Orientational Sampling Schemes Based on Four Dimensional Polytopes

Salvatore Mamone

Giuseppe Pileio

and

Malcolm H. Levitt

School of Chemistry, University of Southampton, University Road, Southampton SO17 1BJ, UK

Author to whom correspondence should be addressed.

Symmetry2010,2(3), 1423-1449;https://doi.org/10.3390/sym2031423

Submission received: 4 February 2010 /Revised: 27 May 2010 /Accepted: 30 June 2010 /Published: 7 July 2010

(This article belongs to the Special IssueFeature Papers: Symmetry Concepts and Applications)

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Abstract

The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may be used in the numerical computation of solid-state nuclear magnetic resonance spectra, and in spherical tensor analysis procedures.

Keywords:

polytope;polychora;group theory;Schläfli symbols;powder averaging;orientational sampling;solid-state nuclear magnetic resonance

1. Introduction

In general, physical properties areanisotropic, meaning that they depend on the orientation of the object of interest in three-dimensional space, defined with respect to an external reference frame. For example, the magnetic resonance response of solid samples depends on the orientation of the molecules with respect to the applied magnetic field [1,2]. Similar considerations apply to many other physical quantities and spectroscopic properties.

If the physical system is macroscopically isotropic (for example, a finely-divided powdered solid), all molecular orientations are encountered with equal probability. The physical response of such systems is an average over all molecular orientations.

Suppose that a computational method exists for estimating the value of a particular macroscopic observable for a single molecular orientation. To estimate the powder response, it is necessary to average the results of such computations over a large number of distinct orientations. This is calledpowder averaging, and is a common procedure in, for example, the computation of solid-state magnetic resonance observables [3,4,5,6]. In general, the computational cost of powder averaging is proportional to the number of sampled orientations. It is clearly desirable to use an orientational sampling scheme that gives an acceptable approximation to the isotropic result using the minimum number of orientations. The problem ofoptimum orientational sampling has been a recurring feature of the solid-state nuclear magnetic resonance (NMR) literature for many years [3,4,5,6].

In addition, there are experimental procedures that require repetition of an experiment for a set of different physical orientations of the system (or parts of the system), in order to estimate the values of anisotropic physical quantities. Physical manipulations of this kind are found, for example, in the NMR of microscopically oriented samples such as single crystals or oriented materials [1].

There are also experiments of this type in which the sample remains fixed in space, but the orientations of the nuclear spin polarizations are manipulated using applied radio-frequency pulse sequences. For example, in the class of experiments known as spherical tensor analysis [7,8,9], the orientational space of the nuclear spins is sampled in order to derive the spherical tensor components of the quantum statistical operator describing the state of the nuclear spin ensemble. In all such experimental procedures, it is desirable that the orientation sampling scheme is as efficient as possible.

1.1. Gaussian Spherical Quadrature

An approach to the orientational sampling problem, using the concept ofGaussian spherical quadrature, was described by Edénet al. in 1998 [4]. This approach may be summarized as follows: Anorientational sampling scheme

S

consists of a finite set

Ω^{S}

N_{S}

distinct orientations

Ω_{j}^{S}

in three-dimensional space, and a set

w^{S}

ofweights

w_{j}^{S}

with the property

\sum_{j = 1}^{N_{S}} w_{j}^{S} = 1

. Both sets have the same number of elements

N_{S}

. The isotropic average

〈Q〉

of a physical observableQ is estimated by computingQ for each orientational sampling point

Ω_{j}^{S}

and superposing the results according to:

{〈Q〉}_{est}^{S} = \sum_{j = 1}^{N_{S}} w_{j}^{S} Q (Ω_{j}^{S})

(1)

The performance of a sampling scheme may be characterized by itsspherical moments, which are defined as follows:

σ_{ℓ m m^{'}}^{S} = \sum_{j = 1}^{N_{S}} w_{j}^{S} D_{m m^{'}}^{ℓ} (Ω_{j}^{S})

(2)

Here

D_{m m^{'}}^{ℓ} (Ω_{j}^{S})

is an element of the Wigner matrix [10] of integer rankℓ, evaluated at orientation

Ω_{j}^{S}

. The Wigner matrices are representations of the group of the three-dimensional rotations

S O (3)

, with the Wigner matrices of integer rankℓ spanning the irreducible representation of

S O (3)

of dimension

2 ℓ + 1

. If the rotation

Ω_{j}^{S}

is parametrized using the three Euler angles

{α_{j}^{S}, β_{j}^{S}, γ_{j}^{S}}

, representing consecutive rotations about thez,y andz-axes of 3D space, all Wigner matrix elements may be written as follows:

D_{m m^{'}}^{ℓ} (Ω_{j}^{S}) = e^{- i m^{'} α_{j}^{S}} d_{m m^{'}}^{ℓ} (β_{j}^{S}) e^{- i m γ_{j}^{S}}

(3)

where

d_{m m^{'}}^{ℓ} (β_{j}^{S})

is an element of the reduced Wigner matrix and the indicesm and

m^{'}

span the integers in the range

- ℓ, \dots, ℓ

. By definition, the zero-rank spherical moment is given by

σ_{000}^{S} = 1

As discussed by Edénet al. [4], orientational sampling schemes may be constructed which have vanishing spherical moments over a range of ranks,i.e.

σ_{ℓ m m^{'}}^{S} = 0 for 1 \leq ℓ \leq ℓ_{\max}^{S}

(4)

Schemes of this kind often provide a good approximation for the isotropic average of an observableQ, using a sampling set

S

of relatively small size. Their performance is particularly good ifQ is a smooth function of orientation

Ω

. This is calledGaussian spherical quadrature since it describes a numerical approach to integration of a function over three-dimensional space that is analogous to Gaussian numerical integration on a line interval. The Wigner functions play the same role as orthogonal polynomials in the case of Gaussian line integration.

In general, sampling schemes with large values of

ℓ_{\max}^{S}

provide a more accurate isotropic average than schemes with small values of

ℓ_{\max}^{S}

, but require a larger number of elements

N_{S}

for their realization. The central problem in Gaussian spherical quadrature is to achieve large values of

ℓ_{\max}^{S}

with as small

N_{S}

as possible.

1.2. Two-angle Sampling and Regular Polyhedra

In many physical situations, the observable of interestQ depends on only two of the three Euler angles defining the orientation in three-dimensional space. This situation arises, for example, in the ordinary NMR of static solids, where the rotational angle of the sample around the static magnetic field has no influence on NMR observables. This is also true for some classes of NMR experiments in rotating solids, as discussed in Reference [4].

Consider an experiment, or computational procedure, of this type, in which the observable of interest does not depend on the third Euler angle

γ

. In such cases, the only relevant spherical moments of an orientational sampling scheme have

m^{'} = 0

. The known relationships between Wigner functions of the type

D_{m 0}^{ℓ} (Ω)

and the spherical harmonics

Y_{ℓ m} (θ, ϕ)

allows the relevant spherical moments to be written as follows:

σ_{ℓ m 0}^{S} = \sqrt{\frac{4 π}{2 ℓ + 1}} \sum_{j = 1}^{N_{S}} w_{j}^{S} Y_{ℓ m} {(β_{j}^{S}, α_{j}^{S})}^{*}

(5)

where

^{*}

means complex conjugation. The problem of two-angle orientational sampling is therefore closely related to the problem of Gaussian quadrature on the surface of a sphere, using spherical harmonics as the orthogonal basis functions. The correspondence of the Euler angles

{α, β}

to the polar angles

{θ, ϕ}

of a point on the surface of a sphere is as follows:

\begin{matrix} α \leftrightarrow ϕ \\ β \leftrightarrow θ \end{matrix}

(6)

For small values of

ℓ_{\max}^{S}

, efficient two-angle sampling schemes may be constructed from the vertices of the regular three-dimensional polyhedra. As discussed below, the point symmetry groups of such polyhedra ensure that many of the spherical moments

σ_{ℓ m 0}^{S}

vanish. For example, the 12 vertices of the icosahedron may be used to construct an orientational sampling set with

N_{S} = 12

, all

w_{j}^{S} = 1 / 12

, and spherical moments

σ_{ℓ m 0}^{S} = 0

for

1 \leq ℓ \leq 5

. All spherical moments with odd values ofℓ vanish for this set as well. These favourable properties are well-known in nuclear magnetic resonance and have led to numerous applications [11,12].

