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Quaternion

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The quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. The idea for quaternions occurred to him while he was walking along the Royal Canal on his way to a meeting of the Irish Academy, and Hamilton was so pleased with his discovery that he scratched the fundamental formula of quaternion algebra,

i^2=j^2=k^2=ijk=-1,
(1)

into the stone of the Brougham bridge (Mishchenko and Solovyov 2000). The set of quaternions is denoted H, H, or Q_8, and the quaternions are a single example of a more general class of hypercomplex numbers discovered by Hamilton. While the quaternions are not commutative, they are associative, and they form a group known as the quaternion group.

By analogy with the complex numbers being representable as a sum of real and imaginary parts, a·1+bi, a quaternion can also be written as a linear combination

H=a·1+bi+cj+dk.
(2)

The quaternion a+bi+cj+dk is implemented as Quaternion[a, b, c, d] in the Wolfram Language package Quaternions` where however a, b, c, and d must be explicit real numbers. Note also that NonCommutativeMultiply (i.e., **) must be used for multiplication of these objects rather than usual multiplication (i.e., *).

A variety of fractals can be explored in the space of quaternions. For example, fixing j=k=0 gives the complex plane, allowing the Mandelbrot set. By fixing j or k at different values, three-dimensional quaternionic fractals have been produced (Sandin et al. , Meyer 2002, Holdaway 2006).

The quaternions can be represented using complex 2×2 matrices

H=[z w; -w^_ z^_]=[a+ib c+id; -c+id a-ib],
(3)

where z and w are complex numbers, a, b, c, and d are real, and z^_ is the complex conjugate of z.

Quaternions can also be represented using the complex 2×2 matrices

U=[1 0; 0 1]
(4)
I=[i 0; 0 -i]
(5)
J=[0 1; -1 0]
(6)
K=[0 i; i 0]
(7)

(Arfken 1985, p. 185). Note that here U is used to denote the identity matrix, not I. The matrices are closely related to the Pauli matrices sigma_x, sigma_y, and sigma_z combined with the identity matrix.

From the above definitions, it follows that

I^2=-U
(8)
J^2=-U
(9)
K^2=-U.
(10)

Therefore I, J, and K are three essentially different solutions of the matrix equation

X^2=-U,
(11)

which could be considered the square roots of the negative identity matrix. A linear combination of basis quaternions with integer coefficients is sometimes called a Hamiltonian integer.

In R^4, the basis of the quaternions can be given by

i=[0 1 0 0; -1 0 0 0; 0 0 0 1; 0 0 -1 0]
(12)
j=[0 0 0 -1; 0 0 -1 0; 0 1 0 0; 1 0 0 0]
(13)
k=[0 0 -1 0; 0 0 0 1; 1 0 0 0; 0 -1 0 0]
(14)
1=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1].
(15)

The quaternions satisfy the following identities, sometimes known as Hamilton's rules,

i^2=j^2=k^2=-1
(16)
ij=-ji=k
(17)
jk=-kj=i
(18)
ki=-ik=j.
(19)

They have the following multiplication table.

1ijk
11ijk
ii-1k-j
jj-k-1i
kkj-i-1

The quaternions +/-1, +/-i, +/-j, and +/-k form a non-Abelian group of order eight (with multiplication as the group operation).

The quaternions can be written in the form

a=a_1+a_2i+a_3j+a_4k.
(20)

The quaternion conjugate is given by

a^_=a_1-a_2i-a_3j-a_4k.
(21)

The sum of two quaternions is then

a+b=(a_1+b_1)+(a_2+b_2)i+(a_3+b_3)j+(a_4+b_4)k,
(22)

and the product of two quaternions is

ab=(a_1b_1-a_2b_2-a_3b_3-a_4b_4)+(a_1b_2+a_2b_1+a_3b_4-a_4b_3)i+(a_1b_3-a_2b_4+a_3b_1+a_4b_2)j+(a_1b_4+a_2b_3-a_3b_2+a_4b_1)k.
(23)

The quaternion norm is therefore defined by

n(a)=sqrt(a^_a)=sqrt(a_1^2+a_2^2+a_3^2+a_4^2).
(24)

In this notation, the quaternions are closely related to four-vectors.

Quaternions can be interpreted as a scalar plus a vector by writing

a=a_1+a_2i+a_3j+a_4k=(a_1,a),
(25)

where a=[a_2a_3a_4]. In this notation, quaternion multiplication has the particularly simple form

q_1q_2=(s_1,v_1)·(s_2,v_2)
(26)
=(s_1s_2-v_1·v_2,s_1v_2+s_2v_1+v_1xv_2).
(27)

Division is uniquely defined (except by zero), so quaternions form a division algebra. The inverse (reciprocal) of a quaternion is given by

a^(-1)=(a^_)/([n(a)]^2),
(28)

and the norm is multiplicative

n(ab)=n(a)n(b).
(29)

In fact, the product of two quaternion norms immediately gives the Euler four-square identity.

