Dördey sayfasının kaynağını görüntüle

[[Dosya:William Rowan Hamilton Plaque - geograph.org.uk - 347941.jpg|küçükresim|William Rowan Hamilton]]
[[Matematik]]te, '''dördeyler''' (ya da kvaterniyon, kuaternion, dördübir), [[karmaşık sayı]]ları bir gerçel, üç sanal boyuta genişleten [[sayı sistemi]]dir. İlk defa [[İrlanda]]lı matematikçi Sir [[William Rowan Hamilton]] tarafından 1843 yılında tanımlanmış ve 3 boyutlu uzaydaki matematiğe uygulanmışlardır. Kuaterniyonlar [[değişme özelliği]]ne (''ab'' = ''ba'') sahip değildir. Her ne kadar pek çok uygulamada [[vektör]]ler ve [[matris]]ler dördeylerin yerini almışsa da, kuramsal ve uygulamalı matematikte hala kullanılmaktadırlar. Başlıca kullanım alanı, 3 boyutlu uzayda dönme hareketinin hesaplanmasıdır.

Dördey cebiri genellikle '''H''' (Hamilton) ile gösterilir. [[Clifford cebiri]] sınıflandırması ''C''ℓ<sub>0,2</sub>('''R''') = ''C''ℓ<sup>0</sup><sub>3,0</sub>('''R''') olarak da gösterilirler. '''H''' cebirinin analizde önemli bir yeri vardır. Çünkü, [[Frobenius teoremi (gerçel bölüm cebirleri)|Frobenius teoremi]]'ne göre, [[gerçel sayılar]] cismini [[althalka]] olarak içeren sonlu-boyutlu dört [[bölüm cebiri]]nden bir tanesidir (diğerleri gerçel sayılar, [[karmaşık sayılar]] ve [[sekizey]]ler (octonions)).

== Tanım ==
Dördeyler bir [[Halka (Cebir)|halka]] olarak tanımlanır. Kümesi:

:<math>\mathbb{H}=\{a+bi+cj+dk | a,b,c,d\in\mathbb{R}\}</math>.

olarak verilir. Burada kullanılan toplama şu şekilde tanımlıdır:

:<math>(a_1+b_1i+c_1j+d_1k)+(a_2+b_2i+c_2j+d_2k)\,</math>
:::<math>=(a_1+a_2)+(b_1+b_2)i+(c_1+c_2)j+(d_1+d_2)k\,</math>

Çarpma ise

:<math>(a_1+b_1i+c_1j+d_1k)(a_2+b_2i+c_2j+d_2k)\,</math>

ifadesinin dağıtma kuralı kullanılarak açılmasıyla ve aşağıdaki bağıntılar yardımıyla tanımlanır.

:<math> i^2 = j^2 = k^2 = ijk = -1,\,</math>

Her dördey tektir ve temel dördeylerin, yani 1, ''i'', ''j'' ve ''k'' nin gerçel doğrusal birleşimidir.

Dördeyler halkası, çarpma işleminin değişmeli olmaması yüzünden bir cisim değildir. Bir [[Bölüm Halkası|bölüm halkasıdır]].

Aynı zamanda, dördeyler, gerçel sayılar üzerinde bir bölüm cebiri oluşturur. Gerçel sayılar ve
karmaşık sayılarla birlikte, gerçelleri içeren [[Birleşmelilik|birleşmeli]] üç bölüm cebirinden biridir.
===Taban ögelerinin çarpımı===
denklikler
:<math>i^2=j^2=k^2=ijk=-1</math>,

burada ''i'', ''j'' ve ''k'' '''H '''nın taban ögeleridir,''i'', ''j'' ve ''k ''nın tüm olası çarpanlarını belirtir .

