View source for Studies of Euler diagrams/transformations

{{EuDi}}
__NOTOC__
These are pairs of functions in the same clan ([[Equivalence classes of Boolean functions#NP|NP equivalence class]]), so they can be expressed in terms of each other.<br>
<span style="opacity: .5; font-size: 80%;">The clan numbers refer to the [[4-ary Boolean functions; clans in rational order|rational ordering]]. (Which will at some point be replaced by a better one.)</span>

The transformation from one to the other is a {{w|signed permutation}}, which means that arguments are negated and permuted.<br>
<small>It can be just a set of negated places or just a permutation. (These cases are marked with N, P or NP respecitively.)</small> 


=={{anchor|dakota_tinora}}<span style="opacity: .5;">clan 84:</span> {{boolfname|dakota}} and {{boolflink|tinora}} &nbsp; (NP)==

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This Euler diagram has no symmetry. Therefore the transformation of one into the other is unique.<br>
The one from left to right is
<math>
\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}
</math>. &nbsp; It means the following:
* The old red border becomes the new blue border, and the orientation changes. <small>(The spikes pointed upward, now they point downward.)</small> 
* The old green border becomes the new red border. <small>(The orientation stays the same.)</small>
* The old blue border becomes the new green border, and the orientation changes. <small>(The spikes pointed left, now they point right.)</small>
* The yellow border remains unchanged.

{| style="text-align: center;"
| [[File:EuDi; dakota.svg|400px|thumb|{{boolfname|dakota}}]]
|style="padding-left: 50px;"| [[File:EuDi; tinora.svg|400px|thumb|{{boolfname|tinora}}]]
|}

The transformation from right to left is
<math>
\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}
</math>, 
the inverse of the one shown above.
{{Studies of Euler diagrams/transformations/dakota tinora/dh}}
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==={{anchor|dagoro_darimi}}<span style="opacity: .5;">clan 157:</span> {{boolflink|dagoro}} and {{boolflink|darimi}} &nbsp; (NP)===

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{{boolfname|Dagoro}} is a gap variant of {{boolfname|tinora}} (shown above on the right).

These two functions do not have the same set of (relevant) arguments. But that is not a problem.<br>
One can see {{boolfname|dagoro}} as a 5-ary function with an irrelevant input ''E'', and {{boolfname|darimi}} has an irrelevant input ''A''.<br>

The transformation from left to right is
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} & {\color{Brown}~4} \\
{\color{Orange}\neg 3} & {\color{Blue}~2} & {\color{ForestGreen}~1} & {\color{Brown}~4} & {\color{Red}~0}
\end{pmatrix}
</math>.
It means the following:
* The old red border becomes the new yellow border, and the orientation changes. <small>(The spikes pointed left, now they point right.)</small>
* The old green border becomes the new blue border.
* The old blue border becomes the new green border.
* The old yellow border becomes the new brown border.
* The inexistent old brown border "becomes" the inexistent new red border. <small>(There could also be a negator in this place.)</small>

{| style="text-align: center;"
| [[File:EuDi; dagoro triangle.svg|401px|thumb|{{boolfname|dagoro}}]]
|style="padding-left: 50px;"| [[File:EuDi; darimi triangle.svg|401px|thumb|{{boolfname|darimi}}]]
|}

The transformation from right to left is
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} & {\color{Brown}~4} \\
{\color{Brown}~4} & {\color{Blue}~2} & {\color{ForestGreen}~1} & {\color{Red}\neg 0} & {\color{Orange}~3}
\end{pmatrix}
</math>, 
the inverse of the one shown above.
{{Template:Studies of Euler diagrams/transformations/dagoro darimi/dh}}
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=={{anchor|tamino_niliko}}<span style="opacity: .5;">clan 109:</span> {{boolfname|tamino}} and {{boolflink|niliko}} &nbsp; (NP)==

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The diagrams in this <abbr title="equivalence class">EC</abbr> are mirror symmetric.
That means that there are two transformations between each pair of functions.<br>
The one used here to get from left to right is
<math>
\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}
</math>.
The one from right to left is the inverse
<math>
\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}
</math>.

