# Orientational Sampling Schemes Based on Four Dimensional Polytopes

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Gaussian Spherical Quadrature

#### 1.2. Two-angle Sampling and Regular Polyhedra

#### 1.3. Three-Angle Sampling and Regular 4-Polytopes

## 2. Group Theory and Symmetry Averaging

#### 2.1. Groups, Representations and Characters

#### 2.2. Subgroup Averaging

#### 2.3. Average of a Function in n-dimensional Space

#### 2.4. Average of a Function Over the Polytope Vertices

## 3. Polyhedral Averaging in Three Dimensions

#### 3.1. Proper and Improper Rotations

#### 3.2. Representations and Characters of $O\left(3\right)$ Isometries

#### 3.3. Regular Convex Polyhedra

#### 3.4. Spherical Moments of the Regular Polyhedra

## 4. Polytopic Averaging in Four Dimensions

#### 4.1. Quaternions

#### 4.2. Unit Quaternions and 3D Rotations

#### 4.3. Proper and Improper Rotations

#### 4.4. Representation and Characters of $O\left(4\right)$ Isometries

#### 4.5. Regular Convex 4-Polytopes

#### 4.6. Spherical Moments of the Regular 4-Polytopes

## 5. Euler Angles

## 6. Conclusions

## Acknowledgements

## References

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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 2.**Spherical rank profiles for the regular convex 3D polyhedra. Open circles indicate that all $(2\ell +1)$ spherical moments ${\sigma}_{\ell m0}^{\mathcal{S}}$ of integer rank ℓ 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 rank ℓ.

**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 ${(2\ell +1)}^{2}$ spherical moments ${\sigma}_{\ell m{m}^{\prime}}^{\mathcal{S}}$ of integer rank ℓ 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 rank ℓ.

**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 ${R}_{(a,b,c)}\left(\xi \right)$ indicates a rotation through the angle $\xi $ about the axis $(a,b,c)$. The symbol ${\tilde{R}}_{(a,b,c)}\left(\xi \right)$ indicates the improper operation constructed by the proper rotation ${R}_{(a,b,c)}\left(\xi \right)$ followed by the inversion operation. ${R}_{}\left(0\right)$ is the identity operation and ${\tilde{R}}_{}\left(0\right)$ is the inversion operation. The symbol $\tau =2\mathrm{cos}(\pi /5)=(\sqrt{5}+1)/2$ indicates the golden ratio.

Symmetry group | h | Symmetry operations |

${T}_{d}$ | 24 | $\left[1\right]R\left(0\right);\left[8\right]{R}_{(1,1,1)}(2\pi /3);\left[3\right]{R}_{(1,0,0)}\left(\pi \right);$ $\left[6\right]{\tilde{R}}_{(1,0,0)}(\pi /2);\left[6\right]{\tilde{R}}_{(1,1,0)}\left(\pi \right)$ |

${O}_{h}$ | 48 | $\left[1\right]R\left(0\right);\left[8\right]{R}_{(1,1,1)}(2\pi /3);\left[3\right]{R}_{(1,0,0)}\left(\pi \right);$ $\left[6\right]{R}_{(1,0,0)}(\pi /2);\left[6\right]{R}_{(1,1,0)}\left(\pi \right);$ $\left[1\right]\tilde{R}\left(0\right);\left[8\right]{\tilde{R}}_{(1,1,1)}(2\pi /3);\left[3\right]{\tilde{R}}_{(1,0,0)}\left(\pi \right);$ $\left[6\right]{\tilde{R}}_{(1,0,0)}(\pi /2);\left[6\right]{\tilde{R}}_{(1,1,0)}\left(\pi \right);$ |

${I}_{h}$ | 120 | $\left[1\right]R\left(0\right);\left[12\right]{R}_{(1,0,0)}(2\pi /5);\left[12\right]{R}_{(1,0,0)}(4\pi /5);$ $\left[20\right]{R}_{(2,0,{\tau}^{2})}(2\pi /3);\left[15\right]{R}_{(\tau ,0,1)}\left(\pi \right);$ $\left[1\right]\tilde{R}\left(0\right);\left[12\right]{\tilde{R}}_{(1,0,0)}(2\pi /5);\left[12\right]{\tilde{R}}_{(1,0,0)}(4\pi /5);$ $\left[20\right]{\tilde{R}}_{(2,0,{\tau}^{2})}(2\pi /3);\left[15\right]{\tilde{R}}_{(\tau ,0,1)}\left(\pi \right);$ |

**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 ${R}_{{\mathbf{q}}_{l},{\mathbf{q}}_{r}}$ for a proper rotation and ${\tilde{R}}_{{\mathbf{q}}_{l},{\mathbf{q}}_{r}}$ for an improper rotation, see Equations 42 and 43. The quaternions $\{\mathbf{q}1,\mathbf{q}2\dots \mathbf{q}15\}$ are given explicitly in the last section.

