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<title>GAP (HAPcryst) - Chapter 2: Bits and Pieces</title>
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<div class="chlinktop">

<div

<p id="mathjaxlink" class="pcenter"><a href="chap2.html">[MathJax off]</a></p>
<p><a id="X86FBE5B77C2F9442" name="X86FBE5B77C2F9442"></a></p>
<div class="ChapSects"><a href="chap2_mj.html#X86FBE5B77C2F9442">2 <span class="Heading">Bits and Pieces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8019925B8294F5B4">2.1 <span class="Heading">Matrices and Vectors</span></a>
</span>
<div ="ContSSBlock">
<span class"ontSS>
nbsp; chap2_mj.html#X7D58A1848182EC26">2.11-1SignRat/a>
<span class="ContSS"><divclassContSect>spanclass=tocline<pan="nocss">;<span =chap2_mjhtml#X8019925B8294F5B4.< =Heading/pan/ajava.lang.StringIndexOutOfBoundsException: Index 176 out of bounds for length 176
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7BB083A57C474F45">2.1-3 IsSquareMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X78C932A48515EF10">2.1-4 DimensionSquareMat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X86BD4FE4871379AD">2.2 <span class="Heading">Affine Matrices OnRight</span></a>
</span
<div class="ContSSBlock">
<span class="ContSS"><br /><spanspan="chap2_mj.#X78C932A48515EF10">1-4DimensionSquareMata<s>
< class</< class=nocss&;&;</spana ="chap2_mj.#X80DD4F2286D49F8D">2.2-2 BasisChangeAffineMatOnRight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X81F3C49580958FB6">2.2-3 TranslationOnRightFromVector</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X84A0B0637F269E37">2.3 <span class="Heading">Geometry</span></a>
java.lang.StringIndexOutOfBoundsException: Range [40, 41) out of bounds for length 7
<divc="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A94DAE679AD73E3">2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup<span="ContSS>
< class="
ocss&; /><a href="chap2_mj.htmlX81F3C49580958FB6">2.2-3 TranslationOnRightFromVector</a></span>
<class">< /><>2.3-2 <span class="Heading">Inequalities</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X80C365AA87BDDAFA">2.3-3 BisectorInequalityFromPointPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8790464D86D189F4">2.3-4 WhichSideOfHyperplane</a></span>
<spanclass""><br/>< class"nocss>; chap2_mj.#"23- RelativePositionPointAndPolygon
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7B14774981F80108">2.4 <span class="Heading">Space Groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7E2F20607F278B70">2.4-1 PointGroupRepresentatives</a></span>
</div></div>
</div>

<h3>2 <span class="Heading">Bits and Pieces</span></h3>

<p><span c="ContSS">br/>spanclassnocss>nbsp </span>a ="chap2_mjhtml#">2.3-4 </a>/>

<p><a id="X8019925B8294F5B4" name="X8019925B8294F5B4"></a></p>

<h4>2.1 <iv class"">< class"ocline>spanclassnocss"> </span><a href="chap2_mj.html#X7B14774981F80108">2.4 <span class="Heading">Space Groups</span></a>

<p><a id="X7D58A1848182EC26" namespan="ContSS>
"&;nbsp;/span>href=chap2_mjhtml#X7E2F20607F278B702-PointGroupRepresentatives<>>

<h5>2.1-1 SignRat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SignRat</code>( <var class="Arg">x</var> )</td><td class"tdright">(&bsp;method )</td></tr></table</div>
<p>Returns: sign of the

<p

<>2.1-2VectorModOneh5>

<div class=java.lang.StringIndexOutOfBoundsException: Range [11, 12) out of bounds for length 0
<pReturnsRational of the length enties <spanclassSimpleMath\[,))<span/p

<p>For a rational vector <var class="Arg">v</var>, this returns the vector with all entries taken "mod 1".</p>


<div class="example"><pre>
<< classArg> Standard classGAP>currently only <codeSignIntcode)/>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">SignRat(9/(-2));</span>
-1
<span 
 110, 1/,5/6 15]
</pre><="func"><table="func"="100%">trtdclass"< ="func">̶VectorModOne/(< class=Arg"v<>)/d><tdtdright&;&;<tdtr>d>

<p><a id="X7BB083A57C474F45" name="X7BB083A57C474F45"></a></p>

<h5>2.1-3 IsSquareMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSquareMat</code>( <var class="Arg">matrix</var> )</td><td class="tdright">( method )</td></tr>/table>/iv
pReturns code="keyw> >matrixvar> a matrixcode=""/>.<p

<p><a id="X78C932A48515EF10" name="X78C932A48515EF10"></a></p>

<h5>2.1-4 DimensionSquareMat</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DimensionSquareMat</code>( <var class="Arg">matrix</var> )[11,1,561/ ]
<>Returns:  oflines the <var="Arg">atrix</> if it issquare and codeclass="">failcodeotherwise</java.lang.StringIndexOutOfBoundsException: Index 136 out of bounds for length 136


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=[[1,2,3],[4,5,6],[9,6,12]];</span>
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 9, 6, 12 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSquareMat(m);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionSquareMat(m);</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">DimensionSquareMat([[1,2],[1,2,3]]);</span>
Error is square from
</pre></div>

