# Critically-Finite Dynamics on the Icosahedron

## Abstract

**:**

## 1. Overview

## 2. Preliminaries: Icosahedral Geometry, Invariants, and Equivariants

## 3. Rational Maps on CP^{1} as Branched Self-Covers

## 4. Icosahedral Maps with Periodic Critical Orbits

**Fact**

**1.**

**Proposition**

**1.**

**Proof.**

**Fact**

**2.**

**Fact**

**3.**

**Fact**

**4.**

**Fact**

**5.**

**Fact**

**6.**

**Fact**

**7.**

## 5. Periodic Cycles

## 6. Special Behavior on a Fundamental Domain

## 7. Dynamical Polyhedra

#### 7.1. Constructing Edges

- Select equally-spaced points $({b}_{k}^{1},\dots ,{b}_{k}^{{n}_{k}})$ along each ${Q}_{0}^{k}$.
- Compute the elements in ${f}^{-1}\left({b}_{k}^{\ell}\right)$ for $\ell =1,\dots ,{n}_{k}$.
- Extract from ${f}^{-1}\left({b}_{k}^{\ell}\right)$ the inverse image points $({a}_{k}^{1},\dots ,{a}_{k}^{{n}_{k}})$ that minimize the sum of the distances to ${p}_{1}$ and ${p}_{2}$.
- Taylor expand f about ${a}_{k}^{\ell}$ for $\ell =1,\dots ,{n}_{k}$.
- Obtain the desired single-valued branch ${\gamma}_{k}^{\ell}$ of ${f}^{-1}$ by inverting the Taylor series for f at ${a}_{k}^{\ell}$. The choice of branch yields$${\gamma}_{k}^{\ell}\left({b}_{k}^{\ell}\right)={a}_{k}^{\ell}.$$
- Compute the “pulled” segmented paths that span $({p}_{1},{p}_{2})$:$$\begin{array}{cc}\hfill {P}_{1}^{1}=& \left(\right)open="("\; close=")">{\gamma}_{1}^{1}\left({Q}_{0}^{1}\right),\dots ,{\gamma}_{1}^{{n}_{1}}\left({Q}_{0}^{1}\right)\hfill \end{array}$$

**Proposition**

**2.**

**Proof.**

#### 7.2. Combinatorial Classification

## 8. Periodic Table

the map g discussed in Section 6 | period 2, achiral |

period 3 | |

period 5 | |

period 5 |

the map h discussed in Section 6 | period 2, achiral |

period 2 | |

period 3 | |

period 3 |

period 3 | |

period 5 | |

period 5 |

period 3 | |

period 3 | |

period 5 |

period 3 | |

period 3 | |

period 5 |

period 3 | |

period 5 | |

period 5 |

period 5 | |

period 5 |

period 2 |

period 2 |

period 5 |

period 2 |

period 2 |

period 3 |

period 5 |

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Fundamental triangle and its image under g: On the left, we see the collection of 120 triangles into which the sphere is carved by the 15 lines of reflective icosahedral symmetry. One such 235 triangle is filled with line segments parallel to the hypotenuse and having a color gradient running perpendicular to the hypotenuse. Any one of the 235 triangles is fundamental relative to the extended icosahedral action $\tilde{\mathcal{I}}$. That is, applying the elements of $\tilde{\mathcal{I}}$ to the triangle will tile the sphere as the plot shows. The image on the right plots the result of applying g to each of the segments in the shaded triangle on the left. Exactly 31 of the 235 triangles are covered by the image—thus revealing g’s topological and algebraic degrees.

**Figure 5.**Homotopy between select proto-edges and the image of a proto-edge for a pentagon (

**left**) and a triangle (

**right**) in a ${B}_{62}$ under one of the 31-maps f. The images of the proto-edges ${P}_{0}$ and ${T}_{0}$ (blue arcs) are respectively homotopic to a chain of pushed versions of ${P}_{0}$ and ${T}_{0}$ (black arcs). In this sample, the pushed edges associated with $f\left({P}_{0}\right)$ are PPT (pentagon–pentagon–triangle) starting with $f\left({p}_{1}\right)$ while those associated with $f\left({T}_{0}\right)$ are PPTPT beginning at $f\left({t}_{1}\right)$.

P | T | Q | ||
---|---|---|---|---|

P | ⟶ | 6 | 15 | 20 |

T | ⟶ | 3 | 10 | 12 |

Q | ⟶ | 8 | 8 | 15 |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 6 | 15 | 20 | $12\xb7P\u27f512\xb76+20\xb73+30\xb78=12\xb731$ | ||

T | ⟶ | 3 | 10 | 12 | $20\xb7T\u27f512\xb715+20\xb710+30\xb78=20\xb731$ | ||

Q | ⟶ | 8 | 8 | 15 | $30\xb7Q\u27f512\xb720+20\xb712+30\xb715=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 6 | 5 | 10 | $12\xb7P\u27f512\xb76+20\xb79+30\xb74=12\xb731$ | ||

T | ⟶ | 9 | 10 | 18 | $20\xb7T\u27f512\xb75+20\xb710+30\xb712=20\xb731$ | ||

Q | ⟶ | 4 | 12 | 15 | $30\xb7Q\u27f512\xb710+20\xb718+30\xb715=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 1 | 5 | 5 | $12\xb7P\u27f512\xb71+20\xb73+30\xb710=12\xb731$ | ||

