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{{short description|Angle between two planes in space}}
{{About|the geometry concept|the aeronautics concept|Dihedral (aeronautics)|other uses|Dihedral (disambiguation){{!}}Dihedral}}
[[File:Dihedral angle.svg|thumbnail|Angle between two half-planes (α, β, pale blue) in a third plane (red) perpendicular to line of intersection.]]
{{Angles}}

A '''dihedral angle''' is the [[angle]] between two [[Plane–plane intersection|intersecting planes]] or [[half-plane]]s. It is a plane angle formed on a third plane, perpendicular to the [[line (geometry)|line]] of intersection between the two planes or the common [[edge (geometry)|edge]] between the two half-planes. In [[higher dimension]]s, a dihedral angle represents the angle between two [[hyperplane]]s. In [[chemistry]], it is the clockwise angle between half-planes through two sets of three [[atoms]], having two atoms in common.

==Mathematical background==
When the two intersecting planes are described in terms of [[Cartesian coordinates]] by the two equations 
:<math> a_1 x + b_1 y + c_1 z + d_1 = 0 </math> 
:<math>a_2 x + b_2 y + c_2 z + d_2 = 0 </math>
the dihedral angle, <math>\varphi</math> between them is given by:
:<math>\cos \varphi = \frac{\left\vert a_1 a_2 + b_1 b_2 + c_1 c_2 \right\vert}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}</math>
and satisfies <math>0\le \varphi \le \pi/2.</math> It can easily be observed that the angle is independent of <math>d_1</math> and <math>d_2
</math>.

Alternatively, if {{math|'''n'''<sub>A</sub>}} and {{math|'''n'''<sub>B</sub>}} are [[normal vector]] to the planes, one has
:<math>\cos \varphi =  \frac{ \left\vert\mathbf{n}_\mathrm{A} \cdot \mathbf{n}_\mathrm{B}\right\vert}{|\mathbf{n}_\mathrm{A} | |\mathbf{n}_\mathrm{B}|}</math>
where {{math|'''n'''<sub>A</sub>&nbsp;·&nbsp;'''n'''<sub>B</sub>}} is the [[dot product]] of the vectors and {{math|{{abs|'''n'''<sub>A</sub>}} {{abs|'''n'''<sub>B</sub>}}}} is the product of their lengths.<ref>{{cite web |title=Angle Between Two Planes |url=https://math.tutorvista.com/geometry/angle-between-two-planes.html |website=TutorVista.com |access-date=2018-07-06 |archive-date=2020-10-28 |archive-url=https://web.archive.org/web/20201028133313/https://math.tutorvista.com/geometry/angle-between-two-planes.html |url-status=dead }}</ref>
<!-- [[File:Spherical bond dihedral angle.png|thumb|Dihedral angle of three vectors, defined as an exterior spherical angle. The longer and shorter black segments are arcs of the great circles passing through '''b'''<sub>1</sub> and '''b'''<sub>2</sub> and through '''b'''<sub>2</sub> and '''b'''<sub>3</sub>, respectively.]] -->

The absolute value is required in above formulas, as the planes are not changed when changing all coefficient signs in one equation, or replacing one normal vector by its opposite.

However the [[absolute value]]s can be and should be avoided when considering the dihedral angle of two [[half plane]]s whose boundaries are the same line. In this case, the half planes can be described by a point {{mvar|P}} of their intersection, and three vectors {{math|'''b'''<sub>0</sub>}}, {{math|'''b'''<sub>1</sub>}} and {{math|'''b'''<sub>2</sub>}} such that {{math|''P'' + '''b'''<sub>0</sub>}}, {{math|''P'' + '''b'''<sub>1</sub>}} and {{math|''P'' + '''b'''<sub>2</sub>}} belong respectively to the intersection line, the first half plane, and the second half plane. The ''dihedral angle of these two half planes'' is defined by 
:<math>
\cos\varphi = \frac{ (\mathbf{b}_0 \times \mathbf{b}_1) \cdot (\mathbf{b}_0 \times \mathbf{b}_2)}{|\mathbf{b}_0 \times \mathbf{b}_1| |\mathbf{b}_0 \times \mathbf{b}_2|}</math>,
and satisfies <math>0\le\varphi <\pi.</math> In this case, switching the two half-planes gives the same result, and so does replacing <math>\mathbf b_0</math> with <math>-\mathbf b_0.</math> In chemistry (see below), we define a dihedral angle such that replacing <math>\mathbf b_0</math> with <math>-\mathbf b_0</math> changes the sign of the angle, which can be between {{math|−{{pi}}}} and {{math|{{pi}}}}.

