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Hamilton’s Quaternions

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Abstract

1. Sir William Rowan Hamilton was born in Dublin in 1805, and at the age of five was already reading Latin, Greek and Hebrew. He entered Trinity College Dublin in 1823, and while still an undergraduate was, in 1827, appointed Andrewes Professor of Astronomy at that university, and Director of the Dunsink Observatory with the title “Royal Astronomer of Ireland.” In that same year he began to develop geometric optics on extremal principles and in 1834/35 extended these ideas to dynamics, with the introduction of the principle of least action, the Hamiltonian function, and his canonical equations of motion. He was knighted in 1835 and was President of the Royal Irish Academy from 1837 to 1845. His great discovery of quaternions was made in 1843. He died in 1865 at Dunsink.

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... 2.5.4],[15, Secs. 3 and 4]and to the general literature[61,28,59,55]. Let {e, i, j, k} be the standard basis of R 4 , where e = (1, 0, 0, 0), i = (0, 1, 0, 0), j = (0, 0, 1, 0), k = (0, 0, 0, 1). ...
... Another convenient feature of the quaternions is that they can be used to parametrise rotations in 3 and 4 dimensions; compare[61]and[15]. In R 3 , any rotation can be parametrised 2 We usually identify a quaternion q = (q 0 , q 1 , q 2 , q 3 ) with the corresponding row vector (q 0 , q 1 , q 2 , q 3 ). ...
... In particular, the rotations of SO(3, Q) can be parametrised by integer quaternions as explained below. In R 4 , a pair of quaternions is needed to parametrise a rotation[61,36]. These quaternions are unique up to positive scaling factors and a common sign change. ...
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In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in crystallography. As many results are algebraic in nature, we also generalise them to Z\mathbb{Z}-modules embedded in Rd\mathbb{R}^d.
... A simple motivation for considering this speciai case is the following. It is known [2] that every orientation-preserving orthogonal mapping of IL4 may be expressed as x H uxü where u, v E H , IuJ = [VI = 1. When one wishes to investigate fixed points of a slightly more general affine map x H -axb-' + cb-' the equation (1) arises. ...
... Then substitution of (8) and then (2) shows that zaz + bz + zc + d = zaz + zay-' + y-'az + y-'ay-' + bz + by-' + zc + y-'c + d = zay-' + y-'az + y-'ay-' + y-'c + by-' = y-'((az + c)y + y(za + 6) + a)y-' which is equal to O by (7). Conversely, if zaz + bz + zc + d = O, it follows that (7) holds. ...
... is dense in H4. To see this, suppose that a, b,c, d are given, and let z be any solution to (2). By Theorem 1, the condition for (7) not to have a unique solution is Re (az + c) = -Re (za + b), luz + cl = Iza + 61. ...
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In this note we give an explicit solution to the linear equation in the quaternions H, a z + zb = C . (1) We apply this to draw some conclusions about the quadratic equation For a general history of the quaternions and referentes to work on the solu-tion of quaternionic polynomial equations, see (21.
... Proof. This follows easily from the corresponding result on orthogonal matrices in R 4 , as every non-zero similarity is the product of an orthogonal transformation and a non-zero homothety, see [13, 4] for details. The result on the determinants is standard. ...
... For some purposes, it is convenient to refer to the standard matrix representation of the linear map x → pxq, as defined via (pxq) t = M (p, q)x t . Details can be found in [13, 4]. Following [8], we consider the lattice ...
... Proof. This is a straight-forward calculation, e.g., with the matrix representation from [13, 4], taking into account that nr( p) = (nr(p)) ′ and expressing the coefficients in terms of traces and norms. ...
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Similar sublattices of the root lattice A4 are possible [J.H. Conway, E.M. Rains, N.J.A. Sloane, On the existence of similar sublattices, Can. J. Math. 51 (1999) 1300–1306] for each index that is the square of a non-zero integer of the form m2+mn−n2. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
... Proof. This follows easily from the corresponding result on orthogonal matrices in R 4 , as every non-zero similarity is the product of an orthogonal transformation and a non-zero homothety, see [13, 4] for details. The result on the determinants is standard. ...
