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Quaternion

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(Redirected fromQuaternions)
Noncommutative extension of the complex numbers
This article is about quaternions in mathematics. For other uses, seeQuaternion (disambiguation).
Quaternion multiplication table
↓ × →1ijk
11ijk
ii−1k−j
jj−k−1i
kkj−i−1
Left column shows the left factor, top row shows the right factor. Also,ab=ba{\displaystyle a\mathbf {b} =\mathbf {b} a} and−b=(−1)b{\displaystyle -\mathbf {b} =(-1)\mathbf {b} } fora∈R{\displaystyle a\in \mathbb {R} },b=i,j,k{\displaystyle \mathbf {b} =\mathbf {i} ,\mathbf {j} ,\mathbf {k} }.
Cayley Q8 graph showing the six cycles of multiplication byi,j andk. (If the image is opened in theWikimedia Commons by clicking twice on it, cycles can be highlighted by hovering over or clicking on them.)

Inmathematics, thequaternionnumber system extends thecomplex numbers. Quaternions were first described by the Irish mathematicianWilliam Rowan Hamilton in 1843[1][2] and applied tomechanics inthree-dimensional space. The set of all quaternions is conventionally denoted by H {\displaystyle \ \mathbb {H} \ } ('H' forHamilton), or ifblackboard bold is not available, byH. Quaternions are not quite afield because, in general, multiplication of quaternions is notcommutative. Quaternions provide a definition of the quotient of twovectors in a three-dimensional space.[3][4] Quaternions are generally represented in the form

a+bi+cj+dk,{\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} ,}

where the coefficientsa,b,c,d arereal numbers, and1,i,j,k are thebasis vectors orbasis elements.[5]

Quaternions are used inpure mathematics, but also have practical uses inapplied mathematics, particularly forcalculations involving three-dimensional rotations, such as inthree-dimensional computer graphics,computer vision, robotics,magnetic resonance imaging[6] andcrystallographic texture analysis.[7] They can be used alongside other methods of rotation, such asEuler angles androtation matrices, or as an alternative to them, depending on the application.

In modern terms, quaternions form a four-dimensionalassociativenormeddivision algebra over the real numbers, and therefore a ring, also adivision ring and adomain. It is a special case of aClifford algebra,classified asCl0,2⁡(R)≅Cl3,0+⁡(R).{\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} It was the first noncommutative division algebra to be discovered.

According to theFrobenius theorem, the algebraH{\displaystyle \mathbb {H} } is one of only two finite-dimensionaldivision rings containing a propersubringisomorphic to the real numbers; the other being the complex numbers. These rings are alsoEuclidean Hurwitz algebras, of which the quaternions are the largestassociative algebra (and hence the largest ring). Further extending the quaternions yields thenon-associativeoctonions, which is the lastnormed division algebra over the real numbers. The next extension gives thesedenions, which havezero divisors and so cannot be a normed division algebra.[8]

Theunit quaternions give agroup structure on the3-sphereS3 isomorphic to the groupsSpin(3) andSU(2), i.e. theuniversal cover group ofSO(3). The positive and negative basis vectors form the eight-elementquaternion group.

Graphical representation of products of quaternion units as 90° rotations in the planes of 4-dimensional space spanned by two of{1,i,j,k}. The left factor can be viewed as being rotated by the right factor to arrive at the product. Visuallyi ⋅ j = −(j ⋅ i).
  • Inblue:
    • 1 ⋅ i = i (1/i plane)
    • i ⋅ j = k (i/k plane)
  • Inred:
    • 1 ⋅ j = j (1/j plane)
    • j ⋅ i = −k (j/k plane)

History

[edit]
Main article:History of quaternions
Quaternion plaque onBrougham (Broom) Bridge,Dublin, which reads:

     Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered
the fundamental formula for
quaternion multiplication
     i2 =j2 =k2 =i j k = −1
& cut it on a stone of this bridge

Quaternions were introduced by Hamilton in 1843.[9] Important precursors to this work includedEuler's four-square identity (1748) andOlinde Rodrigues'parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra.[10][11][a]Gauss had discovered quaternions in 1819, but this work was not published until 1900.[12][13]

Hamilton knew that the complex numbers could be interpreted aspoints in aplane, and he was looking for a way to do the same for points in three-dimensionalspace. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication and division. He could not figure out how to calculate the quotient of the coordinates of two points in space. In fact,Ferdinand Georg Frobenius laterproved in 1877 that for a division algebra over the real numbers to be finite-dimensional and associative, it cannot be three-dimensional, and there are only three such division algebras:R,C{\displaystyle \mathbb {R,C} } (complex numbers) andH{\displaystyle \mathbb {H} } (quaternions) which have dimension 1, 2, and 4 respectively.[citation needed]

The great breakthrough in quaternions finally came on Monday 16 October 1843 inDublin, when Hamilton was on his way to theRoyal Irish Academy to preside at a council meeting. As he walked along the towpath of theRoyal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the defining formula for the quaternions into the stone ofBrougham Bridge with his pocket knife:

i2=j2=k2=ijk=−1{\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {i\;j\;k} =-1}

Although the carving has since faded away, there has been an annual pilgrimage since 1989, called theHamilton Walk, for scientists and mathematicians who process from theDunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery.

On the following day, Hamilton wrote a letter to his friend and fellow mathematician,J.T. Graves, describing the train of thought that led to his discovery. The letter was later published in a letter to thePhilosophical Magazine;[1] Hamilton states:

And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth.[1]

Hamilton called a quadruple with these rules of multiplication aquaternion, and he devoted most of the remainder of his life to studying and teaching them.Hamilton's treatment is moregeometric than the modern approach, which emphasizes quaternions'algebraic properties. He founded a school of "quaternionists", and he tried to popularize quaternions in several books. The last and longest of his books,Elements of Quaternions,[14] was 800 pages long; it was edited byhis son and published shortly after his death.[citation needed]

After Hamilton's death, the Scottish mathematical physicistPeter Tait became the chief exponent of quaternions. At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such askinematics in space andMaxwell's equations, were described entirely in terms of quaternions. There was even a professional research association, theQuaternion Association, devoted to the study of quaternions and otherhypercomplex number systems.[15]

From the mid-1880s, quaternions began to be displaced byvector analysis, which had been developed byJosiah Willard Gibbs,Oliver Heaviside, andHermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature on quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics andphysics. A side-effect of this transition is thatHamilton's work is difficult to comprehend for many modern readers. Hamilton's original definitions are unfamiliar and his writing style was wordy and difficult to follow.[citation needed]

However, quaternions have had a revival since the late 20th century, primarily due to their utility indescribing spatial rotations. The representations of rotations by quaternions are more compact and quicker to compute than the representations bymatrices. In addition, unlike Euler angles, they are not susceptible to "gimbal lock". For this reason, quaternions are used incomputer graphics,[16][b]computer vision,robotics,[18]nuclear magnetic resonance image sampling,[6]control theory,signal processing,attitude control,physics,bioinformatics,molecular dynamics,computer simulations, andorbital mechanics. For example, it is common for theattitude control systems of spacecraft to be commanded in terms of quaternions. Quaternions also have contributed tonumber theory, because of their relationships with thequadratic forms.[19]

Quaternions in physics

[edit]

Hamilton had introducedbiquaternions in hisLectures on Quaternions, and these were used byLudwik Silberstein in 1914 to exhibit theLorentz transformations ofspecial relativity.[20] This representation of Lorentz transformations was also used byCornelius Lanczos in 1949.[21]

The finding of 1924 that inquantum mechanics thespin of an electron and other matter particles (known asspinors) can be described using quaternions (in the form of the famousPauli spin matrices) furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the "Plate trick").[22][23] As of 2018[update], their use has not yet overtakenrotation groups.[c]

W. K. Clifford[25] (1845 − 1879) introduced his algebras as a tensor product (”compound of algebras”) of quaternion algebras (and its even sub-algebra), a concept introduced byB. Peirce[26] (1809 − 1880).R. Lipschitz[27] (1832 − 1903) rediscovered independently the even subalgebra. In 1922,C. L. E. Moore[28] (1876 − 1931) was to callLipschitz’ algebras ”hyperquaternions”. The term ”hyperquaternion” designates nowadays both the tensor product ofn{\displaystyle n} quaternion algebrasH⊗n{\displaystyle \mathbb {H} ^{\otimes n}} and its even subalgebraH⊗n−1⊗C{\displaystyle \mathbb {H} ^{\otimes n-1}\otimes \mathbb {C} }.[29]

Examples of hyperquaternions are:H,H⊗2=H⊗RH{\displaystyle \mathbb {H} ,\mathbb {H} ^{\otimes 2}=\mathbb {H} \otimes _{\mathbb {R} }\mathbb {H} } (isomorphic to the Clifford algebraCl3,1(R){\displaystyle Cl_{3,1}\mathbb {(R)} } and to4×4{\displaystyle 4\times 4} real matricesM(4,R){\displaystyle M(4,\mathbb {R} )}) leading to applications inspecial relativity. Its even subalgebra isH⊗C{\displaystyle \mathbb {H} \otimes \mathbb {C} } (biquaternions).[30][31]

Another example isH⊗3=M(4,H){\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yielding aquaternionic matrix and its even subalgebraH⊗2⊗RC{\displaystyle \mathbb {H} ^{\otimes 2}\otimes _{\mathbb {R} }\mathbb {C} } (Dirac algebra).[32][33][34]

Definition

[edit]

Aquaternion is anexpression of the form

a+bi+cj+dk,{\displaystyle a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} ,}

wherea,b,c,d, are real numbers, andi,j,k, aresymbols that can be interpreted as unit-vectors pointing along the three spatial axes. In practice, if one ofa,b,c,d is 0, the corresponding term is omitted; ifa,b,c,d are all zero, the quaternion is thezero quaternion, denoted 0; if one ofb,c,d equals 1, the corresponding term is written simplyi,j, ork.

