<p>The sum of vectors with the same color in the same square is allways the <a href="https://en.wikipedia.org/wiki/en:Null_vector" class="extiw" title="w:en:Null vector">null vector</a>. 
</p><p>The vectors of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e6377ca086aedf49b5e9f4a375b512a4235224" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.289ex; height:3.176ex;" alt="{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}"></span> are represented by subsets of {ABCD}.
</p><p><b>Example: The green fields in the undermost square (number 18):</b> <sup><small><a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Jupiter_square_magic_sums_vectorial.svg/2000px-Jupiter_square_magic_sums_vectorial.svg.png">(zoom in)</a></small></sup></p>
<ul><li>In terms of sets: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\Delta }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mtext> </mtext><mi mathvariant="normal">Δ<!-- Δ --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle ~\Delta }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aaf016162bbe8d46de76e0ae923034d2d386840" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.517ex; height:2.176ex;" alt="{\displaystyle ~\Delta }"></span> stands for the <a href="https://en.wikipedia.org/wiki/en:symmetric_difference" class="extiw" title="w:en:symmetric difference">symmetric difference</a>:</li></ul><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{~~B~~~~\}~\Delta ~\{AB~~~~\}~\Delta ~\{~~B~~D\}~\Delta ~\{AB~~D\}~~~=~~~\{~~~~~~~~\}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo fence="false" stretchy="false">{</mo><mtext> </mtext><mtext> </mtext><mi>B</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo fence="false" stretchy="false">}</mo><mtext> </mtext><mi mathvariant="normal">Δ<!-- Δ --></mi><mtext> </mtext><mo fence="false" stretchy="false">{</mo><mi>A</mi><mi>B</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo fence="false" stretchy="false">}</mo><mtext> </mtext><mi mathvariant="normal">Δ<!-- Δ --></mi><mtext> </mtext><mo fence="false" stretchy="false">{</mo><mtext> </mtext><mtext> </mtext><mi>B</mi><mtext> </mtext><mtext> </mtext><mi>D</mi><mo fence="false" stretchy="false">}</mo><mtext> </mtext><mi mathvariant="normal">Δ<!-- Δ --></mi><mtext> </mtext><mo fence="false" stretchy="false">{</mo><mi>A</mi><mi>B</mi><mtext> </mtext><mtext> </mtext><mi>D</mi><mo fence="false" stretchy="false">}</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo fence="false" stretchy="false">{</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo fence="false" stretchy="false">}</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \{~~B~~~~\}~\Delta ~\{AB~~~~\}~\Delta ~\{~~B~~D\}~\Delta ~\{AB~~D\}~~~=~~~\{~~~~~~~~\}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fce41665418034d717161281d5911bcb3bb2f782" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:55.825ex; height:2.843ex;" alt="{\displaystyle \{~~B~~~~\}~\Delta ~\{AB~~~~\}~\Delta ~\{~~B~~D\}~\Delta ~\{AB~~D\}~~~=~~~\{~~~~~~~~\}}"></span>
</p>
<ul><li>In terms of vectors: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ~\oplus }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mtext> </mtext><mo>⊕<!-- ⊕ --></mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle ~\oplus }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576a6f429c1e73eb91c04ca78768b94acc2b6dcc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:2.389ex; height:2.176ex;" alt="{\displaystyle ~\oplus }"></span> stands for the addition <a href="https://en.wikipedia.org/wiki/en:Modular_arithmetic" class="extiw" title="w:en:Modular arithmetic">modulo 2</a> (1⊕1=0):</li></ul><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}0\\1\\0\\1\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\1\\\end{pmatrix}}~~~=~~~{\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow><mo>⊕<!-- ⊕ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow><mo>⊕<!-- ⊕ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow><mo>⊕<!-- ⊕ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}0\\1\\0\\1\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\1\\\end{pmatrix}}~~~=~~~{\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e56c0e5210addfc4f4866fbd341eb4ed372a06e9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -5.671ex; width:45.007ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\0\\\end{pmatrix}}\oplus {\begin{pmatrix}0\\1\\0\\1\\\end{pmatrix}}\oplus {\begin{pmatrix}1\\1\\0\\1\\\end{pmatrix}}~~~=~~~{\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}}"></span>
</p><p><br><b>Related files:</b>
</p>