Symmetry group | h | Symmetry operations |

${\mathcal{A}}_{4}$ | 120 | $\left[1\right]{R}_{\mathbf{q}1,\mathbf{q}1};\left[15\right]{R}_{\mathbf{q}3,\mathbf{q}3};\left[20\right]{R}_{\mathbf{q}8,\mathbf{q}8};\left[24\right]{R}_{\mathbf{q}11,\mathbf{q}12}$ $\left[10\right]{\tilde{R}}_{\mathbf{q}4,\mathbf{q}4};\left[30\right]{\tilde{R}}_{\mathbf{q}5,\mathbf{q}5};\left[20\right]{\tilde{R}}_{\mathbf{q}9,\mathbf{q}10}$ |

${\mathcal{B}}_{4}$ | 384 | $\left[1\right]{R}_{\mathbf{q}1,\mathbf{q}1};\left[1\right]{R}_{\mathbf{q}1,-\mathbf{q}1};\left[6\right]{R}_{\mathbf{q}2,\mathbf{q}2};\left[12\right]{R}_{\mathbf{q}2,\mathbf{q}1};\left[12\right]{R}_{\mathbf{q}2,\mathbf{q}3};\left[24\right]{R}_{\mathbf{q}4,\mathbf{q}4};$ $\left[12\right]{R}_{\mathbf{q}6,\mathbf{q}6};\left[12\right]{R}_{\mathbf{q}6,-\mathbf{q}6};\left[32\right]{R}_{\mathbf{q}7,\mathbf{q}7};\left[32\right]{R}_{\mathbf{q}7,-\mathbf{q}7};\left[48\right]{R}_{\mathbf{q}6,\mathbf{q}4};$ $\left[4\right]{\tilde{R}}_{\mathbf{q}1,\mathbf{q}1};\left[4\right]{\tilde{R}}_{\mathbf{q}1,-\mathbf{q}1};\left[24\right]{\tilde{R}}_{\mathbf{q}2,\mathbf{q}1};\left[12\right]{\tilde{R}}_{\mathbf{q}4,\mathbf{q}4};\left[12\right]{\tilde{R}}_{\mathbf{q}4,-\mathbf{q}4};$ $\left[48\right]{\tilde{R}}_{\mathbf{q}6,\mathbf{q}4};\left[24\right]{\tilde{R}}_{\mathbf{q}6,\mathbf{q}6};\left[32\right]{\tilde{R}}_{\mathbf{q}7,\mathbf{q}7};\left[32\right]{\tilde{R}}_{\mathbf{q}7,-\mathbf{q}7}$ |

${\mathcal{F}}_{4}$ | 1152 | $\left[1\right]{R}_{\mathbf{q}1,\mathbf{q}1};\left[1\right]{R}_{\mathbf{q}1,-\mathbf{q}1};\left[12\right]{R}_{\mathbf{q}2,\mathbf{q}1};\left[18\right]{R}_{\mathbf{q}2,-\mathbf{q}2};\left[96\right]{R}_{\mathbf{q}2,\mathbf{q}7};$ $\left[72\right]{R}_{\mathbf{q}4,\mathbf{q}4};\left[144\right]{R}_{\mathbf{q}6,-\mathbf{q}4};\left[36\right]{R}_{\mathbf{q}6,\mathbf{q}6};\left[36\right]{R}_{\mathbf{q}6,-\mathbf{q}6};\left[16\right]{R}_{\mathbf{q}7,\mathbf{q}1};$ $\left[16\right]{R}_{\mathbf{q}7,-\mathbf{q}1};\left[32\right]{R}_{\mathbf{q}7,\mathbf{q}7};\left[32\right]{R}_{\mathbf{q}7,-\mathbf{q}7};\left[32\right]{R}_{\mathbf{q}7,\mathbf{q}8};\left[32\right]{R}_{\mathbf{q}7,-\mathbf{q}8};$ $\left[12\right]{\tilde{R}}_{\mathbf{q}1,\mathbf{q}1};\left[72\right]{\tilde{R}}_{\mathbf{q}2,\mathbf{q}1};\left[12\right]{\tilde{R}}_{\mathbf{q}2,\mathbf{q}2};\left[96\right]{\tilde{R}}_{\mathbf{q}2,\mathbf{q}7};\left[12\right]{\tilde{R}}_{\mathbf{q}4,\mathbf{q}4};$ $\left[12\right]{\tilde{R}}_{\mathbf{q}4,-\mathbf{q}4};\left[72\right]{\tilde{R}}_{\mathbf{q}6,\mathbf{q}4};\left[96\right]{\tilde{R}}_{\mathbf{q}6,\mathbf{q}5};\left[96\right]{\tilde{R}}_{\mathbf{q}6,-\mathbf{q}5};\left[96\right]{\tilde{R}}_{\mathbf{q}7,\mathbf{q}1}$ |