A quaternionq=a+bi+cj+dk{\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} }, can be decomposed into itsscalar part⁠a{\displaystyle a}⁠ (sometimesreal part) and itsvector partq−a=bi+cj+dk{\displaystyle q-a=b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} } (sometimesimaginary part). A quaternion that equals its real part (that is, its vector part is zero) is called ascalar quaternion (sometimesreal quaternion or simplyscalar), and is identified with the corresponding real number. That is, the real numbers areembedded in the quaternions.[d] A quaternion that equals its vector part is called avector quaternion (sometimesright quaternion).

Quaternions form a 4-dimensionalvector space over the real numbers, with{1,i,j,k}{\displaystyle \left\{1,\mathbf {i} ,\mathbf {j} ,\mathbf {k} \right\}} as abasis, by the component-wise addition

(a1+b1i+c1j+d1k)+(a2+b2i+c2j+d2k)=(a1+a2)+(b1+b2)i+(c1+c2)j+(d1+d2)k,{\displaystyle {\begin{aligned}&(a_{1}+b_{1}\mathbf {i} +c_{1}\mathbf {j} +d_{1}\mathbf {k} )+(a_{2}+b_{2}\mathbf {i} +c_{2}\mathbf {j} +d_{2}\mathbf {k} )\\[3mu]&\qquad =(a_{1}+a_{2})+(b_{1}+b_{2})\mathbf {i} +(c_{1}+c_{2})\mathbf {j} +(d_{1}+d_{2})\mathbf {k} ,\end{aligned}}}

and the component-wise scalar multiplication

λ(a+bi+cj+dk)=λa+(λb)i+(λc)j+(λd)k.{\displaystyle \lambda (a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} )=\lambda a+(\lambda b)\mathbf {i} +(\lambda c)\mathbf {j} +(\lambda d)\mathbf {k} .}

A multiplicative group structure, called theHamilton product, denoted by juxtaposition, can be defined on the quaternions in the following way:

Thus the quaternions form adivision algebra.

Multiplication of basis elements

[edit]
Multiplication table
×1ijk
11ijk
ii−1k−j
jj−k−1i
kkj−i−1
Non commutativity is emphasized by colored squares

The multiplication with1 of the basis elementsi,j, andk is defined by the fact that1 is amultiplicative identity, that is,

i1=1i=i,j1=1j=j,k1=1k=k.{\displaystyle \mathbf {i} \,1=1\,\mathbf {i} =\mathbf {i} ,\qquad \mathbf {j} \,1=1\,\mathbf {j} =\mathbf {j} ,\qquad \mathbf {k} \,1=1\,\mathbf {k} =\mathbf {k} .}

The products of other basis elements are

i2=j2=k2=−1,ij=−ji=k,jk=−kj=i,ki=−ik=j.{\displaystyle {\begin{aligned}\mathbf {i} ^{2}&=\mathbf {j} ^{2}=\mathbf {k} ^{2}=-1,\\[5mu]\mathbf {i\,j} &=-\mathbf {j\,i} =\mathbf {k} ,\qquad \mathbf {j\,k} =-\mathbf {k\,j} =\mathbf {i} ,\qquad \mathbf {k\,i} =-\mathbf {i\,k} =\mathbf {j} .\end{aligned}}}

Combining these rules,

ijk=−1.{\displaystyle {\begin{aligned}\mathbf {i\,j\,k} &=-1.\end{aligned}}}

Center

[edit]

Thecenter of anoncommutative ring is the subring of elementsc such thatcx =xc for everyx. The center of the quaternion algebra is the subfield of scalar quaternions. In fact, it is a part of the definition that the scalar quaternions belong to the center. Conversely, ifq =a +bi +cj +dk belongs to the center, then0=iq−qi=2cij+2dik=2ck−2dj,{\displaystyle 0=\mathbf {i} \,q-q\,\mathbf {i} =2c\,\mathbf {ij} +2d\,\mathbf {ik} =2c\,\mathbf {k} -2d\,\mathbf {j} ,}

andc =d = 0. A similar computation withj instead ofi shows that one has alsob = 0. Thusq =a is a scalar quaternion.

The quaternions form a division algebra. This means that the non-commutativity of multiplication is the only property that makes quaternions different from a field. This non-commutativity has some unexpected consequences, among them that apolynomial equation over the quaternions can have more distinct solutions than the degree of the polynomial. For example, the equationz2 + 1 = 0, has infinitely many quaternion solutions, which are the quaternionsz =bi +cj +dk such thatb2 +c2 +d2 = 1. Thus theseimaginary units form aunit sphere in the three-dimensional space of quaternion vectors.

Hamilton product

[edit]

For two elementsa1 +b1i +c1j +d1k anda2 +b2i +c2j +d2k, their product, called theHamilton product (a1 +b1i +c1j +d1k) (a2 +b2i +c2j +d2k), is determined by the products of the basis elements and thedistributive law. The distributive law makes it possible to expand the product so that it is a sum of products of basis elements. This gives the following expression:

a1a2+a1b2i+a1c2j+a1d2k+b1a2i+b1b2i2+b1c2ij+b1d2ik+c1a2j+c1b2ji+c1c2j2+c1d2jk+d1a2k+d1b2ki+d1c2kj+d1d2k2{\displaystyle {\begin{alignedat}{4}&a_{1}a_{2}&&+a_{1}b_{2}\mathbf {i} &&+a_{1}c_{2}\mathbf {j} &&+a_{1}d_{2}\mathbf {k} \\{}+{}&b_{1}a_{2}\mathbf {i} &&+b_{1}b_{2}\mathbf {i} ^{2}&&+b_{1}c_{2}\mathbf {ij} &&+b_{1}d_{2}\mathbf {ik} \\{}+{}&c_{1}a_{2}\mathbf {j} &&+c_{1}b_{2}\mathbf {ji} &&+c_{1}c_{2}\mathbf {j} ^{2}&&+c_{1}d_{2}\mathbf {jk} \\{}+{}&d_{1}a_{2}\mathbf {k} &&+d_{1}b_{2}\mathbf {ki} &&+d_{1}c_{2}\mathbf {kj} &&+d_{1}d_{2}\mathbf {k} ^{2}\end{alignedat}}}

Now the basis elements can be multiplied using the rules given above to get:[9]

a1a2−b1b2−c1c2−d1d2+(a1b2+b1a2+c1d2−d1c2)i+(a1c2−b1d2+c1a2+d1b2)j+(a1d2+b1c2−c1b2+d1a2)k{\displaystyle {\begin{alignedat}{4}&a_{1}a_{2}&&-b_{1}b_{2}&&-c_{1}c_{2}&&-d_{1}d_{2}\\{}+{}(&a_{1}b_{2}&&+b_{1}a_{2}&&+c_{1}d_{2}&&-d_{1}c_{2})\mathbf {i} \\{}+{}(&a_{1}c_{2}&&-b_{1}d_{2}&&+c_{1}a_{2}&&+d_{1}b_{2})\mathbf {j} \\{}+{}(&a_{1}d_{2}&&+b_{1}c_{2}&&-c_{1}b_{2}&&+d_{1}a_{2})\mathbf {k} \end{alignedat}}}

Scalar and vector parts

[edit]

A quaternion of the forma + 0i + 0j + 0k, wherea is a real number, is called ascalar quaternion (sometimesreal quaternion), and a quaternion of the form0 +bi +cj +dk, whereb,c, andd are real numbers, and at least one ofb,c, ord is nonzero, is called avector quaternion (sometimesright quaternion. Ifa +bi +cj +dk is any quaternion, thena is called itsscalar part andbi +cj +dk is called itsvector part. Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector spaceR3.{\displaystyle \mathbb {R} ^{3}.}[e]

Hamilton also called vector quaternionsright quaternions[37][38] and real numbers (considered as quaternions with zero vector part)scalar quaternions.

