-486-49565-5.
*N.W. McLachlan, ''Theory and Application of Mathieu Functions'', New York: Dover, 1964.
*{{cite book
| surname = Teschl
| given = Gerald
|authorlink=Gerald Teschl
| title = Ordinary Differential Equations and Dynamical Systems
| publisher=[[American Mathematical Society]]
| place = [[Providence, Rhode Island|Providence]]
| year = 2012
| isbn= 978-0-8218-8328-0
| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
*M.S.P. Eastham, "The Spectral Theory of Periodic Differential Equations", Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973. ISBN 978-0-7011-1936-2.

==External links==
* {{springer|title=Floquet theory|id=p/f040640}}

[[Category:Dynamical systems|*]]
[[Category:Differential equations|*]]

Jacobi operator

2014-01-21T13:09:25Z

130.60.188.209: /* References */

When an [[igneous]] rock cools, it acquires a '''thermoremanent magnetization (TRM)''' from the Earth's field. TRM can be much larger than it would be if exposed to the same field at room temperature (see [[remanence#Isothermal remanence|isothermal remanence]]). This remanence can also be very stable, lasting without significant change for millions of years. TRM is the main reason that [[paleomagnetists]] are able to deduce the direction and magnitude of the ancient Earth's field.<ref>{{harvnb|Stacey|Banerjee|1974}}</ref>

== History ==
As early as the eleventh century, the Chinese were aware that a piece of [[iron]] could be magnetized by heating until it was red hot and then [[quenched]] in water. While quenching it was oriented in the Earth's field to get the desired polarity. In 1600, [[William Gilbert (astronomer)|William Gilbert]] published ''[[De Magnete]]'' (1600), a report of a series of meticulous experiments in magnetism. In it, he described the quenching of a steel rod in the direction of the Earth's field, and he may have been aware of the Chinese work.<ref name=Temple>{{harvnb|Temple|2006|pp=169–171}}</ref>

In the early 20th century, a few investigators found that [[igneous rocks]] had a [[remanence]] that was much more intense than remanence acquired in the Earth's field without heating; that heating rocks in the Earth's magnetic field could magnetize them in the direction of the field; and that the Earth's field had reversed its direction in the past.<ref>{{harvnb|Glen|1982}}</ref>

== TRM in paleomagnetism ==
=== Demagnetization of TRM ===
It has long been known that a TRM can be removed if it is heated above the [[Curie temperature]] <math>\scriptstyle T_\text{C}</math> of the minerals carrying it. A TRM can also be partially demagnetized by heating up to some lower temperature <math>\scriptstyle T_1</math> and cooling back to room temperature. A common procedure in [[paleomagnetism]] is ''stepwise demagnetization'', in which the sample is heated to a series of temperatures <math>\scriptstyle T_1, T_2, \ldots</math>, cooling to room temperature and measuring the remaining remanence in between each heating step. The series of remanences can be plotted in a variety of ways, depending on the application.

=== Partial TRM ===
If a rock is later re-heated (as a result of burial, for example), part or all of the TRM can be replaced by a new remanence. If it is only part of the remanence, it is known as ''partial thermoremanent magnetization (pTRM)''. Because numerous experiments have been done modeling different ways of acquiring remanence, pTRM can have other meanings. For example, it can also be acquired in the laboratory by cooling in zero field to a temperature <math>\scriptstyle T_1</math> (below the [[Curie temperature]]), applying a magnetic field and cooling to a temperature <math>\scriptstyle T_2</math>, then cooling the rest of the way to room temperature in zero field.

== Ideal TRM behavior ==
=== The Thellier laws ===
The ideal TRM is one that can record the magnetic field in such a way that both its direction and intensity can be measured by some process in the lab. Thellier<ref>{{harvnb|Thellier|1938}}</ref> showed that this could be done if pTRM's satisfied four laws. Suppose that A and B are two non-overlapping temperature intervals. Suppose that <math>\scriptstyle M_\text{A}</math> is a pTRM that is acquired by cooling the sample to room temperature, only switching the field <math>\scriptstyle H</math> on while the temperature is in interval A; <math>\scriptstyle M_\text{B}</math> has a similar definition. The ''Thellier laws'' are
*''Linearity'': <math>\scriptstyle M_\text{A}(H)</math> and <math>\scriptstyle M_\text{B}(H)</math> are proportional to <math>\scriptstyle H</math> when <math>\scriptstyle H</math> is not much larger than the present Earth's field.
*''Reciprocity'': <math>\scriptstyle M_\text{A}</math> can be removed by heating through temperature interval <math>\scriptstyle A</math>, and <math>\scriptstyle M_\text{B}</math> through <math>\scriptstyle B</math>.
*''Independence'': <math>\scriptstyle M_\text{A}</math> and <math>\scriptstyle M_\text{B}</math> are independent.
*''Additivity'': If <math>\scriptstyle M_{\text{A}\cup\text{B}}</math> is acquired by turning the field on in both temperature intervals, <math>\scriptstyle M_{\text{A}\cup \text{B}}=M_\text{A}+M_\text{B}</math>.
If these laws hold for any non-overlapping temperature intervals <math>\scriptstyle A</math> and <math>\scriptstyle B</math>, the sample satisfies the Thellier laws.<ref name=Dunlop>{{harvnb|Dunlop|Özdemir|1997}}</ref>

=== A simple model for the Thellier laws ===
Suppose that a sample has a lot of magnetic minerals, each of which has the following property: It is [[superparamagnetic]] until the temperature reaches a ''blocking temperature'' <math>\scriptstyle T_\text{B}</math> that is independent of magnetic field for small fields. No irreversible changes occur at temperatures below <math>\scriptstyle T_\text{B}</math>. If the resulting TRM is heated in zero field, it becomes superparamagnetic again at an ''unblocking temperature'' <math>\scriptstyle T_\text{UB}</math> that is equal to <math>\scriptstyle T_\text{B}</math>. Then it is easy to verify that reciprocity, independence and additivity hold. It only remains for linearity to be satisfied for all the Thellier laws to be obeyed.

=== The Néel model for single-domain TRM ===
[[Louis Néel]] developed a physical model that showed how real magnetic minerals could have the above properties. It applies to particles that are [[single-domain (magnetism)|single-domain]], having a uniform magnetization that can only rotate as a unit.<ref name=Neel>{{harvnb|Néel|1955}}</ref>

== See also ==
*[[Rock magnetism]]

==Notes==
{{Reflist|3}}

==References==
{{Refbegin}}
*{{cite book
|last1 = Dunlop
|first1 = David J.
|last2 = Özdemir
|first2 = Özden
|title = Rock Magnetism: Fundamentals and Frontiers
|publisher = [[Cambridge Univ. Press]]
|year = 1997
|isbn = 0-521-32514-5
}}
*{{cite book
|last = Glen
|first = William
|title = The Road to Jaramillo: Critical Years of the Revolution in Earth Science
|publisher = [[Stanford University Press]]
|year = 1982
|isbn = 0-8047-1119-4
}}
*{{cite journal
|last = Néel
|first = Louis
|author-link = Louis Néel
|title = Some theoretical aspects of rock magnetism
|journal = [[Advances in Physics]]
|volume = 4
|pages = 191–243
|year = 1955
|bibcode = 1955AdPhy...4..191N |doi = 10.1080/00018735500101204 }}
*{{Cite book
|last1 = Stacey
|first1 = Frank D.
|last2 = Banerjee
|first2 = Subir K.
|title = The Physical Principles of Rock Magnetism
|publisher = [[Elsevier]]
|year = 1974
|isbn = 0-444-41084-8
}}
*{{cite book
|last = Temple
|first = Robert
|title = The Genius of China
|publisher = [[Andre Deutsch]]
|year = 2006
|isbn = 0-671-62028-2
}}
{{Refend}}

[[Category:Rock magnetism]]
[[Category:Magnetic ordering]]

Integration by substitution

2014-01-21T13:05:29Z

130.60.188.209: /* References */

{{about|calculating the area of a triangle|calculating a square root|Heron's method}}
[[Image:Triangle with notations 2.svg|thumb|198px|A triangle with sides ''a'', ''b'', and ''c''.]]
In [[geometry]], '''Heron's (or Hero's) formula''', named after [[Hero of Alexandria]],<ref>{{cite web|title=Fórmula de Herón para calcular el área de cualquier triángulo|url=http://recursostic.educacion.es/descartes/web/materiales_didacticos/formula_heron/formula_de_Heron.htm|language=Spanish|accessdate=30 June 2012}}</ref> states that the [[area]] ''T'' of a [[triangle]] whose sides have lengths ''a'', ''b'', and ''c'' is

:<math>T = \sqrt{s(s-a)(s-b)(s-c)}</math>

where ''s'' is the [[semiperimeter]] of the triangle:

:<math>s=\frac{a+b+c}{2}.</math>

Heron's formula can also be written as:

:<math>T=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}</math>
:<math>T=\frac{1}{4}\sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}</math>
:<math>T=\frac{1}{4}\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}</math>
:<math>T=\frac{1}{4}\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}</math>

Heron's formula is distinguished from other formulas for the area of a triangle, such as half the base times the height or half the modulus of a cross product of two sides, by requiring no arbitrary choice of side as base or vertex as origin.