örneğin  {{Kayma|1=−1 = ''ijk''}} nın sağ çarpanlarının her ikisi de ''k''  ile verilir
:<math>\begin{align}
-k & = i j k k = i j (k^2) = i j (-1), \\
 k & = i j.
\end{align}</math>
Diğer tüm olası çarpanlar  benzer yöntemlerle belirlenebilir
:<math>\begin{alignat}{2}
ij & = k, & \qquad ji & = -k, \\
jk & = i, & kj & = -i, \\
ki & = j, & ik & = -j,
\end{alignat}</math>
olan satır çarpanı sol faktörü teşkil eder ve bir tablo olarak ifade edilebilir, bu yazının üstünde gösterildiği gibi kendilerinin sütunlari sağ faktörü teşkil eder.
=== Hamilton çarpımı ===
iki {{Kayma|''a''<sub>1</sub> + ''b''<sub>1</sub>''i'' + ''c''<sub>1</sub>''j'' + ''d''<sub>1</sub>''k''}} elementler için ve {{Kayma|''a''<sub>2</sub> + ''b''<sub>2</sub>''i'' + ''c''<sub>2</sub>''j'' + ''d''<sub>2</sub>''k''}}, burada çarpıma, '''Hamilton çarpımı''' ({{Kayma|''a''<sub>1</sub> + ''b''<sub>1</sub>''i'' + ''c''<sub>1</sub>''j'' + ''d''<sub>1</sub>''k''}}) ({{Kayma|''a''<sub>2</sub> + ''b''<sub>2</sub>''i'' + ''c''<sub>2</sub>''j'' + ''d''<sub>2</sub>''k''}}) denir, taban ögeler ve [[dağılımsal kanun]]un çarpımları ile tanımlanıyor.Dağılım kanunu onu çarpımın açılımı için olası yapar böylece bu taban ögelerin çarpımlarının bir toplamıdır. Bu aşağıdaki bağıntılarla veriliyor:
:<math>a_1a_2 + a_1b_2i + a_1c_2j + a_1d_2k</math>
:<math>{}+ b_1a_2i + b_1b_2i^2 + b_1c_2ij + b_1d_2ik</math>
:<math>{}+ c_1a_2j + c_1b_2ji + c_1c_2j^2 + c_1d_2jk</math>
:<math>{}+ d_1a_2k + d_1b_2ki + d_1c_2kj + d_1d_2k^2.</math>
Şimdi taban elemanları kullanılarak elde etmek için yukarıda verilen kuralları  çoğaltılabilir:<ref name="SeeHazewinkel">Hazewinkel (2004), ss. 12.</ref>
:<math>a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2</math>
:<math>{}+ (a_1b_2 + b_1a_2 + c_1d_2 - d_1c_2)i</math>
:<math>{}+ (a_1c_2 - b_1d_2 + c_1a_2 + d_1b_2)j</math>
:<math>{}+ (a_1d_2 + b_1c_2 - c_1b_2 + d_1a_2)k.</math>
=== Sıralı liste formu ===
'''H'''nın  1, ''i'', ''j'', ''k'' tabanları kullanılıyor [[tuple|dört kat]]ının bir kümesi olarak '''H''' yazmak için mümkün kılar:
:<math>\mathbf{H} = \{(a, b, c, d) \mid a, b, c, d \in \mathbf{R}\}.</math>
ise taban  ögeleri:
:<math>
\begin{align}
1 & = (1, 0, 0, 0), \\
i & = (0, 1, 0, 0), \\
j & = (0, 0, 1, 0), \\
k & = (0, 0, 0, 1),
\end{align}
</math>
ve toplam ve çarpım için formüller:

:<math>(a_1,\ b_1,\ c_1,\ d_1) + (a_2,\ b_2,\ c_2,\ d_2) = (a_1 + a_2,\ b_1 + b_2,\ c_1 + c_2,\ d_1 + d_2).</math>
:<math>\begin{align}
(a_1,\ b_1,\ c_1,\ d_1)&(a_2,\ b_2,\ c_2,\ d_2) = \\
& = (a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2, \\
& {} \qquad a_1b_2 + b_1a_2 + c_1d_2 - d_1c_2, \\
& {} \qquad a_1c_2 - b_1d_2 + c_1a_2 + d_1b_2, \\
& {} \qquad a_1d_2 + b_1c_2 - c_1b_2 + d_1a_2).
\end{align}</math>

==Dördey değişkenlerinin bir fonksiyonu==
{{Ana|Dördey analizi}}
bir [[karmaşık analiz]]in fonksiyonları gibi, bir dördey değişkenin fonksiyonları kullanışlı fizik modelleri önerir.Örneğin, Maxwell tarafından tanıtılan orijinal elektrik ve manyetik alanlar bir dördey değişkenin fonksiyonları idi.

