# Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4)

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## Abstract

**:**

## 1. Introduction

## 2. Aggregates of Tetrahedra

#### 2.1. Aggregates about a Common Edge

#### 2.2. Aggregates about a Common Vertex

#### 2.3. Periodic, Helical Aggregates

**like**chiralities are used, one obtains 5-fold symmetry; when

**unlike**chiralities are used, one obtains 3-fold symmetry. In addition to rotational symmetry, these structures are given a linear period, which we quantify here as the number of appended tetrahedra necessary to return to an initial angular position on the helix. For a modified BC helix with a period of m tetrahedra, we use the term m-BC helix. Accordingly, the procedure described above produces 3- and 5-BC helices, which are shown in Figure 6. (See [5] for a proof of these structures’ symmetries and periodicities).

## 3. Curiosities

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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- Boerdijk, A.H. Some remarks concerning close-packing of equal spheres. Philips Res. Rep.
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**Figure 1.**“Twisting” tetrahedra centered about a common central edge to close up gaps between adjacent tetrahedra. When this operation is performed, the angle β = arccos ((3ϕ − 1)/4) is produced in the projection of a “face junction”. (

**a**) Five tetrahedra arranged about a common edge. In this arrangement, small gaps exist between the faces of adjacent tetrahedra. Each tetrahedron is to be rotated by α

_{5}about an axis passing between the midpoints of its central and peripheral edges. (

**b**) The tetrahedra after rotation. In this arrangement, the faces of adjacent tetrahedra have been brought into contact with one another. (

**c**) A projection of a “face junction” between two coincident faces in Figure 1b. The angular displacement between the faces is β

_{5}= β.

**Figure 2.**Arranging and rotating three tetrahedra (as done in the n = 5 case) to “close up” gaps between adjacent tetrahedra. When this operation is performed, the angle ${\beta}_{3}=\frac{2\pi}{3}-\beta $ is produced in the projection of a “face junction”. (

**a**) Three tetrahedra arranged about a common edge. In this arrangement, large gaps exist between the faces of adjacent tetrahedra. Each tetrahedron is to be rotated by α

_{3}about an axis passing between the midpoints of its central and peripheral edges. (

**b**) The tetrahedra after rotation. In this arrangement, the faces of adjacent tetrahedra have been brought into contact with one another. (

**c**) A projection of a “face junction” between two coincident faces in Figure 2b. The angular displacement between the faces is β

_{3}.

**Figure 3.**When 20 tetrahedra are organized into an icosahedral arrangement, gaps between adjacent tetrahedra may be “closed” by performing a rotation of each tetrahedron by α

_{20}about an axis passing from the central vertex through each tetrahedron’s exterior face. When this is done, an angle of β is produced in the projection of faces in a “face junction”. (

**a**) Twenty tetrahedra arranged with icosahedral symmetry about a common central vertex. In this arrangement, gaps exist between faces of adjacent tetrahedra. Each tetrahedron is to be rotated by α

_{20}about an axis passing from the central vertex through its exterior face. (

**b**) The tetrahedra after rotation. Like in the cases above, the faces of adjacent tetrahedra have been brought into contact. (

**c**) A projection of the “face junction” between two coincident faces in Figure 3b. The angular displacement between the faces is β

_{20}= β

_{5}= β.

**Figure 4.**Assembly of a modified BC helix. (

**a**) A segment of a modified BC helix with face f

_{k}identified on tetrahedron T

_{k}. (

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

_{k}on T

_{k}. (

**c**) Finally, T

_{k+1}is obtained by rotating ${T}_{k}^{\prime}$ through the angle β about the axis nk.

**Figure 5.**BC helix projections and face junction. (

**a**) A “face junction” between tetrahedra of a 3-or 5-BC helix. (

**b**) A projection of the 5-BC helix along its central axis, showing five-fold symmetry. (

**c**) A projection of the 3-BC helix along its central axis, showing three-fold symmetry.

**Figure 6.**Canonical and modified Boerdijk–Coxeter helices. (

**a**) A right-handed BC helix. (

**b**) A “5-BC helix” may be obtained by appending and rotating tetrahedra by β using the same chirality of the underlying helix. (

**c**) A “3-BC helix” may be obtained by appending and rotating tetrahedra by β using the opposite chirality of the underlying helix.

**Figure 7.**Side-by-side comparison of face junctions. All face junctions may be obtained by translation of the (projected) tetrahedra of the 3- and 5-BC helix (pictured center left) by integer- and golden-ratio-based multiples of $\delta =\u2329\frac{a}{2{\varphi}^{2}\sqrt{6}},0\u232a$, where a is the tetrahedron edge length, the displacement between tetrahedra of a face junction for 5 tetrahedra about a common edge (pictured center right). From left to right, the displacements between the tetrahedra are $-2\delta $, 0, $\delta $, and $\left(3\varphi +1\right)\delta $.

**Table 1.**Plane class numbers for aggregates described in Section 2.

Structure | ${\mathit{\alpha}}_{\mathit{n}}$ | ${\mathit{\beta}}_{\mathit{n}}$ | Plane Classes | |
---|---|---|---|---|

Before | After | |||

3 tetrahedra, common edge | $arccos\left(\frac{1}{\sqrt{6}}\right)$ | $\frac{2\pi}{3}-arccos\left(\frac{3\varphi -1}{4}\right)$ | 12 | 9 |

4 tetrahedra, common edge | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | 16 | 4 |

5 tetrahedra, common edge | $arccos\left(\frac{{\varphi}^{2}}{\sqrt{2\left(\varphi +2\right)}}\right)$ | $arccos\left(\frac{3\varphi -1}{4}\right)$ | 20 | 10 |

20 tetrahedra, common vertex | $arccos\left(\frac{{\varphi}^{2}}{2\sqrt{2}}\right)$ | $arccos\left(\frac{3\varphi -1}{4}\right)$ | 60 | 10 |

n-tetrahedron 3-BC helix | n/a | $arccos\left(\frac{3\varphi -1}{4}\right)$ | $3n+1$ | 9 |

n-tetrahedron 5-BC helix | n/a | $arccos\left(\frac{3\varphi -1}{4}\right)$ | $3n+1$ | 10 |

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**MDPI and ACS Style**

Fang, F.; Irwin, K.; Kovacs, J.; Sadler, G.
Cabinet of Curiosities: The Interesting Geometry of the Angle *β* = arccos ((3*ϕ* − 1)/4). *Fractal Fract.* **2019**, *3*, 48.
https://doi.org/10.3390/fractalfract3040048

**AMA Style**

Fang F, Irwin K, Kovacs J, Sadler G.
Cabinet of Curiosities: The Interesting Geometry of the Angle *β* = arccos ((3*ϕ* − 1)/4). *Fractal and Fractional*. 2019; 3(4):48.
https://doi.org/10.3390/fractalfract3040048

**Chicago/Turabian Style**

Fang, Fang, Klee Irwin, Julio Kovacs, and Garrett Sadler.
2019. "Cabinet of Curiosities: The Interesting Geometry of the Angle *β* = arccos ((3*ϕ* − 1)/4)" *Fractal and Fractional* 3, no. 4: 48.
https://doi.org/10.3390/fractalfract3040048