... For some purposes, it is convenient to refer to the standard matrix representation of the linear map x → pxq, as defined via (pxq) t = M (p, q)x t . Details can be found in [13, 4]. Following [8], we consider the lattice ...
... Proof. This is a straight-forward calculation, e.g., with the matrix representation from [13, 4], taking into account that nr( p) = (nr(p)) ′ and expressing the coefficients in terms of traces and norms. ...
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Similar sublattices of the root lattice A4A_4 are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form m2+mnn2m^2 + mn - n^2. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q(5))\mathbb{H} (\mathbb{Q} (\sqrt{5})) and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
... Another possibly more arithmetically convenient representation of shape gauge group Sim 3 is as the Im(H) stable linear functions over the division algebra of quaternions H. This sections follows the notation of [6]. For this representation we send 3-vectors to pure imaginary quaternions R 3 x i = xê 1 + yê 2 + zê 3 → xi + yj + zk = x i ∈ Im(H) so that the following transformations are precisely the similarity transformations of Im(H) R 3 (they are isomorphic as vector spaces) ...
... Arithematic and geometry of the quaternions. Hamilton's quaternions are a 4 dimensional division algebra of R [6]. They obey all the axioms of a field save that multiplication is not commutative. ...
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... In contrast, determinants were known; they were introduced by Seki Takakazu (Kowa) and (independently) Gottfried Wilhelm Leibniz in the late seventeenth century. We refer to Koecher and Remmert [23], Stillwell [30], and Dickson [8] for historical details. Euler's intention was to solve the corresponding system of twelve quadratic equations in nine unknowns A; B; C; ; I , which exhibits a similar structure to the one in his problema curiosum, and probably explains how Euler came to this task. ...
... The norm of a quaternion is the determinant of its Cayley matrix, which is the sum of the four squares of the coefficients at 1; I; J; K. This and further information about quaternions and their history can be found in Koecher and Remmert [23]. ...
... For an extensive treatment on quaternions, we refer to [15,7,14,13]. It suffices to look at the primitive cubic lattice when studying the coincidences of the threedimensional cubic lattices because of the following well-known result [11,2]. ...
... We now embed the cubic lattices in the Hurwitz ring  of integer quaternions and we employ Cayley's parametrization of SO(3) (cf. [15]). That is, for every R ∈ SOC(Γ) = SO(3, É), there exists a primitive quaternion q (which is unique up to a sign) so that for all x ∈ Im (À), R(x) = qxq −1 = qxq/|q| 2 . ...
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... Of course this is only one of many ways to embed C in H; in fact, every nonreal q lies in a unique C λ (C), which is just the 2-dimensional real subspace R + Rq generated by 1 and q. For more details on the elementary properties of quaternions one may consult [11], [15] and the references there. ...
... which is an orthogonal mapping centered at s. (It is well known [15] that all orthogonal mappings of H are of this form.) Given r > 0, let ...
... We recall some results on quaternions which can be found in [26,29,10]. Let K be a real field. ...
... It is a group epimorphism whose kernel is Re \ {0}; cf. [26,Ch. 3.6]. ...
... Accordingly, generalized Clifford configurations are objects of conformal geometry since circles are mapped to circles. In the current context, it is therefore natural to identify the four-dimensional Euclidean space Ê 4 with the algebra of quaternions À (Koecher & Remmert 1991). Thus, we adopt the quaternionic representation Then, the following properties and identities may be established. ...
... Secondly, Cayley's theorem (Koecher & Remmert 1991) states that any element Ω of the orthogonal group O(4) is represented by either ...
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... Every nonreal q lies in a unique Cx(C), which is just the %-dimensional real subspace P + Pq generated by 1 and q. For more details on the elementary properties of quaternions one may consult [lo], [13] and the references there. Quaternionac linear maps and Mobiw transformations. ...