If a quaternion is divided up into its scalar part and its vector part, that is,

q=(r,v→), q∈H, r∈R, v→∈R3,{\displaystyle \mathbf {q} =(r,\,{\vec {v}}),\ \mathbf {q} \in \mathbb {H} ,\ r\in \mathbb {R} ,\ {\vec {v}}\in \mathbb {R} ^{3},}

then the formulas for addition, multiplication, and multiplicative inverse are

(r1,v→1)+(r2,v→2)=(r1+r2,v→1+v→2),(r1,v→1)(r2,v→2)=(r1r2−v→1⋅v→2,r1v→2+r2v→1+v→1×v→2),(r,v→)−1=(rr2+v→⋅v→, −v→r2+v→⋅v→){\displaystyle {\begin{aligned}(r_{1},\,{\vec {v}}_{1})+(r_{2},\,{\vec {v}}_{2})&=(r_{1}+r_{2},\,{\vec {v}}_{1}+{\vec {v}}_{2}),\\[5mu](r_{1},\,{\vec {v}}_{1})(r_{2},\,{\vec {v}}_{2})&=(r_{1}r_{2}-{\vec {v}}_{1}\cdot {\vec {v}}_{2},\,r_{1}{\vec {v}}_{2}+r_{2}{\vec {v}}_{1}+{\vec {v}}_{1}\times {\vec {v}}_{2}),\\[5mu](r,\,{\vec {v}})^{-1}&=\left({\frac {r}{r^{2}+{\vec {v}}\cdot {\vec {v}}}},\ {\frac {-{\vec {v}}}{r^{2}+{\vec {v}}\cdot {\vec {v}}}}\right)\end{aligned}}}

where "⋅{\displaystyle {}\cdot {}}" and "×{\displaystyle \times }" denote respectively thedot product and thecross product.

Conjugation, the norm, and reciprocal

[edit]

Conjugation of quaternions is analogous to conjugation of complex numbers and to transposition (also known as reversal) of elements of Clifford algebras. To define it, letq=a+bi+cj+dk{\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} } be a quaternion. Theconjugate ofq is the quaternionq∗=a−bi−cj−dk{\displaystyle q^{*}=a-b\,\mathbf {i} -c\,\mathbf {j} -d\,\mathbf {k} }. It is denoted byq∗,qt,q~{\displaystyle {\tilde {q}}}, orq.[9] Conjugation is aninvolution, meaning that it is its owninverse, so conjugating an element twice returns the original element. The conjugate of a product of two quaternions is the product of the conjugatesin the reverse order. That is, ifp andq are quaternions, then(pq)∗ =q∗p∗, notp∗q∗.

The conjugation of a quaternion, in contrast to the complex setting, can be expressed with multiplication and addition of quaternions:

q∗=−12(q+iqi+jqj+kqk).{\displaystyle q^{*}=-{\tfrac {1}{2}}(q+\mathbf {i} \,q\,\mathbf {i} +\mathbf {j} \,q\,\mathbf {j} +\mathbf {k} \,q\,\mathbf {k} ).}

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part ofp is⁠1/2⁠(p +p∗), and the vector part ofp is⁠1/2⁠(p −p∗).

Thesquare root of the product of a quaternion with its conjugate is called itsnorm and is denoted‖q‖ (Hamilton called this quantity thetensor ofq, but this conflicts with the modern meaning of "tensor"). In formulas, this is expressed as follows:

‖q‖=qq∗=q∗q=a2+b2+c2+d2{\displaystyle \lVert q\rVert ={\sqrt {qq^{*}}}={\sqrt {q^{*}q}}={\sqrt {a^{2}+b^{2}+c^{2}+d^{2}}}}

This is always a non-negative real number, and it is the same as the Euclidean norm onH{\displaystyle \mathbb {H} } considered as the vector spaceR4{\displaystyle \mathbb {R} ^{4}}. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, ifα is real, then

‖αq‖=|α|‖q‖.{\displaystyle \lVert \alpha q\rVert =\left|\alpha \right|\,\lVert q\rVert .}

This is a special case of the fact that the norm ismultiplicative, meaning that

‖pq‖=‖p‖‖q‖{\displaystyle \lVert pq\rVert =\lVert p\rVert \,\lVert q\rVert }

for any two quaternionsp andq. Multiplicativity is a consequence of the formula for the conjugate of a product.Alternatively it follows from the identity

det(a+ibid+cid−ca−ib)=a2+b2+c2+d2,{\displaystyle \det {\begin{pmatrix}a+ib&id+c\\id-c&a-ib\end{pmatrix}}=a^{2}+b^{2}+c^{2}+d^{2},}

(wherei denotes the usualimaginary unit) and hence from the multiplicative property ofdeterminants of square matrices.

This norm makes it possible to define thedistanced(p,q) betweenp andq as the norm of their difference:

d(p,q)=‖p−q‖.{\displaystyle d(p,q)=\lVert p-q\rVert .}

This makesH{\displaystyle \mathbb {H} } ametric space.Addition and multiplication arecontinuous in regard to the associatedmetric topology.This follows with exactly the same proof as for the real numbersR{\displaystyle \mathbb {R} } from the fact thatH{\displaystyle \mathbb {H} } is a normed algebra.

Unit quaternion

[edit]
Main article:Versor

Aunit quaternion is a quaternion of norm one. Dividing a nonzero quaternionq by its norm produces a unit quaternionUq called theversor ofq:

Uq=q‖q‖.{\displaystyle \mathbf {U} q={\frac {q}{\lVert q\rVert }}.}

Every nonzero quaternion has a uniquepolar decompositionq=‖q‖⋅Uq,{\displaystyle q=\lVert q\rVert \cdot \mathbf {U} q,} while the zero quaternion can be formed from any unit quaternion.

Using conjugation and the norm makes it possible to define thereciprocal of a nonzero quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product ofq{\displaystyle q} andq∗/‖q‖2{\displaystyle q^{*}/\left\Vert q\right\|^{2}} is 1 (for either order of multiplication). So thereciprocal ofq is defined to be

q−1=q∗‖q‖2.{\displaystyle q^{-1}={\frac {q^{*}}{\lVert q\rVert ^{2}}}.}

Since the multiplication is non-commutative, the quotient quantitiesp q−1 orq−1p are different (except ifp andq have parallel vector parts): the notation⁠p/q⁠ is ambiguous and should not be used.

Algebraic properties

[edit]
Cayley graph ofQ8. The red arrows represent multiplication on the right byi, and the green arrows represent multiplication on the right byj.

The setH{\displaystyle \mathbb {H} } of all quaternions is a vector space over the real numbers withdimension 4.[f] Multiplication of quaternions isassociative anddistributive overvector addition, but it is notcommutative. Therefore, the quaternions, H ,{\displaystyle \ \mathbb {H} \ ,} are a non-commutative, associative algebra over thereal numbers. Even though H {\displaystyle \ \mathbb {H} \ } embeds multiplecopies of the complex numbers, it is not itself anassociative algebra over thecomplex numbers.

Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to afield except that it admits non-commutative multiplication.Finite-dimensionalassociative division algebras over the real numbers are very rare: TheFrobenius theorem states that there are exactly three. Those are R ,{\displaystyle \ \mathbb {R} \ ,} C ,{\displaystyle \ \mathbb {C} \ ,} and H .{\displaystyle \ \mathbb {H} \ .} Thenorm makes the quaternions into anormed algebra, and normed division algebras over the real numbers are also very rare:Hurwitz's theorem says that there are only four: R ,{\displaystyle \ \mathbb {R} \ ,} C ,{\displaystyle \ \mathbb {C} \ ,} H ,{\displaystyle \ \mathbb {H} \ ,} and O {\displaystyle \ \mathbb {O} \ } (theoctonions). The quaternions are also an example of acomposition algebra and of a unitalBanach algebra.

Three-dimensional graph of Q8. Red, green and blue arrows represent multiplication byi,j, andk, respectively. Multiplication by negative numbers is omitted for clarity.

Because the product of any two basis vectors is plus or minus another basis vector, the set{±1, ±i, ±j, ±k} forms a group under multiplication. Thisnon-abelian group is called the quaternion group and is denotedQ8.[39] The realgroup ring ofQ8 is a ringR[Q8]{\displaystyle \mathbb {R} [\mathrm {Q} _{8}]} which is also an eight-dimensional vector space overR.{\displaystyle \mathbb {R} .} It has one basis vector for each element ofQ8.{\displaystyle \mathrm {Q} _{8}.} The quaternions are isomorphic to thequotient ring ofR[Q8]{\displaystyle \mathbb {R} [\mathrm {Q} _{8}]} by theideal generated by the elements1 + (−1),i + (−i),j + (−j), andk + (−k). Here the first term in each of the sums is one of the basis elements1,i,j, andk, and the second term is one of basis elements−1, −i, −j, and−k, not the additive inverses of1,i,j, andk.