It is not possible to construct sampling sets with

ℓ_{\max}^{S} > 5

from the vertices of the regular 3D polyhedra. However Lebedev and co-workers [13,14,15] have constructed schemes with large values of

ℓ_{\max}^{S}

by using well-chosen orientational sampling points and non-uniform weights. Alternative methods are also available, which do not have such well-defined mathematical properties, but which perform well in many circumstances, for example the REPULSION approach of Bak and Nielsen, which uses numerical optimization under a repulsive electrostatic potential to distribute many points evenly on the surface of a 3D sphere [3].

1.3. Three-Angle Sampling and Regular 4-Polytopes

There are numerous cases where the observable of interest depends on all three Euler angles defining the orientation

Ω

. Some examples from the field of solid-state nuclear magnetic resonance are discussed in Reference [4,5]. In such cases, it is important that the spherical moments

σ_{ℓ m m^{'}}^{S}

vanish for all

{(2 ℓ + 1)}^{2}

combinations ofm and

m^{'}

within a given rankℓ, and not just the special components with

m^{'} = 0

As described in Reference [4], it is possible to construct three-angle orientational sampling sets with the appropriate properties by (i) taking a two-angle sampling set with the property

σ_{ℓ m 0}^{S} = 0

for

1 \leq ℓ \leq ℓ_{\max}^{S}

, and (ii) repeating each sampling point while stepping the third angle through

(ℓ_{\max}^{S} + 1)

regularly-spaced subdivisions of

2 π

. This generates a three-angle sampling set with the desired property

σ_{ℓ m m^{'}}^{S} = 0

for all

{m, m^{'}}

and

1 \leq ℓ \leq ℓ_{\max}^{S}

. For example, an icosahedral two-angle set with

N_{S} = 12

may easily be extended to a three-angle set with

N_{S} = 72

and

ℓ_{\max}^{S} = 5

. The Lebedev two-angle sets may be extended in analogous fashion. The problems with this approach are (i) it is not efficient, requiring large numbers of orientational samples for modest values of

ℓ_{\max}^{S}

and (ii) it does not treat the Euler angles

α

and

γ

in the same way.

Since efficient two-angle sampling schemes may be derived from the vertices of regular polyhedra, which fall on a sphere in 3D space, it is natural to speculate that efficient three-angle sampling schemes may be derived from the vertices of regular solids in four dimensions, which fall on a sphere in 4D space. The regular 4D solids are known asregular 4-polytopes orregular polychora [16] and have been studied extensively by mathematicians, in particular Coxeter [17].

Suppose that a 4-polytope is constructed with the vertices lying on the surface of a 4D sphere with unit radius. Each vertex may be converted into a rotation operation in 3D space by identifying it as a unit vector of the following form:

q = (\begin{matrix} \cos \frac{ξ}{2} \\ n_{x} \sin \frac{ξ}{2} \\ n_{y} \sin \frac{ξ}{2} \\ n_{z} \sin \frac{ξ}{2} \end{matrix})

(7)

where

ξ

is the rotation angle and

n = (n_{x}, n_{y}, n_{z})

is the unit rotation axis in 3-space,

n \cdot n = 1

. Hence, uniformly distributed rotations in 3-space may be constructed from the vertices of regular 4-polytopes deducing the corresponding 3D rotation angles and rotation axes from Equation7. There is one important complication: unit vectors of the form

q

and

- q

correspond to rotations differing by an angle of

2 π

, which have the same physical effect on ordinary 3D objects, or on quantum states with integer spin. Hence, a 4-polytope which has the inversion amongst its symmetry operations gives rise to only half the number of physically distinct 3D rotations as its number of vertices. As discussed below, this property applies to all the regular 4-polytopes, with one exception.

Suppose now that a set ofN 3D rotations is constructed from the vertices of a regular 4-polytope, and that all of the sampling weights are uniform,

w_{j}^{S} = N^{- 1}

j \in {1, 2 \dots N}

. Many of the spherical moments, defined in Equation2, are expected to vanish, through symmetry. The question is: for which ranksℓ do all spherical moments of the form

σ_{ℓ m m^{'}}^{S}

vanish? Although this question has been answered in part using the theory of spherical designs [18], it is also possible to treat this problem by relatively simple group theoretical arguments that may be more accessible to non-mathematicians. However, the application of group theory to this problem is made more difficult by the fact that the symmetry operations of the regular 4-polytopes, and the character tables of the corresponding symmetry groups, are distributed over several sources [19,20,21,22,23]. In this article we collate the symmetry operations and their characters for the regular 4-polytopes in the

{(2 ℓ + 1)}^{2}

-dimensional representations spanned by the Wigner matrices

D^{ℓ} (Ω)

. We derive by group theory the vanishing spherical moments for 3D rotation sets derived from each of the regular 3D and 4D solids. Explicit tables of Euler angles are given, based on the vertices of the regular 4-polytopes. These results should be useful for workers in a wide range of physical sciences, especially magnetic resonance, where one such scheme is already in use [9,24].

2. Group Theory and Symmetry Averaging

2.1. Groups, Representations and Characters

A minimal introduction to group theory is now given in order to establish the notation. For more details, consult the standard texts, for example [25,26,27,28].

An abstract group

{G, \circ}

is a collection of elementsG for which a particular associative operation ∘ combines any two elements to give another element in the group. A valid group must include an identity elementE such that

G \circ E = G

, and all elements must have an inverse

G^{- 1}

such that

G \circ G^{- 1} = E

. Any subset of a group which itself satisfies the group axioms above is called a subgroup.

Groups can be represented by matrices. Ann-dimensional linear representation

Γ

of a group

G

assigns an invertible

n \times n

(real or complex) matrix

M^{Γ} (G)

to each group elementG, so that the group operation ∘ corresponds to the operation of matrix multiplication:

M^{Γ} (G_{1} \circ G_{2}) = M^{Γ} (G_{1}) \cdot M^{Γ} (G_{2})

(8)

A representation is said to beirreducible if it is not possible to find a basis in whichall the matrix representatives of the group elements have the same block diagonal form.

The explicit matrix representations

M^{Γ} (G)

are dependent on the choice of the basis vectors. However, for a given representation

Γ

, thecharacters, defined as the traces of the matrix representations

χ^{Γ} (G) = Tr \{M^{Γ} (G)\}

(9)

are independent of the basis. Two group elementsG and

G^{'}

are said to belong to the sameclass

C

if they are related through a similarity transformation of the form

G = A G^{'} A^{- 1}

whereA also belongs to the group

G

. All elements in the same class have the same character for all representations

Γ

,i.e.

χ^{Γ} (G) = χ_{C}^{Γ} for all G \in C

(10)

2.2. Subgroup Averaging

Suppose now that the group

G

contains a finite subgroup

g

containing

h (g)

elements. A representation

Γ

of the group

G

is also a representation of the subgroup

g

. The finite group orthogonality theorem [28] implies that the number of independent linear combinations of basis vectors spanning the representation

Γ

which are invariant underall of the subgroup operations

G \in g

is given by

a^{Γ} (g) = h {(g)}^{- 1} \sum_{G \in g} χ^{Γ} (G) = h {(g)}^{- 1} \sum_{C} h_{C} (g) χ_{C}^{Γ}

(11)

where

h_{C} (g)

is the number of elements of

g

that belong to the class

C

. The last two formulations on the right-hand side of11 are equivalent since all elements in the same class have the same character. This equation leads to the following property:

\sum_{C} h_{C} (g) χ_{C}^{Γ} = 0 \Rightarrow \sum_{G \in g} M^{Γ} (G) = 0

(12)

The sum of matrices in the representation

Γ

vanishes if the characters sum to zero over all classes of

g

, taking into account the number of subgroup elements

h_{C} (g)

in each class.