==Example==

Let ΔABC be the triangle with sides ''a''=7, ''b''=4 and ''c''=5.
The semiperimeter is   <math>s=\tfrac{1}{2}(a+b+c)=\tfrac{1}{2}(7+4+5)=8</math> , and the area is

:  <math>T = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}= \sqrt{8 \cdot (8-7) \cdot (8-4) \cdot (8-5)}</math>
::<math>=\sqrt{8 \cdot 1 \cdot 4 \cdot 3}=\sqrt{96}=4\sqrt{6} \approx 9.8</math></div>

== History ==

The formula is credited to [[Hero of Alexandria|Heron (or Hero) of Alexandria]], and a proof can be found in his book, ''Metrica'', written ''c.'' A.D. 60. It has been suggested that [[Archimedes]] knew the formula over two centuries earlier, and since ''Metrica'' is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.<ref>{{MathWorld |urlname=HeronsFormula |title=Heron's Formula}}</ref>

A formula equivalent to Heron's namely:

:<math>T=\frac1{2}\sqrt{a^2c^2-\left(\frac{a^2+c^2-b^2}{2}\right)^2}</math>, where <math>a \ge b \ge c</math>
was discovered by the Chinese independently of the Greeks. It was published in ''Shushu Jiuzhang'' (“[[Mathematical Treatise in Nine Sections]]”), written by [[Qin Jiushao]] and published in A.D. 1247.

== Proof ==

A modern proof, which uses [[algebra]] and is quite unlike the one provided by Heron (in his book Metrica), follows. Let ''a'', ''b'', ''c'' be the sides of the triangle and ''A'', ''B'', ''C'' the [[angle]]s opposite those sides. We have

:<math>\cos \widehat C = \frac{a^2+b^2-c^2}{2ab}</math>

by the [[law of cosines]]. From this proof get the algebraic statement:

:<math>\sin \widehat C = \sqrt{1-\cos^2 \widehat C} = \frac{\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}.</math>

The [[altitude (triangle)|altitude]] of the triangle on base ''a'' has length ''b''·sin(''C''), and it follows

: <math>
\begin{align}
T & = \frac{1}{2} (\mbox{base}) (\mbox{altitude}) \\
& = \frac{1}{2} ab\sin \widehat C \\
& = \frac{1}{4}\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2} \\
& = \frac{1}{4}\sqrt{(2a b -(a^2 +b^2 -c^2))(2a b +(a^2 +b^2 -c^2))} \\
& = \frac{1}{4}\sqrt{(c^2 -(a -b)^2)((a +b)^2 -c^2)} \\
& = \sqrt{\frac{(c -(a -b))(c +(a -b))((a +b) -c)((a +b) +c)}{16}} \\
& = \sqrt{\frac{(b + c - a)}{2}\frac{(a + c - b)}{2}\frac{(a + b - c)}{2}\frac{(a + b + c)}{2}} \\
& = \sqrt{\frac{(a + b + c)}{2}\frac{(b + c - a)}{2}\frac{(a + c - b)}{2}\frac{(a + b - c)}{2}} \\
& = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}.
\end{align}
</math>

The [[difference of two squares]] factorization was used in two different steps.

==Proof using the Pythagorean theorem==
[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude ''h'' cutting base ''c'' into ''d'' + (''c'' − ''d'').]]Heron's original proof made use of [[cyclic quadrilateral]]s, while other arguments appeal to [[trigonometry]] as above, or to the [[incenter]] and one [[excircle]] of the triangle [http://www.math.dartmouth.edu/~doyle/docs/heron/heron.txt]. The following argument reduces Heron's formula directly to the [[Pythagorean theorem]] using only elementary means.

We wish to prove <math>4T^2=4s(s-a)(s-b)(s-c).</math> The left-hand side equals
:<math>4 T^2 = (c h)^2 = c^2(b^2-d^2) = (c b)^2 - (c d)^2</math>
while the right-hand side equals
:<math>4s(s-a)(s-b)(s-c) = [s(s-a)+(s-b)(s-c)]^2 - [s(s-a)-(s-b)(s-c)]^2</math>
via the identity <math>(p+q)^2-(p-q)^2=4pq.</math> It therefore suffices to show
:<math>cb=s(s-a)+(s-b)(s-c)</math>
and
:<math>cd=s(s-a)-(s-b)(s-c).</math>

Substituting <math>2s=(a+b+c)</math> into the former,
:<math>s(s-a)+(s-b)(s-c)=\frac{1}{4}(a+b+c)(-a+b+c) + \frac{1}{4}(a-b+c)(a+b-c) = \frac{1}{4}[(b+c)^2-a^2] + \frac{1}{4}[a^2-(b-c)^2] = \frac{1}{4}[(b+c)^2 - (b-c)^2] = cb</math>
as desired. Similarly, the latter expression becomes
:<math>s(s-a)-(s-b)(s-c)=\frac{1}{4}[(b+c)^2-a^2] - \frac{1}{4}[a^2-(b-c)^2] = \frac{1}{2}(b^2+c^2-a^2).</math>
Using the Pythagorean theorem twice, <math>b^2=d^2+h^2</math> and <math>a^2=(c-d)^2+h^2,</math> allows us to simplify the expression to
:<math>\frac{1}{2}(b^2+c^2-a^2) = \frac{1}{2}[d^2+c^2-(c-d)^2] = cd.</math>
The result follows.

==Proof using the [[Law of cotangents]] and the [[Proofs of trigonometric identities#Miscellaneous -- the triple cotangent identity|triple cotangent identity]]==
[[File:Herontriangle2.svg|thumb|270px|right||Geometrical significance of ''s-a'', ''s-b'', and ''s-c''. See the [[Law of cotangents]] for the reasoning behind this.]]
From the first part of the [[Law of cotangents]] proof,<ref>The second part of the Law of cotangents proof depends on Heron's formula itself, but this article depends only on the first part.</ref> we have that the triangle's area is both
:<math>
\begin{align}
T &= r\big((s-a) + (s-b) + (s-c)\big) = r^2\left(\frac{s-a}{r} + \frac{s-b}{r} + \frac{s-c}{r}\right) \\[8pt]
&= r^2\big(\cot(A/2) + \cot(B/2) + \cot(C/2)\big) \\[8pt]
\end{align}
</math>
and
:<math>T = rs</math>
 
but, since the sum of the half-angles is <math>\tfrac{\pi}{2}</math>, the [[Proofs of trigonometric identities#Miscellaneous -- the triple cotangent identity|triple cotangent identity]] applies, so the first of these is
:<math>
\begin{align}
T &= r^2\left(\cot(A/2)\ \cot(B/2)\ \cot(C/2)\right) = r^2\ \frac{s-a}{r}\ \frac{s-b}{r}\ \frac{s-c}{r} \\[8pt]
&= \frac{(s-a) (s-b) (s-c)}{r} \\[8pt]
\end{align}
</math>
 
Combining the two, we get
:<math>T^2 = s (s-a) (s-b) (s-c)</math>
from which the result follows.

== Numerical stability ==
Heron's formula as given above is [[Numerical stability|numerically unstable]] for triangles with a very small angle. A stable alternative <ref>{{cite book|title=Floating-Point Computation, Prentice-Hall|author=P. Sterbenz|year=1973}}</ref>
<ref>{{cite web|url=http://www.cs.berkeley.edu/~wkahan/Triangle.pdf|title=Miscalculating Area and Angles of a Needle-like Triangle|author=W. Kahan|date=24 March 2000}}</ref> involves arranging the lengths of the sides so that <math>a \ge b \ge c</math> and computing
:<math>T = \frac{1}{4}\sqrt{(a+(b+c)) (c-(a-b)) (c+(a-b)) (a+(b-c))}.</math>
The brackets in the above formula are required in order to prevent numerical instability in the evaluation.

==Other area formulas resembling Heron's formula==

{{Main|Triangle#Formulas resembling Heron's formula}}

Several other triangle area formulas have the same functional form as Heron's formula but with the sides replaced by either the [[median (geometry)|medians]], the reciprocals of the [[Altitude (triangle)|altitudes]], or the [[sine]]s of the angles.

== Generalizations ==
Heron's formula is a special case of [[Brahmagupta's formula]] for the area of a [[cyclic quadrilateral]]. Heron's formula and Brahmagupta's formula are both special cases of [[Bretschneider's formula]] for the area of a [[quadrilateral]]. Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero.

Heron's formula is also a special case of the [[trapezoid#Area|formula]] for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.

Expressing Heron's formula with a [[Cayley–Menger determinant]] in terms of the squares of the [[distance]]s between the three given vertices,
:<math> T = \frac{1}{4} \sqrt{- \begin{vmatrix}
0 & a^2 & b^2 & 1 \\
a^2 & 0 & c^2 & 1 \\
b^2 & c^2 & 0 & 1 \\
1 & 1 & 1 & 0
\end{vmatrix} } </math>
illustrates its similarity to [[Tartaglia's formula]] for the [[volume]] of a [[Simplex|three-simplex]].

Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by [[David P. Robbins]].<ref>D. P. Robbins, "Areas of Polygons Inscribed in a Circle", Discr. Comput. Geom. 12, 223-236, 1994.</ref>

=== Heron-type formula for the volume of a tetrahedron ===
If ''U'', ''V'', ''W'', ''u'', ''v'', ''w'' are lengths of edges of the tetrahedron (first three form a triangle; ''u'' opposite to ''U'' and so on), then<ref>W. Kahan, "What has the Volume of a Tetrahedron to do with Computer Programming Languages?", [http://www.cs.berkeley.edu/~wkahan/VtetLang.pdf], pp. 16-17.</ref>
:<math>
\text{volume} = \frac{\sqrt {\,( - a + b + c + d)\,(a - b + c + d)\,(a + b - c + d)\,(a + b + c - d)}}{192\,u\,v\,w}</math>

where

: <math>
\begin{align} a & = \sqrt {xYZ} \\ b & = \sqrt {yZX} \\ c & = \sqrt {zXY} \\ d & = \sqrt {xyz} \\ X & = (w - U + v)\,(U + v + w) \\ x & = (U - v + w)\,(v - w + U) \\ Y & = (u - V + w)\,(V + w + u) \\ y & = (V - w + u)\,(w - u + V) \\ Z & = (v - W + u)\,(W + u + v) \\ z & = (W - u + v)\,(u - v + W). \end{align}
</math>

==See also==
* [[Heronian triangle]]

== Notes ==
<references/>

== References ==
* {{cite book
| author=Heath, Thomas L.
| title=A History of Greek Mathematics (Vol II)
| publisher=Oxford University Press
| year=1921
| pages=321–323}}

== External links ==
*[http://www.cut-the-knot.org/pythagoras/herons.shtml A Proof of the Pythagorean Theorem From Heron's Formula] at [[cut-the-knot]]
*[http://www.mathopenref.com/heronsformula.html Interactive applet and area calculator using Heron's Formula]
*[http://www.math.dartmouth.edu/~doyle/docs/heron/heron.txt J.H. Conway discussion on Heron's Formula]
*{{MathPages|id=home/kmath196/kmath196|title=Heron's Formula and Brahmagupta's Generalization}}
*[http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Umberger/MATH7200/HeronFormulaProject/GeometricProof/geoproof.html A Geometric Proof of Heron's Formula]

{{DEFAULTSORT:Heron's Formula}}
[[Category:Triangle geometry]]
[[Category:Articles containing proofs]]
[[Category:Area]]
[[Category:Theorems in plane geometry]]

Hausdorff density

2012-12-10T11:11:57Z

130.60.188.103:

This lists the [[character table]]s for the more common [[point groups in three dimensions|molecular point groups]] used in the study of [[molecular symmetry]]. These tables are based on the [[group theory|group-theoretical]] treatment of the [[symmetry]] operations present in common [[molecule]]s, and are useful in molecular [[spectroscopy]] and [[quantum chemistry]]. Information regarding the use of the tables, as well as more extensive lists of them, can be found in the references.<ref>{{cite book | last = Drago | first = Russell S. | title = Physical Methods in Chemistry | publisher = W.B. Saunders Company | year = 1977 | isbn = 0-7216-3184-3}}</ref><ref>{{cite book | last=Cotton | first = F. Albert | title = Chemical Applications of Group Theory | publisher = John Wiley & Sons: New York | year = 1990 | isbn = 0-471-51094-7}}</ref><ref>{{cite web | last = Gelessus | first = Achim | title = Character tables for chemically important point groups | publisher = Jacobs University, Bremin; Computational Laboratory for Analysis, Modeling, and Visualization | date=2007-07-12 | url=http://symmetry.jacobs-university.de/ | accessdate=2007-07-12 }}</ref><ref name="ShirtsFixJCE">{{cite journal | last=Shirts | first=Randall B. | title=Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables | journal=[[Journal of Chemical Education]] | volume=84 | issue=1882 | publisher=[[American Chemical Society]] | year=2007 | url=http://jchemed.chem.wisc.edu/Journal/Issues/2007/Nov/abs1882.html | accessdate= 2007-10-16 | doi=10.1021/ed084p1882 | pages=1882|bibcode = 2007JChEd..84.1882S }}</ref><ref>{{cite web | url=http://www.webqc.org/symmetry.php | title=POINT GROUP SYMMETRY CHARACTER TABLES | last= Vanovschi | first=Vitalii | accessdate=2008-10-29 | publisher=WebQC.Org}}</ref>

== Notation ==
For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the [[Order (group theory)|order of the group]] (number of invariant symmetry operations). The finite group notation used is: Zn: [[cyclic group]] of order ''n'', Dn: [[dihedral group]] isomorphic to the symmetry group of an ''n''–sided regular polygon, Sn: [[symmetric group]] on ''n'' letters, and An: [[alternating group]] on ''n'' letters.

The character tables then follow for all groups. The rows of the character tables correspond to the irreducible representations of the group, with their conventional names in the left margin. The naming conventions are as follows:

* ''A'' and ''B'' are singly degenerate representations, with the former transforming symmetrically around the principal axis of the group, and the latter asymmetrically. ''E'', ''T'', ''G'', ''H'', ... are doubly, triply, quadruply, quintuply, ... degenerate representations.
* ''g'' and ''u'' subscripts denote symmetry and antisymmetry, respectively, with respect to a center of inversion. Subscripts "1" and "2" denote symmetry and antisymmetry, respectively, with respect to a nonprincipal rotation axis. Higher numbers denote additional representations with such asymmetry.
* Single prime ( ' ) and double prime ( <nowiki>''</nowiki> ) superscripts denote symmetry and antisymmetry, respectively, with respect to a horizontal mirror plane σh, one perpendicular to the principal rotation axis.

All but the two rightmost columns correspond to the [[symmetry operation]]s which are invariant in the group. In the case of sets of similar operations with the same characters for all representations, they are presented as one column, with the number of such similar operations noted in the heading.

The body of the tables contain the characters in the respective irreducible representations for each respective symmetry operation, or set of symmetry operations.

The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (''x'', ''y'' and ''z''), rotations about those three coordinates (''Rx'', ''Ry'' and ''Rz''), and functions of the quadratic terms of the coordinates(''x''2, ''y''2, ''z''2, ''xy'', ''xz'', and ''yz'').

The symbol ''i'' used in the body of the table denotes the [[imaginary unit]]: ''i'' 2 = −1. Used in a column heading, it denotes the operation of inversion. A superscripted uppercase "C" denotes [[Complex conjugate|complex conjugation]].

== Character tables ==

=== Nonaxial symmetries ===
These groups are characterized by a lack of a proper rotation axis, noting that a <math>C_1</math> rotation is considered the identity operation. These groups have [[Involution (mathematics)|involutional]] symmetry: the only nonidentity operation, if any, is its own inverse.

In the group <math>C_1</math>, all functions of the Cartesian coordinates and rotations about them transform as the <math>A</math> irreducible representation.

{| class="wikitable" centered

! Point Group !! Canonical Group !! Order !! Character Table
|-
| <math>C_1</math> || <math>Z_1</math> || <math>1</math>
||
{| class="wikitable" centered
| || <math>E</math>
|-
| <math>A</math> || <math>1</math>
|}

|-
| <math>C_i</math> || <math>Z_2</math> || 2
||
{| class="wikitable" centered
| || <math>E</math> || <math>i</math> || ||
|-
| <math>A_g</math> || <math>1</math> || <math>1</math>
| <math>R_x</math>, <math>R_y</math>, <math>R_z</math>
| <math>x^2</math>, <math>y^2</math>, <math>z^2</math>, <math>xy</math>, <math>xz</math>, <math>yz</math>
|-
| <math>A_u</math> || <math>1</math> || <math>-1</math> || <math>x</math>, <math>y</math>, <math>z</math> ||
|}
|-
| <math>C_s</math> || <math>Z_2</math> || <math>2</math>
||
{| class="wikitable" centered
| || <math>E</math> || <math>\sigma_h</math> || ||
|-
| <math>A'</math> || <math>1</math> || <math>1</math>
| <math>x</math>, <math>y</math>, <math>R_z</math>
| <math>x^2</math>, <math>y^2</math>, <math>z^2</math>, <math>xy</math>
|-
| <math>A''</math> || <math>1</math> || <math>-1</math>
| <math>z</math>, <math>R_x</math>, <math>R_y</math> || <math>yz</math>, <math>xz</math>
|}
|-
|}

=== Cyclic symmetries ===
The families of groups with these symmetries have only one rotation axis.