Quaternions and three-dimensional geometry

[edit]

The vector part of a quaternion can be interpreted as a coordinate vector inR3;{\displaystyle \mathbb {R} ^{3};} therefore, the algebraic operations of the quaternions reflect the geometry ofR3.{\displaystyle \mathbb {R} ^{3}.} Operations such as the vector dot and cross products can be defined in terms of quaternions, and this makes it possible to apply quaternion techniques wherever spatial vectors arise. A useful application of quaternions has been to interpolate the orientations of key-frames in computer graphics.[16]

For the remainder of this section,i,j, andk will denote both the three imaginary[40] basis vectors ofH{\displaystyle \mathbb {H} } and a basis forR3.{\displaystyle \mathbb {R} ^{3}.} Replacingi by−i,j by−j, andk by−k sends a vector to itsadditive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion. For this reason, conjugation is sometimes called thespatial inverse.

For two vector quaternionsp =b1i +c1j +d1k andq =b2i +c2j +d2k theirdot product, by analogy to vectors inR3,{\displaystyle \mathbb {R} ^{3},} is

p⋅q=b1b2+c1c2+d1d2.{\displaystyle p\cdot q=b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}.}

It can also be expressed in a component-free manner as

p⋅q=12(p∗q+q∗p)=12(pq∗+qp∗).{\displaystyle p\cdot q=\textstyle {\frac {1}{2}}(p^{*}q+q^{*}p)=\textstyle {\frac {1}{2}}(pq^{*}+qp^{*}).}

This is equal to the scalar parts of the productspq∗,qp∗,p∗q, andq∗p. Note that their vector parts are different.

Thecross product ofp andq relative to the orientation determined by the ordered basisi,j, andk is

p×q=(c1d2−d1c2)i+(d1b2−b1d2)j+(b1c2−c1b2)k.{\displaystyle p\times q=(c_{1}d_{2}-d_{1}c_{2})\mathbf {i} +(d_{1}b_{2}-b_{1}d_{2})\mathbf {j} +(b_{1}c_{2}-c_{1}b_{2})\mathbf {k} .}

(Recall that the orientation is necessary to determine the sign.) This is equal to the vector part of the productpq (as quaternions), as well as the vector part of−q∗p∗. It also has the formula

p×q=12(pq−qp).{\displaystyle p\times q=\textstyle {\tfrac {1}{2}}(pq-qp).}

For thecommutator,[p,q] =pq −qp, of two vector quaternions one obtains

[p,q]=2p×q,{\displaystyle [p,q]=2p\times q,}

which gives the commutation relationship

qp=pq−2p×q.{\displaystyle qp=pq-2p\times q.}

In general, letp andq be quaternions and write

p=ps+pv,q=qs+qv,{\displaystyle {\begin{aligned}p&=p_{\text{s}}+p_{\text{v}},\\[5mu]q&=q_{\text{s}}+q_{\text{v}},\end{aligned}}}

whereps andqs are the scalar parts, andpv andqv are the vector parts ofp andq. Then we have the formula

pq=(pq)s+(pq)v=(psqs−pv⋅qv)+(psqv+qspv+pv×qv).{\displaystyle pq=(pq)_{\text{s}}+(pq)_{\text{v}}=(p_{\text{s}}q_{\text{s}}-p_{\text{v}}\cdot q_{\text{v}})+(p_{\text{s}}q_{\text{v}}+q_{\text{s}}p_{\text{v}}+p_{\text{v}}\times q_{\text{v}}).}

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of vector quaternions. It also shows that two quaternions commute if and only if their vector parts are collinear. Hamilton[41] showed that this product computes the third vertex of a spherical triangle from two given vertices and their associated arc-lengths, which is also an algebra of points inElliptic geometry.

Unit quaternions can be identified with rotations inR3{\displaystyle \mathbb {R} ^{3}} and were calledversors by Hamilton.[41] Also seeQuaternions and spatial rotation for more information about modeling three-dimensional rotations using quaternions.

SeeHanson (2005)[42] for visualization of quaternions.

Matrix representations

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Just as complex numbers can berepresented as matrices, so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition andmatrix multiplication. One is to use 2 × 2 complex matrices, and the other is to use 4 × 4real matrices. In each case, the representation given is one of a family of linearly related representations. These areinjectivehomomorphisms fromH{\displaystyle \mathbb {H} } to thematrix ringsM(2,C) andM(4,R), respectively.

Representation as complex 2 × 2 matrices

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The quaterniona +bi +cj +dk can be represented using a complex2 × 2 matrix as

[−a+bic+di−c+dia−bi].{\displaystyle {\begin{bmatrix}{\phantom {-}}a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.}

This representation has the following properties:

The representation inM(2,ℂ) is not unique: A different convention, that preserves the direction of cyclic ordering between the quaternions and the Pauli matrices, is to choose1↦I,i↦−i σ1=−σ2 σ3 ,j↦−i σ2=−σ3 σ1 ,k↦−i σ3=−σ1 σ2 ,{\displaystyle 1\mapsto \mathbf {I} ,\quad \mathbf {i} \mapsto -i\ \sigma _{1}=-\sigma _{2}\ \sigma _{3}\ ,\quad \mathbf {j} \mapsto -i\ \sigma _{2}=-\sigma _{3}\ \sigma _{1}\ ,\quad \mathbf {k} \mapsto -i\ \sigma _{3}=-\sigma _{1}\ \sigma _{2}\ ,}

This gives an alternative representation,[44]

a+b i+c j+d k↦[a−d i−c−b ic−b i−a+d i] .{\displaystyle a+b\ \mathbf {i} +c\ \mathbf {j} +d\ \mathbf {k} \mapsto {\begin{bmatrix}a-d\ i&-c-b\ i\\c-b\ i&{\phantom {-}}a+d\ i\end{bmatrix}}~.}

Representation as real 4 × 4 matrices

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Using 4 × 4 real matrices, that same quaternion can be written as

[a−b−c−dba−dccda−bd−cba]=a[1000010000100001]+b[0−1001000000−10010]+c[00−10000110000−100]+d[000−100−1001001000].{\displaystyle {\begin{aligned}\left[{\begin{array}{rrrr}a&-b&-c&-d\\b&a&-d&c\\c&d&a&-b\\d&-c&b&a\end{array}}\right]&=a\left[{\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]+b\left[{\begin{array}{rrrr}0&-1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&1&0\end{array}}\right]\\[10mu]&\qquad +c\left[{\begin{array}{rrrr}0&0&-1&0\\0&0&0&1\\1&0&0&0\\0&-1&0&0\end{array}}\right]+d\left[{\begin{array}{rrrr}0&0&0&-1\\0&0&-1&0\\0&1&0&0\\1&0&0&0\end{array}}\right].\end{aligned}}}

However, the representation of quaternions inM(4,ℝ) is not unique. For example, the same quaternion can also be represented as

[ad−b−c−dac−bb−ca−dcbda]=a[1000010000100001]+b[00−10000−110000100]+c[000−100100−1001000]+d[0100−1000000−10010].{\displaystyle {\begin{aligned}\left[{\begin{array}{rrrr}a&d&-b&-c\\-d&a&c&-b\\b&-c&a&-d\\c&b&d&a\end{array}}\right]&=a\left[{\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{array}}\right]+b\left[{\begin{array}{rrrr}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{array}}\right]\\[10mu]&\qquad +c\left[{\begin{array}{rrrr}0&0&0&-1\\0&0&1&0\\0&-1&0&0\\1&0&0&0\end{array}}\right]+d\left[{\begin{array}{rrrr}0&1&0&0\\-1&0&0&0\\0&0&0&-1\\0&0&1&0\end{array}}\right].\end{aligned}}}

There are 48 distinct matrix representations of this form, in which one of the matrices represents the scalar part and the other three are all skew-symmetric. More precisely, there are 48 sets of quadruples of matrices with these symmetry constraints, such that a function sending1,i,j, andk to the matrices in the quadruple is ahomomorphism, that is, it sends sums and products of quaternions to sums and products of matrices.[45] In this representation, the conjugate of a quaternion corresponds to thetranspose of the matrix. The fourth power of the norm of a quaternion is thedeterminant of the corresponding matrix. The scalar part of a quaternion is one quarter of the matrix trace. As with the 2 × 2 complex representation above, complex numbers can again be produced by constraining the coefficients suitably; for example, as block diagonal matrices with two 2 × 2 blocks by settingc =d = 0.