2.3. Average of a Function inn-dimensional Space

Consider now the case where the group elementsG are transformations acting on the points

x = {x_{1}, x_{2} \dots x_{n}}

of then-dimensional real space

R^{n}

,i.e.

G x = x^{'}

(13)

For each group elementG, there exists a corresponding operator

\hat{G}

acting onfunctions of the coordinate vectors

f (x)

to generate new functions

f^{'} (x)

, defined as follows:

f^{'} (x) = \hat{G} f (x) = f (G^{- 1} x)

(14)

The definition above corresponds to anactive transformation of the objectf [28].

Theaverage function over a finite subgroup

g

G

is defined by

{〈f〉}_{g} = h {(g)}^{- 1} \sum_{G \in g} \hat{G} f

(15)

where the sum is taken over all

h (g)

elements

G \in g

and the same argument

x

is implied on both sides of the equation.

Now suppose we have a set ofm functions

f_{1}^{Γ}, \dots, f_{m}^{Γ}

forming a basis for anm-dimensional representation

Γ

G

. Any operator

\hat{G}

is then represented by an

m \times m

matrix

M^{Γ} (G)

acting on the set of basis functions from the right [29]:

\hat{G} f_{i}^{Γ} (x) = \sum_{j} f_{j}^{Γ} (x) M_{j i}^{Γ} (G)

(16)

Equation12 gives a sufficient condition for the average of each

f_{i}^{Γ}

function to vanish:

\sum_{C} h_{C} (g) χ_{C}^{Γ} = 0 \Rightarrow {〈f_{i}^{Γ}〉}_{g} = 0

(17)

2.4. Average of a Function Over the Polytope Vertices

The average value of a functionf over a finite setP of

N_{0}

points in then-dimensional space is defined as follows:

{〈f〉}_{P} = N_{0}^{- 1} \sum_{v} f (x_{v})

(18)

where

x_{v}

denote the coordinate vectors of the points for

v \in \{1, 2 \dots, N_{0}\}

. A group

G

n -

dimensional transformations is said to act transitively on theP when for any given pair of points

x_{v}, x_{v}^{'} \in P

there is a transformationG which connects such points

x_{v}^{'} = G x_{v}

[25].

The orbit stabilizer and Lagrange theorems for finite groups [25] relate the average of a functionf overP to the average over any finite group

g_{P}

acting transitively on the set:

N_{0}^{- 1} \sum_{v} f (x_{v}) = h {(g_{P})}^{- 1} \sum_{G \in g_{P}} f (G^{- 1} x_{1}) = h {(g_{P})}^{- 1} \sum_{G \in g_{P}} \hat{G} f (x_{1})

(19)

The right-hand side corresponds to Equation15, evaluated at any point

x_{1}

in the set. From Equation17, the average vanishes if the functionf is one of the basis functions of the representation

Γ

, and the characters of the given finite group sum

g_{P}

to zero for that representation:

\sum_{C} h_{C} (g_{P}) χ_{C}^{Γ} = 0 \Rightarrow {〈f_{i}^{Γ}〉}_{P} = 0

(20)

Equation20 is the central result of this section. The point symmetry group of ann-dimensional regular polytope is a finite group which acts transitively on the polytope vertices. It is a subgroup of the (infinite) orthogonal group

O (n)

, which is the group of all then-dimensional space transformations in with a single fixed point and which preserve distance between transformed points. Using Equation20, the averaging properties of a function over the vertices of a polytope may be deduced from the characters of the symmetry elements and the classes of its symmetry point group. This result is now applied to the spherical moments of the regular solids.

3. Polyhedral Averaging in Three Dimensions

Three dimensional polytopes are known as polyhedra. In this section we discuss the averaging properties of the regular polyhedra with respect to spherical harmonics. Although this topic has been treated before in Reference [11], a recapitulation is useful for framing the discussion of four-dimensional symmetries. In addition, the treatment in Reference [11] did not exploit all the available symmetries, as discussed below.

3.1. Proper and Improper Rotations

Theproper rotations in three dimensions may be defined in various ways. For example, the symbol

R_{n} (ξ)

indicates a rotation through the angle

ξ

about a unit rotation axis

n

whose direction is defined by the polar angles

{θ, ϕ}

. The identity operation

R_{} (0)

does not need any specification of the rotation axis. Any rotation in 3D space may be decomposed into the product of three consecutive rotations around the cartesian reference axes, for example:

R_{n} (ξ) = R_{z} (α) R_{y} (β) R_{z} (γ)

, where the rotations are applied in sequence from right to left. For a given rotationR the three Euler angles

Ω_{R} = {α, β, γ}

and the set

{ξ, θ, ϕ}

are related [10]. Specifically, the rotation angle

ξ

is related to the Euler angles as follows:

\cos \frac{ξ}{2} = \cos \frac{β}{2} \cos \frac{α + γ}{2}

(21)

Theimproper rotations in three-dimensional space may be expressed in various ways. In this article, we use the set of improper rotations, denoted

{\tilde{R}}_{n} (ξ)

. Each improper rotation corresponds to a proper rotation

R_{n} (ξ)

followed by an inversion through the reference frame origin (roto-inversion). By definition, the inversion operation corresponds to the improper rotation

{\tilde{R}}_{} (0)

, where the rotation axis does not need to be specified in this case.

Two other improper rotations are often used in the literature: the reflection

σ_{h}

in the planeh, and the roto-reflection

S_{m}

which is a rotation through

2 π / m

followed by reflection in the plane perpendicular to the rotation axis. Reflections and roto-reflections correspond to improper rotations as follows:

σ_{h} = {\tilde{R}}_{n^{'}} (π)

where

n^{'}

is perpendicular to the planeh, and

S_{m} = {\tilde{R}}_{n^{″}} (π + 2 π / m)

where

n^{″}

is the rotation axis defined by

S_{m}

for

m \geq 3

. Clearly

S_{2} = {\tilde{R}}_{} (0)

and

S_{1} = {\tilde{R}}_{n^{″}} (π)

3.2. Representations and Characters of $O (3)$ Isometries

The set of

2 ℓ + 1

spherical harmonics of rank-ℓ is defined as follows:

Y_{ℓ m} (Θ, Φ) = \sqrt{\frac{2 ℓ + 1}{4 π} \frac{(ℓ - m)!}{(ℓ + m)!}} P_{ℓ}^{m} (\cos Θ) e^{i m Φ}

(22)

whereℓ andm are integers with

| m | \leq ℓ

and

P_{ℓ}^{m}

is the associated Legendre polynomial [10]. This set of functions is a basis for the

(2 ℓ + 1)

-dimensional irreducible representation of the

O (3)

group. The action of any

O (3)

operationG on these functions defines an operator

\hat{G}

which is represented by a

(2 ℓ + 1) \times (2 ℓ + 1)

matrix

M^{ℓ} (G)

\hat{G} Y_{ℓ m} (Θ, Φ) = \sum_{m^{'}} Y_{ℓ m^{'}} (Θ, Φ) M_{m^{'} m}^{ℓ} (G)

(23)

In the case of a proper rotationR, the matrix representative is given by the rank-ℓ Wigner matrix:

M^{ℓ} (R) = D^{ℓ} (Ω_{R})

(24)

In the case of an improper rotation

\tilde{R}

, the sign of the matrix changes for odd rankℓ:

M^{ℓ} (\tilde{R}) = {(- 1)}^{ℓ} D^{ℓ} (Ω_{R})

(25)

The character of a proper rotation for the rank-ℓ representation is equal to the trace of the corresponding Wigner matrix,

χ_{D}^{(ℓ)}

, which depends on the rotation angle

ξ

only [10, pp. 99–100]:

χ^{(ℓ)} \{R_{n} (ξ)\} = χ_{D}^{(ℓ)} (ξ)

(26)

where

χ_{D}^{(ℓ)} (ξ) = \frac{\sin \{(2 ℓ + 1) \frac{ξ}{2}\}}{\sin \frac{ξ}{2}}

(27)

This evaluates to

χ_{D}^{(ℓ)} (ξ) = 2 ℓ + 1

when the rotation angle

ξ

is an integer multiple of

2 π

The character of an improper rotation is the same as for the corresponding proper rotation, but with a change in sign for odd values ofℓ:

χ^{(ℓ)} \{{\tilde{R}}_{n} (ξ)\} = {(- 1)}^{ℓ} χ_{D}^{(ℓ)} (ξ)

(28)

3.3. Regular Convex Polyhedra

The five regular convex polyhedra have been known since the Greeks. Their names and properties are listed inFigure 1. This figure also provides the Schläfli symbols [17] of the form

{p, q}

, wherep indicates the number of edges of the regular polygonal face, andq is the number of faces meeting at one vertex. For example, the cube has Schläfli symbol

{4, 3}

, while the regular octahedron has the Schläfli symbol

{3, 4}

. Polyhedra with Schläfli symbols

{p, q}

and

{q, p}

are geometrical reciprocals of each other and belong to the same symmetry group, since the reciprocation operation corresponds to the mutual exchange of faces and vertices. The five Platonic solids therefore belong to only three symmetry point groups: (i)

T_{d}

, represented by the tetrahedron; (ii)

O_{h}

, populated by the cube and the octahedron; and (iii)

I_{h}

, populated by the icosahedron and the dodecahedron. The symmetry point groups of the regular polyhedra are given explicitly inTable 1.