==== Cyclic groups (''C''n) ====
The cyclic groups are denoted by ''C''n. These groups are characterized by an ''n''-fold proper rotation axis ''C''n. The ''C''1 group is covered in the [[#Nonaxial groups|nonaxial groups]] section.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical Group !! Order !! Character Table
|-
| ''C''2 || Z2 || 2
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''2  || colspan="2" |  
|-
| A || 1 || 1 || ''Rz'', ''z''
| ''x''2, ''y''2, ''z''2, ''xy''
|-
| B || 1 || −1 || ''Rx'', ''Ry'', ''x'', ''y''
| ''xz'', ''yz''
|}
|-
| ''C''3 || Z3 || 3
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''3  || ''C''32
| colspan="2" | ''θ'' = ''e''2π''i'' /3
|-
| A || 1 || 1 || 1 || ''Rz'', ''z''
| ''x''2 + ''y''2
|-
| E || 1 1
| ''θ''  ''θ''C
| ''θ''C ''θ'' 
| (''Rx'', ''Ry''), (''x'', ''y'')
| (''x''2 - ''y''2, ''xy''), (''xz'', ''yz'')
|-
|}
|-
| ''C''4 || Z4 || 4
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''4 
| ''C''2  || ''C''43
| colspan="2" |  
|-
| A || 1 || 1 || 1 || 1 ||''Rz'', ''z''
| ''x''2 + ''y''2, ''z''2
|-
| B || 1 || −1 || 1 || −1 ||  
| ''x''2 − ''y''2, ''xy''
|-
| E || 1 1 || ''i'' −''i'' || −1 −1 || −''i'' ''i''
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
|}
|-
| ''C''5 || Z5 || 5
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| ''C''5  || ''C''52
| ''C''53 || ''C''54
| colspan="2" | ''θ'' = ''e''2π''i'' /5
|-
| A || 1 || 1 || 1 || 1 || 1 || ''Rz'', ''z''
| ''x''2 + ''y''2, ''z''2
|-
| E1
| 1 1
| ''θ''  ''θ''C
| ''θ''2 (''θ''2)C
| (''θ''2)C ''θ''2
| ''θ''C ''θ'' 
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
| E2
| 1 1
| ''θ''2 (''θ''2)C
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
| (''θ''2)C ''θ''2
|   || (''x''2 - ''y''2, ''xy'')
|-
|}
|-
| ''C''6 || Z6 || 6
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| ''C''6  || ''C''3 
| ''C''2  || ''C''32
| ''C''65
| colspan="2" | ''θ'' = ''e''2π''i'' /6
|-
| A || 1 || 1 || 1 || 1 || 1 || 1 || ''Rz'', ''z''
| ''x''2 + ''y''2, ''z''2
|-
| B || 1 || −1 || 1 || −1 || 1 || −1 ||   ||  
|-
| E1
| 1 1
| ''θ''  ''θ''C
| −''θ''C −''θ'' 
| −1 −1
| −''θ''  −''θ''C
| ''θ''C −''θ'' 
| (''Rx'', ''Ry''), (''x'', ''y'')
| (''xz'', ''yz'')
|-
| E2
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
|   || (''x''2 − ''y''2, ''xy'')
|-
|}
|-
| ''C''8 || Z8 || 8
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| ''C''8  || ''C''4 
| ''C''83 || ''C''2 
| ''C''85 || ''C''43
| ''C''87
| colspan="2" | ''θ'' = ''e''2π''i'' /8
|-
| A || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || ''Rz'', ''z''
| ''x''2 + ''y''2, ''z''2
|-
| B || 1 || −1 || 1 || −1 || 1 || −1 || 1 || −1 ||   ||  
|-
| E1
| 1 1
| ''θ''  ''θ''C
| ''i'' −''i''
| −''θ''C −''θ'' 
| −1 −1
| −''θ''  −''θ''C
| −''i'' ''i''
| ''θ''C ''θ'' 
| (''Rx'', ''Ry''), (''x'', ''y'')
| (''xz'', ''yz'')
|-
| E2
| 1 1
| ''i'' −''i''
| −1 −1
| −''i'' ''i''
| 1 1
| ''i'' −''i''
| −1 −1
| −''i'' ''i''
|   || (''x''2 − ''y''2, ''xy'')
|-
| E3
| 1 1
| −''θ''  −''θ''C
| ''i'' −''i''
| ''θ''C ''θ'' 
| −1 −1
| ''θ''  ''θ''C
| −''i'' ''i''
| −''θ''C −''θ'' 
|   ||  
|-
|}
|-
|}

==== Reflection groups (''C''nh) ====
The reflection groups are denoted by ''C''nh. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''n; ii) a mirror plane ''σh'' normal to ''C''n. The ''C''1''h'' group is the same as the ''C''s group in the [[#Nonaxial groups|nonaxial groups]] section.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !! Order !! Character Table
|-
| ''C''2''h'' || Z2 × Z2 || 4
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''2 
| ''i'' || ''σh '' || colspan="2" |  
|-
| Ag || 1 || 1 || 1 || 1 || ''Rz''
| ''x''2, ''y''2, ''z''2, ''xy''
|-
| Bg || 1 || −1 || 1 || −1 || ''Rx'', ''Ry''
| ''xz'', ''yz''
|-
| Au || 1 || 1 || −1 || −1 || ''z'' ||  
|-
| Bu || 1 || −1 || −1 || 1 || ''x'', ''y'' ||  
|-
|}
|-
| ''C''3''h'' || Z6 || 6
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''3 
| ''C''32 || ''σh ''
| ''S''3  || ''S''35
| colspan="2" | ''θ'' = ''e''2π''i'' /3
|-
| A' || 1 || 1 || 1 || 1 || 1 || 1 || ''Rz''
| ''x''2 + ''y''2, ''z''2
|-
| E' || align="center" | 1 1
| ''θ''  ''θ''C
| ''θ''C ''θ'' 
| 1 1
| ''θ''  ''θ''C
| ''θ''C ''θ'' 
| (''x'', ''y'') || (''x''2 − ''y''2, ''xy'')
|-
| A<nowiki>''</nowiki> || 1 || 1 || 1 || −1 || −1 || −1 || ''z'' ||  
|-
| E<nowiki>''</nowiki> || align="center" | 1 1
| ''θ''  ''θ''C
| ''θ''C ''θ'' 
| −1 −1
| −''θ''  −''θ''C
| −''θ''C −''θ'' 
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
|}
|-
| ''C''4''h'' || Z2 × Z4 || 8
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''4 
| ''C''2  || ''C''43
| ''i'' || ''S''43 || ''σh ''
| ''S''4  || colspan="2" |  
|-
| Ag || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| ''Rz'' || ''x''2 + ''y''2, ''z''2
|-
| Bg || 1 || −1 || 1 || −1 || 1 || −1 || 1 || −1
|   || ''x''2 − ''y''2, ''xy''
|-
| Eg || 1 1 || ''i'' −''i'' || −1 −1
| −''i'' ''i'' || 1 1 || ''i'' −''i''
| −1 −1 || −''i'' ''i''
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| Au || 1 || 1 || 1 || 1 || −1 || −1 || −1 || −1 || ''z'' ||  
|-
| Bu || 1 || −1 || 1 || −1 || −1 || 1 || −1 || 1 ||   ||  
|-
| Eu || 1 1 || ''i'' −''i'' || −1 −1
| −''i'' ''i'' || −1 −1 || −''i'' ''i''
| 1 1 || ''i'' −''i''
| (''x'', ''y'') ||  
|-
|}
|-
| ''C''5''h'' || Z10 || 10
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| ''C''5  || ''C''52
| ''C''53 || ''C''54
| ''σh '' || ''S''5 
| ''S''57 || ''S''53
| ''S''59
| colspan="2" | ''θ'' = ''e''2π''i'' /5
|-
| A' || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || ''Rz''
| ''x''2 + ''y''2, ''z''2
|-
| E1'
| 1 1
| ''θ''  ''θ''C
| ''θ''2 (''θ''2)C
| (''θ''2)C ''θ''2
| ''θ''C ''θ'' 
| 1 1
| ''θ''  ''θ''C
| ''θ''2 (''θ''2)C
| (''θ''2)C ''θ''2
| ''θ''C ''θ'' 
| (''x'', ''y'') ||  
|-
| E2'
| 1 1
| ''θ''2 (''θ''2)C
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
| (''θ''2)C ''θ''2
| 1 1
| ''θ''2 (''θ''2)C
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
| (''θ''2)C ''θ''2
|   || (''x''2 - ''y''2, ''xy'')
|-
| A<nowiki>''</nowiki> || 1 || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 || −1
| ''z'' ||  
|-
| E1<nowiki>''</nowiki>
| 1 1
| ''θ''  ''θ''C
| ''θ''2 (''θ''2)C
| (''θ''2)C ''θ''2
| ''θ''C ''θ'' 
| −1 −1
| −''θ''  -''θ''C
| −''θ''2 −(''θ''2)C
| −(''θ''2)C −''θ''2
| −''θ''C −''θ'' 
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| E2<nowiki>''</nowiki>
| 1 1
| ''θ''2 (''θ''2)C
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
| (''θ''2)C ''θ''2
| −1 −1
| −''θ''2 −(''θ''2)C
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
| −(''θ''2)C −''θ''2
|   ||  
|-
|}
|-
| ''C''6''h'' || Z2 × Z6 || 12
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| ''C''6  || ''C''3 
| ''C''2  || ''C''32
| ''C''65 || ''i'' || ''S''35
| ''S''65 || ''σh ''
| ''S''6  || ''S''3 
| colspan="2" | ''θ'' = ''e''2π''i'' /6
|-
| Ag || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| ''Rz'' || ''x''2 + ''y''2, ''z''2
|-
| Bg || 1 || −1 || 1 || −1 || 1 || −1
| 1 || −1 || 1 || −1 || 1 || −1
|   ||  
|-
| E1g
| 1 1
| ''θ''  ''θ''C
| −''θ''C −''θ'' 
| −1 −1
| −''θ''  −''θ''C
| ''θ''C ''θ'' 
| 1 1
| ''θ''  ''θ''C
| −''θ''C −''θ'' 
| −1 −1
| −''θ''  −''θ''C
| ''θ''C ''θ'' 
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| E2g
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
|   || (''x''2 − ''y''2, ''xy'')
|-
| Au || 1 || 1 || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 || −1 || −1
| ''z'' ||  
|-
| Bu || 1 || −1 || 1 || −1 || 1 || −1
| −1 || 1 || −1 || 1 || −1 || 1
|   ||  
|-
| E1u
| 1 1
| ''θ''  ''θ''C
| −''θ''C −''θ'' 
| −1 −1
| −''θ''  −''θ''C
| ''θ''C ''θ'' 
| −1 −1
| −''θ''  −''θ''C
| ''θ''C ''θ'' 
| 1 1
| ''θ''  ''θ''C
| −''θ''C −''θ'' 
| (''x'', ''y'') ||  
|-
| E2u
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
| 1 1
| −''θ''C −''θ'' 
| −''θ''  −''θ''C
| −1 −1
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
| −1 −1
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
|   ||  
|}
|-
|}