==In polymer physics==
In some scientific areas such as [[polymer physics]], one may consider a chain of points and links between consecutive points. If the points are sequentially numbered and located at positions {{math|'''r'''<sub>1</sub>}}, {{math|'''r'''<sub>2</sub>}}, {{math|'''r'''<sub>3</sub>}}, etc. then bond vectors are defined by {{math|'''u'''<sub>1</sub>}}={{math|'''r'''<sub>2</sub>}}−{{math|'''r'''<sub>1</sub>}}, {{math|'''u'''<sub>2</sub>}}={{math|'''r'''<sub>3</sub>}}−{{math|'''r'''<sub>2</sub>}}, and {{math|'''u'''<sub>i</sub>}}={{math|'''r'''<sub>i+1</sub>}}−{{math|'''r'''<sub>i</sub>}}, more generally.<ref name="kroger">{{cite book|last=Kröger|first=Martin|title=Models for polymeric and anisotropic liquids|year=2005|publisher=Springer|isbn=3540262105}}</ref> This is the case for [[kinematic chain]]s or [[amino acid]]s in a [[protein structure]]. In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If {{math|'''u'''<sub>1</sub>}}, {{math|'''u'''<sub>2</sub>}} and {{math|'''u'''<sub>3</sub>}} are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval {{math|(−{{pi}}, {{pi}}]}}. This dihedral angle is defined by<ref>{{cite journal
|last1=Blondel |first1= Arnaud |last2=Karplus |first2=Martin |date= 7 Dec 1998 |title= New formulation for derivatives of torsion angles and improper torsion angles in molecular mechanics: Elimination of singularities |journal= Journal of Computational Chemistry
|volume= 17 |issue= 9 |pages= 1132–1141
|doi= 10.1002/(SICI)1096-987X(19960715)17:9<1132::AID-JCC5>3.0.CO;2-T
}}</ref>
:<math>\begin{align}
\cos \varphi&=\frac{ (\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)}{|\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|}\\
\sin \varphi&=\frac{ \mathbf{u}_2 \cdot((\mathbf{u}_1 \times \mathbf{u}_2) \times (\mathbf{u}_2 \times \mathbf{u}_3))}{|\mathbf{u}_2|\, |\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|},
\end{align}</math>
or, using the function [[atan2]],
:<math>\varphi=\operatorname{atan2}(\mathbf{u}_2 \cdot((\mathbf{u}_1 \times \mathbf{u}_2) \times (\mathbf{u}_2 \times \mathbf{u}_3)), |\mathbf{u}_2|\,(\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)).</math>

This dihedral angle does not depend on the orientation of the chain (order in which the point are considered) — reversing this ordering consists of replacing each vector by its opposite vector, and exchanging the indices 1 and 3. Both operations do not change the cosine, but change the sign of the sine. Thus, together, they do not change the angle.

A simpler formula for the same dihedral angle is the following (the proof is given below)
:<math>\begin{align}
\cos \varphi&=\frac{ (\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)}{|\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|}\\
\sin \varphi&=\frac{ |\mathbf{u}_2|\,\mathbf{u}_1 \cdot(\mathbf{u}_2 \times \mathbf{u}_3)}{|\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|},
\end{align}</math>
or equivalently,
:<math>\varphi=\operatorname{atan2}(
|\mathbf{u}_2|\,\mathbf{u}_1 \cdot(\mathbf{u}_2 \times \mathbf{u}_3) ,
(\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)).</math>

