Dördey

Matematikte, dördeyler (ya da kvaterniyon, kuaternion, dördübir), karmaşık sayıları bir gerçel, üç sanal boyuta genişleten sayı sistemidir. İlk defa İrlandalı matematikçi Sir William Rowan Hamilton tarafından 1843 yılında tanımlanmış ve 3 boyutlu uzaydaki matematiğe uygulanmışlardır. Kuaterniyonlar değişme özelliğine (ab = ba) sahip değildir. Her ne kadar pek çok uygulamada vektörler ve matrisler 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 Cℓ0,2(R) = Cℓ03,0(R) olarak da gösterilirler. H cebirinin analizde önemli bir yeri vardır. Çünkü, Frobenius teoremi'ne göre, gerçel sayılar cismini althalka olarak içeren sonlu-boyutlu dört bölüm cebirinden bir tanesidir (diğerleri gerçel sayılar, karmaşık sayılar ve sekizeyler (octonions)).Dördeyler halkası, çarpma işleminin değişmeli olmaması yüzünden bir cisim değildir. Bir bölüm halkasıdır.olan satır çarpanı sol faktörü teşkil eder ve bir tablo olarak ifade edilebilir,bu yazının üstünde gösterildiği gibi kendilerinin sütunlari sağ faktörü teşkil eder.iki  elementler için ve , burada çarpıma, Hamilton çarpımı () () denir, taban ögeler ve dağılımsal kanunun çarpımları ile tanımlanıyor.Dağılım kanunu onu çarpımın açılımı için olası yapar böylece bu taban ögelerin çarpımlarının bir toplamıdır. Bu aşağıdaki bağıntılarla veriliyor:bir karmaşık analizin fonksiyonları gibi,  bir dördey değişkenin fonksiyonları kullanışlı fizik modelleri önerir.Örneğin, Maxwell tarafından tanıtılan orijinal elektrik ve manyetik alanlar bir dördey değişkenin fonksiyonları idi.Hamilton, William Rowan. On quaternions, or on a new system of imaginaries in algebra. Philosophical Magazine. Vol. 25, n 3. p. 489–495. 1844.Hamilton (1866) Elements of Quaternions  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..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 1-4020-2690-0Tait, Peter Guthrie (1886), "". M.A. Sec. R.S.E. Encyclopaedia Britannica, Ninth Edition, 1886, Vol. XX, pp. 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 //r84Macfarlane, Alexander (1906), "Vector analysis and quaternions", 4th ed. 1st thousand. New York, J. Wiley & Sons; [etc., etc.]. LCCN es 16000048Finkelstein, David, Josef M. Jauch, Samuel Schiminovich, and David Speiser (1962), "Foundations of quaternion quantum mechanics". J. Mathematical Phys. 3, pp. 207–220, MathSciNet.Du Val, Patrick (1964), "Homographies, quaternions, and rotations". Oxford, Clarendon Press (Oxford mathematical monographs). LCCN 64056979 //r81Crowe, 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 0-19-855372-2Altmann, Simon L. (1989), "Hamilton, Rodrigues, and the Quaternion Scandal". Mathematics Magazine. Vol. 62, No. 5. p. 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-XTrifonov, Vladimir (1995), "A Linear Solution of the Four-Dimensionality Problem", Europhysics Letters, 32 (8) 621–626,Ward, J. P. (1997), "Quaternions and Cayley Numbers: Algebra and Applications", Kluwer Academic Publishers. ISBN 0-7923-4513-4Kantor, I. L. and Solodnikov, A. S. (1989), "Hypercomplex numbers, an elementary introduction to algebras", Springer-Verlag, New York,  ISBN 0-387-96980-2Gürlebeck, 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 0-471-96200-7Kuipers, Jack (2002), "Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality" (reprint edition), Princeton University Press. ISBN 0-691-10298-8Conway, 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.Kravchenko, Vladislav (2003),  "Applied Quaternionic Analysis", Heldermann Verlag ISBN 3-88538-228-8.Hanson, Andrew J.  (2006), "Visualizing Quaternions", Elsevier: Morgan Kaufmann; San Francisco. ISBN 0-12-088400-3Trifonov, Vladimir (2007), "Natural Geometry of Nonzero Quaternions", International Journal of Theoretical Physics, 46 (2) 251–257,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.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. 21–57."Geometric Tools documentation" (frame ; body) includes several papers focusing on computer graphics applications of quaternions.  Covers useful techniques such as spherical linear interpolation.Patrick-Gilles Maillot  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" (frame; body ) includes free C++ source code for a complete quaternion class suitable for computer graphics work, under a very liberal license.Quaternion Math and Conversions  Great page explaining basic math with links to straight forward rotation conversion formulae.- Technical report and 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); ; E-print arxiv:0506177 .Johan E. Mebius, A matrix-based proof of the quaternion representation theorem for four-dimensional rotations. , arXiv General Mathematics 2005.Johan E. Mebius,  Derivation of the Euler–Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations. , arXiv General Mathematics 2007.David Erickson, Defence Research and Development Canada (DRDC), Complete derivation of rotation matrix from unitary quaternion representation in DRDC TR 2005-228 paper. Drdc-rddc.gc.caAlberto Martinez, University of Texas Department of History, "Negative Math, How Mathematical Rules Can Be Positively Bent",Utexas.eduMorier-Genoud, Sophie, and Valentin Ovsienko. "Well, Papa, can you multiply triplets?", arxiv.org   describes how the quaternions can be made into a skew-commutative algebra graded by .R. Ghiloni, V. Moretti, A. Perotti (2013) "Continuous slice functional calculus in quaternionic Hilbert spaces, " Rev.Math.Phys. 25 1350006. An expository paper about continuous functional calculus in quanternionic Hilbert spaces useful in rigorous quaternionic quantum mechanics.