n | angle | angle | name |

3 | 2π / 3 | 120° | triangle |

4 | π / 2 | 90° | square |

5 | 2π / 5 | 72° | pentagon |

n | 2π / n | 360 / n | n-gon |

But suppose you want to position points around the surface of a sphere so that they're evenly spaced.

What is the angle between adjacent points? Is it possible to position the points so that the angle is the same between all adjacent points on the sphere?

Let's look at some examples. For n=4, you have a tetrahedron.

According to wikipedia, the angle between adjacent points is arccos(-1/3) or about 109.4712°.

For n=6, you have an octahedron. The angle between adjacent points is π / 2 or 90°.

To summarize and expand:

n | faces | tri. faces | edges | tri. edges | angle | angle | name |

4 | 4 | 4 | 6 | 6 | arccos(-1/3) | 109.4712° | tetrahedron |

6 | 8 | 8 | 12 | 12 | π / 2 | 90° | octahedron |

8 | 6 | 12 | 12 | 18 | multiple angles | cube | |

12 | 20 | 20 | 30 | 30 | 2 arctan(2/(1+√5)) | 63.435° | icosahedron |

n | ? | ? | ? | ? | ? | ? |

So what's the answer? I don't know, but one thing we have learned is that a cube's points are not all evenly spaced. The angles between them follow a regular pattern. There are twelve edges with an angle between the points of 2 arcsin(1/√3) ≈ 70.528°, and if you include all triangle edges, there are six edges with an angle between the points of 2 arcsin(√2/√3) ≈ 109.4712°.

Continue reading Part 2

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