${\mathcal{H}}_{4}$ | 14 400 | $\left[1\right]{R}_{\mathbf{q}1,\mathbf{q}1};\left[1\right]{R}_{\mathbf{q}1,-\mathbf{q}1};\left[60\right]{R}_{\mathbf{q}1,\mathbf{q}3};\left[40\right]{R}_{\mathbf{q}1,\mathbf{q}13};\left[40\right]{R}_{\mathbf{q}1,-\mathbf{q}13};$ $\left[24\right]{R}_{\mathbf{q}1,\mathbf{q}14};\left[24\right]{R}_{\mathbf{q}1,-\mathbf{q}14};\left[24\right]{R}_{\mathbf{q}1,\mathbf{q}15};\left[24\right]{R}_{\mathbf{q}1,-\mathbf{q}15};\left[450\right]{R}_{\mathbf{q}3,\mathbf{q}3};$ $\left[1200\right]{R}_{\mathbf{q}3,\mathbf{q}13};\left[720\right]{R}_{\mathbf{q}3,\mathbf{q}14};\left[720\right]{R}_{\mathbf{q}3,\mathbf{q}15};\left[400\right]{R}_{\mathbf{q}13,\mathbf{q}13};\left[400\right]{R}_{\mathbf{q}13,-\mathbf{q}13};$ $\left[480\right]{R}_{\mathbf{q}13,\mathbf{q}14};\left[480\right]{R}_{\mathbf{q}13,-\mathbf{q}14};\left[480\right]{R}_{\mathbf{q}13,\mathbf{q}15};\left[480\right]{R}_{\mathbf{q}13,-\mathbf{q}15};\left[144\right]{R}_{\mathbf{q}14,\mathbf{q}14};$ $\left[144\right]{R}_{\mathbf{q}14,-\mathbf{q}14}\left[288\right]{R}_{\mathbf{q}14,\mathbf{q}15};\left[288\right]{R}_{\mathbf{q}14,-\mathbf{q}15};\left[144\right]{R}_{\mathbf{q}15,\mathbf{q}15};\left[144\right]{R}_{\mathbf{q}15,-\mathbf{q}15}$ $\left[60\right]{\tilde{R}}_{\mathbf{q}1,\mathbf{q}1};\left[60\right]{\tilde{R}}_{-\mathbf{q}1,\mathbf{q}1};\left[1800\right];{\tilde{R}}_{\mathbf{q}3,\mathbf{q}1};\left[1200\right]{\tilde{R}}_{\mathbf{q}13,\mathbf{q}1};\left[1200\right]{\tilde{R}}_{-\mathbf{q}13,\mathbf{q}1};$ $\left[720\right]{\tilde{R}}_{\mathbf{q}14,\mathbf{q}1};\left[720\right]{\tilde{R}}_{-\mathbf{q}14,\mathbf{q}1};\left[720\right]{\tilde{R}}_{\mathbf{q}15,\mathbf{q}1};\left[720\right]{\tilde{R}}_{-\mathbf{q}15,\mathbf{q}1};$ |

$\mathbf{q}1=\left(1,0,0,0\right);\mathbf{q}2=\left(0,0,0,1\right);\mathbf{q}3=\left(0,1,0,0\right);$ | ||

$\mathbf{q}4=\left(0,0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right);\mathbf{q}5=\left(\frac{1}{\sqrt{2}},0,0,\frac{1}{\sqrt{2}}\right);\mathbf{q}6=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0,0\right);$ | ||

$\mathbf{q}7=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right);\mathbf{q}8=\left(\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right);\mathbf{q}9=\left(\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}},-\frac{\sqrt{5}}{2\sqrt{2}}\right);$ | ||