Each 4×4 matrix representation of quaternions corresponds to a multiplication table of unit quaternions. For example, the last matrix representation given above corresponds to the multiplication table

×ad−b−c
aad−b−c
−d−dac−b
bb−ca−d
ccbda

which is isomorphic — through{a↦1,b↦i,c↦j,d↦k}{\displaystyle \{a\mapsto 1,\,b\mapsto i,\,c\mapsto j,\,d\mapsto k\}} — to

×1k−i−j
11k−i−j
−k−k1j−i
ii−j1−k
jjik1

Constraining any such multiplication table to have the identity in the first row and column and for the signs of the row headers to be opposite to those of the column headers, then there are 3 possible choices for the second column (ignoring sign), 2 possible choices for the third column (ignoring sign), and 1 possible choice for the fourth column (ignoring sign); that makes 6 possibilities. Then, the second column can be chosen to be either positive or negative, the third column can be chosen to be positive or negative, and the fourth column can be chosen to be positive or negative, giving 8 possibilities for the sign. Multiplying the possibilities for the letter positions and for their signs yields 48. Then replacing1 witha,i withb,j withc, andk withd and removing the row and column headers yields a matrix representation ofa +bi +cj +dk.

Lagrange's four-square theorem

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Main article:Lagrange's four-square theorem

Quaternions are also used in one of the proofs of Lagrange's four-square theorem innumber theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such ascombinatorial design theory. The quaternion-based proof usesHurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of theEuclidean algorithm.

Quaternions as pairs of complex numbers

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Main article:Cayley–Dickson construction

Quaternions can be represented as pairs of complex numbers. From this perspective, quaternions are the result of applying theCayley–Dickson construction to the complex numbers. This is a generalization of the construction of the complex numbers as pairs of real numbers.

LetC2{\displaystyle \mathbb {C} ^{2}} be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements1 andj. A vector inC2{\displaystyle \mathbb {C} ^{2}} can be written in terms of the basis elements1 andj as

(a+bi)1+(c+di)j.{\displaystyle (a+bi)1+(c+di)\mathbf {j} .}

If we definej2 = −1 andij = −ji, then we can multiply two vectors using the distributive law. Usingk as an abbreviated notation for the productij leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaterniona +b i +cj +dk. If we write the elements ofC2{\displaystyle \mathbb {C} ^{2}} as ordered pairs and quaternions as quadruples, then the correspondence is

(a+bi,c+di)↔(a,b,c,d).{\displaystyle (a+bi,\,c+di)\leftrightarrow (a,\,b,\,c,\,d).}

Square roots

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Square roots of −1

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In the complex numbers,C,{\displaystyle \mathbb {C} ,} there are exactly two numbers,i and−i, that give −1 when squared. InH{\displaystyle \mathbb {H} } there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unitsphere inR3.{\displaystyle \mathbb {R} ^{3}.} To see this, letq =a +bi +cj +dk be a quaternion, and assume that its square is −1. In terms ofa,b,c, andd, this means

a2−b2−c2−d2=−1,x|2ab=0,2ac=0,2ad=0.{\displaystyle {\begin{aligned}a^{2}-b^{2}-c^{2}-d^{2}&=-1,{\vphantom {x^{|}}}\\[3mu]2ab&=0,\\[3mu]2ac&=0,\\[3mu]2ad&=0.\end{aligned}}}

To satisfy the last three equations, eithera = 0 orb,c, andd are all 0. The latter is impossible becausea is a real number and the first equation would imply thata2 = −1. Therefore,a = 0 andb2 +c2 +d2 = 1. In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere.

Only negative real quaternions have infinitely many square roots. All others have just two (or one in the case of 0).[citation needed][g]

As a union of complex planes

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Eachantipodal pair of square roots of −1 creates a distinct copy of the complex numbers inside the quaternions. Ifq2 = −1, then the copy is theimage of the function

a+bi↦a+bq.{\displaystyle a+bi\mapsto a+bq.}

This is aninjectivering homomorphism fromC{\displaystyle \mathbb {C} } toH,{\displaystyle \mathbb {H} ,} which defines a fieldisomorphism fromC{\displaystyle \mathbb {C} } onto itsimage. The images of the embeddings corresponding toq and −q are identical.

Every non-real quaternion generates asubalgebra of the quaternions that is isomorphic toC,{\displaystyle \mathbb {C} ,} and is thus a planar subspace ofH:{\displaystyle \mathbb {H} \colon } writeq as the sum of its scalar part and its vector part:

q=qs+q→v.{\displaystyle q=q_{s}+{\vec {q}}_{v}.}

Decompose the vector part further as the product of its norm and itsversor:

q=qs+‖q→v‖⋅Uq→v=qs+‖q→v‖q→v‖q→v‖.{\displaystyle q=q_{s}+\lVert {\vec {q}}_{v}\rVert \cdot \mathbf {U} {\vec {q}}_{v}=q_{s}+\|{\vec {q}}_{v}\|\,{\frac {{\vec {q}}_{v}}{\|{\vec {q}}_{v}\|}}.}

(This is not the same asqs+‖q‖⋅Uq{\displaystyle q_{s}+\lVert q\rVert \cdot \mathbf {U} q}.) The versor of the vector part ofq,Uq→v{\displaystyle \mathbf {U} {\vec {q}}_{v}}, is a right versor with –1 as its square. A straightforward verification shows thata+bi↦a+bUq→v{\displaystyle a+bi\mapsto a+b\mathbf {U} {\vec {q}}_{v}}defines an injectivehomomorphism ofnormed algebras fromC{\displaystyle \mathbb {C} } into the quaternions. Under this homomorphism,q is the image of the complex numberqs+‖q→v‖i{\displaystyle q_{s}+\lVert {\vec {q}}_{v}\rVert i}.

AsH{\displaystyle \mathbb {H} } is theunion of the images of all these homomorphisms, one can view the quaternions as apencil of planes intersecting on thereal line. Each of thesecomplex planes contains exactly one pair ofantipodal points of the sphere of square roots of minus one.

Commutative subrings

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The relationship of quaternions to each other within the complex subplanes ofH{\displaystyle \mathbb {H} } can also be identified and expressed in terms of commutativesubrings. Specifically, since two quaternionsp andq commute (i.e.,p q =q p) only if they lie in the same complex subplane ofH{\displaystyle \mathbb {H} }, the profile ofH{\displaystyle \mathbb {H} } as a union of complex planes arises when one seeks to find all commutative subrings of the quaternionring.

Square roots of arbitrary quaternions

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Any quaternionq=(r,v→){\displaystyle \mathbf {q} =(r,\,{\vec {v}})} (represented here in scalar–vector representation) has at least one square rootq=(x,y→){\displaystyle {\sqrt {\mathbf {q} }}=(x,\,{\vec {y}})} which solves the equationq2=(x,y→)2=q{\displaystyle {\sqrt {\mathbf {q} }}^{\,2}=(x,\,{\vec {y}})^{2}=\mathbf {q} }. Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions

q=(r,v→)=±(12(‖q‖+r), v→‖v→‖12(‖q‖−r)),{\displaystyle {\sqrt {\mathbf {q} }}={\sqrt {(r,\,{\vec {v}})}}=\pm \left({\sqrt {{\tfrac {1}{2}}{\bigl (}{\|\mathbf {q} \|+r}{\bigr )}}},\ {\frac {\vec {v}}{\|{\vec {v}}\|}}{\sqrt {{\tfrac {1}{2}}{\bigl (}{\|\mathbf {q} \|-r}{\bigr )}}}\right),}

where‖v→‖=v→⋅v→{\textstyle \|{\vec {v}}\|={\sqrt {{\vec {v}}\cdot {\vec {v}}}}} is the norm ofv→{\displaystyle {\vec {v}}} and‖q‖=q∗q=r2+‖v→‖2{\textstyle \|\mathbf {q} \|={\sqrt {\mathbf {q} ^{*}\mathbf {q} }}={\sqrt {r^{2}+\|{\vec {v}}\|^{2}}}} is the norm ofq{\displaystyle \mathbf {q} }. For any scalar quaternionq{\displaystyle \mathbf {q} }, this equation provides the correct square roots ifv→/‖v→‖{\textstyle {\vec {v}}/\|{\vec {v}}\|} is interpreted as an arbitrary unit vector.

Therefore, nonzero, non-scalar quaternions, or positive scalar quaternions, have exactly two roots, while 0 has exactly one root (0), and negative scalar quaternions have infinitely many roots, which are the vector quaternions located on{0}×S2(−r){\displaystyle \{0\}\times S^{2}{\bigl (}{\sqrt {-r}}{\bigr )}}, i.e., where the scalar part is zero and the vector part is located on the2-sphere with radius−r{\displaystyle {\sqrt {-r}}}.

