https://rigidgeometricalgebra.org/wiki/index.php?title=Reverses&feed=atom&action=history
Reverses - Revision history
2024-03-29T14:26:05Z
Revision history for this page on the wiki
MediaWiki 1.40.0
https://rigidgeometricalgebra.org/wiki/index.php?title=Reverses&diff=286&oldid=prev
Eric Lengyel at 06:51, 1 February 2024
2024-02-01T06:51:08Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:51, 1 February 2024</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For any element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$, which we denote by $$\mathbf{\tilde <del style="font-weight: bold; text-decoration: none;">x</del>}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{<del style="font-weight: bold; text-decoration: none;">234</del>}$$ is $$\mathbf <del style="font-weight: bold; text-decoration: none;">e_4 </del>\wedge \mathbf <del style="font-weight: bold; text-decoration: none;">e_3 </del>\wedge \mathbf <del style="font-weight: bold; text-decoration: none;">e_2</del>$$, which we would write as $$-\mathbf e_{<del style="font-weight: bold; text-decoration: none;">234</del>}$$since <del style="font-weight: bold; text-decoration: none;">432 </del>is an odd permutation of <del style="font-weight: bold; text-decoration: none;">234</del>. In general, the reverse of an element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ is given by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For any element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$, which we denote by $$\mathbf{\tilde <ins style="font-weight: bold; text-decoration: none;">u</ins>}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{<ins style="font-weight: bold; text-decoration: none;">423</ins>}$$ is $$\mathbf <ins style="font-weight: bold; text-decoration: none;">e_3 </ins>\wedge \mathbf <ins style="font-weight: bold; text-decoration: none;">e_2 </ins>\wedge \mathbf <ins style="font-weight: bold; text-decoration: none;">e_4</ins>$$, which we would write as $$-\mathbf e_{<ins style="font-weight: bold; text-decoration: none;">423</ins>}$$since <ins style="font-weight: bold; text-decoration: none;">324 </ins>is an odd permutation of <ins style="font-weight: bold; text-decoration: none;">423</ins>. In general, the reverse of an element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ is given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\mathbf{\tilde <del style="font-weight: bold; text-decoration: none;">x</del>} = (-1)^{\operatorname{gr}(\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>)(\operatorname{gr}(\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>) - 1)/2}\,\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\mathbf{\tilde <ins style="font-weight: bold; text-decoration: none;">u</ins>} = (-1)^{\operatorname{gr}(\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>)(\operatorname{gr}(\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>) - 1)/2}\,\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Symmetrically, for any element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{<del style="font-weight: bold; text-decoration: none;">x</del>}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the [[antiwedge product]]). In general, the antireverse of an element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ is given by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Symmetrically, for any element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{<ins style="font-weight: bold; text-decoration: none;">u</ins>}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the [[antiwedge product]]). In general, the antireverse of an element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ is given by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{<del style="font-weight: bold; text-decoration: none;">x</del>}}} = (-1)^{\operatorname{ag}(\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>)(\operatorname{ag}(\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>) - 1)/2}\,\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ .</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{<ins style="font-weight: bold; text-decoration: none;">u</ins>}}} = (-1)^{\operatorname{ag}(\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>)(\operatorname{ag}(\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>) - 1)/2}\,\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ .</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The reverse and antireverse of any element $$\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>$$ are related by</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The reverse and antireverse of any element $$\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>$$ are related by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{<del style="font-weight: bold; text-decoration: none;">x</del>}}} = (-1)^{\operatorname{gr}(\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>)\operatorname{ag}(\mathbf <del style="font-weight: bold; text-decoration: none;">x</del>)}(-1)^{n(n-1)/2}\,\mathbf{\tilde <del style="font-weight: bold; text-decoration: none;">x</del>}$$ ,</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{<ins style="font-weight: bold; text-decoration: none;">u</ins>}}} = (-1)^{\operatorname{gr}(\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>)\operatorname{ag}(\mathbf <ins style="font-weight: bold; text-decoration: none;">u</ins>)}(-1)^{n(n-1)/2}\,\mathbf{\tilde <ins style="font-weight: bold; text-decoration: none;">u</ins>}$$ ,</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse</div></td></tr>
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Eric Lengyel
https://rigidgeometricalgebra.org/wiki/index.php?title=Reverses&diff=275&oldid=prev
Eric Lengyel at 08:01, 22 January 2024
2024-01-22T08:01:34Z
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<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:01, 22 January 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l21">Line 21:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following table lists the reverse and antireverse for all of the basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following table lists the reverse and antireverse for all of the basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Image:Reverses.svg|<del style="font-weight: bold; text-decoration: none;">820px</del>]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Image:Reverses.svg|<ins style="font-weight: bold; text-decoration: none;">720px</ins>]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See Also ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See Also ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Complements]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Complements]]</div></td></tr>
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Eric Lengyel
https://rigidgeometricalgebra.org/wiki/index.php?title=Reverses&diff=252&oldid=prev
Eric Lengyel at 00:49, 10 January 2024
2024-01-10T00:49:36Z
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<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:49, 10 January 2024</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The reverse and antireverse of any element $$\mathbf x$$ are related by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The reverse and antireverse of any element $$\mathbf x$$ are related by</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}\,\mathbf{\tilde x}$$ <del style="font-weight: bold; text-decoration: none;">.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)<ins style="font-weight: bold; text-decoration: none;">}(-1)^{n(n-1)/2</ins>}\,\mathbf{\tilde x}$$ <ins style="font-weight: bold; text-decoration: none;">,</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where $$n$$ is the number of dimensions in the algebra. </ins>To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,</div></td></tr>
</table>
Eric Lengyel
https://rigidgeometricalgebra.org/wiki/index.php?title=Reverses&diff=60&oldid=prev
Eric Lengyel: Created page with "''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations. For any element $$\mathbf x$$ that is the wedge product of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\math..."
2023-07-15T06:14:21Z
<p>Created page with "''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations. For any element $$\mathbf x$$ that is the <a href="/wiki/index.php?title=Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a> of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\math..."</p>
<p><b>New page</b></p><div>''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.<br />
<br />
For any element $$\mathbf x$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\mathbf e_{234}$$since 432 is an odd permutation of 234. In general, the reverse of an element $$\mathbf x$$ is given by<br />
<br />
:$$\mathbf{\tilde x} = (-1)^{\operatorname{gr}(\mathbf x)(\operatorname{gr}(\mathbf x) - 1)/2}\,\mathbf x$$ .<br />
<br />
Symmetrically, for any element $$\mathbf x$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf x$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the [[antiwedge product]]). In general, the antireverse of an element $$\mathbf x$$ is given by<br />
<br />
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{ag}(\mathbf x)(\operatorname{ag}(\mathbf x) - 1)/2}\,\mathbf x$$ .<br />
<br />
The reverse and antireverse of any element $$\mathbf x$$ are related by<br />
<br />
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}\,\mathbf{\tilde x}$$ .<br />
<br />
To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse<br />
<br />
:$$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,<br />
<br />
and similarly for the antireverse.<br />
<br />
The following table lists the reverse and antireverse for all of the basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.<br />
<br />
[[Image:Reverses.svg|820px]]<br />
<br />
== See Also ==<br />
<br />
* [[Complements]]</div>
Eric Lengyel