$\mathbf{q}10=\left(\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}},\frac{\sqrt{5}}{2\sqrt{2}}\right);\mathbf{q}11=\left(\frac{\tau}{2},\frac{{\tau}^{-1}}{2},\frac{1}{2},0\right);\mathbf{q}12=\left(-\frac{{\tau}^{-1}}{2},-\frac{\tau}{2},\frac{1}{2},0\right);$ | ||

$\mathbf{q}13=\left(\frac{1}{2},\frac{{\tau}^{-1}}{2},\frac{\tau}{2},0\right);\mathbf{q}14=\left(\frac{\tau}{2},\frac{1}{2},\frac{{\tau}^{-1}}{2},0\right);\mathbf{q}15=\left(\frac{{\tau}^{-1}}{2},\frac{\tau}{2},\frac{1}{2},0\right)$ |

**Table 3.**The coordinates of the six convex regular 4-polytopes vertices in standard orientation, as reported in Reference [19]. The double round parentheses $\left(\phantom{\rule{-0.166667em}{0ex}}\right(\left)\phantom{\rule{-0.166667em}{0ex}}\right)$ indicate that all even permutations of the quartet are taken. The symbols $\tau $ and $\eta $ take the values $\tau =2\mathrm{cos}(\pi /5)=(\sqrt{5}+1)/2$ and $\eta =\sqrt{5}/4$. 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 hypertetrahedron S and the 120 vertices of the hypericosahedron I. All the polytopes are centred at the origin of the coordinate system, with the vertices lying on the hypersphere of radius 1.

Name | Vertex Coordinates |

5-cell or hypertetrahedron | $S=\left\{\right(1,0,0,0),(-1/4,\eta ,\eta ,\eta ),(-1/4,-\eta ,-\eta ,\eta ),$ |

$(-1/4,-\eta ,\eta ,-\eta ),(-1/4,\eta ,-\eta ,-\eta )\}$ | |

16-cell or hyperoctahedron | $V=\left(\phantom{\rule{-0.166667em}{0ex}}\right(\pm 1,0,0,0\left)\phantom{\rule{-0.166667em}{0ex}}\right)$ |

8-cell or hypercube | $W=\left(\phantom{\rule{-0.166667em}{0ex}}\right(\pm 1/2,\pm 1/2,\pm 1/2,\pm 1/2\left)\phantom{\rule{-0.166667em}{0ex}}\right)$ |

24-cell | $T=V\cup W$ |

600-cell or hypericosahedron | $I=T\cup \frac{1}{2}\left(\phantom{\rule{-0.166667em}{0ex}}(\pm \tau ,\pm 1,\pm {\tau}^{-1},0)\phantom{\rule{-0.166667em}{0ex}}\right)$ |

120-cell or hyperdodecahedron | $J=({2}^{-1/2},{2}^{-1/2},0,0)\ast S\ast I$ |

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

$\alpha $ | $\beta $ | $\gamma $ |

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 vertices V of the 16-cell whose cartesian coordinates are given in Table 3. The 8 vertices are reduced to 4 sets of Euler angles because each quaternion pair $\{\mathbf{q},-\mathbf{q}\}$ corresponds to the same geometrical 3D rotation.

$\alpha $ | $\beta $ | $\gamma $ |

0 | 0 | 0 |

0 | 180 | 0 |

180 | 0 | 0 |

180 | 180 | 0 |

**Table 6.**The set of Euler angles (in degrees) corresponding to the 16 vertices W of the 8-cell whose cartesian coordinates are given in Table 3. The 16 vertices are reduced to 8 sets of Euler angles because each quaternion pair $\{\mathbf{q},-\mathbf{q}\}$ corresponds to the same geometrical 3D rotation.

$\alpha $ | $\beta $ | $\gamma $ |

0 | 90 | 90 |

0 | 90 | 270 |

90 | 90 | 0 |

90 | 90 | 180 |

180 | 90 | 90 |

180 | 90 | 270 |

270 | 90 | 0 |

270 | 90 | 180 |

**Table 7.**The set of Euler angles (in degrees) corresponding to the 24 vertices T of the 24-cell whose cartesian coordinates are given in Table 3. The 24 vertices are reduced to 12 sets of Euler angles because each quaternion pair $\{\mathbf{q},-\mathbf{q}\}$ corresponds to the same geometrical 3D rotation.

