Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems

Katrusiak, Andrzej; Le, Hien Quy

doi:10.3390/sym16101366

Open AccessArticle

Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems

by

Andrzej Katrusiak

^1,*

and

Hien Quy Le

^1,2,3

¹

Faculty of Chemistry, Adam Mickiewicz University, Uniwersytetu Poznańskiego 8, 61-614 Poznań, Poland

²

Institute of Physics, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

³

Laboratory of X-ray Nanoscience and Technology, Center for Photon Science, Paul Scherrer Institute, Forschungsstrasse 111, 5232 Villigen, Switzerland

^*

Author to whom correspondence should be addressed.

Symmetry 2024, 16(10), 1366; https://doi.org/10.3390/sym16101366

Submission received: 13 September 2024 / Revised: 10 October 2024 / Accepted: 12 October 2024 / Published: 14 October 2024

(This article belongs to the Section Physics)

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Abstract

:

Quaternion and biquaternion symmetry transformations have been applied to non-Cartesian reference systems of direct and reciprocal crystal lattices. The transformations performed directly in the sets of crystal reference axes simplify the calculations, eliminate the need for orthogonalization, permit the use of crystallographic vectors for defining the directions of rotations and perform the computations directly in the crystal coordinates. The applications of the general quaternion transformations are envisioned for physical, chemical, crystallographic and engineering applications. The general quaternion multiplication rules for any symmetry-unrestricted lattices have been derived for the triclinic crystallographic system and have been applied to the biquaternion representations of all point-group symmetry elements, including the crystallographic hexagonal system. Cayley multiplication matrices for point-groups, based on the biquaternion symbols of proper and improper symmetry elements, have been exemplified.

Keywords:

rotations; proper and improper symmetry; general transformation; non-Cartesian system; crystal lattice; quaternion; biquaternion; computations optimization; Cayley table

Graphical Abstract

1. Introduction

The inception of quaternions [1,2,3,4] significantly simplified the calculations in technical and scientific applications, particularly those involving rotations. Apart from the geometry-related applications of quaternions, there is a steady progress in simplifying various types of scientific problems, for example in quantum and relativistic physics. A remarkable example of the significance of quaternions is their derivation from Pauli matrices occurring in the Schrödinger–Pauli equation. Quaternions are suitable for constructing the algorithms of rotations and they considerably reduce the computing power required for real-time calculations for artificial intelligence and computer graphics. The main advantage of quaternions, compared to the transformations performed with matrices, is the simplicity and direct connection between the quaternion form explicitly derived from the rotation direction and angle. This simplicity contrasts with the matrix representation of rotations, requiring the elaborate decomposition of rotations into Eulerian angles. Consequently, the quaternion-based algorithm designs have been used to optimize computations in algorithms designed for quick computations in autonomous cars, drones, geolocation, games, animations or demanding technical modelling (e.g., see References [5,6]). Quaternions are applied in various fields of scientific research, too. In diffractometry and crystallography, the quaternions are occasionally used for orthogonal tasks, such as positioning diffractometer shafts [7,8] and for refining orientation matrices during diffraction measurements [9]. However, these diffractometric applications are performed exclusively in the Cartesian laboratory reference system for 4-axes and 6-axes diffractometers equipped either with Eulerian or κ cradles for the sample-crystal rotations [10,11,12]. Likewise, for crystal structure-related problems, such as the description of disorientations between crystal lattices [13,14], comparison of independent molecular fragments [15,16,17,18] and symmetry-operation representations [19,20], the computations in orthogonal reference systems were involved. Most recently, we have presented the quaternion representation of all point-group symmetry operations in traditional crystallographic triclinic, monoclinic, orthorhombic, tetragonal and cubic systems, as well as for the trigonal system in the rhombohedral setting [21].

Presently, we describe the method of expanding the quaternion representations of symmetry operations to the hexagonal setting, which complements all point-group symmetry operations in all seven traditional crystallographic systems. It has been achieved by deriving the quaternion multiplication rules for a general unrestricted reference system, corresponding to the crystallographic triclinic system. This general formalism extends the application of quaternions to non-symmetric operations in all crystallographic direct and reciprocal-lattice reference systems. We also show that the algebra of quaternions is insufficient for generally completing the multiplication tables involving all, proper and improper, symmetry operations. However, we demonstrate that for all crystallographic systems, the symmetry operations can be obtained by combining the non-orthonormal quaternions with the old concept of biquaternions [22]. Such non-Cartesian biquaternions afford the general representation of point-symmetry groups. The crystallographic applications belong to the most demanding and, at the same time, most explored examples of non-Cartesian reference systems in nature. On the other hand, no successful representation of quaternions for representing all point-symmetry operations has been reported so far. The presently reported generalized quaternion and biquaternion representations of transformations devised for non-Cartesian reference systems fill this gap and open new alleys for significantly increasing the efficiency of computations and modelling not only in solid-state physics, chemistry and other materials sciences, but also in engineering and technological applications. We believe that this complete biquaternion representation of proper and improper symmetry operations, as well as any non-symmetric transformations, directly in the non-Cartesian systems, i.e., with no need of orthogonalization, will soon replace other more complicated calculation methods.

2. Discussion

2.1. Basic Algebra and Definitions

According to Hamilton [2], the quaternion is defined as q = s + ui + vj + wk, where s, u, v and w are real numbers, and imaginary versors i, j and k fulfil the following conditions:

i² = j² = k² = i·j·k = −1.

(1)

Equation (1) implies that the quaternion multiplication rules can be rewritten as

ij = k; jk = i; ki = j.

(2)

Generally, the quaternion can be written as q = s + v, where v is an imaginary 3-component vector v = ui + vj + wk. The multiplication between quaternions q₁ and q₂ is represented by the following equation:

q₁·q₂ = s₁·s₂ − v₁·v₂ + s₁·v₂ + s₂·v₁ + v₁ × v₂,

(3)

where indices refer to quaternions q₁ and q₂, symbol ‘·’ indicates the scalar multiplication and ‘×’ the vector multiplication of vectors.