==== Pyramidal groups (''C''nv) ====
The pyramidal groups are denoted by ''C''nv. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''n; ii) ''n'' mirror planes ''σv'' which contain ''C''n. The ''C''1''v'' group is the same as the ''C''s group in the [[#Nonaxial groups|nonaxial groups]] section.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !!Order !! Character Table
|-
| ''C''2''v'' || Z2 × Z2 (=D2) || 4
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''2 
| ''σv ''
| ''σv' ''
| colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || ''z''
| ''x''2, ''y''2, ''z''2
|-
| A2 || 1 || 1 || −1 || −1 || ''Rz'' || ''xy''
|-
| B1 || 1 || −1 || 1 || −1 || ''Ry'', ''x'' || ''xz''
|-
| B2 || 1 || −1 || −1 || 1 || ''Rx'', ''y'' || ''yz''
|-
|}
|-
| ''C''3''v'' || D3 || 6
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || 2 ''C''3 
| 3 ''σv ''
| colspan="2" |  
|-
| A1 || 1 || 1 || 1 || ''z''
| ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || −1 || ''Rz'' ||  
|-
| E || 2 || −1 || 0 || (''Rx'', ''Ry''), (''x'', ''y'')
| (''x''2 − ''y''2, ''xy''), (''xz'', ''yz'')
|-
|}
|-
| ''C''4''v'' || D4 || 8
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || 2 ''C''4 
| ''C''2  || 2 ''σv ''
| 2 ''σd '' || colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || 1
| ''z'' || ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || −1 || −1 || ''Rz'' ||  
|-
| B1 || 1 || −1 || 1 || 1 || −1
|   || ''x''2 − ''y''2
|-
| B2 || 1 || −1 || 1 || −1 || 1 ||   || ''xy''
|-
| E || 2 || 0 || −2 || 0 || 0
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
|}
|-
| ''C''5''v'' || D5 || 10
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''5  || 2 ''C''52
| 5 ''σv '' || colspan="2" | ''θ'' = 2π/5
|-
| A1 || 1 || 1 || 1 || 1 || ''z''
| ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || −1 || ''Rz'' ||  
|-
| E1 || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
| E2 || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
|}
|-
| ''C''6''v'' || D6 || 12
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''6  || 2 ''C''3 
| ''C''2  || 3 ''σv ''
| 3 ''σd '' || colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || 1 || 1
| ''z'' || ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || 1 || −1 || −1 || ''Rz'' ||  
|-
| B1 || 1 || −1 || 1 || −1 || 1 || −1 ||   ||  
|-
| B2 || 1 || −1 || 1 || −1 || −1 || 1 ||   ||  
|-
| E1 ||2 || 1 || −1 || −2 || 0 || 0
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
| E2 ||2 || −1 || −1 || 2 || 0 || 0 ||  
| (''x''2 − ''y''2, ''xy'')
|}
|-
|}

==== Improper rotation groups (''S''n) ====
The improper rotation groups are denoted by ''S''n. These groups are characterized by an ''n''-fold improper rotation axis ''S''n, where ''n'' is necessarily even. The ''S''2 group is the same as the ''C''s group in the [[#Nonaxial groups|nonaxial groups]] section.

The S8 table reflects the 2007 discovery of errors in older references.<ref name="ShirtsFixJCE"/> Specifically, (''Rx'', ''Ry'') transform not as E1 but rather as E3.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !! Order !! Character Table
|-
| ''S''4 || Z4 || 4
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''S''4 
| ''C''2  || ''S''43
| colspan="2" |  
|-
| A || 1 || 1 || 1 || 1 ||''Rz'',  
| ''x''2 + ''y''2, ''z''2
|-
| B || 1 || −1 || 1 || −1 || ''z''
| ''x''2 − ''y''2, ''xy''
|-
| E || 1 1 || ''i'' −''i'' || −1 −1
| −''i'' ''i''
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
|}
|-
| ''S''6 || Z6 || 6
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| ''S''6  || ''C''3 
| ''i'' || ''C''32 || ''S''65
| colspan="2" | ''θ'' = ''e''2π''i'' /6
|-
| Ag || 1 || 1 || 1 || 1 || 1 || 1 || ''Rz''
| ''x''2 + ''y''2, ''z''2
|-
| Eg
| 1 1
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
| 1 1
| ''θ''C ''θ'' 
| ''θ''  ''θ''C
| (''Rx'', ''Ry'')
| (''x''2 − ''y''2, ''xy''), (''xz'', ''yz'')
|-
| Au || 1 || −1 || 1 || −1 || 1 || −1 || ''z'' ||  
|-
| Eu
| 1 1
| −''θ''C −''θ'' 
| ''θ''  ''θ''C
| −1 −1
| ''θ''C ''θ'' 
| −''θ''  −''θ''C
| (''x'', ''y'') ||  
|-
|}
|-
| ''S''8 || Z8 || 8
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| ''S''8  || ''C''4 
| ''S''83 || ''i''
| ''S''85 || ''C''42
| ''S''87
| colspan="2" | ''θ'' = ''e''2π''i'' /8
|-
| A || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || ''Rz''
| ''x''2 + ''y''2, ''z''2
|-
| B || 1 || −1 || 1 || −1 || 1 || −1 || 1 || −1 || ''z'' ||  
|-
| E1
| 1 1
| ''θ''  ''θ''C
| ''i'' −''i''
| −''θ''C −''θ'' 
| −1 −1
| −''θ''  −''θ''C
| −''i'' ''i''
| ''θ''C ''θ'' 
| (''x'', ''y'') || (''xz'', ''yz'')
|-
| E2
| 1 1 || ''i'' −''i'' || −1 −1
| −''i'' ''i'' || 1 1 || ''i'' −''i''
| −1 −1 || −''i'' ''i''
|   || (''x''2 − ''y''2, ''xy'')
|-
| E3
| 1 1
| −''θ''C −''θ'' 
| −''i'' ''i''
| ''θ''  ''θ''C
| −1 −1
| ''θ''C ''θ'' 
| ''i'' −''i''
| −''θ'' −''θ''C
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
|}
|-
|}

=== Dihedral symmetries ===
The families of groups with these symmetries are characterized by 2-fold proper rotation axes normal to a principal rotation axis.

==== Dihedral groups (''D''n) ====
The dihedral groups are denoted by ''D''n. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''n; ii) ''n'' 2-fold proper rotation axes ''C''2 normal to ''C''n. The ''D''1 group is the same as the ''C''2 group in the [[cyclic groups]] section.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !!Order !! Character Table
|-
| ''D''2 || Z2 × Z2 (=D2) || 4
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''2 (''z'')
| ''C''2 (''x'')
| ''C''2 (''y'') || colspan="2" |  
|-
| A || 1 || 1 || 1 || 1 ||  
| ''x''2, ''y''2, ''z''2
|-
| B1 || 1 || 1 || −1 || −1 || ''Rz'', ''z'' || ''xy''
|-
| B2 || 1 || −1 || −1 || 1 || ''Ry'', ''y'' || ''xz''
|-
| B3 || 1 || −1 || 1 || −1 || ''Rx'', ''x'' || ''yz''
|-
|}
|-
| ''D''3 || D3 || 6
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || 2 ''C''3 
| 3 ''C''2  || colspan="2" |  
|-
| A1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || −1 || ''Rz'', ''z'' ||  
|-
| E || 2 || −1 || 0 || (''Rx'', ''Ry''), (''x'', ''y'')
| (''x''2 − ''y''2, ''xy''), (''xz'', ''yz'')
|-
|}
|-
| ''D''4 || D4 || 8
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || 2 ''C''4 
| ''C''2  || 2 ''C''2' 
| 2 ''C''2<nowiki>''</nowiki> 
| colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || −1 || −1 || ''Rz'', ''z'' ||  
|-
| B1 || 1 || −1 || 1 || 1 || −1
|   || ''x''2 − ''y''2
|-
| B2 || 1 || −1 || 1 || −1 || 1 ||   || ''xy''
|-
| E || 2 || 0 || −2 || 0 || 0
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
|}
|-
| ''D''5 || D5 || 10
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''5  || 2 ''C''52
| 5 ''C''2  || colspan="2" | ''θ''=2π/5
|-
| A1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || −1 || ''Rz'', ''z'' ||  
|-
| E1 || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
| E2 || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
|}
|-
| ''D''6 || D6 || 12
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''6  || 2 ''C''3 
| ''C''2  || 3 ''C''2' 
| 3 ''C''2<nowiki>''</nowiki> 
| colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || 1 || −1 || −1
| ''Rz'', ''z'' ||  
|-
| B1 || 1 || −1 || 1 || −1 || 1 || −1 ||   ||  
|-
| B2 || 1 || −1 || 1 || −1 || −1 || 1 ||   ||  
|-
| E1 ||2 || 1 || −1 || −2 || 0 || 0
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
| E2 ||2 || −1 || −1 || 2 || 0 || 0 ||  
| (''x''2 − ''y''2, ''xy'')
|}
|-
|}

==== Prismatic groups (''D''nh) ====
The prismatic groups are denoted by ''D''nh. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''n; ii) ''n'' 2-fold proper rotation axes ''C''2 normal to ''C''n; iii) a mirror plane ''σh'' normal to ''C''n and containing the ''C''2s. The ''D''1''h'' group is the same as the ''C''2''v'' group in the [[#Pyramidal groups|pyramidal groups]] section.