T | ⟶ | 3 | 10 | 12 | $20\xb7T\u27f512\xb75+20\xb710+30\xb712=20\xb731$ | ||

Q | ⟶ | 10 | 12 | 21 | $30\xb7Q\u27f512\xb75+20\xb712+30\xb721=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 11 | 15 | 25 | $12\xb7P\u27f512\xb711+20\xb79+30\xb72=12\xb731$ | ||

T | ⟶ | 9 | 10 | 18 | $20\xb7T\u27f512\xb715+20\xb710+30\xb78=20\xb731$ | ||

Q | ⟶ | 2 | 8 | 9 | $30\xb7Q\u27f512\xb725+20\xb718+30\xb79=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 6 | 15 | 20 | $12\xb7P\u27f512\xb76+20\xb79+30\xb74=12\xb731$ | ||

T | ⟶ | 9 | 19 | 27 | $20\xb7T\u27f512\xb715+20\xb719+30\xb72=20\xb731$ | ||

Q | ⟶ | 4 | 2 | 5 | $30\xb7Q\u27f512\xb720+20\xb727+30\xb75=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 6 | 5 | 10 | $12\xb7P\u27f512\xb76+20\xb73+30\xb78=12\xb731$ | ||

T | ⟶ | 3 | 1 | 3 | $20\xb7T\u27f512\xb75+20\xb71+30\xb718=20\xb731$ | ||

Q | ⟶ | 8 | 18 | 25 | $30\xb7Q\u27f512\xb710+20\xb73+30\xb725=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 1 | 0 | 0 | $12\xb7P\u27f512\xb71+20\xb73+30\xb710=12\xb731$ | ||

T | ⟶ | 3 | 1 | 3 | $20\xb7T\u27f512\xb70+20\xb71+30\xb720=20\xb731$ | ||

Q | ⟶ | 10 | 20 | 29 | $30\xb7Q\u27f512\xb70+20\xb73+30\xb729=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 11 | 20 | 30 | $12\xb7P\u27f512\xb711+20\xb79+30\xb72=12\xb731$ | ||

T | ⟶ | 9 | 19 | 27 | $20\xb7T\u27f512\xb720+20\xb719+30\xb70=20\xb731$ | ||

Q | ⟶ | 2 | 0 | 1 | $30\xb7Q\u27f512\xb730+20\xb727+30\xb71=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 1 | 5 | 5 | $12\xb7P\u27f512\xb71+20\xb73+30\xb710=12\xb731$ | ||

T | ⟶ | 3 | 1 | 3 | $20\xb7T\u27f512\xb75+20\xb71+30\xb718=20\xb731$ | ||

Q | ⟶ | 10 | 18 | 27 | $30\xb7Q\u27f512\xb75+20\xb73+30\xb727=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 11 | 15 | 25 | $12\xb7P\u27f512\xb711+20\xb79+30\xb72=12\xb731$ | ||

T | ⟶ | 9 | 19 | 27 | $20\xb7T\u27f512\xb715+20\xb719+30\xb72=20\xb731$ | ||

Q | ⟶ | 2 | 2 | 3 | $30\xb7Q\u27f512\xb725+20\xb727+30\xb73=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 6 | 15 | 20 | $12\xb7P\u27f512\xb76+20\xb79+30\xb74=12\xb731$ | ||

T | ⟶ | 9 | 10 | 18 | $20\xb7T\u27f512\xb715+20\xb710+30\xb78=20\xb731$ | ||

Q | ⟶ | 4 | 8 | 11 | $30\xb7Q\u27f512\xb720+20\xb718+30\xb711=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 1 | 5 | 5 | $12\xb7P\u27f512\xb71+20\xb70+30\xb712=12\xb731$ | ||

T | ⟶ | 0 | 1 | 0 | $20\xb7T\u27f512\xb75+20\xb71+30\xb718=20\xb731$ | ||

Q | ⟶ | 12 | 18 | 29 | $30\xb7Q\u27f512\xb75+20\xb70+30\xb729=30\xb731$ |

P | T | Q | $\mathit{a}\xb7\mathit{X}\u27f5$ Total Number of Times X Faces Are Covered | ||||
---|---|---|---|---|---|---|---|

P | ⟶ | 6 | 5 | 10 | $12\xb7P\u27f512\xb76+20\xb76+30\xb76=12\xb731$ | ||

T | ⟶ | 6 | 4 | 9 | $20\xb7T\u27f512\xb75+20\xb74+30\xb716=20\xb731$ | ||

Q | ⟶ | 6 | 16 | 21 | $30\xb7Q\u27f512\xb710+20\xb79+30\xb721=30\xb731$ |

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## Share and Cite

**MDPI and ACS Style**

Crass, S.
Critically-Finite Dynamics on the Icosahedron. *Symmetry* **2020**, *12*, 177.
https://doi.org/10.3390/sym12010177

**AMA Style**

Crass S.
Critically-Finite Dynamics on the Icosahedron. *Symmetry*. 2020; 12(1):177.
https://doi.org/10.3390/sym12010177

**Chicago/Turabian Style**

Crass, Scott.
2020. "Critically-Finite Dynamics on the Icosahedron" *Symmetry* 12, no. 1: 177.
https://doi.org/10.3390/sym12010177