It is apparent from Equation (3) that the quaternion multiplication rules of Equations (1) and (2) are valid only for the Cartesian reference systems, where all three reference axes are mutually orthogonal. When assuming a non-Cartesian system, Equations (1) and (2) are no longer valid and new appropriate products must be calculated according to Equation (3). The most general least-restricted reference system corresponds to the crystallographic triclinic system, where no relations binding unit-cell parameters a, b, c, α, β and γ are imposed by symmetry elements. In the triclinic setting, we request that unit quaternion vectors i, j and k be collinear with the unit-cell edges, so vector r = (i, j, k) runs along the unit-cell diagonal in the imaginary space; both scalars s₁ and s₂ are equal to zero (Equation (3)). Thus, for the triclinic unit-cell parameters a ≠ b ≠ c ≠ a and angles α ≠ β ≠ γ ≠ α, all unrestricted to 90° or any other value, we obtain the following:

i² = −i∙i = −|i||i| = −|i|² = −a²
j² = −j∙j = −|j||j| = −|j|² = −b²
k² = −k∙k = −|k||k| = −|k|² = −c²

(4a)

ij = −ab cos γ + i × j,     ji = −ab cos γ − i × j
jk = −bc cos α + j × k,       kj = −bc cos α − j × k
ki = −ca cos β + k × i,       ik = −ca cos β − k × i

(4b)

It can be established that ij = (ji)*, jk = (kj)*, ki = (ik)*, where asterisks denote the conjugation. After some algebraic transformations (cf. Appendix A), the multiplication rule for quaternions in any 3-axis coordinates can be written in the following form:

i j = - a b \cos γ + \frac{a b s i n^{2} γ}{Ω c} k + \frac{b (\cos γ \cos α - \cos β)}{Ω} i + \frac{a (\cos γ \cos β - \cos α)}{Ω} j j k = - b c \cos α + \frac{b c s i n^{2} α}{Ω a} i + \frac{c (\cos α \cos β - \cos γ)}{Ω} j + \frac{b (\cos α \cos γ - \cos β)}{Ω} k k i = - c a \cos β + \frac{c a s i n^{2} β}{Ω b} j + \frac{a (\cos β \cos γ - \cos α)}{Ω} k + \frac{c (\cos β \cos α - \cos γ)}{Ω} i

(4c)

where

Ω = \sqrt{1 - {c o s}^{2} α - {c o s}^{2} β - {c o s}^{2} γ + 2 \cos α \cos β \cos γ} .

(4d)

Parameter

Ω

is connected to the unit-cell volume (V) by the formula V = abc

Ω

. For a normal Cartesian reference system, where a = b = c = 1 and α = β = γ = 90°, all cosines are equal to 0, all sines are equal to 1 and

Ω

= 1. Then, Equation (4) reduce to Equations (1) and (2), which define the quaternions in Cartesian coordinates. For a given lattice described by parameters a, b, c, α, β, and γ, simplify to the form convenient for practical calculations, as illustrated in several examples below. We have used the same number for all of Equation (4) in order to stress that these are the same definitions of quaternions, depending on the reference system defined by parameters.

The crystallographic symmetry operations in the quaternion representations were recently presented for point-groups of all crystallographic systems, except those in the hexagonal setting [21]. The quaternion formula for rotating imaginary vector r by angle φ about vector v is

r′ = q(φ,v)·r·q*(φ,v)

(5)

where quaternion q(φ,v) is

q(φ,v) = cos(φ/2) + n·sin(φ/2)

(6)

and n = v/|v| is the unit vector parallel to the rotation-axis vector v. In Equation (6), vector n specifies the fractions of versors i, j and k of the imaginary space. Owing to the quaternion definition expanded to the non-Cartesian systems in Equation (4), Equation (6) is valid for any rotations, not only those connected to symmetry operations.

The normalization of vector v is mandatory. It is easy in the Cartesian systems; however, for the triclinic reference systems, the length of vector v = [uvw] is

|v| = \sqrt{(u, v, w) G {(u, v, w)}^{T}}

(7)

where superscript T denotes transposition and

G

is the metric tensor:

G = (\begin{matrix} a^{2} & a b \cos γ & a c \cos β \\ a b \cos γ & b^{2} & b c \cos α \\ a c \cos β & b c \cos α & c^{2} \end{matrix})

(8)

Noteworthy, there are non-Cartesian systems where the application of matrix

G

is required only for the transformations, either of the proper or improper rotations, the directions of which do not coincide with the symmetric directions of the system. In other words, two or three vector components involve versors, which are not symmetry-related. This does not apply to any crystallographic symmetry operations, because by definition their components must be symmetry-related. Owing to this feature, the quaternions representing symmetry operations have simple forms. For example, the inversion centre of the triclinic system is not connected with any direction but reverses the sign of all components; for the standard setting of the monoclinic system, the 2-fold axis and the mirror plane are connected to the [y] direction only; in the orthorhombic system, the rotations are either about axes [x], [y] or [z]. The mixed components of the rotation vectors appear in the tetragonal system (diagonal directions [110] and

[1 \bar{1} 0]

), hexagonal family (diagonals [120], [210] and

[\bar{1} 10])

and cubic system (six flat diagonals of type [110] and four space-diagonals of type [111]), but the directions of these rotation axes run along diagonals of regular polygons and a regular polyhedron (only in the cubic system is the space-diagonal symmetric), so their edges are equal and the directions of the diagonals are fixed relative to the crystal reference system. The same applies to symmetric directions in the reciprocal space and to the relation between direct and reciprocal spaces. However, the vectors for non-symmetric directions of rotations require to be normalized using the metric tensor

G

given in Equation (8).

2.2. Symmetry Operations in Hexagonal Lattice

The symmetry operations in the hexagonal family systems, as previously shown for all symmetry operations in the other crystallographic systems [21], do not require the information about the length of the unit vectors, i.e., the unit-cell parameters a, b and c. Therefore, Equation (4) can be simplified, by assuming that a = b = c = 1, α = β = 90° and γ = 120°, to the form

i^{2} = j^{2} = k^{2} = - 1 i j = 1 / 2 + \sqrt{3} / 2 k = {(j i)}^{*} j k = (2 \sqrt{3} / 3) i + (\sqrt{3} / 3) j = {(k j)}^{*} k i = (\sqrt{3} / 3) i + (2 \sqrt{3} / 3) j = {(i k)}^{*} .

(9)

Likewise, valid remain the improper rotations, performed according to transformation type:

r′ = −q(φ,v)·r·q*(φ,v)

(10)

(leftward, due to the combination of the natural rotation with the inversion centre), analogous to improper-rotation symmetry elements

\bar{1},

m,

\bar{3}

,

\bar{4}

and

\bar{6}

[21].