The D8''h'' table reflects the 2007 discovery of errors in older references.<ref name="ShirtsFixJCE"/> Specifically, symmetry operation column headers 2S8 and 2S83 were reversed in the older references.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !!Order !! Character Table
|-
| ''D''2''h''
| Z2×Z2×Z2 (=Z2×D2) || 8
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || ''C''2 
| ''C''2 (x)
| ''C''2 (y) || ''i''
| ''σ(xy)  ''
| ''σ(xz)  ''
| ''σ(yz)  '' || colspan="2" |  
|-
| Ag || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2, ''y''2, ''z''2
|-
| B1g || 1 || 1 || −1 || −1 || 1 || 1 || −1 || −1
| ''Rz'' || ''xy''
|-
| B2g || 1 || −1 || −1 || 1 || 1 || −1 || 1 || −1
| ''Ry'' || ''xz''
|-
| B3g || 1 || −1 || 1 || −1 || 1 || −1 || −1 || 1
| ''Rx'' || ''yz''
|-
| Au || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 ||   ||  
|-
| B1u || 1 || 1 || −1 || −1
| −1 || −1 || 1 || 1 || ''z'' ||  
|-
| B2u || 1 || −1 || −1 || 1
| −1 || 1 || −1 || 1 || ''y'' ||  
|-
| B3u || 1 || −1 || 1 || −1
| −1 || 1 || 1 || −1 || ''x'' ||  
|-
|}
|-
| ''D''3''h'' || D6 || 12
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || 2 ''C''3 
| 3 ''C''2  || ''σh ''
| 2 ''S''3  || 3 ''σv ''
| colspan="2" |  
|-
| A1' || 1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2, ''z''2
|-
| A2' || 1 || 1 || −1 || 1 || 1 || −1 || ''Rz'' ||  
|-
| E' || 2 || −1 || 0 || 2 || −1 || 0 || (''x'', ''y'')
| (''x''2 − ''y''2, ''xy'')
|-
| A1<nowiki>''</nowiki> || 1 || 1 || 1 || −1 || −1 || −1
|   ||  
|-
| A2<nowiki>''</nowiki> || 1 || 1 || −1 || −1 || −1 || 1
| ''z'' ||  
|-
| E<nowiki>''</nowiki> || 2 || −1 || 0 || −2 || 1 || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
|}
|-
| ''D''4''h'' || Z2×D4 || 16
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E || 2 ''C''4 
| ''C''2  || 2 ''C''2' 
| 2 ''C''2<nowiki>''</nowiki>  || ''i''
| 2 ''S''4  || ''σh ''
| 2 ''σv ''
| 2 ''σd '' || colspan="2" |  
|-
| A1g || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2, ''z''2
|-
| A2g || 1 || 1 || 1 || −1 || −1
| 1 || 1 || 1 || −1 || −1
| ''Rz'' ||  
|-
| B1g || 1 || −1 || 1 || 1 || −1
| 1 || −1 || 1 || 1 || −1
|   || ''x''2 − ''y''2
|-
| B2g || 1 || −1 || 1 || −1 || 1
| 1 || −1 || 1 || −1 || 1
|   || ''xy''
|-
| Eg || 2 || 0 || −2 || 0 || 0 || 2 || 0 || −2 || 0 || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| A1u || 1 || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 || −1
|   ||  
|-
| A2u || 1 || 1 || 1 || −1 || −1
| −1 || −1 || −1 || 1 || 1
| ''z'' ||  
|-
| B1u || 1 || −1 || 1 || 1 || −1
| −1 || 1 || −1 || −1 || 1
|   ||  
|-
| B2u || 1 || −1 || 1 || −1 || 1
| −1 || 1 || −1 || 1 || −1
|   ||  
|-
| Eu || 2 || 0 || −2 || 0 || 0 || −2 || 0 || 2 || 0 || 0
| (''x'', ''y'') ||  
|-
|}
|-
| ''D''5''h'' || D10 || 20
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''5  || 2 ''C''52
| 5 ''C''2 
| ''σh '' || 2 ''S''5 
| 2 ''S''53 || 5 ''σv ''
| colspan="2" | ''θ''=2π/5
|-
| A1' || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2, ''z''2
|-
| A2' || 1 || 1 || 1 || −1 || 1 || 1 || 1 || −1
| ''Rz'' ||  
|-
| E1' || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0 || 2
| 2 cos(''θ'') || 2 cos(2''θ'') || 0 || (''x'', ''y'') ||  
|-
| E2' || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0 || 2
| 2 cos(2''θ'') || 2 cos(''θ'') || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
| A1<nowiki>''</nowiki> || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1
|   ||  
|-
| A2<nowiki>''</nowiki> || 1 || 1 || 1 || −1
| −1 || −1 || −1 || 1
| ''z'' ||  
|-
| E1<nowiki>''</nowiki> || 2 || 2 cos(''θ'')
| 2 cos(2''θ'') || 0 || −2 || −2 cos(''θ'')
| −2 cos(2''θ'') || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| E2<nowiki>''</nowiki> || 2 || 2 cos(2''θ'')
| 2 cos(''θ'') || 0 || −2 || −2 cos(2''θ'')
| −2 cos(''θ'') || 0 ||   ||  
|-
|}
|-
| ''D''6''h''
| Z2×D6 || 24
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''6  || 2 ''C''3 
| ''C''2  || 3 ''C''2' 
| 3 ''C''2<nowiki>''</nowiki>  || ''i''
| 2 ''S''3  || 2 ''S''6 
| ''σh '' || 3 ''σd ''
| 3 ''σv '' || colspan="2" |  
|-
| A1g || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2, ''z''2
|-
| A2g || 1 || 1 || 1 || 1 || −1 || −1
| 1 || 1 || 1 || 1 || −1 || −1
| ''Rz'' ||  
|-
| B1g || 1 || −1 || 1 || −1 || 1 || −1
| 1 || −1 || 1 || −1 || 1 || −1
|   ||  
|-
| B2g || 1 || −1 || 1 || −1 || −1 || 1
| 1 || −1 || 1 || −1 || −1 || 1
|   ||  
|-
| E1g || 2 || 1 || −1 || −2 || 0 || 0
| 2 || 1 || −1 || −2 || 0 || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| E2g || 2 || −1 || −1 || 2 || 0 || 0
| 2 || −1 || −1 || 2 || 0 || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
| A1u || 1 || 1 || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 || −1 || −1
|   ||  
|-
| A2u || 1 || 1 || 1 || 1 || −1 || −1
| −1 || −1 || −1 || −1 || 1 || 1
| ''z'' ||  
|-
| B1u || 1 || −1 || 1 || −1 || 1 || −1
| −1 || 1 || −1 || 1 || −1 || 1
|   ||  
|-
| B2u || 1 || −1 || 1 || −1 || −1 || 1
| −1 || 1 || −1 || 1 || 1 || −1
|   ||  
|-
| E1u || 2 || 1 || −1 || −2 || 0 || 0
| −2 || −1 || 1 || 2 || 0 || 0
| (''x'', ''y'') ||  
|-
| E2u || 2 || −1 || −1 || 2 || 0 || 0
| −2 || 1 || 1 || −2 || 0 || 0
|   ||  
|}
|-
| ''D''8''h'' || Z2×D8 || 32
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''8  || 2 ''C''83
| 2 ''C''4  || ''C''2 
| 4 ''C''2' 
| 4 ''C''2<nowiki>''</nowiki>  || ''i''
| 2 ''S''83 || 2 ''S''8 
| 2 ''S''4 
| ''σh ''
| 4 ''σd ''
| 4 ''σv ''
| colspan="2" | ''θ''=21/2
|-
| A1g || 1 || 1 || 1 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2, ''z''2
|-
| A2g || 1 || 1 || 1 || 1 || 1 || −1 || −1
| 1 || 1 || 1 || 1 || 1 || −1 || −1 || ''Rz'' ||  
|-
| B1g || 1 || −1 || −1 || 1 || 1 || 1 || −1
| 1 || −1 || −1 || 1 || 1 || 1 || −1 ||   ||  
|-
| B2g || 1 || −1 || −1 || 1 || 1 || −1 || 1
| 1 || −1 || −1 || 1 || 1 || −1 || 1 ||   ||  
|-
| E1g || 2 || ''θ'' || −''θ'' || 0 || −2 || 0 || 0
| 2 || ''θ'' || −''θ'' || 0 || −2 || 0 || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| E2g || 2 || 0 || 0 || −2 || 2 || 0 || 0
| 2 || 0 || 0 || −2 || 2 || 0 || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
| E3g || 2 || −''θ'' || ''θ'' || 0 || −2 || 0 || 0
| 2 || −''θ'' || ''θ'' || 0 || −2 || 0 || 0
|   ||  
|-
| A1u || 1 || 1 || 1 || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 || −1 || −1 || −1 ||   ||  
|-
| A2u || 1 || 1 || 1 || 1 || 1 || −1 || −1
| −1 || −1 || −1 || −1 || −1 || 1 || 1 || ''z'' ||  
|-
| B1u || 1 || −1 || −1 || 1 || 1 || 1 || −1
| −1 || 1 || 1 || −1 || −1 || −1 || 1 ||   ||  
|-
| B2u || 1 || −1 || −1 || 1 || 1 || −1 || 1
| −1 || 1 || 1 || −1 || −1 || 1 || −1
|   ||  
|-
| E1u || 2 || ''θ'' || −''θ'' || 0 || −2 || 0 || 0
| −2 || −''θ'' || ''θ'' || 0 || 2 || 0 || 0
| (''x'', ''y'') ||  
|-
| E2u || 2 || 0 || 0 || −2 || 2 || 0 || 0
| −2 || 0 || 0 || 2 || −2 || 0 || 0 ||   ||  
|-
| E3u || 2 || −''θ'' || ''θ'' || 0 || −2 || 0 || 0
| −2 || ''θ'' || −''θ'' || 0 || 2 || 0 || 0
|   ||  
|}
|-
|}