=== Üstel, logaritma ve kuvvet ===
bir dördey veriliyor,
:''q'' = ''a'' + ''bi'' + ''cj'' + ''dk'' = ''a'' + '''v''',
üstel
:<math>\exp(q) = \sum_{n=0}^\infty \frac{q^n}{n!}=e^{a} \left(\cos \|\mathbf{v}\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \sin \|\mathbf{v}\|\right) </math>
olarak hesaplanıyor ve
:<math>\ln(q) = \ln \|q\| + \frac{\mathbf{v}}{\|\mathbf{v}\|} \arccos \frac{a}{\|q\|}</math>.<ref>{{Web kaynağı |url=http://www.lce.hut.fi/~ssarkka/pub/quat.pdf |başlık=Lce.hut.fi |erişimtarihi=30 Temmuz 2014 |arşivurl=https://web.archive.org/web/20130926165618/http://www.lce.hut.fi/~ssarkka/pub/quat.pdf |arşivtarihi=26 Eylül 2013 |ölüurl=evet}}</ref>
bir dördeyin kutupsal çözülümünü aşağıdaki gibi yazabiliriz
:<math>q=\|q\|e^{\hat{n}\theta} = \|q\| \left(\cos(\theta) + \hat{n} \sin(\theta)\right),</math>

burada [[açı]] θ ve birim vektör <math>\hat{n}</math> ile tanımlanıyor:
:<math>a=\|q\|\cos(\theta)</math>
ve
:<math>\mathbf{v}=\hat{n} \|\mathbf{v}\|=\hat{n}\|q\|\sin(\theta).</math>
Herhangi birim dördey  <math>e^{\hat{n}\theta}</math>.olan kutupsal biçim içinde ifade edilebilir

bir keyfi (gerçek) üstel <math>\alpha</math>  için bir  yükselen  dördeyin [[Power (mathematics)|kuvveti]] ile veriliyor:
:<math>q^\alpha=\|q\|^\alpha e^{\hat{n}\alpha\theta} = \|q\|^\alpha \left(\cos(\alpha\theta) + \hat{n} \sin(\alpha\theta)\right).</math>

==Ayrıca bakınız==
{{Div sütunu}}
* [[3-küre]]
* [[Birleşmeli cebir]]
* [[Bikuaterniyon]]
* [[Clifford cebiri]]
* [[Karmaşık sayı]]
* [[Kuaterniyonlar ve Euler açıları arasındaki dönüşüm]]
* [[Bölme cebiri]]
* [[Dual Dördey]]
* [[Euler açıları]]
* [[Dış cebir]]
* [[Geometrik cebir]]
* [[Hurwitz Dördey]]
* [[Hurwitz Dördey düzeni]]
* [[Hiperbolik Dördey]]
* [[hiperkarmaşık sayı]]
* [[Lénárt küresi]]
* [[Oktonyon]]
* [[Pauli matrisleri]]
* [[Kuaterniyon grubu]]
* [[Kuaterniyon değişkeni]]
* [[Kuaterniyonik matris]]
* [[Kuaterniyonlar ve mekansal dönme]]
* [[Dönme operatörü (vektör uzayı)]]
* [[4-boyutlu Öklid uzayında dönmeler]]
* [[Slerp]]
* [[Bölünmüş-Dördey]]
* [[Teserakt]]
{{Div sütunu bitiş}}