{| style="text-align: center;"
| [[File:EuDi; tamino.svg|400px|thumb|{{boolfname|tamino}}]]
|style="padding-left: 50px;"| [[File:EuDi; niliko (symmetric).svg|400px|thumb|{{boolfname|niliko}}]]
|}
{{Collapsible END}}


=={{anchor|dukeli_netuno}}<span style="opacity: .5;">clan 203:</span> {{boolflink|dukeli}} and {{boolfname|netuno}} &nbsp; (NP)==

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Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for {{boolfname|netuno}}.<br>
The transformation from the chosen diagram of {{boolfname|dukeli}} to the one of {{boolfname|netuno}} <u>above</u> is
<math>
\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}
</math>.
Its inverse is
<math>
\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}
</math>.

{| style="text-align: center;"
|rowspan="2"| [[File:EuDi; dukeli.svg|300px|thumb|{{boolfname|dukeli}}]]
|style="padding-left: 50px;"| [[File:EuDi; netuno mirrored.svg|300px|thumb|{{boolfname|netuno}}]]
|-
|style="padding-left: 50px;"| [[File:EuDi; netuno.svg|300px|thumb|{{boolfname|netuno}}]]
|}

The transformation from the chosen diagram of {{boolfname|dukeli}} to the one of {{boolfname|netuno}} <u>below</u> is
<math>
\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}
</math>.
Its inverse is
<math>
\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}
</math>.

This whole NP equivalence class with 192 functions (represented by 384 diagrams) can be found in [[Studies of Euler diagrams/dukeli NP|{{boolfname|dukeli}} NP]].
{{Studies of Euler diagrams/transformations/dukeli netuno/dh}}
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=={{anchor|nagini_medusa}}<span style="opacity: .5;">clan 349:</span> {{boolfname|nagini}} and {{boolflink|medusa}} &nbsp; (P)==

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These diagrams have the symmetry of a rectangle, which can be flipped in four ways.<br>
This means that there are four transformations between each pair of functions in it.<br>
The one used here to get from left to right is
<math>
\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}
</math>.
The one from right to left is the inverse
<math>
\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}
</math>.

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

{| style="text-align: center;"
| [[File:EuDi; nagini flat.svg|500px|thumb|{{boolfname|nagini}}]]
|style="padding-left: 50px;"| [[File:EuDi; medusa flat (symmetric).svg|500px|thumb|{{boolfname|medusa}}]]
|}
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=={{anchor|potero_makoto}}<span style="opacity: .5;">clan 15:</span> {{boolflink|potero}} and {{boolfname|makoto}} &nbsp; (NP)==

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Each function can be represented by two mirror symmetric Euler diagrams. Here both are shown for {{boolfname|makoto}}.<br>
The transformation between the chosen diagram of {{boolflink|potero}} and the one of {{boolfname|makoto}} above is
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\
{\color{ForestGreen}\neg 1} & {\color{Red}\neg 0} & {\color{Blue}~2}
\end{pmatrix}
</math>. <small>(It is self-inverse, so it works left to right, and back.)</small>

{| style="text-align: center;"
|rowspan="2"| [[File:EuDi; potero (shapes).svg|300px|thumb|{{boolfname|potero}}]]
|style="padding-left: 50px;"| [[File:EuDi; makoto (shapes).svg|300px|thumb|{{boolfname|makoto}}]]
|-
|style="padding-left: 50px;"| [[File:EuDi; makoto mirrored (shapes).svg|300px|thumb|{{boolfname|makoto}}]]
|}

The transformation from the chosen diagram of {{boolflink|potero}} to the one of {{boolfname|makoto}} below is
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\
{\color{Blue}\neg 2} & {\color{Red}~0} & {\color{ForestGreen}~1}
\end{pmatrix}
</math>.
Its inverse is
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\
{\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Red}\neg 0}
\end{pmatrix}
</math>.
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==={{anchor|potero_potula}} {{boolfname|potero}} and {{boolflink|potula}} &nbsp; (N)===

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These two diagrams differ only in the orientation of border ''A''.
The self-inverse transformation between them is
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\
{\color{Red}\neg 0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\
\end{pmatrix}
</math>.