A rotation about the unit vector n^^ by an angle theta can be computed using the quaternion

q=(s,v)=(cos(1/2theta),n^^sin(1/2theta))
(30)

(Arvo 1994, Hearn and Baker 1996). The components of this quaternion are called Euler parameters. After rotation, a point p=(0,p) is then given by

p^'=qpq^(-1)=qpq^_,
(31)

since n(q)=1. A concatenation of two rotations, first q_1 and then q_2, can be computed using the identity

q_2(q_1pq^__1)q^__2=(q_2q_1)p(q^__1q^__2)=(q_2q_1)pq_2q_1^_
(32)

(Goldstein 1980).

See also

Biquaternion, Cayley-Klein Parameters, Complex Number, Division Algebra, Euler Parameters, Four-Vector, Hamiltonian Integer, Hypercomplex Number, Octonion, Quaternion Conjugate, Quaternion Group, Quaternion Norm Explore this topic in the MathWorld classroom

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References

Altmann, S. L. Rotations, Quaternions, and Double Groups. Oxford, England: Clarendon Press, 1986.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Arvo, J. Graphics Gems II. New York: Academic Press, pp. 351-354 and 377-380, 1994.Baker, A. L. Quaternions as the Result of Algebraic Operations. New York: Van Nostrand, 1911.Conway, J. H. and Guy, R. K. The Book of Numbers. New York:Springer-Verlag, pp. 230-234, 1996.Conway, J. and Smith, D. On Quaternions and Octonions. Wellesley, MA: A K Peters, 2001.Crowe, M. J. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. New York: Dover, 1994.Dickson, L. E. Algebras and Their Arithmetics. New York: Dover, 1960.Downs, L. "CS184: Using Quaternions to Represent Rotation." http://www-inst.eecs.berkeley.edu/~cs184/sp08/lectures/05-3DTRansformations.pdf.Du Val, P. Homographies, Quaternions, and Rotations. Oxford, England: Oxford University Press, 1964.Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A; Koecher, M.; Mainzer, M.; and Remmert, R. Numbers. New York:Springer-Verlag, 1990.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, p. 151, 1980.Hamilton, W. R. Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method. Dublin: Hodges and Smith, 1853.Hamilton, W. R. Elements of Quaternions. London: Longmans, Green, 1866.Hamilton, W. R. The Mathematical Papers of Sir William Rowan Hamilton. Cambridge, England: Cambridge University Press, 1967.Hardy, A. S. Elements of Quaternions. Boston, MA: Ginn, Heath, & Co., 1881.Hardy, G. H. and Wright, E. M. "Quaternions." §20.6 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 303-306, 1979.Hearn, D. and Baker, M. P. Computer Graphics: C Version, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 419-420 and 617-618, 1996.Holdaway, L. "Quaternion Traversal." 2006. http://www.bluestarfolly.com/art/quaternion.html.Meyer, D. "Three-Dimensional Fractals (Quaternionic Fractals)." Nov. 10, 2002. http://www.physcip.uni-stuttgart.de/phy11733/index_e.html.Joly, C. J. A Manual of Quaternions. London: Macmillan, 1905.Julstrom, B. A. "Using Real Quaternions to Represent Rotations in Three Dimensions." UMAP Modules in Undergraduate Mathematics and Its Applications, Module 652. Lexington, MA: COMAP, Inc., 1992.Kelland, P. and Tait, P. G. Introduction to Quaternions, 3rd ed. London: Macmillan, 1904.Kuipers, J. B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton, NJ: Princeton University Press, 1998.Mishchenko, A. and Solovyov, Y. "Quaternions." Quantum 11, 4-7 and 18, 2000.Nicholson, W. K. Introduction to Abstract Algebra, 2nd ed. New York: Wiley, 1999.Salamin, G. Item 107 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, pp. 46-47, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/quaternions.html#item107.Sandin, D.; Dang, Y.; Kauffman, L.; and DeFanti, T. "A Diamond of Quaternionic Julia Sets." http://www.evl.uic.edu/hypercomplex/html/diamond.html.Shoemake, K. "Animating Rotation with Quaternion Curves." Computer Graphics 19, 245-254, 1985.Smith, H. J. "Quaternions for the Masses." http://www.geocities.com/hjsmithh/Quatdoc/Qindex.html.Tait, P. G. An Elementary Treatise on Quaternions, 3rd ed., enl. Cambridge, England: Cambridge University Press, 1890.Tait, P. G. "Quaternions." Encyclopædia Britannica, 9th ed. 1886. Reprinted in Tait, P. §CXXIX in Scientific Papers, Vol. 2. pp. 445-456. http://www.ldc.usb.ve/~vtheok/cursos/ci5322/quaternion/quaternions.pdf.Weisstein, E. W. "Books about Quaternions." http://www.ericweisstein.com/encyclopedias/books/Quaternions.html.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

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Quaternion

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Weisstein, Eric W. "Quaternion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quaternion.html

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