Functions of a quaternion variable

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Main article:Quaternionic analysis
The Julia sets and Mandelbrot sets can be extended to the Quaternions, but they must use cross sections to be rendered visually in 3 dimensions. This Julia set is cross sectioned at thex y plane.

Like functions of acomplex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable. Examples of other functions include the extension of theMandelbrot set andJulia sets into 4-dimensional space.[49]

Exponential, logarithm, and power functions

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A function of a quaternion can be defined from a power series with real coefficients. For example, given a quaternion,

q=a+bi+cj+dk=a+v,{\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} =a+\mathbf {v} ,}

the exponential is computed as[50]

exp⁡(q)=∑n=0∞qnn!=ea(cos⁡‖v‖+v‖v‖sin⁡‖v‖),{\displaystyle \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),}

and the logarithm is[50]

ln⁡(q)=ln⁡‖q‖+v‖v‖arccos⁡a‖q‖.{\displaystyle \ln(q)=\ln \|q\|+{\frac {\mathbf {v} }{\|\mathbf {v} \|}}\arccos {\frac {a}{\|q\|}}.}

It follows that the polar decomposition of a quaternion may be written

q=‖q‖en^φ=‖q‖(cos⁡(φ)+n^sin⁡(φ)),{\displaystyle q=\|q\|e^{{\hat {n}}\varphi }=\|q\|\left(\cos(\varphi )+{\hat {n}}\sin(\varphi )\right),}

where theangleφ{\displaystyle \varphi }[h]

a=‖q‖cos⁡(φ){\displaystyle a=\|q\|\cos(\varphi )}

and the unit vectorn^{\displaystyle {\hat {n}}} is defined by:

v=n^‖v‖=n^‖q‖sin⁡(φ).{\displaystyle \mathbf {v} ={\hat {n}}\|\mathbf {v} \|={\hat {n}}\|q\|\sin(\varphi ).}

Any unit quaternion may be expressed in polar form as:

q=exp⁡(n^φ).{\displaystyle q=\exp {({\hat {n}}\varphi )}.}

Thepower of a quaternion raised to an arbitrary (real) exponentx is given by:

qx=‖q‖xen^xφ=‖q‖x(cos⁡(xφ)+n^sin⁡(xφ)).{\displaystyle q^{x}=\|q\|^{x}e^{{\hat {n}}x\varphi }=\|q\|^{x}\left(\cos(x\varphi )+{\hat {n}}\,\sin(x\varphi )\right).}

Geodesic norm

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Thegeodesic distancedg(p,q) between unit quaternionsp andq is defined as:[52]

dg(p,q)=‖ln⁡(p−1q)‖.{\displaystyle d_{\text{g}}(p,q)=\lVert \ln(p^{-1}q)\rVert .}

and amounts to the absolute value of half the angle subtended byp andq along agreat arc of theS3 sphere.This angle can also be computed from the quaterniondot product without the logarithm as:

dg(p,q)=arccos⁡(2(p⋅q)2−1).{\displaystyle d_{\text{g}}(p,q)=\arccos(2(p\cdot q)^{2}-1).}

Three-dimensional and four-dimensional rotation groups

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Main articles:Quaternions and spatial rotation andRotation operator (vector space)

The word "conjugation", besides the meaning given above, can also mean taking an elementa tor a r−1 wherer is some nonzero quaternion. Allelements that are conjugate to a given element (in this sense of the word conjugate) have the same real part and the same norm of the vector part. (Thus the conjugate in the other sense is one of the conjugates in this sense.)[53]

Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy ofR3{\displaystyle \mathbb {R} ^{3}} consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real partcos(φ) is a rotation by an angle2φ, the axis of the rotation being the direction of the vector part. The advantages of quaternions are:[54]

The set of all unit quaternions (versors) forms a 3-sphereS3 and a group (aLie group) under multiplication,double covering the groupSO(3,R){\displaystyle {\text{SO}}(3,\mathbb {R} )} of real orthogonal 3×3 matrices ofdeterminant 1 sincetwo unit quaternions correspond to every rotation under the above correspondence. Seeplate trick.

Further information:Point groups in three dimensions

The image of a subgroup of versors is apoint group, and conversely, the preimage of a point group is a subgroup of versors. The preimage of a finite point group is called by the same name, with the prefixbinary. For instance, the preimage of theicosahedral group is thebinary icosahedral group.

The versors' group is isomorphic toSU(2), the group of complexunitary 2×2 matrices ofdeterminant 1.

Further information:Hurwitz quaternion

LetA be the set of quaternions of the forma +bi +cj +dk wherea, b, c, andd are either allintegers or allhalf-integers. The setA is a ring (in fact adomain) and alattice and is called the ring of Hurwitz quaternions. There are 24 unit quaternions in this ring, and they are the vertices of aregular 24 cell withSchläfli symbol{3,4,3}. They correspond to the double cover of the rotational symmetry group of the regulartetrahedron. Similarly, the vertices of aregular 600 cell with Schläfli symbol{3,3,5} can be taken as the uniticosians, corresponding to the double cover of the rotational symmetry group of theregular icosahedron. The double cover of the rotational symmetry group of the regularoctahedron corresponds to the quaternions that represent the vertices of thedisphenoidal 288-cell.[55]

Quaternion algebras

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Main article:Quaternion algebra

The Quaternions can be generalized into further algebras calledquaternion algebras. TakeF to be any field with characteristic different from 2, anda andb to be elements ofF; a four-dimensional unitary associative algebra can be defined overF with basis 1,i,j, andi j, wherei2 =a,j2 =b andi j = −j i (so(i j)2 = −a b).

Quaternion algebras are isomorphic to the algebra of 2×2 matrices overF or form division algebras overF, depending on the choice ofa andb.

Quaternions as the even part ofCl3,0(R)

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Main article:Spinor § Three dimensions

The usefulness of quaternions for geometrical computations can be generalised to other dimensions by identifying the quaternions as the even partCl3,0+⁡(R){\displaystyle \operatorname {Cl} _{3,0}^{+}(\mathbb {R} )} of the Clifford algebraCl3,0⁡(R).{\displaystyle \operatorname {Cl} _{3,0}(\mathbb {R} ).} This is an associative multivector algebra built up from fundamental basis elementsσ1,σ2,σ3 using the product rules

σ12=σ22=σ32=1,{\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=1,}σmσn=−σnσm(m≠n).{\displaystyle \sigma _{m}\sigma _{n}=-\sigma _{n}\sigma _{m}\qquad (m\neq n).}

If these fundamental basis elements are taken to represent vectors in 3D space, then it turns out that thereflection of a vectorr in a plane perpendicular to a unit vectorw can be written:

r′=−wrw.{\displaystyle r^{\prime }=-w\,r\,w.}

Two reflections make a rotation by an angle twice the angle between the two reflection planes, so

r′′=σ2σ1rσ1σ2{\displaystyle r^{\prime \prime }=\sigma _{2}\sigma _{1}\,r\,\sigma _{1}\sigma _{2}}

corresponds to a rotation of 180° in the plane containingσ1 andσ2. This is very similar to the corresponding quaternion formula,

r′′=−krk.{\displaystyle r^{\prime \prime }=-\mathbf {k} \,r\,\mathbf {k} .}

Indeed, the two structuresCl3,0+⁡(R){\displaystyle \operatorname {Cl} _{3,0}^{+}(\mathbb {R} )} andH{\displaystyle \mathbb {H} } areisomorphic. One natural identification is

1↦1,i↦−σ2σ3,j↦−σ3σ1,k↦−σ1σ2,{\displaystyle 1\mapsto 1,\quad \mathbf {i} \mapsto -\sigma _{2}\sigma _{3},\quad \mathbf {j} \mapsto -\sigma _{3}\sigma _{1},\quad \mathbf {k} \mapsto -\sigma _{1}\sigma _{2},}

and it is straightforward to confirm that this preserves the Hamilton relations

i2=j2=k2=ijk=−1.{\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {i\,j\,k} =-1.}

In this picture, so-called "vector quaternions" (that is, pure imaginary quaternions) correspond not to vectors but tobivectors – quantities withmagnitudes andorientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers becomes clearer, too: in 2D, with two vector directionsσ1 andσ2, there is only one bivector basis elementσ1σ2, so only one imaginary. But in 3D, with three vector directions, there are three bivector basis elementsσ2σ3,σ3σ1,σ1σ2, so three imaginaries.

This reasoning extends further. In the Clifford algebraCl4,0⁡(R),{\displaystyle \operatorname {Cl} _{4,0}(\mathbb {R} ),} there are six bivector basis elements, since with four different basic vector directions, six different pairs and therefore six different linearly independent planes can be defined. Rotations in such spaces using these generalisations of quaternions, calledrotors, can be very useful for applications involvinghomogeneous coordinates. But it is only in 3D that the number of basis bivectors equals the number of basis vectors, and each bivector can be identified as apseudovector.