< mappings <span class="SimpleMath">\(n\)</span> dimensional space are often written as a pair <span class="SimpleMath">\((A,v)\)</pan where<span class"impleMath">\A\/> is  mapping and < class"">\v)/spanis . strong="pkg"><> represents mappings < classSimpleMath>(n+1\<span> <span ="SimpleMath>(+) matrices spanclass="
\M)/> satisfy <span class="SimpleMath">\(M_{{n+1,n+1}}=1\)</span> and <span class="SimpleMath">\(M_{{i,n+1}}=0\)</span> for all <span class="SimpleMath">\(1\leq i \leq n\)</span>.</p>

<p>An

<p>aid"="X86BD4FE4871379ADa>>

<

<, Matrix not calledrelstylesheet="text="manual//java.lang.StringIndexOutOfBoundsException: Range [12, 13) out of bounds for length 12

<

< =" f" width0>>< class< class>87 </code( < classjava.lang.StringIndexOutOfBoundsException: Index 149 out of bounds for length 0
<p>eturns: the linear partof affinematrix<varclass="Arg">mat/java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 0

<p><a id="X80DD4F2286D49F8D" name="X80DD4F2286D49F8D"></a></p>

<h5>2.2-2 BasisChangeAffineMatOnRight</h5>

<div class="func"><table class="unc" width10%">tr>tdclassjava.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
<>:affine  same asvar==Arg>/>pjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80

<>basis < class</> of spanSimpleMath\<span  space  on mappingsthis  <var=""m</>ismatrix   span">\(+)times(>, method returns thejava.lang.StringIndexOutOfBoundsException: Index 323 out of bounds for length 176


<div class="example"><pre>
<span classpR: part the matrix classArgmat>.That, everything  for last and.</>
[ [ 0, 1] [1, 0]java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
<span="GAPpromptspan =
class="
"  chap2_mjhtml>.1-4 </a>/pan
[  12,, 0] [
<spanclass"gap&BasisChangeAffineMatOnRight,)<span
[ [2/, 0 ], [ ,/   [0,, 1 ] ]
</divclass="ContSSBlock">

<p><a id=

<h5 classGAPprompt>/> <span ="GAPinput"c=[,][,];<span

<div class class">ap;</panspan="GAPinput>:[120,][,230],[,,];<>
<p>: Affine</p>

<> a vector classArgclass"
  
2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup



< ="example>pre>
<span="GAPprompt">gap>/span < class"GAPinput>m=TranslationOnRightFromVector(,2,3]);<>
<span class="GAPprompt">gap></span> <ocss&bspnbsp</< hrefchap2_mjhtmlX83392C417B311B6B">2.3-java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 12
[ [ 1, 0  0, 0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  1,  0 ],
  [  1,  2,  3,  1 ] ]
<spanclass/divjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
[ 1,, 0  ],[01, 0] 0,0  ] ]
<span class="GAPprompt">gap></span> <span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</preh52.11ss"GAPprompt"gap</ <spanclass""m:TranslationOnRightFromVector);<>

<

<[[ ,Returns   rationalnumbervar classArg<var strong"">GAP> currently has class"SignInt

<p[  3   

<h5>2.3-1 GramianOfAverageScalarProductFromFiniteMatrixGroup     of same    <span",))/span>>

<div class java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>turnsSymmetricpositivematrixp>

<p>For a finite matrix group <varjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X866942167802E036" name="X866942167802E036"></a></p>

<h522span="HeadingInequalities

<Inequalities<p<a id=X7BB083A57C474F45name"">/>p>

<p><a id="X80C365AA87BDDAFA" name="X80C365AA87BDDAFA"></a></p>

<h5>2.3-3 BisectorInequalityFromPointPair</h5>

<div
<>Returns  of length ="code>()1/code>p>

<p>Calculates 

<pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<

<div class
<divclass=""><able=""="0>>tdleft class><> <class><vararclassArg>neq/ar)/td><td=tdright method;/>/>/table/>
<p>Returns-1 below) 0 (in) or 1 (above).</p>

<p>Letv classArg><var be a vectorof length class"dleft">codeclassfunc">̻ BisectorInequalityFromPointPair(v1,>2</var[,<varclass="Arg"gram>)<tdtd=tdright(nbspmethod;<td</>t/iv

<p>The version not the class">gap&; >gap></span> <span class="GAPinput">ineq:=BisectorInequalityFromPointPair([0,0],[1,0],[[5,4],[4,5]]);</span>
[5,-1,- ]
<p>An
-5
<span class
1
<span <
0
< class>a =" =X838946957FC75C17">/a/>

<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 62

< ="" name"><><1,-2,0]

<iv="func"< classfunc="100%>tr>
>code="func">#82; RelativePositionPointAndPolygoncode( < class=Arg</var> <var="Arg">oly</ar>)</d><td classtdright>  )</></tr</table/div>
<p>: one <code classcode>VERTEX FACET OUTSIDEspan=GAPprompt>gap;/pan < =GAPinput>neq[2,]}[1/,]<span

<p>Let <var class="Arg">poly</var> java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2

<p><[[0 1java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<h4>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7E2F20607F278B70" name="X7E2F20607F278B70"></a></p>

<52.- PointGroupRepresentatives</5>

<div java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width
<p>Returns

<p>Given an<code class="eyw>AffineCrystGroupOnLeftOrRightLet < java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0


<<><a id="X7B14774981F80108" name="X7B14774981F80108"></span="">gapt;</ <panclassGAPinput>splay)/an

<hr  ,    3,  1  ]
<p class="foot">generated by <a hrefspanclass="GAPprompt">gapgt</pan <span<h42. < class=HeadingSaceGroupsspan>/>
</body
<html

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