Cyclic Dreams

Small little piece: "Modular Dreams"

The yellow symbols represent Toki Pona representations of Numerals. From smallest to largest (the symbol's description in parentheses):
- ala (X) = 0
- wan (1) = 1
- tu (2 vertical lines) = 2
- luka (simplified hand) = 5, as in 5 fingers on a hand
- mute (3 vertical lines) = 20, but usually represents numbers past 2, or just many of something
- ale (lemniscate or infinity) = 100, yet usually represents infinite values or refers to all of something

The Zs aren't just Zs, but rather symbols of Cyclic Groups (denoted ℤₙ, where n is a positive integer). In short, modular arithmetic. Instead of going off to infinity, you are bound to a subset of integers between 0 and some positive number, n. n, or any multiple of n, would basically be a 0 in this theoretical group.

In example, ℤ₆ only allows 0, 1, 2, 3, 4, and 5. Using addition, the following becomes apparent:
- 0 is a neutral element, meaning when applied to any other element, it does nothing. 1 + 0 = 1.
- 1 is a normal element, but has an inverse of 5, since 1 + 5 = 6, which cancels out to 0. This inverse duality also happens for 2 and 4, which also add to 6.
- 3 however is its own inverse, as 3 + 3 = 6, which once again cancels out to 0.
And this is what makes a group a group. A group is essentially a neutral element, and a bunch of elements that all have an inverse that could either be itself or another element.

Sweet dreams, fellow learners.

AwkwardHumanand14 othersliked this post