#
Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

**Observation**

**1**

#### 1.1. A Warm-Up: Two Dimensions

**Theorem**

**1.**

**Proof.**

#### 1.2. The Segmented Helix

- L is the distance between any two adjacent joints (between B and C, for example).
- r is the distance between a joint and the helix axis (between B and ${B}_{a}$ for example).
- $\theta $ is the rotation about the helix axis between two consecutive joints.
- c is the length of a chord formed by the projection of the segment between two points projected along the axis of the segmented helix (a chemist may recognize this as the distance between residues on a helical wheel projection).
- d is the distance along the axis of the helix between any two joint axis points (between ${B}_{a}$ and ${C}_{a}$ in Figure 3, for example, rendered as a small black and blue sphere, respectively).
- $\varphi $ is the angle between any vector between two adjacent joints and the axis of the helix. In physical screws used in mechanical engineering, this is analogous to the helix angle [18].
- p is the pitch of the helix, the distance traveled in one complete rotation.
- s is the number of segments in a complete rotation (in general not rational).
- Finally, we find it useful to define the tightness of a segmented helix as travel divided by radius, a number analogous to the extension of a coil spring or slinky. A torus-like segmented helix has zero tightness and a zig-zag has maximum tightness. The letter t represents tightness.

#### 1.3. Sign Conventions for Spatially Located Segmented Helices

- A right-handed coordinate system.
- The helix axis is a normalized vector which never vanishes.
- The travel along the axis (d) is negative when the helix is anti-clockwise (that is, when motion from joint n to joint $n+1$ appears anti-clockwise when $\theta <\pi $, zero when toroidal), and positive when the helix is clockwise.
- $\theta $ is never negative.

## 2. Results

#### 2.1. The Intrinsic Properties of Periodic Chains of Solids

- (1)
- The segments may form a straight line. (For example, see Figure 14).
- (2)
- The segments may be planar about a center, forming a polygon or ring. (For example, see Figure 20).
- (3)
- The segments may form a planar saw-tooth or zig-zag pattern of indefinite extent (For example, see Figure 4).

#### 2.2. Periodic Chains Produce Segmented Helices

**Theorem**

**2**

**Proof.**

**Corollary**

**1**

**Proof.**

#### 2.3. Computing Screws and Segmented Helices from Transformation Matrices

#### 2.4. Computing the Screw Axis from a Transformation Matrix

#### 2.5. Combining a Single Joint with the Matrix

#### 2.6. PointAxis: Computing Segmented Helices from Joints

#### 2.7. A Sketch of the 4-Point Method

- Construct a rigid transformation that places the points conveniently on the z-axis and balanced around the y-axis.
- Compute the bisectors of the angle between object axes $\angle ABC$, called $\overrightarrow{{B}_{b}}$ and the bisecting angle $\overrightarrow{{C}_{b}}$ of $\angle BCD$. If the points are collinear, they are a special case.
- Because these angle bisectors point at the axis of the segmented helix, their cross product is a vector in the direction of the axis. If $\overrightarrow{{B}_{b}}$ and $\overrightarrow{{C}_{b}}$ are parallel or anti-parallel the cross product is not defined and we have special cases.
- Otherwise the vectors $\overrightarrow{{B}_{b}}$ and $\overrightarrow{{C}_{b}}$ are skew, and the algorithm for the closest points on two skew lines provides two axis points ${B}_{a}$ and ${C}_{a}$ on these vectors which are the closest points on those lines and are also points on the helix axis.
- The distance between ${B}_{a}$ and B is the radius, and the distance between ${B}_{a}$ and ${C}_{a}$ is the travel d along the axis.
- The angle between $\overrightarrow{B-{B}_{a}}$ and $\overrightarrow{C-{C}_{a}}$ is $\theta $.

#### 2.8. Rotating into Balance from Face Normal Vectors

**Lemma**

**1**

- The segment $BC$ is centered on the z-axis: $({B}_{x}=0\wedge {B}_{y}=0\wedge {C}_{x}=0\wedge {C}_{y}=0)\wedge {B}_{z}<0\wedge {C}_{z}=-{B}_{z}$.
- The joints A and D are in rotational balance about the y axis, as if they were weights hanging downward: ${A}_{y}={D}_{y}\wedge {A}_{y}\le 0$ and ${A}_{x}=-{D}_{x}\wedge {A}_{z}=-{D}_{z}$.