3.4. Spherical Moments of the Regular Polyhedra

The theorem in Equation20 may be used withTable 1 and the characters given in Equations26 and28 to deduce the vanishing spherical moments of the regular polyhedra. In general, both improper and proper rotations must be taken into account. The treatment in Reference [11] uses only the proper rotations, and gives slightly different results for the groups

O_{h}

and

I_{h}

(see below).

As a first example, consider the tetrahedron. As shown inTable 1, the tetrahedron has three symmetry classes of proper rotations, with number of elements

(1, 8, 3)

and rotation angles

(0, 2 π / 3, π)

respectively. In addition, there are two symmetry classes of improper rotations, with number of elements

(6, 6)

and rotation angles

(π / 2, π)

respectively. The sum of characters for rank

ℓ = 2

is therefore given by

\begin{matrix} \sum_{C} h_{C} (T_{d}) χ_{C}^{(2)} & = & χ_{D}^{(2)} (0) + 8 χ_{D}^{(2)} (2 π / 3) + 3 χ_{D}^{(2)} (π) + 6 {(- 1)}^{2} χ_{D}^{(2)} (π / 2) + 6 {(- 1)}^{2} χ_{D}^{(2)} (π) \\ = & 0 \end{matrix}

(29)

This proves the well-known fact that a tetrahedron averages second-rank spherical harmonics to zero:

σ_{2 m 0} (T_{d}) = 0

(30)

The point symmetry groups of the octahedron and icosahedron contain the inversion element. Each proper rotation is therefore accompanied by an improper rotation through the same angle, as shown inTable 1. It follows that all odd-rank spherical moments harmonics vanish when summed over the vertices of polyhedra with symmetries

O_{h}

and

I_{h}

\begin{matrix} \sum_{C} h_{C} (O_{h}) χ_{C}^{(ℓ)} = \sum_{C} h_{C} (I_{h}) χ_{C}^{(ℓ)} & = & 0 (for odd ℓ) \end{matrix}

(31)

and hence

σ_{ℓ m 0} (O_{h}) = σ_{ℓ m 0} (I_{h}) = 0 (for odd ℓ)

(32)

The treatment of Reference [11] does not predict this result, since only proper rotations were taken into account. The two analyses differ for rank

ℓ = 9

and all odd ranks

ℓ \geq 13

Figure 2 summarizes the spherical rank profiles of the regular convex polyhedra up to rank

ℓ = 30

. Note that even the most symmetrical polyhedra (the icosahedron and the dodecahedron) fail to average the rank

ℓ = 6

terms.

There are 4 regular non-convex polyhedra (star-polyhedra), which all fall in the group

I_{h}

[17]. Four of them have the same vertices of the icosahedron while one has the same vertices as the dodecahedron. All have the same spherical moment characteristics as the icosahedron.

4. Polytopic Averaging in Four Dimensions

In this section we derive the spherical averaging properties of the regular 4-polytopes. In the discussion below, we make extensive use of quaternions [29]. As shown in Equation7, quaternions provide a correspondence between points on a unit sphere in four-dimensional space, and the group of three-dimensional rotations.

4.1. Quaternions

Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing themultiplication of two 4D vectors

q (1)

and

q (2)

according to

(\begin{matrix} q_{1} (2) \\ q_{2} (2) \\ q_{3} (2) \\ q_{4} (2) \end{matrix}) * (\begin{matrix} q_{1} (1) \\ q_{2} (1) \\ q_{3} (1) \\ q_{4} (1) \end{matrix}) = (\begin{matrix} q_{1} (2) q_{1} (1) - q_{2} (2) q_{2} (1) - q_{3} (2) q_{3} (1) - q_{4} (2) q_{4} (1) \\ q_{1} (2) q_{2} (1) + q_{2} (2) q_{1} (1) + q_{3} (2) q_{4} (1) - q_{4} (2) q_{3} (1) \\ q_{1} (2) q_{3} (1) - q_{2} (2) q_{4} (1) + q_{3} (2) q_{1} (1) + q_{4} (2) q_{2} (1) \\ q_{1} (2) q_{4} (1) + q_{2} (2) q_{3} (1) - q_{3} (2) q_{2} (1) + q_{4} (2) q_{1} (1) \end{matrix})

(33)

The adjoint of a quaternion is denoted here

q^{†}

and is defined as follows:

q^{†} = (\begin{matrix} q_{1} \\ - q_{2} \\ - q_{3} \\ - q_{4} \end{matrix})

(34)

and it can be verified that

{q (1) * q (2)}^{†} = q^{†} (2) * q^{†} (1)

The inverse

q^{- 1}

is defined for any non zero quaternion

q

as the unique quaternion that satisfies:

q * q^{- 1} = q^{- 1} * q = (\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix})

(35)

It can be shown that

q^{- 1} = q^{†} / {| | q | |}^{2}

where

| | q | | = \sqrt{\sum_{i} q_{i}^{2}}

4.2. Unit Quaternions and 3D Rotations

The set of 4D unit vectors, together with the quaternion multiplication operation ∗, forms the group of unit quaternions

Q

. The adjoint of a unit quaternion is the same as its inverse:

q^{- 1} = q^{†}

. From Equation7, a unit quaternion and its inverse represent a pair of rotations through opposite angles about the same axis.

The group of unit quaternions

Q

is homomorphic with the group of proper three-dimensional rotations

S O (3)

[30]. The relationship between the product of quaternions and the product of proper 3D rotations is expressed by

R \{q (2) * q (1)\} = R \{q (2)\} \circ R \{q (1)\}

(36)

where

R (q)

is the function which associates a unit quaternion

q

with the corresponding 3D rotation through Equation7. Consider, for example, a rotation through the angle

ξ (1)

about the axis

n (1)

, followed by a rotation through the angle

ξ (2)

about the axis

n (2)

. The overall rotation angle

ξ (2, 1)

is given by

\begin{matrix} \cos \frac{ξ (2, 1)}{2} & = q_{1} (2, 1) = \\ = q_{1} (2) q_{1} (1) - q_{2} (2) q_{2} (1) - q_{3} (2) q_{3} (1) - q_{4} (2) q_{4} (1) \\ = \cos \frac{ξ (2)}{2} \cos \frac{ξ (1)}{2} - n (2) \cdot n (1) \times \sin \frac{ξ (2)}{2} \sin \frac{ξ (1)}{2} \end{matrix}

(37)

from Equation7 and33.