==Notlar==
{{Kaynakça}}

==Dış makaleler ve kaynaklar==
{{Vikisözlük}}

===Kitaplar ve yayınlar===
* [[William Rowan Hamilton|Hamilton, William Rowan]]. [https://web.archive.org/web/20120227160059/http://www.emis.ams.org/classics/Hamilton/OnQuat.pdf On quaternions, or on a new system of imaginaries in algebra]. Philosophical Magazine. Vol. 25, n 3. p.&nbsp;489–495. 1844.
* [[William Rowan Hamilton|Hamilton, William Rowan]] (1853), "''[http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 Lectures on Quaternions] {{Webarşiv|url=https://web.archive.org/web/20060511175326/http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05230001&seq=9 |tarih=11 Mayıs 2006 }}''". Royal Irish Academy.
* Hamilton (1866) ''[http://books.google.com/books?id=fIRAAAAAIAAJ Elements of Quaternions] {{Webarşiv|url=https://web.archive.org/web/20140712121828/http://books.google.com/books?id=fIRAAAAAIAAJ |tarih=12 Temmuz 2014 }}'' [[University of Dublin]] Press. Edited by William Edwin Hamilton, son of the deceased author.
* Hamilton (1899) ''Elements of Quaternions'' volume I, (1901) volume II. Edited by [[Charles Jasper Joly]]; published by [[Longmans, Green & Co.]].
* [[Peter Guthrie Tait|Tait, Peter Guthrie]] (1873), "''An elementary treatise on quaternions''". 2d ed., Cambridge, [Eng.]: The University Press.
* [[Michiel Hazewinkel]], Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
* Maxwell, James Clerk (1873), "''[[A Treatise on Electricity and Magnetism]]''". Clarendon Press, Oxford.
* [[Peter Guthrie Tait|Tait, Peter Guthrie]] (1886), "''{{wayback|url=http://www.ugcs.caltech.edu/~presto/papers/Quaternions-Britannica.ps.bz2|title=Quaternion|tarih=20140808040037}}''". M.A. Sec. R.S.E. [[Encyclopaedia Britannica]], Ninth Edition, 1886, Vol. XX, pp.&nbsp;160–164. (bzipped [[PostScript]] file)
* Joly, Charles Jasper (1905), "''A manual of quaternions''". London, Macmillan and co., limited; New York, The Macmillan company. LCCN 05036137 //r84
* [[Alexander Macfarlane|Macfarlane, Alexander]] (1906), "''Vector analysis and quaternions''", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048
* [[1911 encyclopedia]]: "''[http://www.1911encyclopedia.org/Quaternions Quaternions] {{Webarşiv|url=https://web.archive.org/web/20120507012802/http://www.1911encyclopedia.org/Quaternions |tarih=7 Mayıs 2012 }}''".
* Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "''Foundations of quaternion quantum mechanics''". J. Mathematical Phys. 3, pp.&nbsp;207–220, MathSciNet.
* [[Patrick du Val|Du Val, Patrick]] (1964), "''Homographies, quaternions, and rotations''". Oxford, Clarendon Press (Oxford mathematical monographs). LCCN 64056979 //r81
* Crowe, Michael J. (1967), [[A History of Vector Analysis]]: ''The Evolution of the Idea of a Vectorial System'', University of Notre Dame Press. Surveys the major and minor vector systems of the 19th century (Hamilton, Möbius, Bellavitis, Clifford, Grassmann, Tait, Peirce, Maxwell, Macfarlane, MacAuley, Gibbs, Heaviside).
