# Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations

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## Abstract

**:**

## 1. Introduction

## 2. Description of Discrete Curvature and Twist Methods

#### 2.1. Discrete Curvature

#### 2.2. Twist

## 3. Equivalence between the Discrete Curvature and Twist Methods

## 4. Angle Matching between Discrete Curvature and Twist Transformations

#### 4.1. The Geometric Elements

#### 4.1.1. Basic Definitions

#### 4.1.2. Reflections

#### 4.2. Definitions of Transformations

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

#### 4.3. Equivalence of Transformation Angles

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

#### 4.4. Equivalence of Joint Angles

**Definition**

**3.**

**Remark**

**4.**

**Definition**

**4.**

**Remark**

**5.**

**Theorem**

**2.**

**Proof.**

## 5. Discussion and Outlook

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Computation of Angles for a Few Specific Cases

#### Appendix A.1. Shared Edge Configurations

**Figure A1.**(

**a**) Edge-sharing group of three tetrahedra, including the ${a}_{i}$ from the centre of the shared edge out toward the centres of the respective opposite edges. (The unit vectors ${a}_{i}$ are not to scale—the distance between the centres of two opposite edges is not necessarily a unit length.) (

**b**) Overhead view showing the face planes F and ${F}^{\prime}$ that will be rotated into coincidence by the transformation (curve or twist) defined by ${a}_{1}$ and ${a}_{2}$. D is the dihedral angle of a tetrahedron and A is the angle between their centres (i.e., between adjacent ${a}_{i}$).

#### Appendix A.2. Shared Vertex Configuration, 20 G

**Figure A3.**Example of 2 vertex-sharing tetrahedra with accompanying face planes: (

**a**) shown opaque for visual clarity, particularly where the shared vertex lies on the intersection line of the face planes; and (

**b**) shown partially transparent, so the centroid axes ${a}_{i}$ and angle bisector n can be seen.

#### Appendix A.3. Other Shared Vertex Configurations

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**Figure 1.**Images of: (

**a**) edge-sharing; and (

**b**) vertex sharing local tetrahedral clusters, which fail to locally fill space.

**Figure 2.**Symmetric arrangements of a 20-tetrahedron vertex sharing cluster: (

**a**) with open gaps; and then with gaps closed by: (

**b**) discrete curvature; (

**c**) distortion; and (

**d**) twisting. (

**e**,

**f**) The 2D analogue of the transition from gaps to discrete curvature and then to distortion.

**Figure 3.**Face planes F and ${F}^{\prime}$ symmetric across a mirror plane M, shown (

**a**) obliquely and (

**b**) directly from the $+h$ direction. They share a common vector h, and have normals f and ${f}^{\prime}$. The plane normal to h is H, which contains a number of auxiliary vectors. For visual clarity, the bivectors are shown enlarged, but are understood to have unit magnitude.

**Figure 4.**The discrete curvature transformation, shown in the 3-space of $H{e}_{4}$: (

**a**) the initial configuration; and (

**b**) both the initial and the final state. Shaded planar segments represent bivectors H and ${H}^{\prime}$ before (blue) and after (red) the transformation. Vectors g and ${g}^{\prime}$ rotate in planes parallel to $a{e}_{4}$ and ${a}^{\prime}{e}_{4}$, respectively, until they meet in the $n{e}_{4}$ plane, perpendicular to m. Vector a also rotates up to ${a}_{c}$, so ${\theta}_{F}$ between ${a}_{c}$ and ${g}_{c}$ projects down to ${\theta}_{M}$ between a and n.

**Figure 5.**The twist transformation, in the 3-space of $Hh$. Bivectors F and ${F}^{\prime}$ are shown: (

**a**) in the initial state; and (

**b**) in the final state, where they are equal, having been rotated clockwise around a and ${a}^{\prime}$, respectively, until meeting in ${F}_{t}={F}_{t}^{\prime}$, which contains n. Their normals f and $-{f}^{\prime}$ also rotate around a and ${a}^{\prime}$, to where they meet in the $mh$ plane, perpendicular to n. This exactly matches the behavior of g and ${g}^{\prime}$ in Figure 4 above.