There are several advantages for placing quaternions in this wider setting:[56]

  • Rotors are a natural part of geometric algebra and easily understood as the encoding of a double reflection.
  • In geometric algebra, a rotor and the objects it acts on live in the same space. This eliminates the need to change representations and to encode new data structures and methods, which is traditionally required when augmenting linear algebra with quaternions.
  • Rotors are universally applicable to any element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on.
  • In theconformal model of Euclidean geometry, rotors allow the encoding of rotation, translation and scaling in a single element of the algebra, universally acting on any element. In particular, this means that rotors can represent rotations around an arbitrary axis, whereas quaternions are limited to an axis through the origin.
  • Rotor-encoded transformations make interpolation particularly straightforward.
  • Rotors carry over naturally topseudo-Euclidean spaces, for example, theMinkowski space ofspecial relativity. In such spaces rotors can be used to efficiently representLorentz boosts, and to interpret formulas involving thegamma matrices.[citation needed]

For further detail about the geometrical uses of Clifford algebras, seeGeometric algebra.

Brauer group

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Further information:Brauer group

The quaternions are "essentially" the only (non-trivial)central simple algebra (CSA) over the real numbers, in the sense that every CSA over the real numbers isBrauer equivalent to either the real numbers or the quaternions. Explicitly, theBrauer group of the real numbers consists of two classes, represented by the real numbers and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being amatrix ring over another. By theArtin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the real numbers.

CSAs – finite dimensional rings over a field, which aresimple algebras (have no non-trivial 2-sided ideals, just as with fields) whose center is exactly the field – are a noncommutative analog ofextension fields, and are more restrictive than general ring extensions. The fact that the quaternions are the only non-trivial CSA over the real numbers (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial finite field extension of the real numbers.

Quotations

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I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse tox, y, z, etc.

— William Rowan Hamilton (c. 1848)[57]

Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": And in this sense it has, or at least involves a reference to, four dimensions. ...And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be.

— William Rowan Hamilton (c. 1853)[58]

Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, includingClerk Maxwell.

— W. Thompson, Lord Kelvin (1892)[59]

There was a time, indeed, when I, although recognizing the appropriateness of vector analysis in electromagnetic theory (and in mathematical physics generally), did think it was harder to understand and to work than the Cartesian analysis. But that was before I had thrown off the quaternionic old-man-of-the-sea who fastened himself about my shoulders when reading the only accessible treatise on the subject – Prof. Tait'sQuaternions. But I came later to see that, so far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work. There is not a ghost of a quaternion in any of my papers (except in one, for a special purpose). The vector analysis I use may be described either as a convenient and systematic abbreviation of Cartesian analysis; or else, as Quaternions without the quaternions, ... ."Quaternion" was, I think, defined by an American schoolgirl to be"an ancient religious ceremony". This was, however, a complete mistake: The ancients – unlike Prof. Tait – knew not, and did not worship Quaternions.

— Oliver Heaviside (1893)[60]

Neither matrices nor quaternions and ordinary vectors were banished from these ten [additional] chapters. For, in spite of the uncontested power of the modern Tensor Calculus, those older mathematical languages continue, in my opinion, to offer conspicuous advantages in the restricted field of special relativity. Moreover, in science as well as in everyday life, the mastery of more than one language is also precious, as it broadens our views, is conducive to criticism with regard to, and guards against hypostasy [weak-foundation] of, the matter expressed by words or mathematical symbols.

— Ludwik Silberstein (1924)[61]

... quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist.

— Simon L.Altmann (1986)[page needed]

See also

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Notes

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  1. ^Bradley & Sandifer (2007), p. 193 mentionWilhelm Blaschke's claim in 1959 that "the quaternions were first identified by L. Euler in a letter to Goldbach written on 4 May 1748," and they comment that "it makes no sense whatsoever to say that Euler 'identified' the quaternions in this letter ... this claim is absurd."[11]
  2. ^Tomb Raider (1996) is often cited as the first mass-market computer game to have used quaternions to achieve smooth three-dimensional rotations.[17]
  3. ^A more personal view of quaternions was written byJoachim Lambek in 1995. He wrote in his essayIf Hamilton had prevailed: Quaternions in physics: "My own interest as a graduate student was raised by the inspiring book by Silberstein". He concluded by stating "I firmly believe that quaternions can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics."[24]
  4. ^More properly, thefield of real numbers is isomorphic to a uniquesubring of the quaternions. The field of complex numbers is also isomorphic to three subsets of quaternions.[35]
  5. ^The vector part of a quaternion is apseudovector oraxial vector, not an ordinary orpolar vector.[36] A polar vector can be represented in calculations (for example, for rotation by a "quaternion similarity transform") by a vector quaternion, with no loss of information, but the two should not be confused. The axis of a "binary" (180°) rotation quaternion corresponds to the direction of the represented polar vector in such a case.
  6. ^For comparison, thenatural numbers, N0 ,{\displaystyle \ \mathbb {N} _{0}\ ,}rational numbers, Q ,{\displaystyle \ \mathbb {Q} \ ,} andreal numbers, R ,{\displaystyle \ \mathbb {R} \ ,} all have dimension 1; thecomplex numbers, C ,{\displaystyle \ \mathbb {C} \ ,} have dimension 2; and theoctonions O ,{\displaystyle \ \mathbb {O} \ ,} have dimension 8.
  7. ^The identification of the square roots of minus one inH{\displaystyle \mathbb {H} } was given by Hamilton[46] but was frequently omitted in other texts. By 1971 the sphere was included by Sam Perlis in his three-page exposition included inHistorical Topics in Algebra published by theNational Council of Teachers of Mathematics.[47] More recently, the sphere of square roots of minus one is described inIan R. Porteous's bookClifford Algebras and the Classical Groups (Cambridge, 1995) in proposition 8.13.[48]
  8. ^Books on applied mathematics, such as Corke (2017)[51] often use different notation withφ :=⁠1/2⁠θ — that is,another variableθ = 2φ.