2.3. Non-Symmetric Transformations

For the non-symmetric transformations (i.e., the transformations inconsistent with the symmetry elements of crystallographic systems) in non-Cartesian systems, in general all lattice parameters must be substituted to Equation (4). Below, we exemplify the quaternion representation of such a non-symmetric rotation by an arbitrary angle about the [y] direction in the monoclinic system. Let us rotate vector a by angle −β about the [y] direction in the monoclinic reference system, where we assume the following unit-cell parameters: a = b = 1, c = 2; α = γ = 90° ≠ β (Figure 1).

Example 1.

The substitution of these parameters into the quaternion multiplication rules, specified in Equation (4), gives the following:

i^{2} = j^{2} = - 1 k^{2} = - 4 Ω = \sin β i j = \frac{1}{2 \sin β} k - \frac{\cos β}{\sin β} i = {(j i)}^{*} j k = \frac{2}{\sin β} i - \frac{\cos β}{\sin β} k = (k j)^{*} k i = - 2 \cos β + 2 j \sin β = (i k)^{*}

The rotation of r to r′ is the leftward rotation, because the rotation angle is negative (−β), so transformation type r′ = q*rq can be applied [21]:

r^{'} = (\cos \frac{β}{2} - j \sin \frac{β}{2}) i (\cos \frac{β}{2} + j \sin \frac{β}{2}) = k / 2

(for detailed numerical calculations cf. Example S1 in the Supplementary Materials). Thus, as expected for the assumed relation |a| = |c|/2, the rotation of vector a by angle −β around [y], we obtain vector r′ = [0 0 1/2], as shown in Figure 1.

Example 2.

This next example illustrates the use of reciprocal vectors for quaternion representations of any, not necessarily symmetric, rotations. Hereafter, we will denote the reciprocal vectors by superscript “r” (instead of the usually used asterisk, to avoid confusion with the conjugation of quaternions). It can be requested for the monoclinic lattice to rotate the a = [100] vector about the reciprocal vector

c^{r}

. According to Ewald’s definition of reciprocal vectors,

c^{r} = a \times b / V

and for the monoclinic lattice

|c^{r}| = 1 / (c s i n β)

. The crystal-lattice components of this reciprocal vector

c^{r}

are

c^{r} = [\frac{- \cos β}{ca \sin ² β}, 0, \frac{1}{c^{2} \sin ² β}]

and the unit vector parallel to

c^{r}

is

n^{r}

:

n^{r} = [- \frac{\cos β}{\sin β}, 0, \frac{1}{c \sin β}]

For the example described above (unit-cell parameters: a = b = 1, c = 2; α = γ = 90° ≠ β, as shown in Figure 1),

n^{r} = [- \frac{\cos β}{\sin β}, 0, \frac{1}{2 \sin β}]

. The quaternion for the rightward, 90° rotation around vector

n^{r}

is

q = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} (- i ctg β + \frac{k}{2 \sin β})

The rotation of vector a by 90° around the axis

c^{r}

is

r^{'} = q i q^{*} = \frac{1}{\sqrt{2}} (1 - \frac{\cos β}{\sin β} i + \frac{1}{2 \sin β} k) i \frac{1}{\sqrt{2}} (1 + \frac{\cos β}{\sin β} i - \frac{1}{2 \sin β} k) = j

Thus, the rotation by 90° around

c^{r}

superimposes the image of vector a with vector b along axis [y], as expected for the assumed unit-cell dimensions (Figure 1). It should be noted that the reciprocal-space vectors are products of vector multiplication (cross products or pseudovectors) and the above example illustrates the validity of the quaternion transformations to all pseudovectors applied in crystallography, like Miller indices (hkl) of planes or Bragg indices hkl of diffraction reflections.

In the next examples below, we focus on illustrating the advantages of quaternion transformations easily performed in one step in crystallographic coordinates.

Example 3.

First, let us rotate vector r = [001] by an angle of 180° about direction [110] for an orthorhombic lattice with unit-cell a = b/2 = c = 1 and α = β = γ = 90°. The multiplication rules for quaternions specified in Equation (4), in this system, yield

i^{2} = k^{2} = - 1 j^{2} = - 4 i j = 2 k = (j i)^{*} j k = 2 i = (k j)^{*} k i = j / 2 = (i k)^{*} .

So, the quaternion for the requested rotation is q(180°, [110]) = (i + j)/

\sqrt{5}

.

And the rotated vector r’ can be calculated as

r^{'} = q r q^{*} = [(i + j) / \sqrt{5}] i [- (i + j) / \sqrt{5}] = = - 3 i / 5 + 2 j / 5,

which corresponds to direct-space crystal-lattice coordinates r′ = [−3/5, 2/5, 0].

Example 4.

In order to illustrate the effects of skew lattices for the quaternion transformations, let us consider the non-conventional unit-cell a = b/2 = c = 1, α = β = 90°, γ = 120° (cf. the previous example) and vector r = [100] is to be rotated by angle 180° about the direction [110]. The quaternion multiplication rules of Equation (4) for this system give

i^{2} = k^{2} = - 1 j^{2} = - 4 i j = 1 + \sqrt{3} k = (j i)^{*} j k = (4 \sqrt{3} / 3) i + (\sqrt{3} / 3) j = (k j)^{*} k i = (\sqrt{3} / 3) i + (\sqrt{3} / 3) j = (i k)^{*}

The metric matrix G specified in Equation (8) simplifies to

G = (\begin{matrix} 1 & - 1 & 0 \\ - 1 & 4 & 0 \\ 0 & 0 & 1 \end{matrix})

According to Equation (7), the length of the vector v = [110] is

|v| = \sqrt{(1 1 0) (\begin{matrix} 1 & - 1 & 0 \\ - 1 & 4 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 \\ 1 \\ 0 \end{matrix})} = \sqrt{3} .

The quaternion for the rotation by angle 180° about direction [110] is

q = (i + j) / \sqrt{3} .

The rotation of vector r = [100] by angle 180° about direction [110] leads to (cf. Section S4 in the Supplementary Materials)

r^{'} = q r q^{*} = (i + j) i (- i - j) / 3 = - i,

which corresponds to the crystal-lattice vector

r^{'} = [\bar{1} 00]

.

Example 5.