This can be deduced from previous formulas by using the [[vector quadruple product]] formula, and the fact that a [[scalar triple product]] is zero if it contains twice the same vector:
:<math>
 (\mathbf{u}_1\times\mathbf{u}_2)\times(\mathbf{u}_2\times\mathbf{u}_3) = [(\mathbf{u}_2\times\mathbf{u}_3)\cdot\mathbf{u}_1]\mathbf{u}_2 - [(\mathbf{u}_2\times\mathbf{u}_3)\cdot\mathbf{u}_2]\mathbf{u}_1
   = [(\mathbf{u}_2\times\mathbf{u}_3)\cdot\mathbf{u}_1]\mathbf{u}_2
</math>

Given the definition of the [[cross product]], this means that <math>\varphi</math> is the angle in the clockwise direction of the fourth atom compared to the first atom, while looking down the axis from the second atom to the third. Special cases (one may say the usual cases) are <math>\varphi = \pi</math>, <math>\varphi = +\pi/3</math> and <math>\varphi = -\pi/3</math>, which are called the ''trans'', ''gauche<sup>+</sup>'', and ''gauche<sup>−</sup>'' conformations.

==In stereochemistry==
{{duplication|section=y|dupe=Torsion angle}}
{{main|Torsion angle}}
{| style="margin: 0 auto;"
| [[File:Synantipericlinal.svg|200px]]
| [[File:Newman projection butane -sc.svg|200px]]
|[[File:Sawhorse projection butane -sc.svg|200px]]
|-
|Configuration names<br />according to dihedral angle
| ''syn'' ''n-''[[Butane]] in the<br />''gauche<sup>−</sup>'' conformation (−60°)<br /> [[Newman projection]]
| ''syn'' ''n-''[[Butane]]<br /> sawhorse projection
|}
[[File:Dihedral angles of Butane.svg|alt=|thumb|400x400px|Free energy diagram of [[butane|''n''-butane]] as a function of dihedral angle.]]
A [[torsion angle]], found in [[stereochemistry]], is a particular example of a dihedral angle describing the geometric relation of two parts of a molecule joined by a [[chemical bond]].<ref>{{GoldBookRef|file=T06406|title=Torsion angle}}</ref><ref>{{GoldBookRef|file=D01730|title=Dihedral angle}}</ref> Every set of three non-colinear atoms of a [[molecule]] defines a half-plane. As explained above, when two such half-planes intersect (i.e., a set of four consecutively-bonded atoms), the angle between them is a dihedral angle. Dihedral angles are used to specify the [[Conformational isomerism|molecular conformation]].<ref name="dougherty">{{cite book|last=Anslyn|first=Eric|title=Modern Physical Organic Chemistry|year=2006|publisher=University Science|isbn=978-1891389313|page=95|author2=Dennis Dougherty}}</ref> [[Stereochemical]] arrangements corresponding to angles between 0° and ±90° are called ''syn'' (s), those corresponding to angles between ±90° and 180° ''anti'' (a). Similarly, arrangements corresponding to angles between 30° and 150° or between −30° and −150° are called ''clinal'' (c) and those between 0° and ±30° or ±150° and 180° are called ''periplanar'' (p).

The two types of terms can be combined so as to define four ranges of angle; 0° to ±30° synperiplanar (sp); 30° to 90° and −30° to −90° synclinal (sc); 90° to 150° and −90° to −150° anticlinal (ac); ±150° to 180° antiperiplanar (ap). The synperiplanar conformation is also known as the ''syn''- or ''cis''-conformation; antiperiplanar as ''anti'' or ''trans''; and synclinal as ''gauche'' or ''skew''.

For example, with ''n''-[[butane]] two planes can be specified in terms of the two central carbon atoms and either of the methyl carbon atoms. The ''syn''-conformation shown above, with a dihedral angle of 60° is less stable than the ''anti''-conformation with a dihedral angle of 180°.

For macromolecular usage the symbols T, C, G<sup>+</sup>, G<sup>−</sup>, A<sup>+</sup> and A<sup>−</sup> are recommended (ap, sp, +sc, −sc, +ac and −ac respectively).

