# Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix)

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## Abstract

**:**

## 1. Introduction

## 2. Method of Assembly: Modified BC Helices

## 3. Modified BC Helices: General Formula for Periodicity

#### 3.1. Standard BC Helix, Aperiodic

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Modified BC Helix, Periodic

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Definition**

**1.**

**Corollary**

**1.**

**Theorem**

**3.**

**Proof.**

## 4. Modified BC Helices: Explicit Examples

- (i)
- When the chiralities of the rotation by$\beta $and that of the underlying helix produced by the face sequence$F=\left({f}_{0},{f}_{1},\dots ,{f}_{k}\right)$are
**alike**, one obtains a 5-period philix. - (ii)
- When the chiralities of the rotation by$\beta $and that of the underlying helix produced by the face sequence$F=\left({f}_{0},{f}_{1},\dots ,{f}_{k}\right)$are
**unlike**, one obtains a 3-period philix.

#### 4.1. The 5-BC Helix

#### 4.2. The 3-BC Helix

## 5. Conclusions

## Supplementary Materials

`m-BC-helix-ancillary.nb`, 3D rotatable images of the 3- and 5-period philices. This file can be viewed with the Wolfram Player, available for free at https://www.wolfram.com/player/.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Transformations ${\mathit{A}}_{{\mathit{T}}_{\mathit{k}}}^{{\mathit{f}}_{\mathit{k}}}$ and ${\mathit{B}}_{{\mathit{T}}_{\mathit{k}}}^{{\mathit{f}}_{\mathit{k}}}$

#### Appendix A.1. Transformations Related to the 5-BC Helix

#### Appendix A.2. Transformations Related to the 3-BC Helix

## References

- Coxeter, H.S.M. Regular Complex Polytopes; Cambridge University: Cambridge, UK, 1974. [Google Scholar]
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**1952**, 7, 30. [Google Scholar] - Gray, R.W. Tetrahelix Data. Available online: http://www.rwgrayprojects.com/rbfnotes/helix/helix01.html (accessed on 4 January 2013).
- Fang, F.; Irwin, K.; Kovacs, J.; Sadler, G. Cabinet of curiosities: The interesting geometry of the angle β = arccos((3ϕ − 1)/4). arXiv
**2013**, arXiv:1304.1771. [Google Scholar] - Fuller, B.R. Synergetics: Explorations in the Geometry of Thinking; Macmillan: London, UK, 1975. [Google Scholar]
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**Figure 1.**Canonical and modified Boerdijk–Coxeter helices: (

**a**) a right-handed Boerdijk-Coxeter (BC) helix; (

**b**) a “5-BC helix” may be obtained by appending and rotating tetrahedra through the angle given by Equation (4) using the same chirality of the underlying helix; (

**c**) a “3-BC helix” may be obtained by appending and rotating tetrahedra through the angle given by Equation (4) using the opposite chirality of the underlying helix.

**Figure 2.**Assembly of modified BC helix: (

**a**) a segment of an m-BC helix with face f identified on tetrahedron ${T}_{k}$; (

**b**) an interim tetrahedron, ${T}_{k}^{\prime}$ (shown in blue), is appended (face-to-face) to f on ${T}_{k}$; (

**c**) finally, ${T}_{k+1}$ is obtained by rotating ${T}_{k}^{\prime}$ through the angle $\beta $ about the axis ${n}_{k}$.

**Figure 3.**First step in constructing the standard BC helix: (

**a**) the initial tetrahedron ${T}_{0}$, with face ${f}_{0}$, edges {${\mathbf{a}}_{0},{\mathbf{b}}_{0},{\mathbf{c}}_{0}$}, and normal ${n}_{0}$; (

**b**) ${T}_{0}$ with ${T}_{1}$ and the transformed face, edges, and face normal.

**Figure 4.**${S}_{k}$, the kth rotation of ${S}_{0}={T}_{0}$. ${n}_{k}$ is the face normal, and unit vectors $\{{a}_{k}.{b}_{k},{c}_{k}\}$ are aligned with edges $\{{\mathbf{a}}_{k},{\mathbf{b}}_{k},{\mathbf{c}}_{k}\}$.

**Figure 5.**The periodicity of the 5-BC helix: (

**a**) the vertices of ${T}_{5}$ are the vertices of ${T}_{0}$ translated by ${w}_{5}$; (

**b**) a projection of the 5-BC helix along its central axis.

**Figure 6.**The periodicity of the 3-BC helix: (

**a**) the vertices of ${T}_{3}$ are the vertices of ${T}_{0}$ translated by ${w}_{3}$; (

**b**) a projection of the 3-BC helix along its central axis.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sadler, G.; Fang, F.; Clawson, R.; Irwin, K. Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix). *Mathematics* **2019**, *7*, 1001.
https://doi.org/10.3390/math7101001

**AMA Style**

Sadler G, Fang F, Clawson R, Irwin K. Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix). *Mathematics*. 2019; 7(10):1001.
https://doi.org/10.3390/math7101001

**Chicago/Turabian Style**

Sadler, Garrett, Fang Fang, Richard Clawson, and Klee Irwin. 2019. "Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix)" *Mathematics* 7, no. 10: 1001.
https://doi.org/10.3390/math7101001