For the same lattice (unit-cell a = b/2 = c = 1, α = β = 90°, γ = 120° (cf. the previous Example 4), let us now rotate vector r = [100] by angle 180° about crystal direction

[1 \bar{1} 0]

. Hence,

v = [1 \bar{1} 0] |v| = \sqrt{(1 \bar{1} 0) G {(1 \bar{1} 0)}^{T}} = \sqrt{(1 - 1 0) (\begin{matrix} 1 & - 1 & 0 \\ - 1 & 4 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix})} = \sqrt{7}

The quaternion for the rotation by angle 180° about direction

[1 \bar{1} 0]

is

q = (i - j) / \sqrt{7 .}

The rotation vector r = [100] by angle 180° about direction

[1 \bar{1} 0]

is

r^{'} = q r q^{*} = (i - j) i (- i + j) / 7 = (- 3 i - 4 j) / 7 .

Thus, the rotation of vector r = [100] about direction

[1 \bar{1} 0]

by angle 180° results in vector r′ = [−3/7, −4/7, 0]. The calculations are detailed in Section S5 in the Supplementary Materials.

Example 6.

For somewhat modified unit-cell dimensions a = b/2 = c = 1, α = β = 90°, γ = 120° (cf. the previous example) and vector r = [100] rotated by angle 180° about direction [110], the quaternion multiplication rules generally presented in Equation (4) are

i^{2} = - 4, j^{2} = k^{2} = - 1, i j = 1 + \sqrt{3} k = (j i)^{*}, j k = (i + j) / \sqrt{3} = (k j)^{*}, k i = (i + 4 j) / \sqrt{3} = (i k)^{*} .

The metric matrix

G

of Equation (8) becomes

G = (\begin{matrix} 4 & - 1 & 0 \\ - 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

According to Equation (7), the length of the vector v = [110] is

|v| = \sqrt{(1 1 0) (\begin{matrix} 4 & - 1 & 0 \\ - 1 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 \\ 1 \\ 0 \end{matrix})} = \sqrt{3} .

The quaternion for the rotation by angle 180° about direction [110] is

q = (i + j) / \sqrt{3 .}

The rotation of vector r = [100] by angle 180° about direction [110] leads to (cf. Section S4 in the Supplementary Materials)

r^{'} = q r q^{*} = (i + j) i (- i - j) / 3 = i - j

corresponding to the crystal-lattice vector

r^{'} = [1 \bar{1} 0]

.

2.4. Trigonal System in Hexagonal Setting

All point-group symmetry elements for the crystallographic trigonal system in the hexagonal reference system are listed in Table 1. We have generally applied the Hermann–Mauguin notation of symmetry elements (1, 2, 3-fold natural axes and their improper analogues

\bar{1},

m,

\bar{3}

), with the subscripts defining the directions parallel to the rotation axes and perpendicular to the mirror planes. The hexagonal unit-cell coordinates imply the quaternion multiplications defined in Equation (9). For generating the improper rotations, the combination of natural rotations (1, 2 and 3-fold axes) and the inversion centre have been used, which implies the leftward rotation. As shown previously for all point-group symmetry operations, two transformation types, qrq* and −qrq*, can be distinguished, which in some cases can be reduced to qr or qrq [21].

Example 7.

For example, the reflection of vector r = [1, −1, 0] in mirror plane m_[100] can be represented as

q (m_{[100]}) \cdot r \cdot q (m_{[100]}) = i (i - j) i = (- 1 - i j) i = - (1 + \frac{1}{2} + \frac{\sqrt{3}}{2} k) i = - \frac{3}{2} i - \frac{\sqrt{3}}{2} k i = = - \frac{3}{2} i - \frac{\sqrt{3}}{2} (\frac{\sqrt{3}}{3} i + \frac{2 \sqrt{3}}{3} j) = - 2 i - j

which corresponds to the transformed vector

[\bar{2} \bar{1} 0]

. For the same mirror plane m_[100] and vector

[\bar{1} \bar{1} 0]

,

i (- i - j) i

gives the transformed vector

[0 \bar{1} 0]

.

2.5. Hexagonal System

All symmetry elements for the hexagonal point-groups and the quaternion representations of the symmetry operations in the hexagonal reference system are listed in Table 2. The same notation as in the previous section has been applied.

Example 8.

The rotations of the vector [100] around the 6-fold axis parallel to the [z] axis (Figure 2) can be performed by calculating r′ = q(60°, [z])·i·q(60°, [z]):

(\frac{\sqrt{3}}{2} + \frac{1}{2} k) i (\frac{\sqrt{3}}{2} - \frac{1}{2} k) = (\frac{\sqrt{3}}{2} i + \frac{1}{2} k i) (\frac{\sqrt{3}}{2} - \frac{1}{2} k) = [\frac{\sqrt{3}}{2} i + \frac{1}{2} (\frac{\sqrt{3}}{3} i + \frac{2 \sqrt{3}}{3} j)] (\frac{\sqrt{3}}{2} - \frac{1}{2} k) = [\frac{2 \sqrt{3}}{3} i + \frac{\sqrt{3}}{3} j] (\frac{\sqrt{3}}{2} - \frac{1}{2} k) = i - \frac{i k}{\sqrt{3}} + \frac{1}{2} j - \frac{\sqrt{3}}{6} j k = i + \frac{1}{\sqrt{3}} (\frac{\sqrt{3}}{3} i + \frac{2 \sqrt{3}}{3} j) + \frac{1}{2} j - \frac{\sqrt{3}}{6} (\frac{2 \sqrt{3}}{3} i + \frac{\sqrt{3}}{3} j) = i + j,

corresponding to vertex [110], as shown in Figure 2.

By continuing the same procedure, we obtain the quaternion coordinates of the remaining vertices:

q (i + j) q^{*} = j q j q^{*} = - i q (- i) q^{*} = - i - j q (- i - j) q^{*} = - j q (- j) q^{*} = i

The coordinates of these vertices correspond to the nodes in the crystal structure and define the direction symbols in the crystal lattice, as indicated in Figure 2.

Example 9.

These transformations derived in Example 8 can be applied for the D_6h-symmetric molecule of benzene (C₆H₆), where the C-C aromatic bond length is 1.39 Å and C-H bond is 1.09 Å long. The hexagonal atomic coordinates (cf. Figure 2) can be easily obtained as

C1 [1.39, 0, 0], H1 [2.48, 0, 0],

C2 [1.39, 1.39, 0], H2 [2.48, 2.48, 0],

C3 [0, 1.39, 0], H3 [0, 2.48, 0],

C4 [−1.39, 0, 0], H4 [−2.48, 0, 0],

C5 [−1.39, −1.39, 0], H5 [−2.48, −2.48, 0],

C6 [0, −1.39, 0], H6 [0, −2.48, 0].