$\alpha $ | $\beta $ | $\gamma $ | $\alpha $ | $\beta $ | $\gamma $ |

0 | 0 | 0 | 180 | 0 | 0 |

0 | 90 | 90 | 180 | 90 | 90 |

0 | 90 | 270 | 180 | 90 | 270 |

0 | 180 | 0 | 180 | 180 | 0 |

90 | 90 | 0 | 270 | 90 | 0 |

90 | 90 | 180 | 270 | 90 | 180 |

**Table 8.**The set of Euler angles (in degrees) corresponding to the 120 vertices I of the 600-cell whose cartesian coordinates are given in Table 3. The 120 vertices are reduced to 60 sets of Euler angles because each quaternion pair $\{\mathbf{q},-\mathbf{q}\}$ corresponds to the same geometrical 3D rotation.

$\alpha $ | $\beta $ | $\gamma $ | $\alpha $ | $\beta $ | $\gamma $ |

0 | 0 | 0 | 180 | 0 | 0 |

0 | 90 | 90 | 180 | 90 | 90 |

0 | 90 | 270 | 180 | 90 | 270 |

0 | 180 | 0 | 180 | 180 | 0 |

20.9052 | 60 | 200.905 | 200.905 | 60 | 20.9052 |

20.9052 | 120 | 159.095 | 200.905 | 60 | 200.905 |

20.9052 | 60 | 20.9052 | 200.905 | 120 | 159.095 |

20.9052 | 120 | 339.095 | 200.905 | 120 | 339.095 |

58.2826 | 36 | 58.2826 | 238.283 | 36 | 58.2826 |

58.2826 | 36 | 238.283 | 238.283 | 36 | 238.283 |

58.2826 | 72 | 121.717 | 238.283 | 72 | 121.717 |

58.2826 | 72 | 301.717 | 238.283 | 72 | 301.717 |

58.2826 | 108 | 58.2826 | 238.283 | 108 | 58.2826 |

58.2826 | 108 | 238.283 | 238.283 | 108 | 238.283 |

58.2826 | 144 | 121.717 | 238.283 | 144 | 121.717 |

58.2826 | 144 | 301.717 | 238.283 | 144 | 301.717 |

90 | 90 | 0 | 270 | 90 | 0 |

90 | 90 | 180 | 270 | 90 | 180 |

121.717 | 36 | 121.717 | 301.717 | 36 | 121.717 |

121.717 | 36 | 301.717 | 301.717 | 36 | 301.717 |

121.717 | 72 | 58.2826 | 301.717 | 72 | 58.2826 |

121.717 | 72 | 238.283 | 301.717 | 72 | 238.283 |

121.717 | 108 | 121.717 | 301.717 | 108 | 121.717 |

121.717 | 108 | 301.717 | 301.717 | 108 | 301.717 |

121.717 | 144 | 58.2826 | 301.717 | 144 | 58.2826 |

121.717 | 144 | 238.283 | 301.717 | 144 | 238.283 |

159.095 | 60 | 159.095 | 339.095 | 60 | 159.095 |

159.095 | 60 | 339.095 | 339.095 | 60 | 339.095 |

159.095 | 120 | 20.9052 | 339.095 | 120 | 20.9052 |

159.095 | 120 | 200.905 | 339.095 | 120 | 200.905 |

**Table 9.**The set of Euler angles (in degrees) corresponding to the 600 vertices J of the 120-cell whose cartesian coordinates are given in Table 3. The 600 vertices are reduced to 300 sets of Euler angles because each quaternion pair $\{\mathbf{q},-\mathbf{q}\}$ corresponds to the same geometrical 3D rotation.

$\alpha $ | $\beta $ | $\gamma $ | $\alpha $ | $\beta $ | $\gamma $ | $\alpha $ | $\beta $ | $\gamma $ | $\alpha $ | $\beta $ | $\gamma $ |

0 | 90 | 0 | 90 | 90 | 90 | 180 | 90 | 0 | 270 | 90 | 90 |

0 | 90 | 180 | 90 | 90 | 270 | 180 | 90 | 180 | 270 | 90 | 270 |

7.25597 | 49.1176 | 70.6909 | 90 | 180 | 0 | 187.256 | 49.1176 | 70.6909 | 270 | 180 | 0 |

7.25597 | 49.1176 | 250.691 | 95.6599 | 75.5225 | 137.470 | 187.256 | 49.1176 | 250.691 | 275.660 | 75.5225 | 137.470 |

7.25597 | 130.882 | 109.309 | 95.6599 | 75.5225 | 317.471 | 187.256 | 130.882 | 109.309 | 275.660 | 75.5225 | 317.471 |

7.25597 | 130.882 | 289.309 | 95.6599 | 104.478 | 42.5298 | 187.256 | 130.882 | 289.309 | 275.660 | 104.478 | 42.5298 |