* Altmann, Simon L. (1986), "''Rotations, quaternions, and double groups''". Oxford [Oxfordshire]: Clarendon Press; New York: [[Oxford University Press]]. LCCN 85013615 ISBN 0-19-855372-2
* Altmann, Simon L. (1989), "''Hamilton, Rodrigues, and the Quaternion Scandal''". Mathematics Magazine. Vol. 62, No. 5. p.&nbsp;291–308, December  1989.
* Adler, Stephen L. (1995), "''Quaternionic quantum mechanics and quantum fields''". New York: Oxford University Press. International series of monographs on physics (Oxford, England) 88. LCCN 94006306 ISBN 0-19-506643-X
* [https://web.archive.org/web/20110805044706/http://members.cox.net/vtrifonov/ Trifonov, Vladimir] (1995), "''A Linear Solution of the Four-Dimensionality Problem''", Europhysics Letters, '''32 (8)''' 621–626, {{doi|10.1209/0295-5075/32/8/001}}
* Ward, J. P. (1997), "''Quaternions and Cayley Numbers: Algebra and Applications''", Kluwer Academic Publishers. ISBN 0-7923-4513-4
* Kantor, I. L. and Solodnikov, A. S. (1989), "''Hypercomplex numbers, an elementary introduction to algebras''", Springer-Verlag, New York, ISBN 0-387-96980-2
* Gürlebeck, Klaus and Sprössig, Wolfgang (1997), "''Quaternionic and Clifford calculus for physicists and engineers''". Chichester; New York: Wiley (Mathematical methods in practice; v. 1). LCCN 98169958 ISBN 0-471-96200-7
* Kuipers, Jack (2002), "''Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality''" (reprint edition), [[Princeton University Press]]. ISBN 0-691-10298-8
* [[John Horton Conway|Conway, John Horton]], and Smith, Derek A. (2003), "''On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry''", A. K. Peters, Ltd. ISBN 1-56881-134-9.
* [[Vladislav Kravchenko|Kravchenko, Vladislav]] (2003), "''Applied Quaternionic Analysis''", Heldermann Verlag ISBN 3-88538-228-8.
* [http://www.cs.indiana.edu/~hanson/quatvis/ Hanson, Andrew J.] {{Webarşiv|url=https://web.archive.org/web/20071227231029/http://www.cs.indiana.edu/~hanson/quatvis/ |tarih=27 Aralık 2007 }} (2006), "''Visualizing Quaternions''", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3
* [https://web.archive.org/web/20110805044706/http://members.cox.net/vtrifonov/ Trifonov, Vladimir] (2007), "''Natural Geometry of Nonzero Quaternions''", International Journal of Theoretical Physics, '''46 (2)''' 251–257, {{doi|10.1007/s10773-006-9234-9}}
* Ernst Binz & Sonja Pods (2008) ''Geometry of Heisenberg Groups'' [[American Mathematical Society]], Chapter 1: "The Skew Field of Quaternions" (23 pages) ISBN 978-0-8218-4495-3.
* Vince, John A. (2008), ''Geometric Algebra for Computer Graphics'', Springer, ISBN 978-1-84628-996-5.
* For molecules that can be regarded as classical rigid bodies [[molecular dynamics]] computer simulation employs quaternions. They were first introduced for this purpose by D.J. Evans, (1977), "On the Representation of Orientation Space", Mol. Phys., vol 34, p 317.
* Zhang, Fuzhen (1997), "Quaternions and Matrices of Quaternions", Linear Algebra and its Applications, Vol. 251, pp.&nbsp;21–57.