Using the notation

D^{ℓ} (q)

to indicate the Wigner matrix of rankℓ evaluated for the 3D rotation corresponding to the unit quaternion

q

, Equation36 implies:

D^{ℓ} \{q (2) * q (1)\} = D^{ℓ} \{q (2)\} \cdot D^{ℓ} \{q (1))\}

(38)

The Wigner matrices of rankℓ form a

2 l + 1

-dimensional representation of the unit quaternion group

Q

. In particular

D^{ℓ} (q^{- 1}) = {[D^{ℓ} (q)]}^{- 1}

and we can use the following properties for the Wigner matrix elements [10, pp.79–80]

D_{m m^{'}}^{ℓ} (q^{†}) = D_{m m^{'}}^{ℓ} (q^{- 1}) = {(- 1)}^{m - m^{'}} D_{- m^{'} - m}^{ℓ} (q)

(39)

The explicit correspondence between the Euler angles and the unit quaternion components is as follows:

\begin{matrix} α + γ & = & 2 \arctan (q_{1}, q_{4}) \\ β & = & \arccos (1 - 2 q_{2}^{2} - 2 q_{3}^{2}) \\ α - γ & = & 2 \arctan (q_{3}, - q_{2}) \end{matrix}

(40)

where

\arctan (x, y)

is equal to

\arctan (y / x)

, determing the quadrant from the sign ofx andy. In the special cases

q_{2} = q_{3} = 0

q_{1} = q_{4} = 0

, only the combinations

α \pm γ

are defined, as follows:

\begin{matrix} \begin{matrix} β = 0 \\ α + γ = 2 \arctan (q_{1}, q_{4}) \end{matrix}\} if q_{2} = q_{3} = 0 \\ \begin{matrix} β = π \\ α - γ = 2 \arctan (q_{3}, - q_{2}) \end{matrix}\} if q_{1} = q_{4} = 0 \end{matrix}

(41)

4.3. Proper and Improper Rotations

Isometries in 4D space are classed as eitherproper (preserving the handedness of the four-dimensional axis system) orimproper (changing the handedness of the axis system). The group of all isometries with one fixed point in four dimensions is called

O (4)

. Any

O (4)

operation may be expressed in terms of two unit quaternions, denoted here

q_{l}

and

q_{r}

[19], as explained below. Proper operations will be denoted by

R_{q_{l}, q_{r}}

and improper operations by

{\tilde{R}}_{q_{l}, q_{r}}

respectively. The action of a proper rotation

R_{q_{l}, q_{r}}

on a point in 4D space

q

may be written as follows:

q^{'} = R_{q_{l}, q_{r}} q = q_{l} * q * q_{r}^{- 1}

(42)

The action of an improper roation

{\tilde{R}}_{q_{l}, q_{r}}

is as follows:

q^{'} = {\tilde{R}}_{q_{l}, q_{r}} q = q_{l} * q^{†} * q_{r}^{- 1}

(43)

The inverse operations are given by

\begin{matrix} {\{R_{q_{l}, q_{r}}\}}^{- 1} & = R_{q_{l}^{- 1}, q_{r}^{- 1}} & \Leftrightarrow & {\{R_{q_{l}, q_{r}}\}}^{- 1} q & = q_{l}^{- 1} * q * q_{r} \end{matrix}

(44)

\begin{matrix} {\{{\tilde{R}}_{q_{l}, q_{r}}\}}^{- 1} & = {\tilde{R}}_{q_{r}^{- 1}, q_{l}^{- 1}} & \Leftrightarrow & {\{{\tilde{R}}_{q_{l}, q_{r}}\}}^{- 1} q & = q_{r}^{- 1} * q^{†} * q_{l} \end{matrix}

(45)

for proper and improper operations respectively.

4.4. Representation and Characters of $O (4)$ Isometries

In this section we give the explicit matrix representations of the

O (4)

operators and their characters in the basis of the Wigner matrices. These results will then be used to establish the spherical averaging properties of the regular 4-polytopes.

According to Equations14,44,38 and39, a proper transformation in

O (4)

defines an operator

{\hat{R}}_{q_{l}, q_{r}}

which acts as follows on the Wigner matrix elements evaluated at any unit quaternion

q

\begin{matrix} {\hat{R}}_{q_{l}, q_{r}} D_{m m^{'}}^{ℓ} (q) & = D_{m m^{'}}^{ℓ} ({\{R_{q_{l}, q_{r}}\}}^{- 1} q) = D_{m m^{'}}^{ℓ} (q_{l}^{- 1} * q * q_{r}) \\ = \sum_{n, n^{'}} D_{m n}^{ℓ} (q_{l}^{- 1}) D_{n n^{'}}^{ℓ} (q) D_{n^{'} m^{'}}^{ℓ} (q_{r}) \\ = \sum_{n, n^{'}} {(- 1)}^{m - n} D_{- n - m}^{ℓ} (q_{l}) D_{n, n^{'}}^{ℓ} (q) D_{n^{'}, m^{'}}^{ℓ} (q_{r}) \\ = \sum_{n, n^{'}} D_{n, n^{'}}^{ℓ} (q) [{(- 1)}^{m - n} D_{- n - m}^{ℓ} (q_{l}) D_{n^{'} m^{'}}^{ℓ} (q_{r})] \end{matrix}

(46)

Similarly according to Equations14, 45,38 and39 an improper transformation in

O (4)

defines an operator

{\hat{\tilde{R}}}_{q_{l}, q_{r}}

which acts as follows:

\begin{matrix} {\hat{\tilde{R}}}_{q_{l}, q_{r}} D_{m m^{'}}^{ℓ} (q) & = D_{m m^{'}}^{ℓ} ({\{{\tilde{R}}_{q_{l}, q_{r}}\}}^{- 1} q) = D_{m m^{'}}^{ℓ} (q_{r}^{- 1} * q^{†} * q_{l}) \\ = \sum_{n, n^{'}} D_{m, n}^{ℓ} (q_{r}^{- 1}) D_{n n^{'}}^{ℓ} (q^{†}) D_{n^{'} m^{'}}^{ℓ} (q_{l}) \\ = \sum_{n, n^{'}} {(- 1)}^{m - n} D_{- n - m}^{ℓ} (q_{r}) {(- 1)}^{n - n^{'}} D_{- n^{'} - n}^{ℓ} (q) D_{n^{'} m^{'}}^{ℓ} (q_{l}) \\ = \sum_{n, n^{'}} {(- 1)}^{m + n^{'}} D_{n^{'} - m}^{ℓ} (q_{r}) {(- 1)}^{n - n^{'}} D_{n n^{'}}^{ℓ} (q) D_{- n m^{'}}^{ℓ} (q_{l}) \\ = \sum_{n, n^{'}} D_{n n^{'}}^{ℓ} (q) {(- 1)}^{m + n} [{(- 1)}^{m + n} D_{n^{'} - m}^{ℓ} (q_{r}) D_{- n m^{'}}^{ℓ} (q_{l})] \end{matrix}

(47)

The action of any

O (4)

operationG on the

{(2 ℓ + 1)}^{2}

Wigner functions

D_{m m^{'}}^{ℓ} (q)

, evaluated for the rotation corresponding to the unit quaternion

q

, defines an operator

\hat{G}

which may be represented as a

{(2 ℓ + 1)}^{2} \times {(2 ℓ + 1)}^{2}

-dimensional matrix

M^{ℓ} (G)

\hat{G} D_{m m^{'}}^{ℓ} (q) = \sum_{n, n^{'}} D_{n n^{'}}^{ℓ} (q) {[M (G)]}_{n n^{'}, m m^{'}}^{ℓ}

(48)

This proves that the Wigner matrices are a basis for the representation of the group

O (4)

. The matrix representations are given by

{[M (R_{q_{l}, q_{r}})]}_{n n^{'}, m m^{'}}^{ℓ} = {(- 1)}^{m - n} D_{- n - m}^{ℓ} (q_{l}) D_{n^{'} m^{'}}^{ℓ} (q_{r})

(49)

for a proper transformation

R_{q_{l}, q_{r}}

and

{[M ({\tilde{R}}_{q_{l}, q_{r}})]}_{n n^{'}, m m^{'}}^{ℓ} = {(- 1)}^{m + n} D_{- n m^{'}}^{ℓ} (q_{l}) D_{n^{'} - m}^{ℓ} (q_{r})

(50)

for an improper transformation

{\tilde{R}}_{q_{l}, q_{r}}

. In both cases the Wigner matrix elements are evaluated for rotations corresponding to the left and right quaternions

q_{l}

and

q_{r}

, as defined for the given

O (4)

operation.