**Figure 6.**Further illustration of the twist transformation, in the: (

**a**) initial; and (

**b**) final states. Bivectors F and ${F}^{\u2033}$ with a dihedral angle of $2{\theta}_{F}$ rotate around a. Vector g is rotated around a down to ${g}_{t}$, whereupon ${\theta}_{M}$ between n and a projects in ${H}_{t}$ to ${\theta}_{F}$ between ${g}_{t}$ and a. Within the H plane the effective dihedral angle (determined by its intersections with ${F}_{t}$ and ${F}_{t}^{\u2033}$) has expanded from $2{\theta}_{F}$ to $2{\theta}_{M}$.

**Table 1.**Properties of three different types of local tetrahedral clusters and their corresponding 4D polyhedra. Symmetrically arranged configurations with gaps are shown in Row 1 (edge sharing) and Row 3 (vertex sharing). Corresponding twisted configurations are shown in Rows 2 and 4. Rows 5 and 6 show overlays of various twisted configurations. In the last two rows note that, for each column, $V=B$ and $F=D$. The symbol $\varphi $ is used for the golden ratio, $\varphi ={\textstyle \frac{1}{2}}(1+\sqrt{5}).$ See the Supplementary Materials for dynamical versions of some images.

Cluster Types | Threefold | Fourfold | Fivefold |
---|---|---|---|

Evenly spaced edge-sharing | |||

Twisted edge-sharing | |||

Evenly spaced vertex-sharing | |||

Twisted vertex-sharing | |||

Twisted vertex-sharing, overlaid with twisted edge-sharing, scaled for face matching | |||

Twisted vertex-sharing, overlaid with twisted vertex-sharing 20G | |||

Transformation angles and joint angles for twist | $E\phantom{\rule{0.166667em}{0ex}}=arccos\frac{1}{\sqrt{6}}$ $V=2arccos\frac{{\varphi}^{2}}{2\sqrt{2}}$ $F\phantom{\rule{0.166667em}{0ex}}=arccos\frac{1}{4}$ | $E\phantom{\rule{0.166667em}{0ex}}=\frac{\pi}{4}$ $V=\frac{\pi}{3}$ $F\phantom{\rule{0.166667em}{0ex}}=\frac{2\pi}{3}$ | $E\phantom{\rule{0.166667em}{0ex}}=arctan\frac{1}{{\varphi}^{3}}$ $V=arccos\frac{{\varphi}^{2}}{2\sqrt{2}}$ $F\phantom{\rule{0.166667em}{0ex}}=arccos\frac{1}{4}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\frac{\pi}{3}$ |

Vertex cap of the 4D polytope | 5-cell $B=2arccos\frac{{\varphi}^{2}}{2\sqrt{2}}$ $D\phantom{\rule{0.166667em}{0ex}}=arccos\frac{1}{4}$ | 16-cell $B=\frac{\pi}{3}$ $D=\frac{2\pi}{3}$ | 600-cell $B=arccos\frac{{\varphi}^{2}}{2\sqrt{2}}$ $D=arccos\frac{1}{4}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\frac{\pi}{3}$ |

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**MDPI and ACS Style**

Fang, F.; Clawson, R.; Irwin, K.
Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations. *Mathematics* **2018**, *6*, 89.
https://doi.org/10.3390/math6060089

**AMA Style**

Fang F, Clawson R, Irwin K.
Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations. *Mathematics*. 2018; 6(6):89.
https://doi.org/10.3390/math6060089

**Chicago/Turabian Style**

Fang, Fang, Richard Clawson, and Klee Irwin.
2018. "Closing Gaps in Geometrically Frustrated Symmetric Clusters: Local Equivalence between Discrete Curvature and Twist Transformations" *Mathematics* 6, no. 6: 89.
https://doi.org/10.3390/math6060089