**Proof Sketch of Lemma**

**1.**

#### 2.9. On the Choice of the Screw Axis Direction

#### 2.10. The 4-Point Method

- By virtue of Lemma 1, without loss of generality, think of any member whose faces and twist generate a non-degenerate helix as being “above” the axis of the helix. Furthermore choose to place the object in this figure so that ${B}_{y}={C}_{y}$, that is, that the members are symmetric about the z-axis. A and D are “balanced” across the $YZ$-plane, and ${A}_{x}=-{D}_{x}$ and ${A}_{y}={D}_{y}$.
- Every joint ($A,B,C$, and D) is the same distance r from the axis H of the helix.
- Every member is in the same angular relation $\varphi $ to the axis of the helix.
- Since every member of a non-degenerate helix cuts across a cylinder around the axis, the midpoint of every member is the same distance from the axis which is, in general, a little a less than r. In particular the midpoint M whose closest point on the helix axis m is on the y-axis and $\parallel \overrightarrow{{M}_{m}}\parallel <\parallel \overrightarrow{{B}_{b}}\parallel $.
- The points (${A}_{a},{B}_{a},{C}_{a},{D}_{a}$) on the axis closest to the joints ($A,B,C,D$) are equidistant about the axis and centered about the y-axis. In particular, $\parallel \overrightarrow{B-{B}_{a}}\parallel =\parallel \overrightarrow{C-{C}_{a}}\parallel $.

#### 2.11. Degenerate Cases

#### 2.12. Standard Case

#### 2.13. The 4-Point Test

**Theorem**

**3**

**Proof. “If”**

**Case**

**“Only if” Case**(equal scalar projections and length imply coincident segmented helix):

#### 2.14. Comparison of Two Methods

#### 2.15. The Joint Face Normal Method

- Given an object with two identified faces, labeled B and C, assume there are normalized vectors $\overrightarrow{{N}_{B}}$ and $\overrightarrow{{N}_{C}}$ from each of these points that are aligned with the axis of the conjoined object attached to that face. These normals might be enforced by the fact that flat faces are joined in the joint plane. However, molecules do not have faces; this conjoining relationship may be enforced some other way.
- The length L of an object, measured from joint point A to joint point B.
- A joint twist $\tau $ defining the change in computed out-vector between objects, measured at the joint face.

#### 2.16. Adjoining Prisms with Linear Algebra, Producing a Transformation Matrix

- Create the transformation that aligns and centers $\overrightarrow{BC}$ on the z-axis.
- Create a translation of B to C.
- Create a rotation of the z-axis to $\overrightarrow{{N}_{B}}$.
- Create a rotation of of $-\overrightarrow{{N}_{C}}$ about $\overrightarrow{{N}_{B}}$.
- Create a rotation of $\tau $ around that axis.

#### 2.17. Completing the Face Normal Computation

#### 2.18. Changing $\tau $ Smoothly Changes Tightness

**Theorem**

**4**

**Proof.**

## 3. Discussion

#### 3.1. Checks and Explorations

#### 3.1.1. Qualitative Observations

#### 3.1.2. A Brute-Force Approach to Finding Helix Angle from Twist

#### 3.2. Implications

#### 3.3. Applying to the Boerdijk–Coxeter Tetrahelix

#### 3.4. Confirming Periodic Twists

#### 3.5. The Platonic Helices

#### Qualitative Descriptions and Interesting Shapes

- The “Blockhelix” (Figure 13) is a cubic rectilinear structure in which all angles are right angles; nonetheless a segmented helix hides inside it which is perhaps not apparent to the human eye at first glance.
- The “Pearlshaft” (Figure 14). Conjoining parallel faces always produces a shaft. This icosahedron, being relatively round, resembles a string of pearls.
- However, shaft-like helices exist which do not join opposite faces. The “Dodecashaft” (Figure 15) is a remarkably tight non-self-intersecting dodecahelix with very narrow gaps between objects. Such a configuration might be formed by nanofibers under pressure.
- The “Dodecadoubler” (Figure 16) presents the appearance of being a double helix, even though in fact it is a single helix with a simple twist of 72 from the “Dodecashaft”.
- The “Dodecacorkscrew” (Figure 17) is a contrasting example of a loose helix, reminiscent of a corkscrew for opening wine bottles.
- The “Quasi-planar” (Figure 18) icosahelix presents a slowly twisting metahelix, so perhaps 10 icosahedron could be said to “lay flat”. If this were a molecule or a physical structure made of less-than-perfectly rigid members, it might be possible to force it into a pure planar configuration, thus wrapping a cylinder or a plane, studding it with icosahedra.
- “Two Strands” (Figure 19) is similar to the “Dodecadoubler” but even more visually striking. It is reminiscent of a depiction of a DNA double helix.
- The “Wheel” (Figure 20) resembles a modern car tire in proportions. All Platonic solids and indeed all shapes have torus-like configurations. In general, they do not “close” perfectly. There is a gap that prevents the final faces from fitting together perfectly. However, a tiny adjustment could be made to the repeated shape to close this gap.