The character of a general 4D rotations in the rank-ℓ representation is obtained by summing the matrix representations given by Equations49 and50 over the indices

m = n

and

m^{'} = n^{'}

. For proper rotations this leads to the following result:

\begin{matrix} χ^{(ℓ)} (R_{q_{l}, q_{r}}) & = & χ_{D}^{(ℓ)} (ξ_{l}) χ_{D}^{(ℓ)} (ξ_{r}) \end{matrix}

(51)

where

ξ_{l}

and

ξ_{r}

are the rotation angles for the pair of 3D rotations corresponding to the left and right quaternions. For improper rotations, on the other hand, we get

\begin{matrix} χ^{(ℓ)} ({\tilde{R}}_{q_{l}, q_{r}}) & = & χ_{D}^{(ℓ)} (ξ_{l, r}) \end{matrix}

(52)

where

ξ_{l, r}

is the rotational angle associated with the quaternion product

q (l, r) = q_{l} * q_{r}

4.5. Regular Convex 4-Polytopes

The six regular convex polytopes are summarized inFigure 3. Each of them is represented by a Schläfli symbol of the form

{p, q, r}

in whichp andq determine the Schläfli symbol

{p, q}

for the 3-dimensional polyhedron that forms the boundary of the figure andr is the number of polyhedra meeting at one edge [17].

Polytopes with Schläfli symbols

{p, q, r}

and

{r, q, p}

are reciprocals of each other and belong to the same symmetry group. The six regular convex 4-polytopes therefore belong to only four symmetry groups. These are (i) the group

A_{4}

(isomorphic to the permutation group of 5 elements,

S_{5}

), populated by the 5-cell (hypertetrahedron); (ii) the group

B_{4}

, populated by the mutually reciprocal 8-cell (hypercube) and 16-cell (hyperoctahedron); (iii) the group

F_{4}

, populated by the 24-cell; and (iv) the group

H_{4}

, populated by the mutually reciprocal 120-cell (hyperdodecahedron) and 600-cell (hypericosahedron).

Table 2 reports the four symmetry groups of the six regular polytopes and their symmetry elements, given in the quaternion form. The numbers of operations in each class are provided, together with one representative operation, using the notation

R_{q_{l}, q_{r}}

for proper transformations and

{\tilde{R}}_{q_{l}, q_{r}}

for improper transformations. In the case of the group

H_{4}

, the symmetry classes and representative operations are given directly in quaternion form in Reference [23]. For the other groups, the information given in the literature [20,21,22] is not directly suitable for this type of analysis. In these cases, the quaternion form of the representative operations and the class structure were obtained by using the information provided in Reference [19] with the help of the symbolic software platformMathematica [31].

4.6. Spherical Moments of the Regular 4-Polytopes

The spherical averaging properties of the regular 4-polytopes may be deduced by using Equation20 together with the sets of symmetry operations (Table 2), and the characters of the 4D rotations, given in Equations51 and52.

As an example, consider the 5-cell, which has symmetry group

A_{4}

. FromTable 2, there are seven symmetry classes. The four classes of proper operations have

(1, 15, 20, 24)

elements respectively. The rotational angles

(ξ_{l}, ξ_{r})

to be used in Equation51 are obtained from Equation7 and have the following values:

((0, 0)

(π, π)

(2 π / 3, 2 π / 3)

(2 π / 5, 6 π / 5))

. The remaining three classes of improper operations have

(10, 30, 20)

elements respectively. The rotational angles

ξ_{l, r}

to be used in Equation52 are obtained from Equations7 and37 and are as follows:

(2 π, π, 2 π / 3)

. The sum of characters for rank

ℓ = 1

is therefore given by

\begin{matrix} \sum_{C} h_{C} (T_{d}) χ_{C}^{(1)} & = & χ_{D}^{(1)} (0) χ_{D}^{(1)} (0) + 15 χ_{D}^{(1)} (π) χ_{D}^{(1)} (π) + 20 χ_{D}^{(1)} (2 π / 3) χ_{D}^{(1)} (2 π / 3) \\ + 24 χ_{D}^{(1)} (2 π / 5) χ_{D}^{(1)} (6 π / 5) + 10 χ_{D}^{(1)} (2 π) + 30 χ_{D}^{(1)} (π) + 20 χ_{D}^{(1)} (2 π / 3) \\ = & 0 \end{matrix}

(53)

This proves that all first-rank spherical moments of a 5-cell are equal to zero:

σ_{1 m m^{'}} (A_{4}) = 0

(54)

The spherical rank profiles of the other regular polytopes may be obtained in this way for anyℓ:Figure 4 summarizes the results up to rank

ℓ = 30

. As in the 3D case, even the 600-cell and 120-cell, which have the highest symmetry, fail to average out the rank-6 Wigner matrices.

This figure is slightly misleading since only integer ranksℓ are shown. Since the groups

B_{4}

F_{4}

and

H_{4}

possess an inversion operation,

R_{q 1, - q 1}

with

q 1 = (1, 0, 0, 0)

, all spherical moments of half-integer rank vanish for these groups. The group

A_{4}

, on the other hand, lacks the inversion, so the spherical moments of half-integer rank do not vanish in this case. The fact that

A_{4}

and

B_{4}

appear to have the same rank profiles inFigure 4 is therefore due to the omission of half-integer ranks. Most applications of orientational averaging only require integer ranks, in which case the properties shown inFigure 4 are appropriate.

There are 10 regular non-convex polytopes (star-polytopes) in four dimensions, which all fall in the group

H_{4}

[17]. Nine of them have the same vertices as the 600-cell, while one has the same vertices as the 120-cell. All have the same spherical moment characteristics as the 600-cell.

Under the reviewing of this paper, an anonymous referee pointed out that the pattern of empty and filled circles inFigure 4 may also be derived using the theory of spherical designs [18]. In general, 4D spherical harmonics of degreek generate a

{(k + 1)}^{2}

-dimensional representation of the group

O (4)

[18]. Such a representation is equivalent to the

{(2 ℓ + 1)}^{2}

-dimensional representation constructed in Equation48, with

k = 2 ℓ

. A sphericalt-design in 4 dimensions is defined as a subset of the hypersphere for which all the 4D spherical harmonics of degrees 1 tot average to 0 [18]. In other words, all the spherical moments of rank from 1 to

ℓ = t / 2

vanish. In Reference [18] the largest valuest of the spherical design have been derived to be 2 for the 5-cell, 3 for the 8-cell, 5 for the 24-cell, and 11 for the 600-cell, which correspond to

ℓ = 1, 1, 2, 5

inFigure 4.

The anonymous referee also pointed out that invariant theory may be used to prove that non-zero spherical moments in the

H_{4}

column inFigure 4 may appear atℓ values corresponding to any sum of 6’s, 10’s and 15’s and for all

ℓ \geq 30

5. Euler Angles

In order to facilitate exploitation of these results, we provide explicit tables of Euler angles derived from the vertices of the regular 4-polytopes. The

z - y - z

convention for the Euler angles is used throughout. All Euler angle sets are derived from 4-polytopes in their standard orientations, as defined inTable 3. Ambiguities of the form given in Equation41 were always resolved by choosing solutions with

γ = 0

. All angles are reduced to the interval 0 to

2 π

by a modulo-

2 π

operation.

Different Euler angle sets with the same spherical averaging properties may be constructed by applying an equal but arbitrary 4D isometry to all the quaternions underlying the set.

The set of Euler angles corresponding to the 5 vertices of the 5-cell is provided inTable 4. As shown inFigure 4, all first-rank spherical moments vanish for this set of Euler angles. Since the 5-cell lacks an inversion operation, the number of orientations is the same as the number of vertices in this case.

The sets of Euler angles corresponding to the 8 vertices of the 16-cell, and the 16 vertices of the 8-cell are provided inTable 5 andTable 6. As shown inFigure 4, all first-rank spherical moments vanish for these sets of Euler angles. The symmetry groups of both polytopes include an inversion operation, so the number of distinguishable orientations is therefore one-half the number of the vertices. Clearly the four rotations specified inTable 5 comprise the most economical way to set all first-rank spherical moments to zero.