{| style="text-align: center;"
| [[File:EuDi; potero (shapes).svg|300px|thumb|{{boolfname|potero}}]]
|style="padding-left: 50px;"| [[File:EuDi; potula (shapes).svg|300px|thumb|{{boolfname|potula}}]]
|}
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==={{anchor|potula_basori}} {{boolfname|potula}} and {{boolflink|basori}} &nbsp; (P)===

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The transformation between the two diagrams is
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} & {\color{Orange}~3} & {\color{Brown}~4} \\
{\color{Red}~0} & {\color{Orange}~3} & {\color{Brown}~4} & {\color{ForestGreen}~1} & {\color{Blue}~2}
\end{pmatrix}
</math>, which is self-inverse.<br>
<small>
From left to right the columns 3 and 4 are irrelevant, and from right to left the columns 1 and 2.<br>
The entries in these columns could be negated and permuted. Counting these leads to 8 possible transformations.<br>
(That is between the two diagrams. Between the two functions it would be 16.)
</small>

{| style="text-align: center;"
| [[File:EuDi; potula (shapes).svg|300px|thumb|{{boolfname|potula}}]]
|style="padding-left: 50px;"| [[File:EuDi; basori (shapes).svg|300px|thumb|{{boolfname|basori}}]]
|}
{{Studies of Euler diagrams/transformations/potula basori/dh}}
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=={{anchor|dobare_dobipi}}<span style="opacity: .5;">clan 19:</span> {{boolflink|dobare}} and {{boolfname|dobipi}} &nbsp; (N)==

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Euler diagrams in this <abbr title="equivalence class">EC</abbr> have 3-fold {{w|Dihedral group|dihedral symmetry}}.
<small>(This is obfuscated by the conventional representation on the right.)</small><br>
Thus there are six transformations from one function to another.<br>
The simplest one between these two is to change the orientation of border ''A'', i.e.
<math>
\begin{pmatrix}
{\color{Red}~0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\
{\color{Red}\neg 0} & {\color{ForestGreen}~1} & {\color{Blue}~2} \\
\end{pmatrix}
</math>.
&nbsp; <small>(Compare [[#potero_potula|''potero'' and ''potula'']].)</small>

{|
| [[File:EuDi; dobare.svg|thumb|x200px|{{boolfname|dobare}}]]
|style="padding-left: 50px;"| [[File:EuDi; dobipi.svg|thumb|x200px|{{boolfname|dobipi}}]]
|}
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=={{anchor|barita_filtrates}}{{boolflink|barita}} [[Studies of Euler diagrams/filtrates#barita|filtrates]] &nbsp; (P)==

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{|
| [[File:EuDi; barita filtrate sediri and miniri.svg|thumb|x250px|{{boolflink|miniri}} and {{boolflink|sediri}}]]
|style="padding-left: 50px;"| [[File:EuDi; barita filtrate sedofu and gepofu.svg|thumb|x250px|{{boolflink|gepofu}} and {{boolflink|sedofu}}]]
|style="padding-left: 50px;"| [[File:EuDi; barita filtrate seduki and putuki.svg|thumb|x250px|{{boolflink|putuki}} and {{boolflink|seduki}}]]
|}
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=={{anchor|bloatless_alternatives}}bloatless alternatives &nbsp; (N)==

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==={{boolflink|demole}}===

{|
| [[File:EuDi; demole bloatless alternative.svg|x140px]]
|style="padding-left: 50px;"|  [[File:EuDi; demole bloatless.svg|x230px]]
|}

==={{boolflink|futare}}===

{|
| [[File:EuDi; batch 5; 10 redrawn alternative.svg|x170px]]
|style="padding-left: 50px;"|  [[File:EuDi; batch 5; 10 redrawn.svg|x250px]]
|}

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