chap2.txt

  
  2 The IntPic package main function
  
  This  chapter consists of two sections, the first of which decribes the main
  function of the package. The second one can be thought just as an example to
  produce a table where the integers appear ordered in a non standard way.
  
  
  2.1 The main function
  
  The  function  IP_TikzArrayOfIntegers  (2.1-1)  is  the main function of the
  IntPic  package.  It  aims  to  produce  tikz  code for displaying arrays of
  integers.
  
  
  2.1-1 Tikz code for arrays of integers
  
  IP_TikzArrayOfIntegers( arg )  function
  
  The arguments (at most 3) are:
  
  1   (optional)
  
            a table of integers. In this case, the length of the rows is the
              maximum of the lengths of the sublists in the table, or
  
            a  list  of integers and, optionally, an integer which indicates
              the  length  of  the  rows;  when  the length of the rows is not
              indicated,  a  compromise  between  the  width and the height is
              tried.
  
  2   a  record  of  options.  One  of  the  fields  of  this  record, named
        highlights,   is  an  array  whose  entries  are  the  numbers  to  be
        highlighted:  one  color per sublist. See details and other options in
        Chapter 5.
  
  When  no  list  nor  table is present, the smallest range containing all the
  integers to be highlighted is taken.
  
    Example  
    gap> rg := [81..89];;
    gap> len := 10;;
    gap> arr := [Filtered(rg,IsPrime),Filtered(rg,u->(u mod 2)=0),
    >         Filtered(rg,u->(u mod 3)=0)];;
    gap> tkz := IP_TikzArrayOfIntegers(rg,len,rec(highlights:=arr));;
  
  
  The  aspect  of  the  string  tkz produced is not very appealing. We show it
  once,  by  asking  it  exlicitly  in  the  next example. In the forthcomming
  examples we keep using two semicolons to avoid showing this kind of strings.
  
    The tkz string  
    gap> tkz;
    "%tikz\n\\begin{tikzpicture}[every node/.style={draw,scale=1pt,\nminimum width\
    =20pt,inner sep=3pt,\nline width=1pt,draw=black}]\n\\matrix[row sep=2pt,column\
     sep=2pt]\n{\\node[fill=-red]{86};&\n\\node[fill=green]{87};&\n\\node[fill=-re\
    d]{88};&\n\\node[fill=red]{89};\\\\\n\\node[fill=green]{81};&\n\\node[fill=-re\
    d]{82};&\n\\node[fill=red]{83};&\n\\node[left color=-red,right color=green]{84\
    };&\n\\node[]{85};\\\\\n};\n\\end{tikzpicture}\n"
  
  
  This  string can be used at the users wish. In particular, it can be sent to
  the standard output using the command Print (Reference: Print).
  
    The tikz code  
    gap> Print(tkz);
    %tikz
    \begin{tikzpicture}[every node/.style={draw,scale=1pt,
    minimum width=20pt,inner sep=3pt,
    line width=1pt,draw=black}]
    \matrix[row sep=2pt,column sep=2pt]
    {\node[fill=-red]{86};&
    \node[fill=green]{87};&
    \node[fill=-red]{88};&
    \node[fill=red]{89};\\
    \node[fill=green]{81};&
    \node[fill=-red]{82};&
    \node[fill=red]{83};&
    \node[left color=-red,right color=green]{84};&
    \node[]{85};\\
    };
    \end{tikzpicture}
  
  
  It  can now be copied and pasted in a LaTeX document (having the appropriate
  packages in the preamble). See Chapter 4 for details and alternatives.
  
  The  next  function  uses  the  previous  one,  but is called with a simpler
  argument.  It  will  hopefully  be useful for simple drawings. The length of
  each  row  and  the  umber  of  columns  varies.  A compromise based on some
  experiments  has  been  established in order to obtain not too large nor too
  high images.
  
  
  2.1-2 Tikz code for arrays, in a simplified way
  
  IP_SimpleTikzArrayOfIntegers( arg )  function
  
  The  argument  is  either  a  list  of integers or a matrix of integers. The
  integers  involved  are  embeded  in  a  range  rg  of  minimum  length  and
  highlighted by using the list of default colors.
  