2.6. Biquaternion for General Application of the Point Symmetry

So far, we have discussed the potential application of quaternions for representing symmetry operations in point-groups. Quaternions simplify the calculation and avoid the use of matrices. However, the different actions required for the proper and improper rotations (

\bar{1}

(inversion centre),

\bar{2}

(mirror plane m),

\bar{3}

(3-fold inversion axis),

\bar{4}

,

\bar{6}

, etc.) hinder the construction of uniform multiplication tables of symmetry operations for all point-groups. The action for proper rotations is defined by Equation (5):

r′ = qrq*,

while for improper rotations, the action is defined by semi-empirical Equation (10):

r′ = −qrq*

The presence of the minus sign in Equation (10) naturally arises from the definition of improper rotations as the combination of proper rotations followed by an inversion.

This problem of different actions can be solved by introducing the square root of −1 as a number, which leads to Hamilton’s less well-known biquaternion concept proposed by him in 1853 [22]. This square root of −1 is considered as an imaginary number, not a quaternion field. Historically, to avoid confusion with i used in quaternions (Equations (1) and (2)), Hamilton denoted it as h. As a number, the commutativity of the scalar field h and quaternion q is assumed

hq = qh.

Accordingly, now Equation (10) for improper rotations (r′ = −qrq*) can be rewritten by replacing −1 with h²:

r′ = h²qrq* = hhqrq*

and by using the commutativity of h, we obtain

r′ = (hq)r(hq*)

(11)

As we treat h as a number, we can extend the concept of the quaternion conjugate to the biquaternion “biconjugate”. Given a biquaternion

w = a + bi + cj + dk

where a, b, c, d each is a complex number (f + gh), h is defined above and f, g are real numbers.

The biconjugate of a biquaternion has exactly the same formula as the conjugate of a quaternion.

w* = a − bi − cj − dk

So, by using this new definition and noting that (hq) and (hq*) are biconjugates of each other, Equations (5) and (10) can be rewritten as

r′ = wrw*

(12)

Because a quaternion conjugate is a special case of a biquaternion biconjugate, this formula is true for both proper and improper rotations. However, compared to Equation (6), the proper and improper rotations are now differentiated as

w_proper = cos(φ/2) + n·sin(φ/2)
w_improper = h·cos(φ/2) + n·h·sin(φ/2)

(13)

One of the simplest point-groups involving improper symmetry elements is point-group C_i (

\bar{1}

), and its multiplication table of biquaternions defined in Equation (13), used in symmetry operations defined in Equation (12), is presented in Table 3. The two symmetry operations, identity and inversion centre, are represented by biquaternions 1 and h, respectively.

The multiplication table of biquaternions applied for the representation of symmetry operations in point-group mm2 is presented in Table 4. The four symmetry elements, identity, two-fold axis 2 along [z] and two mirror planes perpendicular to [x] and [y], are represented by biquaternions 1, k, hi and hj, respectively.

In Table 3 and Table 4, some products of multiplied biquaternions are negative, because the biquaternions and quaternions for the identity operation are either 1 or −1. This follows the quaternion rotation definition in Equation (5), that q = cos(φ/2) + n sin(φ/2), which for φ = 0° gives q = 1, but for φ = 360° gives q = −1. In geometrical terms, although the final position identical to the original one is obtained, the positive and negative q signs indicate the even and odd numbers of full rotations of the object, respectively (rotations by 0·360°, 1·360°, 2·360°, 3·360°, 4·360°, etc.). Interestingly, this parity relation applies to the identity composed of inversion centres (h·h = −1, h·h·h·h = (h·h)² = 1, (h·h)³ = −1, (h·h)⁴ = 1, (h·h)⁵ = −1, etc.), but not to other improper rotations. For example, for the biquaternions for a combination of two mirror planes m_y (Table 4), hj·hj, owing to the commutativity, can be rewritten as h·h·j·j = (−1)·(−1) = 1; hence, any power of (hj·hj)ⁿ = 1. Likewise, all diagonal mirror planes (

\bar{2}

), and inversion axes

\bar{3}

,

\bar{4}

,

\bar{6}

, etc., can be generally encoded as (h·

\bar{N}

^N)²ⁿ = 1. Nonetheless, the transformation definition in Equation (12) eliminates these negative values for the identity operations and it also unequivocally defines the correspondence of the biquaternions and symmetry operations. Hence, the substitution of biquaternions with the corresponding symmetry operations for point-group mm2 yields the multiplication Cayley table (Table 5).

The multiplications of biquaternions applied for the representation of the cyclic point-group

\bar{6}

in Table 6, composed of the powers wⁿ of w = h

\bar{6}

(Table 3), display no sign changes in the multiplication products, neither for the identity operations nor for any powers wⁿ, in accordance with the above discussion for the improper rotations. Hence, the multiplication table of biquaternions can be straightforwardly rewritten into the Cayley multiplication table of symmetry operations of point-group

\bar{6}

(Table 7).

3. Conclusions

In this study, all symmetry operations in the hexagonal coordinates for the trigonal and hexagonal systems have been represented as quaternions. Consequently, quaternion representations of all point-symmetry transformations have been completed for crystallographic reference systems. The quaternion representations for the hexagonal setting require the general multiplication rules of quaternions, which in turn expand the capabilities of quaternion representations to non-symmetric rotations in non-Cartesian reference systems. Thus, any transformations can be performed directly in crystallographic coordinates, which eliminates the orthogonalization and reverse steps and significantly simplifies the numerical calculations. The use of crystallographic reference systems is also convenient for relating the rotation axes to the symbols of crystallographic directions in the direct and reciprocal space and to any vectors expressed in the crystal coordinates. Owing to the simple mathematical basis, quaternions are also convenient for illustrating and teaching symmetry, as an alternative to the matrix representation of symmetry operations. The quaternion representations of symmetry operations acquire a simple form, suitable for using them as symmetry-element symbols. The quaternion representations of symmetric and non-symmetric transformations in the direct and reciprocal spaces open their possible further applications to diffractometric computations, spectroscopic analysis as well as other prospective fields in materials and computer sciences. Furthermore, the application of Hamilton’s biquaternions has successfully unified proper and improper symmetry operations; hence, point symmetry groups can be conveniently described by biquaternions. They can be applied for the symmetric and asymmetric transformations directly in non-Cartesian reference systems. It should be noted that the data treatment directly in the crystal-lattice system has also additional advantages connected to the estimated standard deviations (ESDs), accompanying any experimental data. In the crystal-diffraction experiments, the ESDs of all experimentally measured results are obtained for the lattice reference system. Thus, by confining the transformation to the crystal lattice, one significantly simplifies the propagation-of-errors computations.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/sym16101366/s1, The Supplementary Materials contain the computational details and graphical illustrations for five examples of practical applications of quaternions presented in this article.