The set of 12 Euler angles corresponding to the 24 vertices of the 24-cell is provided inTable 7. As shown inFigure 4, all first and second-rank spherical moments vanish for this Euler angle set.

The sets of 60 and 300 Euler angles corresponding to the vertices of the 600-cell and the 120-cell are provided inTable 8 andTable 9.Figure 4 shows that all spherical moments up to and including rank 5 vanish for these Euler angle sets. The most economical way of annihilating spherical ranks up to and including rank 5 is therefore the 60-angle set inTable 8. This rotation set was previously described in Reference [24], where it was presented without any supporting theory.

It is worth pointing out that the 3D rotations discussed above for the 24-cell and the 600-cell have more inutitive descriptions. The set of Euler angles obtained from the vertices of the 24-cell generates exactly the 12 rotational symmetries of the tetrahedron, compareTable 7 with the last column for the group

T_{d}

inTable 1. Similarly the set of Euler angles obtained from the vertices of the 600-cell generates exactly the 60 rotational symmetries of the icosahedron, compareTable 8 with the last column for the group

I_{h}

inTable 1. The 24 rotational symmetries of the cube (

O_{h}

group) do not corresond to any regular 4-polytope. In fact they are not well distributed in the sense of particle repulsion over the hypersphere in 4D as the other polytopic cases. Regarding this last point, it has been rigorously proven that that some of the regular 4-polytopes (the 5-cell, the 16-cell and the 600-cell) minimize a full class of repulsive potentials over the 4D sphere [32].

6. Conclusions

We expect that these sets of rotations will be useful for the computation of orientational averages in a range of physical sciences, and in experimental procedures such as spherical tensor analysis in nuclear magnetic resonance [7,8,9]. In addition, we anticipate that where necessary, finer sampling of orientational space may be implemented by interpolating between the vertices of the polytopes, or by four-dimensional tiling and honeycomb schemes, such as those described by Coxeter [17].

Finally, we note that highly-symmetric four-dimensional figures have been found by using a computational procedure [33] which is closely related to the REPULSION algorithm on the surface of a sphere [3]. Such methods could be adapted to generate much larger sets of evenly spaced three-dimensional rotations than those described here.

Acknowledgements

This research was supported by EPSRC (UK) and the Basic Technology Research Program (UK). We thank Michael Tayler for discussions. We would like to thanks the referees for constructive and insightful comments.

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Figure 1. The 3D regular convex polyhedra organised according to their symmetry group. Here

N_{0}

is the number of vertices,

N_{1}

is the number of edges and

N_{2}

is the number of faces constituting the solid.

Figure 1. The 3D regular convex polyhedra organised according to their symmetry group. Here

N_{0}

is the number of vertices,

N_{1}

is the number of edges and

N_{2}

is the number of faces constituting the solid.

Figure 2. Spherical rank profiles for the regular convex 3D polyhedra. Open circles indicate that all

(2 ℓ + 1)

spherical moments

σ_{ℓ m 0}^{S}

of integer rankℓ are zero for the set of orientations corresponding to the vertices of the corresponding polyhedron. Closed circles indicate that there is at least one non-zero spherical moment of rankℓ.

Figure 2. Spherical rank profiles for the regular convex 3D polyhedra. Open circles indicate that all

(2 ℓ + 1)

spherical moments

σ_{ℓ m 0}^{S}

Figure 3. A list of the 4D regular convex polytopes organized according to their symmetry group. Here

N_{0}

is the number of vertices,

N_{1}

is the number of edges,

N_{2}

is the number of faces and

N_{3}

is the number of three dimensional cells. The two dimensional graphs indicate the vertex connections.

Figure 3. A list of the 4D regular convex polytopes organized according to their symmetry group. Here

N_{0}

is the number of vertices,

N_{1}

is the number of edges,

N_{2}

is the number of faces and

N_{3}

is the number of three dimensional cells. The two dimensional graphs indicate the vertex connections.

Figure 4. Spherical rank profiles of the regular convex 4-polytopes. Open circles indicate that all

{(2 ℓ + 1)}^{2}

spherical moments

σ_{ℓ m m^{'}}^{S}

of integer rankℓ are zero for the set of orientations derived from the vertices of the corresponding polytope. Closed circles indicate that there is at least one non-zero spherical moment of rankℓ.

Figure 4. Spherical rank profiles of the regular convex 4-polytopes. Open circles indicate that all

{(2 ℓ + 1)}^{2}

spherical moments

σ_{ℓ m m^{'}}^{S}

Table 1. The three symmetry point groups of the regular polyhedra.h is the number of symmetry elements in the group. The last column shows the number of elements in each class (in square parentheses), followed by a single symmetry element of the class, for a polyhedron in standard orientation. The symbol

R_{(a, b, c)} (ξ)

indicates a rotation through the angle

ξ

about the axis

(a, b, c)

. The symbol

{\tilde{R}}_{(a, b, c)} (ξ)

indicates the improper operation constructed by the proper rotation

R_{(a, b, c)} (ξ)

followed by the inversion operation.

R_{} (0)

is the identity operation and

{\tilde{R}}_{} (0)

is the inversion operation. The symbol

τ = 2 \cos (π / 5) = (\sqrt{5} + 1) / 2

indicates the golden ratio.

R_{(a, b, c)} (ξ)

indicates a rotation through the angle

ξ

about the axis

(a, b, c)

. The symbol

{\tilde{R}}_{(a, b, c)} (ξ)

indicates the improper operation constructed by the proper rotation

R_{(a, b, c)} (ξ)

followed by the inversion operation.

R_{} (0)

is the identity operation and

{\tilde{R}}_{} (0)

is the inversion operation. The symbol

τ = 2 \cos (π / 5) = (\sqrt{5} + 1) / 2

indicates the golden ratio.

Symmetry group	h	Symmetry operations
$T_{d}$	24	$[1] R (0); [8] R_{(1, 1, 1)} (2 π / 3); [3] R_{(1, 0, 0)} (π);$ $[6] {\tilde{R}}_{(1, 0, 0)} (π / 2); [6] {\tilde{R}}_{(1, 1, 0)} (π)$
$O_{h}$	48	$[1] R (0); [8] R_{(1, 1, 1)} (2 π / 3); [3] R_{(1, 0, 0)} (π);$ $[6] R_{(1, 0, 0)} (π / 2); [6] R_{(1, 1, 0)} (π);$ $[1] \tilde{R} (0); [8] {\tilde{R}}_{(1, 1, 1)} (2 π / 3); [3] {\tilde{R}}_{(1, 0, 0)} (π);$ $[6] {\tilde{R}}_{(1, 0, 0)} (π / 2); [6] {\tilde{R}}_{(1, 1, 0)} (π);$
$I_{h}$	120	$[1] R (0); [12] R_{(1, 0, 0)} (2 π / 5); [12] R_{(1, 0, 0)} (4 π / 5);$ $[20] R_{(2, 0, τ^{2})} (2 π / 3); [15] R_{(τ, 0, 1)} (π);$ $[1] \tilde{R} (0); [12] {\tilde{R}}_{(1, 0, 0)} (2 π / 5); [12] {\tilde{R}}_{(1, 0, 0)} (4 π / 5);$ $[20] {\tilde{R}}_{(2, 0, τ^{2})} (2 π / 3); [15] {\tilde{R}}_{(τ, 0, 1)} (π);$

Table 2. The four symmetry groups of the 4D regular polytopes.h denotes the total number of symmetry elements. The last column shows the number of elements in each class (in square parentheses), followed by a single symmetry element of the class, for a polytope in standard orientation. The symmetry elements are denoted

R_{q_{l}, q_{r}}

for a proper rotation and

{\tilde{R}}_{q_{l}, q_{r}}

for an improper rotation, see Equations42 and43. The quaternions

{q 1, q 2 \dots q 15}

are given explicitly in the last section.

R_{q_{l}, q_{r}}

for a proper rotation and

{\tilde{R}}_{q_{l}, q_{r}}

for an improper rotation, see Equations42 and43. The quaternions

{q 1, q 2 \dots q 15}

are given explicitly in the last section.

