# Vortex-based Mathematics

For those unfamiliar with vortex-based mathematics, read more about the Rodin Coil. In short, this is a coil winded in a specific method of repetition formed from a vortex planned onto a circle. This page however is about extending vortex-based math to higher dimensions, so it requires some foreknowledge, more info here.

I have discovered how to calculate higher dimensions of the standard base-9 coil. Every odd number squared by 2, can result in a vortex circle. These are unique in that they all have two circular webs. One web oscillates between *the odd number that is squared* numbers. The other passes the remaining numbers only once, with a finite amount of patterns where each pattern can be reversed in direction.

If we take Rodin's base-9 version, this would be 3^{2}. There are 3 numbers in the oscillation web: 3, 6 and 9. And the remaining numbers are only passed once within each pattern. In the case of base-9 this is: 1, 2, 4, 8, 7, 5 and the reversed direction version: 5, 7, 8, 4, 2, 1. There are thus only 2 patterns for base-9, more on this later. Interestingly the lowest possible base, 1^{2} = 1, results in two conflicted webs. Where both the oscillation and patterned web are using only 1 number and 1 pattern.

The first 10 odd-squared bases are: 1, 9, 25, 49, 81, 121, 169, 289, 361, 529. The following is a crude list of all possible patterns for the first 4 bases of odd-squared numbers:

base | result | rest |
---|---|---|

1 | 1 | 1 |

9 | 1, 2, 4, 8, 7, 5 | 9, 3, 6 |

9 | 5, 7, 8, 4, 2, 1 | 9, 3, 6 |

25 | 1, 2, 4, 8, 16, 7, 14, 3, 6, 12, 24, 23, 21, 17, 9, 18, 11, 22, 19, 13 | 25, 5, 10, 15, 20 |

25 | 1, 3, 9, 2, 6, 18, 4, 12, 11, 8, 24, 22, 16, 23, 19, 7, 21, 13, 14, 17 | 25, 5, 10, 15, 20 |

25 | 1, 8, 14, 12, 21, 18, 19, 2, 16, 3, 24, 17, 11, 13, 4, 7, 6, 23, 9, 22 | 25, 5, 10, 15, 20 |

25 | 1, 12, 19, 3, 11, 7, 9, 8, 21, 2, 24, 13, 6, 22, 14, 18, 16, 17, 4, 23 | 25, 5, 10, 15, 20 |

25 | 13, 19, 22, 11, 18, 9, 17, 21, 23, 24, 12, 6, 3, 14, 7, 16, 8, 4, 2, 1 | 25, 5, 10, 15, 20 |

25 | 17, 14, 13, 21, 7, 19, 23, 16, 22, 24, 8, 11, 12, 4, 18, 6, 2, 9, 3, 1 | 25, 5, 10, 15, 20 |

25 | 22, 9, 23, 6, 7, 4, 13, 11, 17, 24, 3, 16, 2, 19, 18, 21, 12, 14, 8, 1 | 25, 5, 10, 15, 20 |

25 | 23, 4, 17, 16, 18, 14, 22, 6, 13, 24, 2, 21, 8, 9, 7, 11, 3, 19, 12, 1 | 25, 5, 10, 15, 20 |

49 | 1, 3, 9, 27, 32, 47, 43, 31, 44, 34, 4, 12, 36, 10, 30, 41, 25, 26, 29, 38, 16, 48, 46, 40, 22, 17, 2, 6, 18, 5, 15, 45, 37, 13, 39, 19, 8, 24, 23, 20, 11, 33 | 49, 7, 14, 21, 28, 35, 42 |

49 | 1, 5, 25, 27, 37, 38, 43, 19, 46, 34, 23, 17, 36, 33, 18, 41, 9, 45, 29, 47, 39, 48, 44, 24, 22, 12, 11, 6, 30, 3, 15, 26, 32, 13, 16, 31, 8, 40, 4, 20, 2, 10 | 49, 7, 14, 21, 28, 35, 42 |

49 | 1, 12, 46, 13, 9, 10, 22, 19, 32, 41, 2, 24, 43, 26, 18, 20, 44, 38, 15, 33, 4, 48, 37, 3, 36, 40, 39, 27, 30, 17, 8, 47, 25, 6, 23, 31, 29, 5, 11, 34, 16, 45 | 49, 7, 14, 21, 28, 35, 42 |

49 | 1, 17, 44, 13, 25, 33, 22, 31, 37, 41, 11, 40, 43, 45, 30, 20, 46, 47, 15, 10, 23, 48, 32, 5, 36, 24, 16, 27, 18, 12, 8, 38, 9, 6, 4, 19, 29, 3, 2, 34, 39, 26 | 49, 7, 14, 21, 28, 35, 42 |

