# Temari Balls, Spheres, SphereHarmonic: From Japanese Folkcraft to Music

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Tonnetz in Music Theory, Triangles as Triads

#### 2.2. Basic Divisions of Temari

_{n}. The magic numbers help retrieve proportions in the process of building Temari balls. With these values, Temari artists approximate spherical icosahedral edge lengths as $(1/6+1/100)$ times the ball circumference [6,28], for example. There are three types of combination divisions: the combination 6 (C6), the combination 8 (C8), and the combination 10 (C10), as shown in Figure 3b–d.

#### 2.3. Concept and Visualization of SphereHarmonic

## 3. Results

#### 3.1. Computing Combinatorial Patterns of Temari Balls

#### 3.2. Prototype of C8

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

^{®}is available from https://github.com/medusamedusa/SphereHarmonic (accessed on 10 August 2022).

## Conflicts of Interest

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**Figure 2.**The prototype of CubeHarmonic developed at the Tohoku University, with the IM3D platform and a virtual-reality screen. Detail from a video’s photogram. Video by M. Mannone with Kitamura Lab. CC-BY license for NIME conference.

**Figure 3.**(

**a**–

**d**):Traditional basic divisions (S8, C6, C8, and C10, from left to right). Large marks on the top denote poles. (

**a’**–

**d’**): Spherical patterns corresponding to (

**a**–

**d**). (

**e’**): A simple textured triangular unit used to construct spherical patterns (

**a’**–

**d’**).

**Figure 9.**Representation of sequential rotations of the hemispheres of C8. L stands for “lower”, and U for “upper”.

Divisions | Number of Triangles | Number of Axes | Amount of Rotations | |
---|---|---|---|---|

Sn | $2n$ | $1+n/2$ | 1 | $2\pi /n$ |

$n/2$ | $\pi $ | |||

C6 | 24 | 6 | $\pi $ | |

C8 | 48 | 9 | 6 | $\pi $ |

3 | $\pi /2$ | |||

C10 | 120 | 15 | $\pi $ |

Divisions | Number of Vertices | Degree of Vertices | Number of Combinations | |
---|---|---|---|---|

Sn | $n+2$ | 2 | n | $\left(\begin{array}{c}2n\\ n\end{array}\right)$ |

n | 4 | $\left(\begin{array}{c}2n\\ 4\end{array}\right)$ | ||

C6 | 10 | 6 | 4 | 4356 |

4 | 6 | 48,400 | ||

C8 | 26 | 12 | 4 | 76,176 |

8 | 6 | 4,096,576 | ||

6 | 8 | 112,911,876 | ||

C10 | 62 | 30 | 4 | 3,132,900 |

20 | 6 | 1,171,008,400 | ||

12 | 10 | 29,828,113,326,144 |

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**MDPI and ACS Style**

Mannone, M.; Yoshino, T. Temari Balls, Spheres, SphereHarmonic: From Japanese Folkcraft to Music. *Algorithms* **2022**, *15*, 286.
https://doi.org/10.3390/a15080286

**AMA Style**

Mannone M, Yoshino T. Temari Balls, Spheres, SphereHarmonic: From Japanese Folkcraft to Music. *Algorithms*. 2022; 15(8):286.
https://doi.org/10.3390/a15080286

**Chicago/Turabian Style**

Mannone, Maria, and Takashi Yoshino. 2022. "Temari Balls, Spheres, SphereHarmonic: From Japanese Folkcraft to Music" *Algorithms* 15, no. 8: 286.
https://doi.org/10.3390/a15080286