===Bağlantılar ve uzman yazıları===
* {{springer|title=Quaternion|id=p/q076770}}
* [http://www.j3d.org/matrix_faq/matrfaq_latest.html Matrix and Quaternion FAQ v1.21] {{Webarşiv|url=https://web.archive.org/web/20071216113340/http://www.j3d.org/matrix_faq/matrfaq_latest.html |tarih=16 Aralık 2007 }} Frequently Asked Questions
* "Geometric Tools documentation" ([http://www.geometrictools.com/Documentation/Documentation.html frame] {{Webarşiv|url=https://web.archive.org/web/20071028013932/http://www.geometrictools.com/Documentation/Documentation.html |tarih=28 Ekim 2007 }}; [https://web.archive.org/web/20120220062435/http://www.geometrictools.com/Documentation/DocumentationBody.html body]) includes several papers focusing on computer graphics applications of quaternions. Covers useful techniques such as spherical linear interpolation.
* [https://sites.google.com/site/patrickmaillot/english Patrick-Gilles Maillot] {{Webarşiv|url=https://web.archive.org/web/20140712110333/https://sites.google.com/site/patrickmaillot/english |tarih=12 Temmuz 2014 }} Provides free [[Fortran]] and C source code for manipulating quaternions and rotations / position in space. Also includes mathematical background on quaternions.
* "Geometric Tools source code" ([https://web.archive.org/web/20100111025321/http://www.geometrictools.com/LibFoundation/Mathematics/Mathematics.html frame]; [http://www.geometrictools.com/LibFoundation/Mathematics/MathematicsBody.html body] {{Webarşiv|url=https://web.archive.org/web/20100111025321/http://www.geometrictools.com/LibFoundation/Mathematics/MathematicsBody.html |tarih=11 Ocak 2010 }}) includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.
* Doug Sweetser, [http://world.std.com/~sweetser/quaternions/qindex/qindex.html Doing Physics with Quaternions] {{Webarşiv|url=https://web.archive.org/web/20031204225515/http://world.std.com/~sweetser/quaternions/qindex/qindex.html |tarih=4 Aralık 2003 }}
* [https://web.archive.org/web/20120515203342/http://www.fho-emden.de/~hoffmann/quater12012002.pdf Quaternions for Computer Graphics and Mechanics (Gernot Hoffman)]
* [http://arxiv.org/pdf/math-ph/0201058 The Physical Heritage of Sir W. R. Hamilton] (PDF)
* D. R. Wilkins, [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html Hamilton’s Research on Quaternions] {{Webarşiv|url=https://web.archive.org/web/20071224082631/http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html |tarih=24 Aralık 2007 }}
* [http://www.unpronounceable.com/julia/ Quaternion Julia Fractals] {{Webarşiv|url=https://web.archive.org/web/20071215071120/http://www.unpronounceable.com/julia/ |tarih=15 Aralık 2007 }} 3D Raytraced Quaternion [[Julia set|Julia Fractals]] by David J. Grossman
* [http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/index.htm Quaternion Math and Conversions] {{Webarşiv|url=https://web.archive.org/web/20070927221021/http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/index.htm |tarih=27 Eylül 2007 }} Great page explaining basic math with links to straight forward rotation conversion formulae.
* John H. Mathews, [https://web.archive.org/web/20121104060352/http://math.fullerton.edu/mathews/c2003/QuaternionBib/Links/QuaternionBib_lnk_3.html Bibliography for Quaternions].
* [https://web.archive.org/web/20060423132728/http://www.gamedev.net/reference/articles/article1095.asp Quaternion powers on GameDev.net]
* Andrew Hanson, [https://web.archive.org/web/20070521234919/http://books.elsevier.com/companions/0120884003/vq/index.html Visualizing Quaternions home page].
* {{Web kaynağı | url = http://ai.stanford.edu/~diebel/attitude.html | başlık = Representing Attitude with Euler Angles and Quaternions: A Reference | soyadı = Diebel | ad = James | tarih =  | eser =  | yayıncı = Stanford University | arşivurl = https://web.archive.org/web/20080705044150/http://ai.stanford.edu/~diebel/attitude.html | arşivtarihi = 5 Temmuz 2008 | ölüurl =evet| erişimtarihi = 22 Aralık 2007 }} - Technical report and [[MATLAB|Matlab]] toolbox summarizing all common attitude representations, with detailed equations and discussion on features of various methods.