Author Contributions

Conceptualization, A.K.; methodology, validation, writing, original draft preparation, writing—review and editing, visualization, A.K. and H.Q.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Polish Ministry of Higher Education, through the statutory fund of the Adam Mickiewicz University in Poznań.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

This project was carried out at the Adam Mickiewicz University in Poznań, during HQL’s Master study. HQL is grateful to the European Education and Culture Executive Agency (EACEA) for financial support of his Master of Science study (program Erasmus Mundus Joint Master Degree in Surface, Electro, Radiation and Photo-Chemistry—SERP+). We acknowledge the support from the Polish Ministry of Education statutory fund.

Conflicts of Interest

The authors declare no conflicts of interest. The funder had no role in the design of this study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A. On Deriving the General Quaternion Multiplication Rules

The general quaternion multiplication rules, which can be applied for multiplying the quaternions for all non-orthogonal reference systems, can be derived for the crystallographic triclinic system. In the triclinic system, there are no restrictions for the unit-cell parameters a, b, c, α, β and γ. The unit versors of the triclinic lattice (a, b, c) can be converted from the crystal fractional coordinates to Cartesian coordinates in various ways [23], but the following upper-triangle orthogonalization matrix has been chosen:

(\begin{matrix} a & b \cos γ & c \cos β \\ 0 & b \sin γ & c \frac{\cos α - \cos β \cos γ}{\sin γ} \\ 0 & 0 & \frac{V}{a b \sin γ} \end{matrix})

w h e r e V = a b c Ω = a b c \sqrt{1 - {c o s}^{2} α - {c o s}^{2} β - {c o s}^{2} γ + 2 c o s α c o s β c o s γ}

is the unit-cell volume (Equation (4d)).

This one of infinitely many possible orthogonalization matrices will be used for representing lattice vectors a, b and c, with the fractional crystal coordinates [100], [010] and [001], respectively, with a set of unit vectors e₁, e₂, e₃ in a Cartesian system.

a = (\begin{matrix} a & b \cos γ & c \cos β \\ 0 & b \sin γ & c \frac{\cos α - \cos β \cos γ}{\sin γ} \\ 0 & 0 & \frac{V}{a b \sin γ} \end{matrix}) (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})

a = (\begin{matrix} a \\ 0 \\ 0 \end{matrix}) = a e_{1}

e_{1} = a / a

b = (\begin{matrix} a & b \cos γ & c \cos β \\ 0 & b \sin γ & c \frac{\cos α - \cos β \cos γ}{\sin γ} \\ 0 & 0 & \frac{V}{a b \sin γ} \end{matrix}) (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix})

b = (\begin{matrix} b \cos γ \\ b \sin γ \\ 0 \end{matrix}) = b e_{1} \cos γ + b e_{2} \sin γ

b = \frac{b a}{a} \cos γ + b e_{2} \sin γ

e_{2} = \frac{a b - b a \cos γ}{a b \sin γ}

a \times b = a b e_{3} \sin γ

c = (\begin{matrix} a & b \cos γ & c \cos β \\ 0 & b \sin γ & c \frac{\cos α - \cos β \cos γ}{\sin γ} \\ 0 & 0 & \frac{V}{a b \sin γ} \end{matrix}) (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix})

c = (\begin{matrix} c \cos β \\ c \frac{\cos α - \cos β \cos γ}{\sin γ} \\ \frac{V}{a b \sin γ} \end{matrix}) = c e_{1} \cos β + c e_{2} \frac{\cos α - \cos β \cos γ}{\sin γ} + \frac{V e_{3}}{a b \sin γ}

Put Equation (4d):

Ω = \sqrt{1 - {c o s}^{2} α - {c o s}^{2} β - {c o s}^{2} γ + 2 \cos α \cos β \cos γ} \to V = a b c Ω c = c e_{1} \cos β + c e_{2} \frac{\cos α - \cos β \cos γ}{\sin γ} + \frac{Ω e_{3}}{\sin γ} c = \frac{c a \cos β}{a} + c (\frac{\cos α - \cos β \cos γ}{\sin γ}) (\frac{a b - b a \cos γ}{a b \sin γ}) + \frac{Ω c}{{ab sin}^{2} γ} a \times b \to a \times b = \frac{a b s i n^{2} γ}{Ω c} c + \frac{b c (\cos γ \cos α - \cos β)}{Ω c} a + \frac{c a (\cos γ \cos β - \cos α)}{Ω c} b

As a, b and c have the same role, by symmetry, we have:

b \times c = \frac{b c s i n^{2} α}{Ω a} a + \frac{c a (\cos α \cos β - \cos γ)}{Ω a} b + \frac{a b (\cos α \cos γ - \cos β)}{Ω a} c c \times a = \frac{c a s i n^{2} β}{Ω b} b + \frac{a b (\cos β \cos γ - \cos α)}{Ω b} c + \frac{b c (\cos β \cos α - \cos γ)}{Ω b} a

Or if we replace a as i, b as j and c as k (for quaternion notation):

i \times j = \frac{a b s i n^{2} γ}{Ω c} k + \frac{b c (\cos γ \cos α - \cos β)}{Ω c} i + \frac{c a (\cos γ \cos β - \cos α)}{Ω c} j j \times k = \frac{b c s i n^{2} α}{Ω a} i + \frac{c a (\cos α \cos β - \cos γ)}{Ω a} j + \frac{a b (\cos α \cos γ - \cos β)}{Ω a} k k \times i = \frac{c a s i n^{2} β}{Ω b} j + \frac{a b (\cos β \cos γ - \cos α)}{Ω b} k + \frac{b c (\cos β \cos α - \cos γ)}{Ω b} i

These equations substituted to the multiplication rule yield the general rules for quaternions of any system.