14.5454 | 84.5204 | 131.110 | 95.6599 | 104.478 | 222.530 | 194.546 | 84.5204 | 131.110 | 275.660 | 104.478 | 222.530 |

14.5454 | 84.5204 | 311.110 | 98.3008 | 41.4096 | 98.3008 | 194.546 | 84.5204 | 311.110 | 278.301 | 41.4096 | 98.3008 |

14.5454 | 95.4796 | 48.8895 | 98.3008 | 41.4096 | 278.301 | 194.546 | 95.4796 | 48.8895 | 278.301 | 41.4096 | 278.301 |

14.5454 | 95.4796 | 228.890 | 98.3008 | 138.590 | 81.6992 | 194.546 | 95.4796 | 228.890 | 278.301 | 138.590 | 81.6992 |

20.9052 | 15.5225 | 20.9052 | 98.3008 | 138.590 | 261.699 | 200.905 | 15.5225 | 20.9052 | 278.301 | 138.590 | 261.699 |

20.9052 | 15.5225 | 200.905 | 105.450 | 69.7882 | 105.450 | 200.905 | 15.5225 | 200.905 | 285.450 | 69.7882 | 105.450 |

20.9052 | 44.4775 | 159.095 | 105.450 | 69.7882 | 285.450 | 200.905 | 44.4775 | 159.095 | 285.450 | 69.7882 | 285.450 |

20.9052 | 44.4775 | 339.095 | 105.450 | 110.212 | 74.5496 | 200.905 | 44.4775 | 339.095 | 285.450 | 110.212 | 74.5496 |

20.9052 | 60 | 110.905 | 105.450 | 110.212 | 254.550 | 200.905 | 60 | 110.905 | 285.450 | 110.212 | 254.550 |

20.9052 | 60 | 290.905 | 109.309 | 49.1176 | 172.744 | 200.905 | 60 | 290.905 | 289.309 | 49.1176 | 172.744 |

20.9052 | 75.5225 | 159.095 | 109.309 | 49.1176 | 352.7441 | 200.905 | 75.5225 | 159.095 | 289.309 | 49.1176 | 352.7441 |

20.9052 | 75.5225 | 339.095 | 109.309 | 130.882 | 7.25597 | 200.905 | 75.5225 | 339.095 | 289.309 | 130.882 | 7.25597 |

20.9052 | 104.478 | 20.9052 | 109.309 | 130.882 | 187.256 | 200.905 | 104.478 | 20.9052 | 289.309 | 130.882 | 187.256 |

20.9052 | 104.478 | 200.905 | 110.905 | 60 | 20.9052 | 200.905 | 104.478 | 200.905 | 290.905 | 60 | 20.9052 |

20.9052 | 120 | 69.0948 | 110.905 | 60 | 200.905 | 200.905 | 120 | 69.0948 | 290.905 | 60 | 200.905 |

20.9052 | 120 | 249.095 | 110.905 | 120 | 159.095 | 200.905 | 120 | 249.095 | 290.905 | 120 | 159.095 |

20.9052 | 135.522 | 20.9052 | 110.905 | 120 | 339.095 | 200.905 | 135.522 | 20.9052 | 290.905 | 120 | 339.095 |

20.9052 | 135.522 | 200.905 | 121.717 | 36 | 31.7175 | 200.905 | 135.522 | 200.905 | 301.717 | 36 | 31.7175 |

20.9052 | 164.478 | 159.095 | 121.717 | 36 | 211.717 | 200.905 | 164.478 | 159.095 | 301.717 | 36 | 211.717 |

20.9052 | 164.478 | 339.095 | 121.717 | 72 | 148.283 | 200.905 | 164.478 | 339.095 | 301.717 | 72 | 148.283 |

31.7175 | 36 | 121.717 | 121.717 | 72 | 328.283 | 211.717 | 36 | 121.717 | 301.717 | 72 | 328.283 |

31.7175 | 36 | 301.717 | 121.717 | 108 | 31.7175 | 211.717 | 36 | 301.717 | 301.717 | 108 | 31.7175 |

31.7175 | 72 | 58.2826 | 121.717 | 108 | 211.717 | 211.717 | 72 | 58.2826 | 301.717 | 108 | 211.717 |

31.7175 | 72 | 238.283 | 121.717 | 144 | 148.283 | 211.717 | 72 | 238.283 | 301.717 | 144 | 148.283 |

31.7175 | 108 | 121.717 | 121.717 | 144 | 328.283 | 211.717 | 108 | 121.717 | 301.717 | 144 | 328.283 |