==== Antiprismatic groups (''D''nd) ====
The antiprismatic groups are denoted by ''D''nd. These groups are characterized by i) an ''n''-fold proper rotation axis ''C''n; ii) ''n'' 2-fold proper rotation axes ''C''2 normal to ''C''n; iii) ''n'' mirror planes ''σd'' which contain ''C''n. The ''D''1''d'' group is the same as the ''C''2''h'' group in the [[#Reflection groups|reflection groups]] section.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !! Order !! Character Table
|-
| ''D''2''d'' || D4 || 8
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  || 2 ''S''4 
| ''C''2  || 2 ''C''2' 
| 2 ''σd '' || colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2, ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || −1 || −1 || ''Rz'' ||  
|-
| B1 || 1 || −1 || 1 || 1 || −1 ||  
| ''x''2 − ''y''2
|-
| B2 || 1 || −1 || 1 || −1 || 1 || ''z'' || ''xy''
|-
| E || 2 || 0 || −2 || 0 || 0
| (''Rx'', ''Ry''), (''x'', ''y'') || (''xz'', ''yz'')
|-
|}
|-
| ''D''3''d'' || D6 || 12
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  || 2 ''C''3 
| 3 ''C''2  || ''i'' 
| 2 ''S''6 
| 3 ''σd ''
| colspan="2" |  
|-
| A1g || 1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2, ''z''2
|-
| A2g || 1 || 1 || −1 || 1 || 1 || −1
| ''Rz'' ||  
|-
| Eg || 2 || −1 || 0 || 2 || −1 || 0
| (''Rx'', ''Ry'')
| (''x''2 − ''y''2, ''xy''), (''xz'', ''yz'')
|-
| A1u || 1 || 1 || 1 || −1 || −1 || −1 ||   ||  
|-
| A2u || 1 || 1 || −1 || −1 || −1 || 1 || ''z'' ||  
|-
| Eu || 2 || −1 || 0 || −2 || 1 || 0 || (''x'', ''y'') ||  
|-
|}
|-
| ''D''4''d'' || D8 || 16
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  || 2 ''S''8 
| 2 ''C''4  || 2 ''S''83
| ''C''2  || 4 ''C''2' 
| 4 ''σd ''
| colspan="2" | ''θ''=21/2
|-
| A1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || 1 || 1 || −1 || −1
| ''Rz'' ||  
|-
| B1 || 1 || −1 || 1 || −1 || 1 || 1 || −1 ||   ||  
|-
| B2 || 1 || −1 || 1 || −1 || 1 || −1 || 1 || ''z'' ||  
|-
| E1 || 2 || ''θ'' || 0 || −''θ'' || −2 || 0 || 0
| (''x'', ''y'') ||  
|-
| E2 || 2 || 0 || −2 || 0 || 2 || 0 || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
| E3 || 2 || −''θ'' || 0 || ''θ'' || −2 || 0 || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
|}
|-
| ''D''5''d'' || D10 || 20
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''C''5  || 2 ''C''52
| 5 ''C''2  || ''i'' 
| 2 ''S''10  || 2 ''S''103
| 5 ''σd ''
| colspan="2" | ''θ''=2π/5
|-
| A1g || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2, ''z''2
|-
| A2g || 1 || 1 || 1 || −1 || 1 || 1 || 1 || −1
| ''Rz'' ||  
|-
| E1g || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
| 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| E2g || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
| 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
| A1u || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 ||   ||  
|-
| A2u || 1 || 1 || 1 || −1
| −1 || −1 || −1 || 1 || ''z'' ||  
|-
| E1u || 2 || 2 cos(''θ'') || 2 cos(2''θ'') || 0
| −2 || −2 cos(2''θ'') || −2 cos(''θ'') || 0
| (''x'', ''y'') ||  
|-
| E2u || 2 || 2 cos(2''θ'') || 2 cos(''θ'') || 0
| −2 || −2 cos(''θ'') || −2 cos(2''θ'') || 0
|   ||  
|-
|}
|-
| ''D''6''d'' || D12 || 24
| align="left" |
{| class="wikitable" style="text-align:center"
|   || E  
| 2 ''S''12  || 2 ''C''6 
| 2 ''S''4  || 2 ''C''3 
| 2 ''S''125 || ''C''2 
| 6 ''C''2'  || 6 ''σd ''
| colspan="2" | ''θ''=31/2
|-
| A1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2, ''z''2
|-
| A2 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || −1 || −1
| ''Rz'' ||  
|-
| B1 || 1 || −1 || 1 || −1 || 1 || −1 || 1 || 1 || −1
|   ||  
|-
| B2 || 1 || −1 || 1 || −1 || 1 || −1 || 1 || −1 || 1
| ''z'' ||  
|-
| E1 || 2 || ''θ'' || 1 || 0 || −1
| −''θ'' || −2 || 0 || 0 || (''x'', ''y'') ||  
|-
| E2 || 2 || 1 || −1 || −2 || −1 || 1 || 2 || 0 || 0 ||  
| (''x''2 − ''y''2, ''xy'')
|-
| E3 || 2 || 0 || −2 || 0 || 2 || 0 || −2 || 0 || 0
|   ||  
|-
| E4 || 2 || −1 || −1 || 2 || −1 || −1 || 2 || 0 || 0
|   ||  
|-
| E5 || 2 || −''θ'' || 1 || 0 || −1
| ''θ'' || −2 || 0 || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
|}
|-
|}

=== [[Polyhedral group|Polyhedral]] symmetries ===
These symmetries are characterized by having more than one proper rotation axis of order greater than 2.

==== Cubic groups ====
These polyhedral groups are characterized by not having a ''C''5 proper rotation axis.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !! Order !! Character Table
|-
| ''[[Tetrahedral group#Chiral tetrahedral symmetry|T]]'' || A4 || 12
| align="left" |
{| style="text-align:center"
|   || E || 4 ''C''3 
| 4 ''C''32
| 3 ''C''2 
| colspan="2" | ''θ''=e2π ''i''/3
|-
| A || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2 + ''z''2
|-
| E || 1 1 || ''θ''  ''θ''C
| ''θ''C ''θ'' 
| 1 1 ||  
| (2 ''z''2 − ''x''2 − ''y''2, ''x''2 − ''y''2)
|-
| T || 3 || 0 || 0 || −1
| (''Rx'', ''Ry'', ''Rz''), (''x'', ''y'', ''z'')
| (''xy'', ''xz'', ''yz'')
|-
|}
|-
| ''[[Tetrahedral group#Achiral tetrahedral symmetry|Td]]'' || S4 || 24
| align="left" |
{| style="text-align:center"
|   || E || 8 ''C''3 
| 3 ''C''2  || 6 ''S''4 
| 6 ''σd ''
| colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2 + ''z''2
|-
| A2 || 1 || 1 || 1 || −1 || −1 ||   ||  
|-
| E || 2 || −1 || 2 || 0 || 0 ||  
| (2 ''z''2 − ''x''2 − ''y''2, ''x''2 − ''y''2)
|-
| T1 || 3 || 0 || −1 || 1 || −1
| (''Rx'', ''Ry'', ''Rz'') ||  
|-
| T2 || 3 || 0 || −1 || −1 || 1
| (''x'', ''y'', ''z'') || (''xy'', ''xz'', ''yz'')
|-
|}
|-
| ''[[Tetrahedral group#Pyritohedral symmetry|Th]]'' || Z2×A4 || 24
| align="left" |
{| style="text-align:center"
|   || E || 4 ''C''3 
| 4 ''C''32
| 3 ''C''2  || ''i''
| 4 ''S''6  || 4 ''S''65
| 3 ''σh ''
| colspan="2" | ''θ''=e2π ''i''/3
|-
| Ag || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2 + ''z''2
|-
| Au || 1 || 1 || 1 || 1 || −1 || −1 || −1 || −1
|   ||  
|-
| Eg || 1 1
| ''θ''  ''θ''C
| ''θ''C ''θ'' 
| 1 1 || 1 1
| ''θ''  ''θ''C
| ''θ''C ''θ'' 
| 1 1
|  
| (2 ''z''2 − ''x''2 − ''y''2, ''x''2 − ''y''2)
|-
| Eu || 1 1
| ''θ''  ''θ''C
| ''θ''C ''θ'' 
| 1 1 || −1 −1
| −''θ''  −''θ''C
| −''θ''C −''θ'' 
| −1 −1
|   ||  
|-
| Tg || 3 || 0 || 0 || −1 || 3 || 0 || 0 || −1
| (''Rx'', ''Ry'', ''Rz'')
| (''xy'', ''xz'', ''yz'')
|-
| Tu || 3 || 0 || 0 || −1 || −3 || 0 || 0 || 1
| (''x'', ''y'', ''z'') ||  
|-
|}
|-
| ''[[Octahedral symmetry#Chiral octahedral symmetry|O]]'' || S4 || 24
| align="left" |
{| style="text-align:center"
|   || E  
| 6 ''C''4 
| 3 ''C''2  (''C''42)
| 8 ''C''3  || 6 ''C''2 
| colspan="2" |  
|-
| A1 || 1 || 1 || 1 || 1 || 1 ||  
| ''x''2 + ''y''2 + ''z''2
|-
| A2 || 1 || −1 || 1 || 1 || −1 ||   ||  
|-
| E || 2 || 0 || 2 || −1 || 0 ||  
| (2 ''z''2 − ''x''2 − ''y''2, ''x''2 − ''y''2)
|-
| T1 || 3 || 1 || −1 || 0 || −1
| (''Rx'', ''Ry'', ''Rz''), (''x'', ''y'', ''z'')
|  
|-
| T2 || 3 || −1 || −1 || 0 || 1
|   || (''xy'', ''xz'', ''yz'')
|-
|}
|-
| ''[[Octahedral symmetry#Achiral octahedral symmetry|Oh]]''
| Z2×S4 || 48
| align="left" |
{| style="text-align:center"
|   || E  
| 8 ''C''3  || 6 ''C''2 
| 6 ''C''4 
| 3 ''C''2  (''C''42)
| ''i'' || 6 ''S''4 
| 8 ''S''6  || 3 ''σh ''
| 6 ''σd ''
| colspan="2" |  
|-
| A1g || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
|   || ''x''2 + ''y''2 + ''z''2
|-
| A2g || 1 || 1 || −1 || −1 || 1 || 1 || −1 || 1 || 1 || −1
|   ||  
|-
| Eg || 2 || −1 || 0 || 0 || 2 || 2 || 0 || −1 || 2 || 0
|  
| (2 ''z''2 − ''x''2 − ''y''2, ''x''2 − ''y''2)
|-
| T1g || 3 || 0 || −1 || 1 || −1 || 3 || 1 || 0 || −1 || −1
| (''Rx'', ''Ry'', ''Rz'')
|  
|-
| T2g || 3 || 0 || 1 || −1 || −1 || 3 || −1 || 0 || −1 || 1
|   || (''xy'', ''xz'', ''yz'')
|-
| A1u || 1 || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 || −1
|   ||  
|-
| A2u || 1 || 1 || −1 || −1 || 1
| −1 || 1 || −1 || −1 || 1
|   ||  
|-
| Eu || 2 || −1 || 0 || 0 || 2 || −2 || 0 || 1 || −2 || 0
|   ||  
|-
| T1u || 3 || 0 || −1 || 1 || −1
| −3 || −1 || 0 || 1 || 1
| (''x'', ''y'', ''z'') ||  
|-
| T2u || 3 || 0 || 1 || −1 || −1
| −3 || 1 || 0 || 1 || −1
|   ||  
|}
|-
|}

