This means that magical archetypes are associated with integers and grouped accordingly. For example, 1,11, 21, and 31 would all be extensions of the Creation archetype. 6, 16, 26, and 36 would be grouped as extensions of the Illusion archetype. Since there is an infinite number of integers, there is an infinite number of possible instances.

What do you get when you add element #616 of the Creation archetype, that has a value of 6151, with element #201 of the Form archetype, that has a value of 2002? 6151+2002 is congruent to 8153mod10. 8153Mod10≡3, so from adding those two instances together, you get an instance of the Alteration archetype. I am getting the number from the sequence number for where that element shows up in the corresponding ordered list. In a list for the remainders of 1 for kmod10, the 616th element would have the value of 6151.

What is are the properties of Creation instance #2, that has the value of 11? You can write 11mod10 as the sum 3+3+3+2≡11mod10. So, Creation instance #2 can have four properties: Alteration(3); Alteration(3); Alteration(3); Form(2). You can also write 11mod10 as 1+2+3+5≡11mod10. Those four properties would be: Creation(1); Form(2); Alteration(3); Conjuration(5).

# Balancing Out Your Magical Classes with Additive Inverses

If you look at a lot of systems of magic, you’ll likely notice that each archetype has an idea of an opposite. This idea of opposites can be thought of as an idea of duality or polarity. That idea of opposites makes whatever magical system you create conceptually balanced. If it isn’t balanced, all sorts of counter-intuitive relationships will emerge. You want to make sure to include that balance in the design of your list. You derive an opposite from an additive inverse. For example, 1-1=0 and 2-2=0. If your magical system takes into consideration the opposite of every corresponding archetype, your system will be well balanced. What number you have to add to another number to get 0 is called your **additive inverse**. For example, the additive inverse of 1 is -1. This concept of duality and polarity will keep your magical system conceptually balanced. To figure out where your opposites would fall in the context of the magical ring, you take a mod of the additive inverse of a positive integer. For example, -1 is the additive inverse of 1, so you would perform the operation -1mod10≡9. This means that the reflection of 1mod10≡1 would be -1mod10≡9. The reflection of 2mod10≡2 would be -2mod10≡8. An antonym of Creation is Destruction and an antonym of Form is Entropy.

# Higher-Dimensional Projections of Magic

Below is an example of a higher-dimensional projection that you can create from the vectors of whatever ontology you make for magic. Keep in mind there is an infinite number of shapes that your magical ontology can take because there are an infinite number of integers and thus an infinite number of magical elements that can go into each archetype. This projection was created recursively. Each “higher” archetype, such as 10, contains within itself a lower archetype, such as 9. 10-1=9, so 9 is in 10. 9-1 is 8, so 8 is in 9. This means inside Magic you have Destruction. Inside Destruction, you have Entropy. The reflection of Destruction, 9, is Creation, 1, so if you draw a path from 9, Destruction, to every 1 archetype until you get back to 9, Destruction, you’ll draw the reflections of Creation and Destruction that are lower forms of the archetype Magic. The same could be said for 9-1. The reflection of Entropy, 8, is Form, 2, so if you draw a path from 8, Entropy, to every 2 archetypes until you get back to 8, Entropy, you’ll draw the reflections of Form and Entropy that are lower forms of the archetype Destruction. When you start somewhere and end at the place you started after some number of paths, you have a circuit.

## Magical Decagon Vectors

From | To | Magical Archetype | Vector Length |
---|---|---|---|

9 | 0 | Magic | 1 |

0 | 1 | Creation | 1 |

1 | 2 | Form | 1 |

2 | 3 | Alteration | 1 |

3 | 4 | Reality | 1 |

4 | 5 | Conjuration | 1 |

5 | 6 | Illusion | 1 |

6 | 7 | Normality | 1 |

7 | 8 | Entropy | 1 |

8 | 9 | Destruction | 1 |

8 | 0 | Magic | 2 |

0 | 2 | Form | 2 |

2 | 4 | Reality | 2 |

4 | 6 | Illusion | 2 |

6 | 8 | Entropy | 2 |

7 | 0 | Magic | 3 |

0 | 3 | Alteration | 3 |

3 | 6 | Illusion | 3 |

6 | 9 | Destruction | 3 |

9 | 2 | Form | 3 |

2 | 5 | Conjuration | 3 |

5 | 8 | Entropy | 3 |

8 | 1 | Creation | 3 |

1 | 4 | Reality | 3 |

4 | 7 | Normality | 3 |

6 | 0 | Magic | 4 |

0 | 4 | Reality | 4 |

4 | 8 | Entropy | 4 |

8 | 2 | Form | 4 |

2 | 6 | Illusion | 4 |

4 | 0 | Magic | 6 |

0 | 6 | Illusion | 6 |

6 | 2 | Form | 6 |

2 | 8 | Entropy | 6 |

8 | 4 | Reality | 6 |

3 | 0 | Magic | 7 |

0 | 7 | Normality | 7 |

7 | 4 | Reality | 7 |

4 | 1 | Creation | 7 |

1 | 8 | Entropy | 7 |

8 | 5 | Conjuration | 7 |

5 | 2 | Form | 7 |

2 | 9 | Destruction | 7 |

9 | 6 | Illusion | 7 |

6 | 3 | Alteration | 7 |

2 | 0 | Magic | 8 |

0 | 8 | Entropy | 8 |

8 | 6 | Illusion | 8 |

6 | 4 | Reality | 8 |

4 | 2 | Form | 8 |

1 | 0 | Magic | 9 |

0 | 9 | Destruction | 9 |

9 | 8 | Entropy | 9 |

8 | 7 | Normality | 9 |

7 | 6 | Illusion | 9 |

6 | 5 | Conjuration | 9 |

5 | 4 | Reality | 9 |

4 | 3 | Alteration | 9 |

3 | 2 | Form | 9 |

2 | 1 | Creation | 9 |

If you take the polar reflection of the above shape by rotating it 180 degrees, and superimpose that on the other non-rotated figure, you will get an even higher dimensional projection on the higher dimensional* Coxeter plane* called a *5-orthoplex*.** A 5-orthoplex is like a tesseract except it is 5-celled instead of 8-celled. If you were to project a lower-dimensional pentagram onto a higher plane, you would get this shape.** I left off the labels because the labels would lie perfectly on top of each other and thus would not be legible.