49 | 1, 24, 37, 6, 46, 26, 36, 31, 9, 20, 39, 5, 22, 38, 30, 34, 32, 33, 8, 45, 2, 48, 25, 12, 43, 3, 23, 13, 18, 40, 29, 10, 44, 27, 11, 19, 15, 17, 16, 41, 4, 47 | 49, 7, 14, 21, 28, 35, 42 |

49 | 1, 38, 23, 41, 39, 12, 15, 31, 2, 27, 46, 33, 29, 24, 30, 13, 4, 5, 43, 17, 9, 48, 11, 26, 8, 10, 37, 34, 18, 47, 22, 3, 16, 20, 25, 19, 36, 45, 44, 6, 32, 40 | 49, 7, 14, 21, 28, 35, 42 |

49 | 33, 11, 20, 23, 24, 8, 19, 39, 13, 37, 45, 15, 5, 18, 6, 2, 17, 22, 40, 46, 48, 16, 38, 29, 26, 25, 41, 30, 10, 36, 12, 4, 34, 44, 31, 43, 47, 32, 27, 9, 3, 1 | 49, 7, 14, 21, 28, 35, 42 |

49 | 10, 2, 20, 4, 40, 8, 31, 16, 13, 32, 26, 15, 3, 30, 6, 11, 12, 22, 24, 44, 48, 39, 47, 29, 45, 9, 41, 18, 33, 36, 17, 23, 34, 46, 19, 43, 38, 37, 27, 25, 5, 1 | 49, 7, 14, 21, 28, 35, 42 |

49 | 45, 16, 34, 11, 5, 29, 31, 23, 6, 25, 47, 8, 17, 30, 27, 39, 40, 36, 3, 37, 48, 4, 33, 15, 38, 44, 20, 18, 26, 43, 24, 2, 41, 32, 19, 22, 10, 9, 13, 46, 12, 1 | 49, 7, 14, 21, 28, 35, 42 |

49 | 26, 39, 34, 2, 3, 29, 19, 4, 6, 9, 38, 8, 12, 18, 27, 16, 24, 36, 5, 32, 48, 23, 10, 15, 47, 46, 20, 30, 45, 43, 40, 11, 41, 37, 31, 22, 33, 25, 13, 44, 17, 1 | 49, 7, 14, 21, 28, 35, 42 |

49 | 47, 4, 41, 16, 17, 15, 19, 11, 27, 44, 10, 29, 40, 18, 13, 23, 3, 43, 12, 25, 48, 2, 45, 8, 33, 32, 34, 30, 38, 22, 5, 39, 20, 9, 31, 36, 26, 46, 6, 37, 24, 1 | 49, 7, 14, 21, 28, 35, 42 |

49 | 40, 32, 6, 44, 45, 36, 19, 25, 20, 16, 3, 22, 47, 18, 34, 37, 10, 8, 26, 11, 48, 9, 17, 43, 5, 4, 13, 30, 24, 29, 33, 46, 27, 2, 31, 15, 12, 39, 41, 23, 38, 1 | 49, 7, 14, 21, 28, 35, 42 |

I've already calculated a larger set, which is too large to paste here. But what interested me most was the result of the amount of possible patterns for each consecutive base. Again for the first 54 odd-squared bases this would be: 1, 2, 8, 12, 18, 40, 48, 128, 108, 220, 200, 162, 336, 240, 432, 640, 504, 1012, 588, 1248, 1624, 960, 1320, 1680, 1728, 1872, 1458, 3280, 3520, 3072, 4000, 3264, 5512, 3888, 5376, 4840, 5000, 4536, 6240, 8704, 6072, 10656, 6000, 7488, 8748, 13612, 8112, 14448, 15664, 8640, 13680, 12288, 16464, 11880. I have yet to find a formula for this pattern, perhaps someone more wise then me can find it.

Here are some drawn webs where the first number is the base and the second the multiplier used to create the pattern:

Some random thoughts: It might be interesting to see what would happen if we mapped these patterns on spheres instead of circles. Especially for the higher bases. Could the pattern count represent the possible paths from a single point within its represented dimension. Where each path also has a mirrored return path. Base-9 in the 1st dimension, having 1 path from a single point. Base-25 in the 2nd dimension, having 4 paths, up/down and left/right. And Base-49 in 3D, having 6 paths, adding in/out. There might also be cryptographical applications considering its likeness to Elliptic curve cryptography.

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