Symmetry group	h	Symmetry operations
$A_{4}$	120	$[1] R_{q 1, q 1}; [15] R_{q 3, q 3}; [20] R_{q 8, q 8}; [24] R_{q 11, q 12}$ $[10] {\tilde{R}}_{q 4, q 4}; [30] {\tilde{R}}_{q 5, q 5}; [20] {\tilde{R}}_{q 9, q 10}$
$B_{4}$	384	$[1] R_{q 1, q 1}; [1] R_{q 1, - q 1}; [6] R_{q 2, q 2}; [12] R_{q 2, q 1}; [12] R_{q 2, q 3}; [24] R_{q 4, q 4};$ $[12] R_{q 6, q 6}; [12] R_{q 6, - q 6}; [32] R_{q 7, q 7}; [32] R_{q 7, - q 7}; [48] R_{q 6, q 4};$ $[4] {\tilde{R}}_{q 1, q 1}; [4] {\tilde{R}}_{q 1, - q 1}; [24] {\tilde{R}}_{q 2, q 1}; [12] {\tilde{R}}_{q 4, q 4}; [12] {\tilde{R}}_{q 4, - q 4};$ $[48] {\tilde{R}}_{q 6, q 4}; [24] {\tilde{R}}_{q 6, q 6}; [32] {\tilde{R}}_{q 7, q 7}; [32] {\tilde{R}}_{q 7, - q 7}$
$F_{4}$	1152	$[1] R_{q 1, q 1}; [1] R_{q 1, - q 1}; [12] R_{q 2, q 1}; [18] R_{q 2, - q 2}; [96] R_{q 2, q 7};$ $[72] R_{q 4, q 4}; [144] R_{q 6, - q 4}; [36] R_{q 6, q 6}; [36] R_{q 6, - q 6}; [16] R_{q 7, q 1};$ $[16] R_{q 7, - q 1}; [32] R_{q 7, q 7}; [32] R_{q 7, - q 7}; [32] R_{q 7, q 8}; [32] R_{q 7, - q 8};$ $[12] {\tilde{R}}_{q 1, q 1}; [72] {\tilde{R}}_{q 2, q 1}; [12] {\tilde{R}}_{q 2, q 2}; [96] {\tilde{R}}_{q 2, q 7}; [12] {\tilde{R}}_{q 4, q 4};$ $[12] {\tilde{R}}_{q 4, - q 4}; [72] {\tilde{R}}_{q 6, q 4}; [96] {\tilde{R}}_{q 6, q 5}; [96] {\tilde{R}}_{q 6, - q 5}; [96] {\tilde{R}}_{q 7, q 1}$
$H_{4}$	14 400	$[1] R_{q 1, q 1}; [1] R_{q 1, - q 1}; [60] R_{q 1, q 3}; [40] R_{q 1, q 13}; [40] R_{q 1, - q 13};$ $[24] R_{q 1, q 14}; [24] R_{q 1, - q 14}; [24] R_{q 1, q 15}; [24] R_{q 1, - q 15}; [450] R_{q 3, q 3};$ $[1200] R_{q 3, q 13}; [720] R_{q 3, q 14}; [720] R_{q 3, q 15}; [400] R_{q 13, q 13}; [400] R_{q 13, - q 13};$ $[480] R_{q 13, q 14}; [480] R_{q 13, - q 14}; [480] R_{q 13, q 15}; [480] R_{q 13, - q 15}; [144] R_{q 14, q 14};$ $[144] R_{q 14, - q 14} [288] R_{q 14, q 15}; [288] R_{q 14, - q 15}; [144] R_{q 15, q 15}; [144] R_{q 15, - q 15}$ $[60] {\tilde{R}}_{q 1, q 1}; [60] {\tilde{R}}_{- q 1, q 1}; [1800]; {\tilde{R}}_{q 3, q 1}; [1200] {\tilde{R}}_{q 13, q 1}; [1200] {\tilde{R}}_{- q 13, q 1};$ $[720] {\tilde{R}}_{q 14, q 1}; [720] {\tilde{R}}_{- q 14, q 1}; [720] {\tilde{R}}_{q 15, q 1}; [720] {\tilde{R}}_{- q 15, q 1};$
$q 1 = (1, 0, 0, 0); q 2 = (0, 0, 0, 1); q 3 = (0, 1, 0, 0);$
$q 4 = (0, 0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}); q 5 = (\frac{1}{\sqrt{2}}, 0, 0, \frac{1}{\sqrt{2}}); q 6 = (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0, 0);$
$q 7 = (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}); q 8 = (\frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}); q 9 = (\frac{1}{2 \sqrt{2}}, \frac{1}{2 \sqrt{2}}, \frac{1}{2 \sqrt{2}}, - \frac{\sqrt{5}}{2 \sqrt{2}});$
$q 10 = (\frac{1}{2 \sqrt{2}}, \frac{1}{2 \sqrt{2}}, \frac{1}{2 \sqrt{2}}, \frac{\sqrt{5}}{2 \sqrt{2}}); q 11 = (\frac{τ}{2}, \frac{τ^{- 1}}{2}, \frac{1}{2}, 0); q 12 = (- \frac{τ^{- 1}}{2}, - \frac{τ}{2}, \frac{1}{2}, 0);$
$q 13 = (\frac{1}{2}, \frac{τ^{- 1}}{2}, \frac{τ}{2}, 0); q 14 = (\frac{τ}{2}, \frac{1}{2}, \frac{τ^{- 1}}{2}, 0); q 15 = (\frac{τ^{- 1}}{2}, \frac{τ}{2}, \frac{1}{2}, 0)$

Table 3. The coordinates of the six convex regular 4-polytopes vertices in standard orientation, as reported in Reference [19]. The double round parentheses

(())

indicate that all even permutations of the quartet are taken. The symbols

τ

and

η

take the values

τ = 2 \cos (π / 5) = (\sqrt{5} + 1) / 2

and

η = \sqrt{5} / 4

. The 600 vertices of the hyperdodecahedron are obtained by multiplying the quaternion

(2^{- 1 / 2}, 2^{- 1 / 2}, 0, 0)

with all possible quaternion products of the 5 vertices of the hypertetrahedronS and the 120 vertices of the hypericosahedronI. All the polytopes are centred at the origin of the coordinate system, with the vertices lying on the hypersphere of radius 1.

Table 3. The coordinates of the six convex regular 4-polytopes vertices in standard orientation, as reported in Reference [19]. The double round parentheses

(())

indicate that all even permutations of the quartet are taken. The symbols

τ

and

η

take the values

τ = 2 \cos (π / 5) = (\sqrt{5} + 1) / 2

and

η = \sqrt{5} / 4

. The 600 vertices of the hyperdodecahedron are obtained by multiplying the quaternion

(2^{- 1 / 2}, 2^{- 1 / 2}, 0, 0)

Name	Vertex Coordinates
5-cell or hypertetrahedron	$S = {(1, 0, 0, 0), (- 1 / 4, η, η, η), (- 1 / 4, - η, - η, η),$
	$(- 1 / 4, - η, η, - η), (- 1 / 4, η, - η, - η)}$
16-cell or hyperoctahedron	$V = ((\pm 1, 0, 0, 0))$
8-cell or hypercube	$W = ((\pm 1 / 2, \pm 1 / 2, \pm 1 / 2, \pm 1 / 2))$
24-cell	$T = V \cup W$
600-cell or hypericosahedron	$I = T \cup \frac{1}{2} ((\pm τ, \pm 1, \pm τ^{- 1}, 0))$
120-cell or hyperdodecahedron	$J = (2^{- 1 / 2}, 2^{- 1 / 2}, 0, 0) * S * I$

Table 4. The set of Euler angles (in degrees) corresponding to the 5 verticesS of the 5-cell whose cartesian coordinates are given inTable 3.

$α$	$β$	$γ$
0	0	0
69.0948	104.478	159.095
110.905	104.478	20.9052
249.095	104.478	339.095
290.905	104.478	200.905

Table 5. The set of Euler angles (in degrees) corresponding to the 8 verticesV of the 16-cell whose cartesian coordinates are given inTable 3. The 8 vertices are reduced to 4 sets of Euler angles because each quaternion pair

{q, - q}