===Proteins===
{{anchor|Dihedral angles of biological molecules}}
[[Image:Protein backbone PhiPsiOmega drawing.svg|thumb|175px|Depiction of a [[protein]], showing where ω, φ, & ψ refer to.]]
A [[Ramachandran plot]] (also known as a Ramachandran diagram or a [''φ'',''ψ''] plot), originally developed in 1963 by [[G. N. Ramachandran]], C. Ramakrishnan, and V. Sasisekharan,<ref>{{cite journal |pages=95–9 |doi=10.1016/S0022-2836(63)80023-6 |title=Stereochemistry of polypeptide chain configurations |year=1963 |last1=Ramachandran |first1=G. N. |last2=Ramakrishnan |first2=C. |last3=Sasisekharan |first3=V. |journal=Journal of Molecular Biology |volume=7 |pmid=13990617}}</ref> is a way to visualize energetically allowed regions for backbone dihedral angles ''ψ'' against ''φ'' of [[amino acid]] residues in [[protein structure]].

In a [[protein]] chain three dihedral angles are defined:
* ω (omega) is the angle in the chain C<sup>α</sup> − C' − N − C<sup>α</sup>,
* φ (phi) is the angle in the chain C' − N − C<sup>α</sup> − C'
* ψ (psi) is the angle in the chain N − C<sup>α</sup> − C' − N (called ''φ′'' by Ramachandran)

The figure at right illustrates the location of each of these angles (but it does not show correctly the way they are defined).<ref>{{cite book |year=1981 |last1=Richardson |first1=J. S. |title=Anatomy and Taxonomy of Protein Structures |chapter=The Anatomy and Taxonomy of Protein Structure |volume=34 |pages=167–339 |doi=10.1016/S0065-3233(08)60520-3 |pmid=7020376 |series=Advances in Protein Chemistry |isbn=9780120342341}}</ref>

The planarity of the [[peptide bond]] usually restricts ''ω'' to be 180° (the typical ''[[Cis-trans isomerism|trans]]'' case) or 0° (the rare ''[[Cis-trans isomerism|cis]]'' case).  The distance between the C<sup>α</sup> atoms in the ''trans'' and ''cis'' [[geometric isomerism|isomers]] is approximately 3.8 and 2.9&nbsp;Å, respectively.  The vast majority of the peptide bonds in proteins are ''trans'', though the peptide bond to the nitrogen of [[proline]] has an increased prevalence of ''cis'' compared to other amino-acid pairs.<ref>{{Cite journal|vauthors=Singh J, Hanson J, Heffernan R, Paliwal K, Yang Y, Zhou Y|date=August 2018|title=Detecting Proline and Non-Proline Cis Isomers in Protein Structures from Sequences Using Deep Residual Ensemble Learning|journal=Journal of Chemical Information and Modeling|volume=58|issue=9|pages=2033–2042|doi=10.1021/acs.jcim.8b00442|pmid=30118602|s2cid=52031431}}</ref>

The side chain dihedral angles are designated with ''χ<sub>n</sub>'' (chi-''n'').<ref>{{Cite web|url=http://www.cryst.bbk.ac.uk/PPS95/course/3_geometry/conform.html|title=Side Chain Conformation}}</ref> They tend to cluster near 180°, 60°, and −60°, which are called the ''trans'', ''gauche<sup>−</sup>'', and ''gauche<sup>+</sup>'' conformations.  The stability of certain sidechain dihedral angles is affected by the values ''φ'' and ''ψ''.<ref>{{cite journal|last1=Dunbrack|first1=RL Jr.|last2=Karplus|first2=M|title=Backbone-dependent rotamer library for proteins. Application to side-chain prediction.|journal=Journal of Molecular Biology|date=20 March 1993|volume=230|issue=2|pages=543–74|pmid=8464064|doi=10.1006/jmbi.1993.1170}}</ref> For instance, there are direct steric interactions between the C''γ'' of the side chain in the ''gauche<sup>+</sup>'' rotamer and the backbone nitrogen of the next residue when ''ψ'' is near −60°.<ref>{{cite journal|last1=Dunbrack|first1=RL Jr|last2=Karplus|first2=M|title=Conformational analysis of the backbone-dependent rotamer preferences of protein sidechains.|journal=Nature Structural Biology|date=May 1994|volume=1|issue=5|pages=334–40|pmid=7664040|doi=10.1038/nsb0594-334|s2cid=9157373}}</ref> This is evident from statistical distributions in [[Backbone-dependent rotamer library|backbone-dependent rotamer libraries]].