#### 3.6. Future Work

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 11.**Node (Coxeter) helix (in black) and rail helices (in red, blue, and yellow), different than joint helices.

Name | Solid | # of Analogs | C-Face # | $\mathit{\tau}$ | Radius | $\mathit{\theta}$ | d | $\mathit{\varphi}$ |
---|---|---|---|---|---|---|---|---|

Tetrahelix | Tet | 6 | 1 | −120 | 0.094 | 131.810 | −0.516 | 161.565 |

Tetratorus | Tet | 3 | 1 | 0 | 0.471 | 70.529 | 0.000 | 90.000 |

Boxbeam | Cub | 4 | 1 | −180 | 0.000 | 0.000 | 1.000 | 0.000 |

Staircase | Cub | 4 | 2 | −180 | 0.000 | 0.000 | 0.707 | 0.000 |

Blockhelix | Cub | 8 | 2 | −90 | 0.236 | 120.000 | −0.577 | 144.736 |

Cubatorus | Cub | 4 | 2 | 0 | 0.500 | 90.000 | 0.000 | 90.000 |

Octabeam | Oct | 3 | 5 | −60 | 0.000 | 0.000 | 1.155 | 0.000 |

Octaspiky | Oct | 6 | 1 | −120 | 0.148 | 146.443 | −0.603 | 154.761 |

Octamedium | Oct | 6 | 2 | −120 | 0.163 | 131.810 | −0.894 | 161.565 |

Octagear | Oct | 3 | 1 | 0 | 0.408 | 109.471 | 0.000 | 90.000 |

Treestar | Oct | 3 | 2 | 0 | 0.816 | 70.529 | 0.000 | 90.000 |

Dodecabeam | Dod | 5 | 8 | −108 | 0.000 | 0.000 | 1.589 | 0.000 |

Dodecadoubler | Dod | 10 | 1 | −144 | 0.113 | 161.301 | −0.805 | 164.550 |

The Alternater | Dod | 10 | 2 | −144 | 0.118 | 149.520 | −1.333 | 170.306 |

Dodecashaft | Dod | 10 | 1 | −72 | 0.351 | 129.657 | −0.543 | 130.501 |

Dodecagear | Dod | 5 | 1 | 0 | 0.491 | 116.565 | 0.000 | 90.000 |

Dodecacorkscrew | Dod | 10 | 2 | −72 | 0.546 | 93.026 | −1.095 | 144.110 |

Dodecadonut | Dod | 5 | 2 | 0 | 1.286 | 63.435 | 0.000 | 90.000 |

Pearlshaft | Ico | 3 | 13 | −60 | 0.000 | 0.000 | 1.589 | 0.000 |

Quasi−planar | Ico | 6 | 8 | 165 | 0.049 | 167.764 | 1.294 | 4.347 |

Two Strands | Ico | 6 | 1 | 120 | 0.137 | 159.446 | 0.499 | 28.340 |

Slow Twist | Ico | 6 | 12 | 120 | 0.169 | 124.309 | 1.454 | 11.641 |

Rock Candy | Ico | 12 | 2 | 120 | 0.204 | 146.443 | 0.830 | 25.239 |

Icosa Tree Star | Ico | 3 | 1 | 0 | 0.304 | 138.190 | 0.000 | 90.000 |

Icosacorkscrew | Ico | 6 | 8 | −75 | 0.512 | 99.253 | −1.037 | 143.042 |

Planar point cluster | Ico | 6 | 2 | 0 | 0.562 | 109.471 | 0.000 | 90.000 |

Big Icosacorkscrew | Ico | 6 | 8 | 45 | 0.803 | 82.064 | 0.756 | 54.343 |

The Wheel | Ico | 3 | 12 | 0 | 2.080 | 41.810 | 0.000 | 90.000 |

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**MDPI and ACS Style**

Read, R.L.
Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices. *Mathematics* **2022**, *10*, 2533.
https://doi.org/10.3390/math10142533

**AMA Style**

Read RL.
Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices. *Mathematics*. 2022; 10(14):2533.
https://doi.org/10.3390/math10142533

**Chicago/Turabian Style**

Read, Robert L.
2022. "Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices" *Mathematics* 10, no. 14: 2533.
https://doi.org/10.3390/math10142533