<ul><li>The colors and gray numbers are taken from <a href="//commons.wikimedia.org/wiki/File:Boolean_partition_matrices_colored.svg#File" class="mw-redirect" title="File:Boolean partition matrices colored.svg">File:Boolean partition matrices colored.svg</a>. There are 35 matrices colored only in red, green, yellow and blue.</li></ul><ul><li>The 4-ary Boolean functions marked with red squares in this graphic are also listed in the central matrix in <a href="//commons.wikimedia.org/wiki/File:Boolean_partition_matrices_top_lines_by_digit_sum.svg#File" class="mw-redirect" title="File:Boolean partition matrices top lines by digit sum.svg">File:Boolean partition matrices top lines by digit sum.svg</a>, where they are displayed by linear bit patterns, instead of squares.</li></ul><ul><li>If the content of this file remains strange to you, you may take a look at the colored squares in <a href="//commons.wikimedia.org/wiki/File:Jupiter_square_magic_sums_numerical.svg" title="File:Jupiter square magic sums numerical.svg">File:Jupiter square magic sums numerical.svg</a>, and convince yourself, that fields with the same color in the same square always sum up to 30. <sup><small><a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Jupiter_square_magic_sums_numerical.svg/2000px-Jupiter_square_magic_sums_numerical.svg.png">(zoom in ...magic sums numerical.svg)</a></small></sup> Here it works in the same way, but the elements are vectors, and fields with the same color in the same square always sum up to the null vector.</li></ul><ul><li>The vectorial magic squares can be found in <a href="//commons.wikimedia.org/wiki/Category:Jupiter_magic_squares_(vectorial)" title="Category:Jupiter magic squares (vectorial)">Category:Jupiter magic squares (vectorial)</a>.</li></ul><p><br><b>Related topics:</b><br></p>
<ul><li>In each of these 35 matrices are three nonzero numbers on red background. <sup><small><a class="external text" href="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Jupiter_square_magic_sums_vectorial.svg/2000px-Jupiter_square_magic_sums_vectorial.svg.png">(zoom in)</a></small></sup><br></li></ul><p>These triples are the blocks of a <a href="https://en.wikipedia.org/wiki/Steiner_system" class="extiw" title="w:Steiner system">Steiner triple system</a> of order 15.<br> 
They appear at the top of page 10 in<br><small>Mats Näslund     
Steiner Triple Systems and Perfect Codes     
Master of Science thesis, February 1993     
<a rel="nofollow" class="external autonumber" href="https://docs.google.com/viewer?a=v&amp;q=cache:D4s5ur89shwJ:citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.52.4560%26rep%3Drep1%26type%3Dpdf+%22steiner+triple+systems+and+perfect+codes%22&amp;hl=de&amp;gl=de&amp;pid=bl&amp;srcid=ADGEESh7RAhUcuhUobllbtCeHBldEZtktnSDENSqlPS3auluBhJoXbYJLmx2a76WjabypM3QL9BVkxtRP0AgSGTsSjyAO6oVD9HscYBX4J476Ao4tJb1vdjBLFqbIafs6cCj5myu8GiM&amp;sig=AHIEtbSBPN7TplwjkTeRdgGUe3Ae1_iDnA">[1]</a></small><br>
in this arrangement:
</p>
<pre>{1  2  3}  {1  4  5}  {1  6  7}  {1  8  9}  {1 10 11}  {1 12 13}  {1 14 15}
{2  4  6}  {2  5  7}  {2  8 10}  {2  9 11}  {2 12 14}  {2 13 15}  {3  4  7}
{3  5  6}  {3  8 11}  {3  9 10}  {3 12 15}  {3 13 14}  {4  8 12}  {4  9 13}
{4 10 14}  {4 11 15}  {5  8 13}  {5  9 12}  {5 10 15}  {5 11 14}  {6  8 14}
{6  9 15}  {6 10 12}  {6 11 13}  {7  8 15}  {7  9 14}  {7 10 13}  {7 11 12}
</pre>
<ul><li>These 35 matrices are related to the square model of <a href="https://en.wikipedia.org/wiki/Finite_geometry" class="extiw" title="w:Finite geometry">PG(3,2)</a>.</li></ul><p><a href="//commons.wikimedia.org/wiki/File:150614-PG-3-2-schoolgirls-arrangement.png" class="image"><img alt="150614-PG-3-2-schoolgirls-arrangement.png" src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/150614-PG-3-2-schoolgirls-arrangement.png/200px-150614-PG-3-2-schoolgirls-arrangement.png" decoding="async" width="200" height="264" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/150614-PG-3-2-schoolgirls-arrangement.png/300px-150614-PG-3-2-schoolgirls-arrangement.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/0/0a/150614-PG-3-2-schoolgirls-arrangement.png 2x" data-file-width="333" data-file-height="439"></a>
</p>
<ul><li>The red fields in these matrices denote the elements of the 35 <a href="https://en.wikipedia.org/wiki/Klein_four-group" class="extiw" title="w:Klein four-group">Klein four</a>-subgroups of the 16 element group Z<sub>2</sub><sup>4</sup>.</li></ul><table style="color:#000; background:#F8F9FA; border:1px solid #BAB;"><tbody><tr><td><a href="//commons.wikimedia.org/wiki/File:Inkscape_Logo.svg" class="image"><img alt="Inkscape Logo.svg" src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Inkscape_Logo.svg/20px-Inkscape_Logo.svg.png" decoding="async" width="20" height="20" srcset="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Inkscape_Logo.svg/30px-Inkscape_Logo.svg.png 1.5x, https://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Inkscape_Logo.svg/40px-Inkscape_Logo.svg.png 2x" data-file-width="128" data-file-height="128"></a>
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