==== Icosahedral groups ====
{{see also|Icosahedral symmetry}}

These polyhedral groups are characterized by having a ''C''5 proper rotation axis.

{| class="wikitable" style="text-align:center"
! Point Group !! Canonical group !!Order !! Character Table
|-
| ''I'' || A5 || 60
| align="left" |
{| style="text-align:center"
|   || E || 12 ''C''5 
| 12 ''C''52
| 20 ''C''3 
| 15 ''C''2 
| colspan="2" | ''θ''=π/5
|-
| A || 1 || 1 || 1 || 1 || 1 || 
| ''x''2 + ''y''2 + ''z''2
|-
| T1 || 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || −1
| (''Rx'', ''Ry'', ''Rz''), (''x'', ''y'', ''z'') ||  
|-
| T2 || 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || −1
|   ||  
|-
| G || 4 || −1 || −1 || 1 || 0 ||   ||  
|-
| H || 5 || 0 || 0 || −1 || 1 ||  
| (2 ''z''2 − ''x''2 − ''y''2, ''x''2 − ''y''2, ''xy'', ''xz'', ''yz'')
|-
|}
|-
| ''Ih'' || Z2×A5 || 120
| align="left" |
{| style="text-align:center"
|   || E || 12 ''C''5 
| 12 ''C''52
| 20 ''C''3 
| 15 ''C''2  || ''i''
| 12 ''S''10 
| 12 ''S''103
| 20 ''S''6 
| 15 ''σ''
| colspan="2" | ''θ''=π/5
|-
| Ag || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 
| ''x''2 + ''y''2 + ''z''2
|-
| T1g || 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || −1
| 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || −1
| (''Rx'', ''Ry'', ''Rz'') ||  
|-
| T2g || 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || −1
| 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || −1 ||   ||  
|-
| Gg || 4 || −1 || −1 || 1 || 0 || 4 || −1 || −1 || 1 || 0
|   ||  
|-
| Hg || 5 || 0 || 0 || −1 || 1 || 5 || 0 || 0 || −1 || 1 ||  
| (2 ''z''2 − ''x''2 − ''y''2, ''x''2 − ''y''2, ''xy'', ''xz'', ''yz'')
|-
| Au || 1 || 1 || 1 || 1 || 1
| −1 || −1 || −1 || −1 || −1
|   ||  
|-
| T1u || 3 || 2 cos(''θ'') || 2 cos(3''θ'') || 0 || −1
| −3 || −2 cos(3''θ'') || −2 cos(''θ'') || 0 || 1
| (''x'', ''y'', ''z'') ||  
|-
| T2u || 3 || 2 cos(3''θ'') || 2 cos(''θ'') || 0 || −1
| −3 || −2 cos(''θ'') || −2 cos(3''θ'') || 0 || 1
|   ||  
|-
| Gu || 4 || −1 || −1 || 1 || 0
| −4 || 1 || 1 || −1 || 0
|   ||  
|-
| Hu || 5 || 0 || 0 || −1 || 1 || −5 || 0 || 0 || 1 || −1
|   ||  
|-
|}
|-
|}

=== Linear (cylindrical) groups ===
These groups are characterized by having a proper rotation axis ''C''∞ around which the symmetry is invariant to ''any'' rotation.

{| class="wikitable" style="text-align:center"
! Point Group !! Character Table
|-
| ''C∞v''
| align="left" |
{| style="text-align:center"
|   || E || 2 ''C''∞Φ
| ...
| ∞ σv 
| colspan="2" |  
|-
| A1=Σ+ || 1 || 1 || ... || 1 || ''z''
| ''x''2 + ''y''2, ''z''2
|-
| A2=Σ− || 1 || 1 || ... || −1 || ''Rz''
|  
|-
| E1=Π || 2 || 2 cos(Φ) || ... || 0
| (''x'', ''y''), (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| E2=Δ || 2 || 2 cos(2Φ) || ... || 0
|   || (''x''2 - ''y''2, ''xy'')
|-
| E3=Φ || 2 || 2 cos(3Φ) || ... || 0
|   ||  
|-
| ... || ... || ... || ... || ... ||   ||  
|-
|}
|-
| ''D∞h''
| align="left" |
{| style="text-align:center"
|   || E || 2 ''C''∞Φ || ...
| ∞ σv  || ''i''
| 2 ''S''∞Φ || ... || ∞ ''C''2 
| colspan="2" |  
|-
| Σg+ || 1 || 1 || ... || 1 || 1 || 1 || ... || 1
|   || ''x''2 + ''y''2, ''z''2
|-
| Σg− || 1 || 1 || ...
| −1 || 1 || 1 || ... || −1
| ''Rz'' ||  
|-
| Πg || 2 || 2 cos(Φ) || ... || 0 ||2 || −2 cos(Φ) || .. || 0
| (''Rx'', ''Ry'') || (''xz'', ''yz'')
|-
| Δg || 2 || 2 cos(2Φ) || ... || 0 || 2 || 2 cos(2Φ) || .. || 0
|   || (''x''2 − ''y''2, ''xy'')
|-
| ... || ... || ... || ... || ... || ... || ... || ... || ... ||   ||  
|-
| Σu+ || 1 || 1 || ...
| 1 || −1 || −1 || ... || −1
| ''z'' ||  
|-
| Σu− || 1 || 1 || ...
| −1 || −1 || −1 || ... || 1
|   ||  
|-
| Πu || 2 || 2 cos(Φ) || ...
| 0 || −2 || 2 cos(Φ) || .. || 0
| (''x'', ''y'') ||  
|-
| Δu || 2 || 2 cos(2Φ) || ...
| 0 || −2 || −2 cos(2Φ) || .. || 0
|   ||  
|-
| ... || ... || ... || ... || ... || ... || ... || ... || ... ||   ||  
|}
|-
|}

== See also ==
*[[Linear combination of atomic orbitals|Linear combination of atomic orbitals molecular orbital method]]
*[[Raman spectroscopy]]
*[[Molecular vibration|Vibrational spectroscopy (molecular vibration)]]
*[[List of small groups]]
*[[Cubic harmonic]]s

== Notes ==
{{Reflist}}

==External links==
*[http://gernot-katzers-spice-pages.com/character_tables/ Character tables for many more point groups] (includes symmetry transformations of Cartesian products up to sixth order)

== Further reading ==
* {{cite book | last = Bunker | first = Philip | coauthors = Jensen, Per | title = Molecular Symmetry and Spectroscopy, Second edition | publisher = NRC Research Press | year = 2006 | location = [[Ottawa]] | isbn = 0-660-19628-X}}

[[Category:Theoretical chemistry]]
[[Category:Physical chemistry]]
[[Category:Group theory]]
[[Category:Finite groups]]