* Charles F. F. Karney, ''Quaternions in molecular modeling'', J. Mol. Graph. Mod. '''25'''(5), 595–604 (January  2007); {{doi|10.1016/j.jmgm.2006.04.002}}; E-print [http://arxiv.org/abs/physics/0506177 arxiv:0506177] {{Webarşiv|url=https://web.archive.org/web/20150904052651/http://arxiv.org/abs/physics/0506177 |tarih=4 Eylül 2015 }}.
* Johan E. Mebius, [http://arxiv.org/abs/math/0501249 A matrix-based proof of the quaternion representation theorem for four-dimensional rotations.] {{Webarşiv|url=https://web.archive.org/web/20150904052651/http://arxiv.org/abs/math/0501249 |tarih=4 Eylül 2015 }}, ''arXiv General Mathematics'' 2005.
* Johan E. Mebius, [http://arxiv.org/abs/math/0701759  Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations.] {{Webarşiv|url=https://web.archive.org/web/20150904052651/http://arxiv.org/abs/math/0701759 |tarih=4 Eylül 2015 }}, ''arXiv General Mathematics'' 2007.
* [[NUI Maynooth]] [http://www.maths.nuim.ie/links/hamilton.shtml Department of Mathematics, Hamilton Walk] {{Webarşiv|url=https://web.archive.org/web/20071122015142/http://www.maths.nuim.ie/links/hamilton.shtml |tarih=22 Kasım 2007 }}.
* [https://web.archive.org/web/20071215235040/http://gpwiki.org/index.php/OpenGL:Tutorials:Using_Quaternions_to_represent_rotation OpenGL:Tutorials:Using Quaternions to represent rotation]
* David Erickson, [[Defence Research and Development Canada]] (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper. [https://web.archive.org/web/20080228100448/http://aiss.suffield.drdc-rddc.gc.ca/uploads/quaternion.pdf Drdc-rddc.gc.ca]
* Alberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent",[https://web.archive.org/web/20120120160608/https://webspace.utexas.edu/aam829/1/m/NegativeMath.html Utexas.edu]
* D. Stahlke, Quaternions in Classical Mechanics [http://www.stahlke.org/dan/phys-papers/quaternion-paper.pdf Stahlke.org] {{Webarşiv|url=https://web.archive.org/web/20120915114758/http://www.stahlke.org/dan/phys-papers/quaternion-paper.pdf |tarih=15 Eylül 2012 }} (PDF)
* Morier-Genoud, Sophie, and Valentin Ovsienko. "Well, Papa, can you multiply triplets?", [http://arxiv.org/abs/0810.5562 arxiv.org] {{Webarşiv|url=https://web.archive.org/web/20150904052651/http://arxiv.org/abs/0810.5562 |tarih=4 Eylül 2015 }}  describes how the quaternions can be made into a skew-commutative algebra graded by {{Kayma|'''Z'''/2 × '''Z'''/2 × '''Z'''/2}}.
* [http://plus.maths.org/content/os/issue32/features/baez/index Curious Quaternions] {{Webarşiv|url=https://web.archive.org/web/20140709070210/http://plus.maths.org/content/os/issue32/features/baez/index |tarih=9 Temmuz 2014 }} by Helen Joyce hosted by [[John Baez]].
* Luis Ibanez "Tutorial on Quaternions" [https://web.archive.org/web/20120204055438/http://www.itk.org/CourseWare/Training/QuaternionsI.pdf Part I] [https://web.archive.org/web/20121005003247/http://www.itk.org/CourseWare/Training/QuaternionsII.pdf Part II] (PDF)
* R. Ghiloni, V. Moretti, A. Perotti (2013) "[http://arxiv.org/pdf/1207.0666.pdf Continuous slice functional calculus in quaternionic Hilbert spaces,] {{Webarşiv|url=https://web.archive.org/web/20150904052651/http://arxiv.org/pdf/1207.0666.pdf |tarih=4 Eylül 2015 }}" Rev.Math.Phys. 25 1350006. An expository paper about continuous functional calculus in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.
* [http://www.cs.indiana.edu/~hanson/quatvis/ Visualizing Quaternions] {{Webarşiv|url=https://web.archive.org/web/20071227231029/http://www.cs.indiana.edu/~hanson/quatvis/ |tarih=27 Aralık 2007 }} by Andrew J. Hanson at Indiana University in Bloomington.

{{Number Systems}}

{{sayılar}}

{{Otorite kontrolü}}

[[Kategori:Dördeyler| ]]