References

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  1. ^abcHamilton described quaternions in a letter toJohn Graves, 17 October 1843. It was published inHamilton (1844).
  2. ^Rozenfelʹd, Boris Abramovich (1988).The history of non-euclidean geometry: Evolution of the concept of a geometric space. Springer. p. 385.ISBN 9780387964584.
  3. ^Hamilton, W.R., Sir (1853).Lectures on Quaternions. Dublin, EI: Hodges & Smith. p. 60.quaternion quotient lines tridimensional space time{{cite book}}: CS1 maint: multiple names: authors list (link)
  4. ^Hardy, A.S. (1881).Elements of Quaternions. Vol. 2. Ginn, Heath, & Co. p. 32.doi:10.1126/science.os-2.75.564.ISBN 9781429701860.PMID 17819877.{{cite book}}:ISBN / Date incompatibility (help);|journal= ignored (help)
  5. ^Curtis, Morton L. (1984),Matrix Groups (2nd ed.), New York:Springer-Verlag, p. 10,ISBN 978-0-387-96074-6
  6. ^abMamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010)."Orientational Sampling Schemes Based on Four Dimensional Polytopes".Symmetry.2 (3):1423–1449.Bibcode:2010Symm....2.1423M.doi:10.3390/sym2031423.
  7. ^Kunze, Karsten; Schaeben, Helmut (November 2004). "The Bingham distribution of quaternions and its spherical radon transform in texture analysis".Mathematical Geology.36 (8):917–943.Bibcode:2004MatGe..36..917K.doi:10.1023/B:MATG.0000048799.56445.59.S2CID 55009081.
  8. ^Smith, Frank (Tony)."Why not sedenion?". Archived fromthe original on 14 January 2024. Retrieved8 June 2018.
  9. ^abcSeeHazewinkel, Gubareni & Kirichenko 2004, p. 12
  10. ^Conway & Smith 2003, p. 9
  11. ^abBradley, Robert E.; Sandifer, Charles Edward (2007).Leonhard Euler: Life, work and legacy. Elsevier. p. 193.ISBN 978-0-444-52728-8.
  12. ^Pujol, J. (2012)."Hamilton, Rodrigues, Gauss, quaternions, and rotations: A historical reassessment".Communications in Mathematical Analysis.13 (2):1–14 – via projecteuclid.org.
  13. ^Gauss, C.F. (c. 1819). "Mutationen des Raumes" [Transformations of space]. In Martin Brendel (ed.).Carl Friedrich Gauss Werke [The works of Carl Friedrich Gauss]. Vol. 8. Prof. Stäckel of Kiel, Germany (article edited by). Göttingen, DE: Königlichen Gesellschaft der Wissenschaften [Royal Society of Sciences] (published 1900). pp. 357–361 – via Google.
  14. ^Hamilton, W.R., Sir (1866).Hamilton, W.E. (ed.).Elements of Quaternions. London: Longmans, Green, & Co.{{cite book}}: CS1 maint: multiple names: authors list (link)
  15. ^The Quaternion Association atMacTutor History of Mathematics archive
  16. ^abShoemake, Ken (1985)."Animating rotation with quaternion curves"(PDF).Computer Graphics.19 (3):245–254.doi:10.1145/325165.325242. Presented atSIGGRAPH '85.
  17. ^Bobick, Nick (July 1998)."Rotating objects using quaternions".Game Developer.
  18. ^McCarthy, J.M. (1990).An Introduction to Theoretical Kinematics. MIT Press.ISBN 978-0-262-13252-7.
  19. ^Hurwitz, A. (1919).Vorlesungen über die Zahlentheorie der Quaternionen. Berlin, DE: J. Springer.JFM 47.0106.01. — concerningHurwitz quaternions
  20. ^Ludwik Silberstein (1914)The Theory of Relativity viaInternet Archive
  21. ^Cornelius Lanczos (1949)The Variational Principles of Mechanics,University of Toronto PressISBN 0-8020-1743-6
  22. ^Huerta, John (27 September 2010)."Introducing the quaternions"(PDF).Archived(PDF) from the original on 2014-10-21. Retrieved8 June 2018 – viaU.C. Riverside (math.ucr.edu).
  23. ^Wood, Charlie (6 September 2018)."The strange numbers that birthed modern algebra".Abstractions (blog). Quanta Magazine.
  24. ^Lambek, J. (1995). "If Hamilton had prevailed: Quaternions in physics".The Mathematical Intelligencer. Vol. 17, no. 4. pp. 7–15.doi:10.1007/BF03024783.
  25. ^Clifford, W.K. (1878). "Applications of Grassmann's Extensive Algebra".American Journal of Mathematics.1 (4). The Johns Hopkins University Press:350–358.doi:10.2307/2369379.JSTOR 2369379./
  26. ^Peirce, B. (1881). "Linear associative algebra".American Journal of Mathematics.4 (1). Johns Hopkins University:221–226.doi:10.2307/2369153.JSTOR 2369153.
  27. ^Lipschitz, R. (1880). "Principes d'un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions".C. R. Acad. Sci. Paris (in French).91:619–621,660–664.
  28. ^Moore, C.L.E. (1922). "Hyperquaternions".Journal of Mathematics and Physics.1 (2):63–77.doi:10.1002/sapm19221263.
  29. ^Girard, P.R.; Clarysse, P.; Pujol, R.; Goutte, R.; Delachartre, P. (2018)."Hyperquaternions: a new tool for physics".Advances in Applied Clifford Algebras.28 (3) 68. Springer:1–14.doi:10.1007/s00006-018-0881-8.
  30. ^Girard, P. R. (2007).Quaternions, Clifford algebras and relativistic physics. Springer Science & Business Media.doi:10.1007/978-3-7643-7791-5.ISBN 978-3-7643-7790-8.
  31. ^Gürlebeck, K.; Sprössig, W. (1997).Quaternionic and Clifford calculus for physicists and engineers. New York: John Wiley & Sons.
  32. ^Girard, P.R.; Clarysse, P.; Pujol, R.; Delachartre, P. (2025)."Hyperquaternionic unitary symplectic groups: A unifying tool for physics".Advances in Applied Clifford Algebras.35 (40) 40. Springer.doi:10.1007/s00006-025-01402-w.
  33. ^Gürlebeck, K.; Habetha, K.; Sprössig, W. (2016).Application of holomorphic functions in two and higher dimensions. Springer.doi:10.1007/978-3-0348-0964-1.ISBN 978-3-0348-0962-7.
  34. ^Sprössig, W. (2020). "Some new aspects in hypercomplex analysis". In Breaz, Daniel; Rassias, Michael Th. (eds.).Advancements in Complex Analysis: From Theory to Practice. Springer. pp. 497–518.arXiv:2006.16386.doi:10.1007/978-3-030-40120-7.ISBN 978-3-030-40119-1.
  35. ^Eves (1976, p. 391)
  36. ^Altmann, Simon L. (1986).Rotations, Quaternions, and Double Groups. Clarendon Press. Ch. 12.ISBN 0-19-855372-2.LCCN 85013615.
  37. ^Hamilton, W.R., Sir (1866). "Article 285".Elements of Quaternions. Longmans, Green, & Company. p. 310.{{cite book}}: CS1 maint: multiple names: authors list (link)
  38. ^Hardy, A.S. (1881).Elements of Quaternions. Vol. 2. Ginn, Heath, & Co. p. 65 or 81 (?).doi:10.1126/science.os-2.75.564.ISBN 9781429701860.PMID 17819877.{{cite book}}:|journal= ignored (help)CS1 maint: ignored ISBN errors (link)
  39. ^"quaternion group".Wolframalpha.com.
  40. ^Gibbs, J. Willard; Wilson, Edwin Bidwell (1901).Vector Analysis. Yale University Press. p. 428.right tensor dyadic
  41. ^abHamilton, W.R. (1844–1850)."On quaternions or a new system of imaginaries in algebra". David R. Wilkins collection.Philosophical Magazine.Trinity College Dublin.
  42. ^"Visualizing Quaternions". Morgan-Kaufmann/Elsevier. 2005. Archived fromthe original on 2007-12-27. Retrieved2007-01-01.
  43. ^"[no title]".Wolframalpha.com. — determinant evaluation via onlinecomputer algebra software.
  44. ^Altmann (1986), p. 212, eqn 5.
  45. ^Farebrother, Richard William; Groß, Jürgen; Troschke, Sven-Oliver (2003)."Matrix representation of quaternions".Linear Algebra and Its Applications.362:251–255.doi:10.1016/s0024-3795(02)00535-9.
  46. ^Hamilton, W.R. (2009) [First published 1899].Elements of Quaternions (2nd ed.). Cambridge University Press. p. 244.ISBN 978-1-108-00171-7.
  47. ^Perlis, Sam (1971)."Capsule 77: Quaternions".Historical Topics in Algebra. Historical Topics for the Mathematical Classroom. Vol. 31. Reston, VA:National Council of Teachers of Mathematics. p. 39.ISBN 9780873530583.OCLC 195566.
  48. ^Porteous, Ian R. (1995). "Chapter 8: Quaternions".Clifford Algebras and the Classical Groups(PDF). Cambridge Studies in Advanced Mathematics. Vol. 50. Cambridge:Cambridge University Press. p. 60.doi:10.1017/CBO9780511470912.009.ISBN 9780521551779.MR 1369094.OCLC 32348823.
  49. ^"[no title cited]"(PDF).bridgesmathart.org. archive. Retrieved19 August 2018.
  50. ^abSärkkä, Simo (June 28, 2007)."Notes on Quaternions"(PDF).Lce.hut.fi. Archived fromthe original(PDF) on 5 July 2017.
  51. ^Corke, Peter (2017).Robotics, Vision, and Control – Fundamental Algorithms in MATLAB.Springer.ISBN 978-3-319-54413-7.
  52. ^Park, F.C.; Ravani, Bahram (1997)."Smooth invariant interpolation of rotations".ACM Transactions on Graphics.16 (3):277–295.doi:10.1145/256157.256160.S2CID 6192031.
  53. ^Hanson, Jason (2011). "Rotations in three, four, and five dimensions".arXiv:1103.5263 [math.MG].
  54. ^Günaşti, Gökmen (2016).Quaternions Algebra, Their Applications in Rotations and Beyond Quaternions (BS). Linnaeus University.
  55. ^"Three-Dimensional Point Groups".www.classe.cornell.edu. Retrieved2022-12-09.
  56. ^"Quaternions and Geometric Algebra".geometricalgebra.net. Archived fromthe original on 2008-09-08. Retrieved2008-09-12. See also:Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007).Geometric Algebra for Computer Science.Morgan Kaufmann.ISBN 978-0-12-369465-2. Archived fromthe original on 2008-09-08. Retrieved2008-09-12.
  57. ^Hamilton, W.R. (1853).Lectures on Quaternions. Dublin, IE: Hodges & Smith. p. 522.
  58. ^Graves, R.P.Life of Sir William Rowan Hamilton. Dublin Hodges, Figgis. pp. 635–636.
  59. ^Thompson, Silvanus Phillips (1910).The life of William Thomson. Vol. 2. London, UK: Macmillan. p. 1138.
  60. ^Heaviside, Oliver (1893).Electromagnetic Theory. Vol. I. London, UK: The Electrician Printing and Publishing Company. pp. 134–135.
  61. ^Silberstein, Ludwik (1924)."Preface to second edition".The Theory of Relativity (2nd ed.).

Further reading

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Books and publications

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Links and monographs

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External links

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