... Consider the following three auxiliary mappings. Given S E W and u,v 6 S 3 = {q: IqI = 11, let which is an orthogonal mapping centered at s. (In fact, it is well known [13] that ail orthogonai mappings of MI are of this form.) Given T > O, let which is the inversion in the 3-sphere of radius r centered at s. Lastly, given ...
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One-parameter families of quaternionic linear-fractional transformations are defined in terms of the exponential mapping from the Lie algebra of PSL2 H. The invariance of loxodromic curves allows us to characterize the fixed points corresponding to the family exp(tX) in terms of the generator X sl2 H. Certain degenerate cases are described; it is shown that for nonplanar loxodromes the generator is unique.
... General introductions to algebraic properties of quaternions can be found in [2] and [6]. ...
... In this section, following [1], [2], and [6], we recall some definitions and elementary results. By K we will denote a finite xtension of Q contained 1 in R. Then D is the ring of algebraic integers of K, and an element α ∈ R is said to be K-rational (resp. ...
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We prove a constructive version of Tits' alternative for groups of quaternions with algebraic coefficients by bounding valuations of their entries considered as elements of a fraction field of an opportunely chosen Dedekind domain.
... Any rotation in 4 dimensions can be parameterized by two quaternions p = (k, ℓ, m, n) and q = (a, b, c, d) in the following way [7,8,9]: ...
... In order to do this we recall that the 3-dimensional rotations can be parameterized by quaternions as well [7,8,9]. The group G of order |G| = 48 generated by the quaternions (±1, 0, 0, 0), 1 √ 2 (±1, ±1, 0, 0), 1 2 (±1, ±1, ±1, ±1) and permutations thereof is a double cover of the cubic symmetry group O of order |O| = 24. ...
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... Then one can show that a rotation is a coincidence rotation if and only if it is a orthogonal matrix with rational entries [4,6,7]. Now any proper rotation in three-dimensional space can be parameterized by quaternions (Cayley's parameterization) [12,13,14,15]: ...
... In addition, for any quaternion r we define the conjugated quaternion byr := (κ, −λ, −µ, −ν). For more details we have to refer to the literature [12,13,14]. Although we will use quaternions extensively in the following no knowledge of quaternions is necessary to understand most of the results. ...
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We consider the symmetries of coincidence site lattices of 3-dimensional cubic lattices. This includes the discussion of the symmetry groups and the Bravais classes of the CSLs. We derive various criteria and necessary conditions for symmetry operations of CSLs. They are used to obtain a complete list of the symmetry groups and the Bravais classes of those CSLs that are generated by a rotation through the angle π\pi.
... by quaternions which were introduced by W.R. Hamilton in the middle of the 19th century after searching for more general number systems than complex numbers (e.g.[33] [34] which both (by definition) are real in case of real quaternions.The associative but not commutative multiplication law in the quaternion alge- ...
... where the group of quaternions as defined by Hamilton in 1843 [23] utilizes the imaginary units that follow the definition i 2 = j 2 = k 2 = ijk = −1 and {q 0 , q 1 , q 2 , q 3 ∈ R}. It is also common to represent the quaternion as two components, the vector component (i, j, and k) and the scalar component (denoted by q 0 ). ...
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... In this case the motion is decomposed into a center of mass translation and a rotation about the center of mass. The first one is treated in the standard way, while the rotations are described using Hamilton's quaternions [51]. The use of a rigid molecule may result in saving computer time by a facter of 5 − 10, as ∆t can be increased while maintaining the same energy conservation. ...
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... Relative to the inner product tr(x¯ y) = 2x|y, where x|y denotes the standard Euclidean inner product, this lattice is the root lattice A 4 ; see [8, 4, 3] for details. This particular description of the root lattice A 4 in R 4 is very convenient for our problem, as it enables us to use the arithmetic of the quaternion algebra H(Q( √ 5 )); see [12] for a detailed introduction to Hamilton's quaternions. For brevity we use from now on the notation ...
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