# Studies of Euler diagrams/transformations

dummy

These are pairs of functions in the same clan (NP equivalence class), so they can be expressed in terms of each other.
The clan numbers refer to the rational ordering. (Which will at some point be replaced by a better one.)

The transformation from one to the other is a signed permutation, which means that arguments are negated and permuted.
As a special case it can be just a set of negated places or just a permutation. (These cases are marked with N, P or NP respecitively.)

## clan 84: dakota and tinora   (NP)

This Euler diagram has no symmetry. Therefore the transformation of one into the other is unique.
The one from left to right is ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {Blue}\neg 2}&{\color {Red}~0}&{\color {ForestGreen}\neg 1}&{\color {Orange}~3}\end{pmatrix}}}$.   It means the following:

• The old red border becomes the new blue border, and the orientation changes. (The spikes pointed upward, now they point downward.)
• The old green border becomes the new red border. (The orientation stays the same.)
• The old blue border becomes the new green border, and the orientation changes. (The spikes pointed left, now they point right.)
• The yellow border remains unchanged.
 dakota tinora

The transformation from right to left is ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {ForestGreen}~1}&{\color {Blue}\neg 2}&{\color {Red}\neg 0}&{\color {Orange}~3}\end{pmatrix}}}$, the inverse of the one shown above.

## clan 109: tamino and niliko   (NP)

The diagrams in this EC are mirror symmetric. That means that there are two transformations between each pair of functions.
The one used here to get from left to right is ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {Blue}\neg 2}&{\color {Orange}~3}&{\color {ForestGreen}~1}&{\color {Red}~0}\end{pmatrix}}}$. The one from right to left is the inverse ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {Orange}~3}&{\color {Blue}~2}&{\color {Red}\neg 0}&{\color {ForestGreen}~1}\end{pmatrix}}}$.

 tamino niliko

## clan 203: dukeli and netuno   (NP)

Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for netuno.
The transformation from the chosen diagram of dukeli to the one of netuno above is ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {Red}~0}&{\color {Orange}~3}&{\color {Blue}\neg 2}&{\color {ForestGreen}\neg 1}\end{pmatrix}}}$. Its inverse is ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {Red}~0}&{\color {Orange}\neg 3}&{\color {Blue}\neg 2}&{\color {ForestGreen}~1}\end{pmatrix}}}$.

 dukeli netuno netuno

The transformation from the chosen diagram of dukeli to the one of netuno below is ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {ForestGreen}\neg 1}&{\color {Blue}\neg 2}&{\color {Orange}~3}&{\color {Red}~0}\end{pmatrix}}}$. Its inverse is ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}&{\color {Orange}~3}\\{\color {Orange}~3}&{\color {Red}\neg 0}&{\color {ForestGreen}\neg 1}&{\color {Blue}~2}\end{pmatrix}}}$.

This whole NP equivalence class with 192 functions (represented by 384 diagrams) can be found in dukeli NP.

## clan 349: nagini and medusa   (P)

These diagrams have the symmetry of a rectangle, which can be flipped in four ways.
This means that there are four transformations between each pair of functions in it.
The one used here to get from left to right is ${\displaystyle {\begin{pmatrix}{\color {Red}0}&{\color {ForestGreen}1}&{\color {Blue}2}&{\color {Orange}3}\\{\color {ForestGreen}1}&{\color {Blue}2}&{\color {Orange}3}&{\color {Red}0}\end{pmatrix}}}$. The one from right to left is the inverse ${\displaystyle {\begin{pmatrix}{\color {Red}0}&{\color {ForestGreen}1}&{\color {Blue}2}&{\color {Orange}3}\\{\color {Orange}3}&{\color {Red}0}&{\color {ForestGreen}1}&{\color {Blue}2}\end{pmatrix}}}$.

The arguments are only permuted, but not negated. So only the colors change, but not the direction of the spikes.

 nagini medusa

## clan 19: dobare and dobipi   (N)

Euler diagrams in this EC have 3-fold dihedral symmetry. (This is obfuscated by the conventional representation on the right.)
Thus there are six transformations from one function to another.
The simplest one between these two is to change the orientation of border A, i.e. ${\displaystyle {\begin{pmatrix}{\color {Red}~0}&{\color {ForestGreen}~1}&{\color {Blue}~2}\\{\color {Red}\neg 0}&{\color {ForestGreen}~1}&{\color {Blue}~2}\\\end{pmatrix}}}$.   (Compare potero and potula.)

 dobare dobipi