References

Rodrigues, O. Des lois géometriques qui regissent les déplacements d’un systéme solide dans l’espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire. J. Math. Pures Appl. 1840, 5, 380–440. [Google Scholar]
Hamilton, W.R. Lectures on Quaternions; Hodges and Smith: Dublin, Ireland, 1853. [Google Scholar]
Altmann, S.L. Rotations, Quaternions, and Double Groups; Clarendon Press: Oxford, UK, 1986; ISBN 0198553722. [Google Scholar]
Hanson, A. Visualizing Quaternions; Morgan-Kaufmann: New York, NY, USA; Elsevier: Amsterdam, The Netherlands, 2006; ISBN 9780080474779. [Google Scholar]
Yang, Q.; Qu, W.; Gao, J.; Wang, Y.; Song, X.; Guo, Y.; Ke, Y. Quaternion-based placement orientation trajectory smoothing method under the Domain of Admissible Orientation. Int. J. Adv. Manuf. Technol. 2023, 128, 491–510. [Google Scholar] [CrossRef]
Barr, A.H.; Currin, B.; Gabriel, S.; Hughes, J.F. Smooth Interpolation of Orientations with Angular Velocity Constraints using Quaternions. Comput. Graph. 1992, 26, 313–320. [Google Scholar] [CrossRef]
Thomas, D.J. Modern equations of diffractometry. Goniometry. Acta Cryst. A 1990, 46, 321–343. [Google Scholar] [CrossRef]
White, K.I.; Bugris, V.; McCarthy, A.; Ravelli, R.B.G.; Csankó, K.; Cassettaf, A.; Brockhauser, S. Calibration of rotation axes for multi-axis goniometers in macromolecular crystallography. J. Appl. Cryst. 2018, 51, 1421–1427. [Google Scholar] [CrossRef] [PubMed]
Clegg, W. Orientation matrix refinement during four-circle diffractometer data collection. Acta Cryst. A 1984, 40, 703–704. [Google Scholar] [CrossRef]
Busing, W.R.; Levy, H.A. Angle calculations for 3- and 4-circle X-ray and neutron diffractometers. Acta Cryst. A 1967, 22, 457–464. [Google Scholar] [CrossRef]
Paciorek, W.A.; Meyer, M.; Chapuis, G. On the geometry of a modern imaging diffractometer. Acta Cryst. A 1999, 55, 543–557. [Google Scholar] [CrossRef] [PubMed]
Dera, P.; Katrusiak, A. Towards general diffractometry. III. Beyond the normal-beam geometry. J. Appl. Cryst. 2001, 34, 27–32. [Google Scholar] [CrossRef]
Grimmer, H. Disorientations and coincidence rotations for cubic lattices. Acta Cryst. A 1974, 30, 685–688. [Google Scholar] [CrossRef]
Bonnet, R. Disorientation between any two lattices. Acta Cryst. A 1980, 36, 116–122. [Google Scholar] [CrossRef]
Mackay, A.L. Quaternion transformation of molecular orientation. Acta Cryst. A 1984, 40, 165–166. [Google Scholar] [CrossRef]
Diamond, R. A note on the rotational superposition problem. Acta Cryst. A 1988, 44, 211–216. [Google Scholar] [CrossRef]
Theobald, D.L. Rapid calculation of RMSDs using a quaternion-based characteristic polynomial. Acta Cryst. A 2005, 61, 478–480. [Google Scholar] [CrossRef] [PubMed]
Hanson, A.J. The quaternion-based spatial-coordinate and orientation-frame alignment problems. Acta Cryst. A 2020, 76, 432–457. [Google Scholar] [CrossRef] [PubMed]
Bernal, J.D. The Analytic Theory of Point Systems. 1923. Unpublished Monograph. Available online: https://www.iucr.org/__data/assets/pdf_file/0008/25559/Bernal_monograph.pdf (accessed on 30 May 2024).
Fritzer, H.P. Molecular symmetry with quaternions. Spectrochim. Acta A 2001, 57, 1919–1930. [Google Scholar] [CrossRef] [PubMed]
Katrusiak, A.; Llenga, S. Crystallographic quaternions. Symmetry 2024, 16, 818. [Google Scholar] [CrossRef]
Hamilton, W.R. On Geometrical Interpretation of Some Results Obtained by Calculation with Biquaternions. In Proceedings of the Royal Irish Academy (1836–1869); JSTOR: New York, NY, USA, 1850; Volume 5, pp. 388–390. Available online: https://www.jstor.org/stable/20489781?seq=1 (accessed on 23 June 2023).
Rollett, J.S. Computing Methods in Crystallography; Pergamon Press: London, UK, 1965; pp. 22–23. [Google Scholar]

Figure 1. Monoclinic direct-lattice axes x, y, z (black) as well as the reciprocal axes (red, superscript ‘r’); green arrows indicate the direct-space lattice angles. The dimensions in this drawing correspond to the example described in the text: a = b = c/2 = 1 (in the crystal coordinates); hence, a^r = b^r = 2c^r.

Figure 2. A hexagonal lattice projected along [z], with hexagonal coordinates of nodes along directions [x], [y] and

[\bar{1} \bar{1} 0]

at z = 0.

Figure 2. A hexagonal lattice projected along [z], with hexagonal coordinates of nodes along directions [x], [y] and

[\bar{1} \bar{1} 0]

at z = 0.

Table 1. Symmetry elements for the trigonal system and their quaternion representations for the hexagonal reference axes (cf. Table 6 in Reference [21]).

Symmetry Element	Quaternion (q)	Quaternion Action
1	1	qr
$\bar{1}$	$-$ 1	qr
2_[100]	i	qrq*
2_[010]	j	qrq*
2_[110]	i + j	qrq*
3_[001]	$1 / 2 + k \sqrt{3} / 2$	qrq*
${\bar{3}}_{[001]}$	$1 / 2 + k \sqrt{3} / 2$	−qrq*
m_[100]	i	qrq
m_[010]	j	qrq
m_[110]	i + j	qrq

Table 2. Symmetry elements for the hexagonal system and the quaternion representations of symmetry operations in the crystal reference setting (cf. Table 1).