31.7175 | 108 | 301.717 | 131.110 | 84.5204 | 14.5454 | 211.717 | 108 | 301.717 | 311.110 | 84.5204 | 14.5454 |

31.7175 | 144 | 58.2826 | 131.110 | 84.5204 | 194.546 | 211.717 | 144 | 58.2826 | 311.110 | 84.5204 | 194.546 |

31.7175 | 144 | 238.283 | 131.110 | 95.4796 | 165.455 | 211.717 | 144 | 238.283 | 311.110 | 95.4796 | 165.455 |

35.8898 | 25.2428 | 35.8898 | 131.110 | 95.4796 | 345.455 | 215.890 | 25.2428 | 35.8898 | 311.110 | 95.4796 | 345.455 |

35.8898 | 25.2428 | 215.890 | 137.470 | 75.5225 | 95.6599 | 215.890 | 25.2428 | 215.890 | 317.471 | 75.5225 | 95.6599 |

35.8898 | 154.757 | 144.110 | 137.470 | 75.5225 | 275.660 | 215.890 | 154.757 | 144.110 | 317.471 | 75.5225 | 275.660 |

35.8898 | 154.757 | 324.110 | 137.470 | 104.478 | 84.3401 | 215.890 | 154.757 | 324.110 | 317.471 | 104.478 | 84.3401 |

42.5298 | 75.5225 | 84.3401 | 137.470 | 104.478 | 264.340 | 222.530 | 75.5225 | 84.3401 | 317.471 | 104.478 | 264.340 |

42.5298 | 75.5225 | 264.340 | 144.110 | 25.2428 | 144.110 | 222.530 | 75.5225 | 264.340 | 324.110 | 25.2428 | 144.110 |

42.5298 | 104.478 | 95.6599 | 144.110 | 25.2428 | 324.110 | 222.530 | 104.478 | 95.6599 | 324.110 | 25.2428 | 324.110 |

42.5298 | 104.478 | 275.660 | 144.110 | 154.757 | 35.8898 | 222.530 | 104.478 | 275.660 | 324.110 | 154.757 | 35.8898 |

48.8895 | 84.5204 | 165.455 | 144.110 | 154.757 | 215.890 | 228.890 | 84.5204 | 165.455 | 324.110 | 154.757 | 215.890 |

48.8895 | 84.5204 | 345.455 | 148.283 | 36 | 58.2826 | 228.890 | 84.5204 | 345.455 | 328.283 | 36 | 58.2826 |

48.8895 | 95.4796 | 14.5454 | 148.283 | 36 | 238.283 | 228.890 | 95.4796 | 14.5454 | 328.283 | 36 | 238.283 |

48.8895 | 95.4796 | 194.546 | 148.283 | 72 | 121.717 | 228.890 | 95.4796 | 194.546 | 328.283 | 72 | 121.717 |

58.2826 | 36 | 148.283 | 148.283 | 72 | 301.717 | 238.283 | 36 | 148.283 | 328.283 | 72 | 301.717 |

58.2826 | 36 | 328.283 | 148.283 | 108 | 58.2826 | 238.283 | 36 | 328.283 | 328.283 | 108 | 58.2826 |

58.2826 | 72 | 31.7175 | 148.283 | 108 | 238.283 | 238.283 | 72 | 31.7175 | 328.283 | 108 | 238.283 |

58.2826 | 72 | 211.717 | 148.283 | 144 | 121.717 | 238.283 | 72 | 211.717 | 328.283 | 144 | 121.717 |

58.2826 | 108 | 148.283 | 148.283 | 144 | 301.717 | 238.283 | 108 | 148.283 | 328.283 | 144 | 301.717 |

58.2826 | 108 | 328.283 | 159.095 | 15.5225 | 159.095 | 238.283 | 108 | 328.283 | 339.095 | 15.5225 | 159.095 |

58.2826 | 144 | 31.7175 | 159.095 | 15.5225 | 339.095 | 238.283 | 144 | 31.7175 | 339.095 | 15.5225 | 339.095 |

58.2826 | 144 | 211.717 | 159.095 | 44.4775 | 20.9052 | 238.283 | 144 | 211.717 | 339.095 | 44.4775 | 20.9052 |

69.0948 | 60 | 159.095 | 159.095 | 44.4775 | 200.905 | 249.095 | 60 | 159.095 | 339.095 | 44.4775 | 200.905 |

