origami

Origami is an Opus Magnum mod based off of the alternating group A₄. It takes the 12 elements and adds four glyphs that act as functions that transmute pairs into elements, as described below. The Glyph of Composition and Augmented Composition are based off of the Cayley graph of the alternating group A₄ and the Glyphs of High and Low Retrieval are based off of various subgroups of A₄. The former two lose one (1) mass per operation while the latter two add one (1) mass per operation. The Glyph of Translation translates between the Senneumetal family and the orimetals. Finally, this mod adds an equivalent to Van Berlo's wheel and at the end of this file we prove an Ex Nihilo construction of the elements.

There is no Glyph of Sacrifice. There are only five glyphs.

Dependencies

• Quintessential, latest
• Brimstone, latest

Optional:

• Neuvolics, latest
• Sennmetals, latest

Atom List

We refer internally to the atoms as the following set of numbers:

{1, 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23}

Externally, they have the Following names and properties:

Number	Name	Abbr.	Verbal Abbrev.	Based On	Properties
1	Rei	Rei	Rei	零	Neutral-type atom
3	Nazum	Naz.	Naz	Ninazu	Strong, undeath-type healing
4	Reyja	Rea.	Rea	Freyja	Colourful, healing, used in cosmetics
7	Chronos	Chr.	Chron	Chronos	Time-like, artificial aging
8	Zaryan	Zar.	Zar	Zarya	Dull, mildly poisonous, dark
11	Iaso	Iaso	Ia	Iaso	Healing, gaseous, used in remedies
12	Tano	Tano	Tan/Tano	Tano	Weak, globular, long-term poison.
15	Maru	Maru	Mar/Maru	Maru	Aqueous, blue, strong solvent, slight poison.
16	Homonculum	Hom.	Hom/Homon	Homonculus	Sacrificial power and strength
19	Lono	Lono	Lon/Lono	Lono	Growth, fertility (agricultural)
20	Lua	Lua	Lua	Lua	Destruction, poison, fluid
23	Tao	Tao	Tao	Tao(ism)	Regulation

Some say that Homonculum generated by Sacrifice has different properties the more you sacrifice...

Function List

A(f,g): (f,g) ↦ k where k is according to the Cayley graph of the alternating group A₄

B(f, g; h): (f, g, h) ↦ A(f,A(g,h))

C(f, g, h): (f, g, h) ↦ k where f, g, h are in {1, 7, 16, 23} and k is the element {1, 7, 16, 23} \ {f, g, h}. Example: C(1, 16, 23) = 7.

D(f, g): (f, g) ↦ k where f, g are in one of {1, 3, 4}, {1. 8, 12}, {1, 11, 19}, {1, 15, 20} and k is the third element of whatever set f, g match into. Example: D(1, 3) = 4, D(11, 19) = 1.

Glyph list

Composition: takes f, g as inputs and outputs A(f, g).

Augmented Composition: takes f, g as inputs, h as a bowl, then outputs B(f, g; h).

High Retrieval takes f, g, h as bowls, outputs C(f, g, h).

Low Retrieval: takes f, g as bowls, outputs D(f, g).

Translation: translates via bowl using the following table as a bijective map:

Senneumetal	Orimetal
Zephiron	Rei
Mitrum	Nazum
Iridium	Reyja
Azulum	Maru
Taceum	Lua
Sordi	Zaryan
Eitros	Tano
Rofor	Lono
Serket	Iaso

Sacrifice: There is no glyph of Sacrifice. There are only five glyphs.

Sacrifice the souls of your loved ones for the guise of immortality. 666 souls or 6 mors required.

Flemming's Wheel

Flemming's wheel consists of the identity element (1), and elements 11 and 12 in the following arrangement:

  11 12 
1       1
  12 11

Ex Nihilo

The minimal generating set for A₄ has size two, and thus we only need two non-identity elements to generate all of A₄, and thus all elements in this mod. We derive the following sequence to get to all major elements:

D(1, 11) = 19 D(1, 12) = 8 A(8, 19) = 20 A(19, 8) = 3 A(19, 20) = 16 A(20, 19) = 7

We then derive the rest of the elements

Identity: A(7, 7) = 1

Subgroup of type Z₂ x Z₂: A(7, 16) = 23

Subgroups of type Z₃:

A(3, 3) = 4 A(19, 19) = 11 A(8, 8) = 12 A(20, 20) = 15

Reference Tables

Below are reference tables for the various glyphs whose behaviour may be hard to memorise.

Translated Cayley table with names of atoms instead of numbers

The rows are the left input (a) and the columns are the right input (b). This table computes A(a,b).

	Rei	Naz.	Rey.	Chr.	Zar.	Iaso	Tano	Maru	Hom.	Lono	Lua	Tao
Rei	Rei	Naz.	Rey.	Chr.	Zar.	Iaso	Tano	Maru	Hom.	Lono	Lua	Tao
Naz.	Naz.	Rey.	Rei	Iaso	Chr.	Zar.	Maru	Hom.	Tano	Tao	Lono	Lua
Rey.	Rey.	Rei	Naz.	Zar.	Iaso	Chr.	Hom.	Tano	Maru	Lua	Tao	Lono
Chr.	Chr.	Tano	Lono	Rei	Maru	Lua	Naz.	Zar.	Tao	Rey.	Iaso	Hom.
Zar.	Zar.	Hom.	Lua	Rey.	Tano	Tao	Rei	Iaso	Lono	Naz.	Chr.	Maru
Iaso	Iaso	Maru	Tao	Naz.	Hom.	Lono	Rey.	Chr.	Lua	Rei	Zar.	Tano
Tano	Tano	Lono	Chr.	Lua	Rei	Maru	Zar.	Tao	Naz.	Hom.	Rey.	Iaso
Maru	Maru	Tao	Iaso	Lono	Naz.	Hom.	Chr.	Lua	Rey.	Tano	Rei	Zar.
Hom.	Hom.	Lua	Zar.	Tao	Rey.	Tano	Iaso	Lono	Rei	Maru	Naz.	Chr.
Lono	Lono	Chr.	Tano	Maru	Lua	Rei	Tao	Naz.	Zar.	Iaso	Hom.	Rey.
Lua	Lua	Zar.	Hom.	Tano	Tao	Rey.	Lono	Rei	Iaso	Chr.	Maru	Naz.
Tao	Tao	Iaso	Maru	Hom.	Lono	Naz.	Lua	Rey.	Chr.	Zar.	Tano	Rei

Glyph of Translation Table

These are bijective (both ways) maps.

Senneumetal	Orimetal
Zephiron	Rei
Mitrum	Nazum
Iridium	Reyja
Azulum	Maru
Taceum	Lua
Sordi	Zaryan
Eitros	Tano
Rofor	Lono
Serket	Iaso