Symmetry Operation	Quaternion (q)	Quaternion Action
1	1	qr
$\bar{1}$	$-$ 1	qr
$2_{[100]}$	$i$	qrq*
$2_{[010]}$	$j$	qrq*
$2_{[110]}$	$i + j$	qrq*
$2_{[210]}$	$2 i / \sqrt{3} + j / \sqrt{3}$	qrq*
$2_{[120]}$	$i / \sqrt{3} + 2 j / \sqrt{3}$	qrq*
$2_{[1 \bar{1} 0]}$	$i / \sqrt{3} - j / \sqrt{3}$	qrq*
$6_{[001]}$	$\sqrt{3} / 2 + k / 2$	qrq*
${\bar{6}}_{[001]}$	$\sqrt{3} / 2 + k / 2$	$-$ qrq*
$m_{[210]}$	$2 i / \sqrt{3} + j / \sqrt{3}$	qrq
$m_{[120]}$	$i / \sqrt{3} + 2 j / \sqrt{3}$	qrq
$m_{[1 \bar{1} 0]}$	$(i - j) / \sqrt{3}$	qrq
$m_{[100]}$	$i$	qrq
$m_{[010]}$	$j$	qrq
$m_{[110]}$	$i + j$	qrq
$m_{[001]}$	$k$	qrq

Table 3. Multiplication table for biquaternions, as specified in Equation (13), in the representation of symmetry operations of point-group

\bar{1}

(C_i).

Table 3. Multiplication table for biquaternions, as specified in Equation (13), in the representation of symmetry operations of point-group

\bar{1}

(C_i).

$\bar{1}$	$1$	$\bar{1}$
1	1	h
$\bar{1}$	h	−1

Table 4. Multiplication table of biquaternions, as specified in Equation (13), used in representing the symmetry operations in point group mm2 (C2v).

mm2	1	2	m_x	m_y
1	1	k	hi	hj
2	k	−1	−hj	hi
m_x	hi	hj	1	k
m_y	hj	−hi	−k	1

Table 5. Multiplication Cayley table of the symmetry operations in point-group mm2 (C2_v).

mm2	1	2	m_x	m_y
1	1	2	m_x	m_y
2	2	1	m_y	m_x
m_x	m_x	m_y	1	2
m_y	m_y	m_x	2	1

Table 6. Multiplication table of biquaternions for the symmetry operations in the cyclic point-group

\bar{6}

(S₃).

Table 6. Multiplication table of biquaternions for the symmetry operations in the cyclic point-group

\bar{6}

(S₃).

$\bar{6}$	$1$	$\bar{6} ¹$	$\bar{6} ²$	$\bar{6} ³$	$\bar{6} ⁴$	$\bar{6} ⁵$
1	$1$	$\frac{h (\sqrt{3} + k)}{2}$	$- \frac{1 + k \sqrt{3}}{2}$	$- h k$	$\frac{- 1 + k \sqrt{3}}{2}$	$\frac{h (- \sqrt{3} + k)}{2}$
$\bar{6} ¹$	$\frac{h (\sqrt{3} + k)}{2}$	$- \frac{1 + k \sqrt{3}}{2}$	$- h k$	$\frac{- 1 + k \sqrt{3}}{2}$	$\frac{h (- \sqrt{3} + k)}{2}$	$1$
$\bar{6} ²$	$- \frac{1 + k \sqrt{3}}{2}$	$- h k$	$\frac{- 1 + k \sqrt{3}}{2}$	$\frac{h (- \sqrt{3} + k)}{2}$	$1$	$\frac{h (\sqrt{3} + k)}{2}$
$\bar{6} ³$	$- h k$	$\frac{- 1 + k \sqrt{3}}{2}$	$\frac{h (- \sqrt{3} + k)}{2}$	$1$	$\frac{h (\sqrt{3} + k)}{2}$	$- \frac{1 + k \sqrt{3}}{2}$
$\bar{6} ⁴$	$\frac{- 1 + k \sqrt{3}}{2}$	$\frac{h (- \sqrt{3} + k)}{2}$	$1$	$\frac{h (\sqrt{3} + k)}{2}$	$- \frac{1 + k \sqrt{3}}{2}$	$- h k$
$\bar{6} ⁵$	$\frac{h (- \sqrt{3} + k)}{2}$	$1$	$\frac{h (\sqrt{3} + k)}{2}$	$- \frac{1 + k \sqrt{3}}{2}$	$- h k$	$\frac{- 1 + k \sqrt{3}}{2}$

Table 7. Multiplication Caley table of the symmetry operations in point group

\bar{6}

(S₃).

Table 7. Multiplication Caley table of the symmetry operations in point group

\bar{6}

(S₃).

$\bar{6}$	$1$	$\bar{6} ¹$	$\bar{6} ²$	$\bar{6} ³$	$\bar{6} ⁴$	$\bar{6} ⁵$
1	$1$	$\bar{6} ¹$	$\bar{6} ²$	$\bar{6} ³$	$\bar{6} ⁴$	$\bar{6} ⁵$
$\bar{6} ¹$	$\bar{6} ¹$	$\bar{6} ²$	$\bar{6} ³$	$\bar{6} ⁴$	$\bar{6} ⁵$	$1$
$\bar{6} ²$	$\bar{6} ²$	$\bar{6} ³$	$\bar{6} ⁴$	$\bar{6} ⁵$	$1$	$\bar{6} ¹$
$\bar{6} ³$	$\bar{6} ³$	$\bar{6} ⁴$	$\bar{6} ⁵$	$1$	$\bar{6} ¹$	$\bar{6} ²$
$\bar{6} ⁴$	$\bar{6} ⁴$	$\bar{6} ⁵$	$1$	$\bar{6} ¹$	$\bar{6} ²$	$\bar{6} ³$
$\bar{6} ⁵$	$\bar{6} ⁵$	$1$	$\bar{6} ¹$	$\bar{6} ²$	$\bar{6} ³$	$\bar{6} ⁴$

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Katrusiak, A.; Le, H.Q. Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems. Symmetry 2024, 16, 1366. https://doi.org/10.3390/sym16101366

AMA Style

Katrusiak A, Le HQ. Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems. Symmetry. 2024; 16(10):1366. https://doi.org/10.3390/sym16101366

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Katrusiak, Andrzej, and Hien Quy Le. 2024. "Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems" Symmetry 16, no. 10: 1366. https://doi.org/10.3390/sym16101366

APA Style

Katrusiak, A., & Le, H. Q. (2024). Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems. Symmetry, 16(10), 1366. https://doi.org/10.3390/sym16101366

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Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems

Abstract

1. Introduction

2. Discussion

2.1. Basic Algebra and Definitions

2.2. Symmetry Operations in Hexagonal Lattice

2.3. Non-Symmetric Transformations

2.4. Trigonal System in Hexagonal Setting

2.5. Hexagonal System

2.6. Biquaternion for General Application of the Point Symmetry

3. Conclusions

Supplementary Materials

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

Appendix A. On Deriving the General Quaternion Multiplication Rules

References

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