69.0948 | 60 | 339.095 | 159.095 | 60 | 69.0948 | 249.095 | 60 | 339.095 | 339.095 | 60 | 69.0948 |

69.0948 | 120 | 20.9052 | 159.095 | 60 | 249.095 | 249.095 | 120 | 20.9052 | 339.095 | 60 | 249.095 |

69.0948 | 120 | 200.905 | 159.095 | 75.5225 | 20.9052 | 249.095 | 120 | 200.905 | 339.095 | 75.5225 | 20.9052 |

70.6909 | 49.1176 | 7.25597 | 159.095 | 75.5225 | 200.905 | 250.691 | 49.1176 | 7.25597 | 339.095 | 75.5225 | 200.905 |

70.6909 | 49.1176 | 187.256 | 159.095 | 104.478 | 159.095 | 250.691 | 49.1176 | 187.256 | 339.095 | 104.478 | 159.095 |

70.6909 | 130.882 | 172.744 | 159.095 | 104.478 | 339.095 | 250.691 | 130.882 | 172.744 | 339.095 | 104.478 | 339.095 |

70.6909 | 130.882 | 352.7441 | 159.095 | 120 | 110.905 | 250.691 | 130.882 | 352.7441 | 339.095 | 120 | 110.905 |

74.5496 | 69.7882 | 74.5496 | 159.095 | 120 | 290.905 | 254.550 | 69.7882 | 74.5496 | 339.095 | 120 | 290.905 |

74.5496 | 69.7882 | 254.550 | 159.095 | 135.522 | 159.095 | 254.550 | 69.7882 | 254.550 | 339.095 | 135.522 | 159.095 |

74.5496 | 110.212 | 105.450 | 159.095 | 135.522 | 339.095 | 254.550 | 110.212 | 105.450 | 339.095 | 135.522 | 339.095 |

74.5496 | 110.212 | 285.450 | 159.095 | 164.478 | 20.9052 | 254.550 | 110.212 | 285.450 | 339.095 | 164.478 | 20.9052 |

81.6992 | 41.4096 | 81.6992 | 159.095 | 164.478 | 200.905 | 261.699 | 41.4096 | 81.6992 | 339.095 | 164.478 | 200.905 |

81.6992 | 41.4096 | 261.699 | 165.455 | 84.5204 | 48.8895 | 261.699 | 41.4096 | 261.699 | 345.455 | 84.5204 | 48.8895 |

81.6992 | 138.590 | 98.3008 | 165.455 | 84.5204 | 228.890 | 261.699 | 138.590 | 98.3008 | 345.455 | 84.5204 | 228.890 |

81.6992 | 138.590 | 278.301 | 165.455 | 95.4796 | 131.110 | 261.699 | 138.590 | 278.301 | 345.455 | 95.4796 | 131.110 |

84.3401 | 75.5225 | 42.5298 | 165.455 | 95.4796 | 311.110 | 264.340 | 75.5225 | 42.5298 | 345.455 | 95.4796 | 311.110 |

84.3401 | 75.5225 | 222.530 | 172.744 | 49.1176 | 109.309 | 264.340 | 75.5225 | 222.530 | 352.7441 | 49.1176 | 109.309 |

84.3401 | 104.478 | 137.470 | 172.744 | 49.1176 | 289.309 | 264.340 | 104.478 | 137.470 | 352.7441 | 49.1176 | 289.309 |

84.3401 | 104.478 | 317.471 | 172.744 | 130.882 | 70.6909 | 264.340 | 104.478 | 317.471 | 352.7441 | 130.882 | 70.6909 |

90 | 0 | 0 | 172.744 | 130.882 | 250.691 | 270 | 0 | 0 | 352.7441 | 130.882 | 250.691 |

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## Share and Cite

**MDPI and ACS Style**

Mamone, S.; Pileio, G.; Levitt, M.H.
Orientational Sampling Schemes Based on Four Dimensional Polytopes. *Symmetry* **2010**, *2*, 1423-1449.
https://doi.org/10.3390/sym2031423

**AMA Style**

Mamone S, Pileio G, Levitt MH.
Orientational Sampling Schemes Based on Four Dimensional Polytopes. *Symmetry*. 2010; 2(3):1423-1449.
https://doi.org/10.3390/sym2031423

**Chicago/Turabian Style**

Mamone, Salvatore, Giuseppe Pileio, and Malcolm H. Levitt.
2010. "Orientational Sampling Schemes Based on Four Dimensional Polytopes" *Symmetry* 2, no. 3: 1423-1449.
https://doi.org/10.3390/sym2031423