Dihedral angles have also been defined by the [[IUPAC]] for other molecules, such as the [[nucleic acids]] ([[DNA]] and [[RNA]]) and for [[polysaccharides]].

==In polyhedra==
{{see also|Table of polyhedron dihedral angles}}

Every polyhedron has a dihedral angle at every edge describing the relationship of the two faces that share that edge. This dihedral angle, also called the ''face angle'', is measured as the [[internal angle]] with respect to the polyhedron. An angle of 0° means the face normal vectors are [[antiparallel vectors|antiparallel]] and the faces overlap each other, which implies that it is part of a [[Degeneracy (mathematics)|degenerate]] polyhedron. An angle of 180° means the faces are parallel, as in a [[List of uniform planar tilings|tiling]]. An angle greater than 180° exists on concave portions of a polyhedron.

Every dihedral angle in a polyhedron that is [[isotoxal figure|isotoxal]] and/or [[isohedral figure|isohedral]] has the same value. This includes the 5 [[Platonic solid]]s, the 13 [[Catalan solid]]s, the 4 [[Kepler–Poinsot polyhedron|Kepler–Poinsot polyhedra]], the 2 convex [[quasiregular polyhedron|quasiregular polyhedra]], and the 2 infinite families of [[bipyramid]]s and [[trapezohedron|trapezohedra]].

==Law of cosines for dihedral angle==
Given 3 faces of a polyhedron which meet at a common vertex P and have edges AP, BP and CP, the cosine of the dihedral angle between the faces containing APC and BPC is:<ref>{{cite web|title=dihedral angle calculator polyhedron|url=http://www.had2know.com/academics/dihedral-angle-calculator-polyhedron.html|website=www.had2know.com|access-date=25 October 2015|archive-url=https://web.archive.org/web/20151125044900/http://www.had2know.com/academics/dihedral-angle-calculator-polyhedron.html|archive-date=25 November 2015|url-status=dead}}</ref>
:<math>\cos\varphi = \frac{ \cos (\angle \mathrm{APB})  - \cos (\angle \mathrm{APC}) \cos (\angle \mathrm{BPC})}{ \sin(\angle \mathrm{APC}) \sin(\angle \mathrm{BPC})}  </math>
This can be deduced from the [[spherical law of cosines]], but can also be found by other means.<ref>{{cite web|title=Formula Derivations from Polyhedra|url=http://www.whistleralley.com/polyhedra/derivations.htm|access-date=4 December 2024}}</ref>

==Higher dimensions==
In {{mvar|m}}-dimensional Euclidean space, the dihedral angle {{tmath|\varphi}} between the two [[hyperplane]]s defined by the equations
<math display=block>\mathbf{n}_\mathrm{A} \cdot \mathbf{x} = c_A</math>
<math display=block>\mathbf{n}_\mathrm{B} \cdot \mathbf{x} = c_B</math>
for vectors {{math|'''n'''{{sub|A}}, '''n'''{{sub|B}}, '''x''' ∈ '''R'''{{sup|''m''}}}} and constants {{math|''c''{{sub|''A''}}}} and {{math|''c''{{sub|''B''}}}}, is given by
<math display=block>\cos \varphi =  \frac{ \left\vert\mathbf{n}_\mathrm{A} \cdot \mathbf{n}_\mathrm{B}\right\vert}{|\mathbf{n}_\mathrm{A} | |\mathbf{n}_\mathrm{B}|}\,.</math>

==See also==
*[[Atropisomer]]

== References ==
{{Reflist}}

==External links==
* [https://web.archive.org/web/20160303221511/http://tips.fm/entry.php?3972-Woodworking-The-Dihedral-Angle The Dihedral Angle in Woodworking at Tips.FM]
* [http://kjmaclean.com/Geometry/Platonic.html Analysis of the 5 Regular Polyhedra] gives a step-by-step derivation of these exact values.

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[[Category:Stereochemistry]]
[[Category:Protein structure]]
[[Category:Euclidean solid geometry]]
[[Category:Angle]]
[[Category:Planes (geometry)]]