    Example  
    gap> d := DivisorsInt(30);
    [ 1, 2, 3, 5, 6, 10, 15, 30 ]
    gap> IP_SimpleTikzArrayOfIntegers(d);;
  
  
    Example  
    gap> d30 := DivisorsInt(30);
    [ 1, 2, 3, 5, 6, 10, 15, 30 ]
    gap> d40 := DivisorsInt(40);
    [ 1, 2, 4, 5, 8, 10, 20, 40 ]
    gap> tkz := IP_SimpleTikzArrayOfIntegers([d30,d40]);;
  
  
  
  2.2 Producing tables
  
  When  the  user  is interested in tables of a certain kind, it may be a good
  idea  to  write  some  code  to produce these tables. The following function
  (whose  code  is  part  of  the  file ip_tables.gi in the gap folder of this
  package) is convenient to deal with numerical semigroups with two generators
  and has been used to produce the images contained in [DFGSL14].
  
  2.2-1 IP_TableWithModularOrder
  
  IP_TableWithModularOrder( o, a, b, depth, height, rep, pos )  function
  
  The  arguments  rep  and  pos are booleans (true or false). When rep is true
  there  is some repetition: the last column is equal to the first, but pushed
  down  some  rows.  When  pos  is  true,  no  rows  below  0  are considered,
  (contradicting depth, if needed).
  
  The  first  five  arguments arguments o, a, b,depth and height are integers.
  What  they  represent  is  described in what follows. There is assigned some
  kind  of  a  referential  on the constructed table and the fist argument, o,
  stands for the origin. A table with b columns (b+1 columns when rep is true)
  is constructed as follows. The row containing the origin is
  
      o+ [0..b-1]*a, if rep is false, or
  
      o+ [0..b]*a, if rep is true
  
  The  remaining rows are obtained by adding b (the upper ones) or subtracting
  b (the others) to these rows.
  
  Note:  when  a < b are co-prime, this construction provides a representation
  of the integers as an array.
  
    Example  
    gap> a := 8;; b := 19;;  
    gap> ns := NumericalSemigroup(a,b);;
    gap> c := ConductorOfNumericalSemigroup(ns);;
    gap> origin := 2*c-1;
    251
    gap> ground := [origin..origin+b-1];;
    gap> 
    gap> height:=2;;
    gap> depth:=8;;
    gap>   xaxis := [origin];;
    gap>   for n in [1..b-1] do
    >     Add(xaxis, origin+n*a);
    >   od;
    gap>   yaxis := [];;
    gap>   for n in [-depth..height] do
    >     Add(yaxis, origin+n*b);
    >   od;
    gap> 
    gap> table := IP_TableWithModularOrder(origin,a,b,depth,height,false,false);;
    gap> arr := [xaxis,yaxis,ground];
    [ [ 251, 259, 267, 275, 283, 291, 299, 307, 315, 323, 331, 339, 347, 355, 
          363, 371, 379, 387, 395 ], 
      [ 99, 118, 137, 156, 175, 194, 213, 232, 251, 270, 289 ], [ 251 .. 269 ] ]
    gap> tkz:=IP_TikzArrayOfIntegers(table,rec(highlights:=arr));;
  
  
  The  next picture is obtained in the same way. The information that only the
  shape  has  interest  is  given by including the option shape_only:=" ". The
  variable tkz should be defined in a similar manner to the following one.
  
    Example  
    gap> tkz:=IP_TikzArrayOfIntegers(table,rec(highlights:=arr,shape_only:=" ",
    >              cell_width := "6",colsep:="1",rowsep:="1",inner_sep:="2",
    >              line_color:="black!20"));;
  
  
  Next, a minimum of changes, just to illustrate the effect of rep and pos.
  
    Example  
    gap> table := IP_TableWithModularOrder(origin,a,b,depth,50,true,true);;
    gap> tkz:=IP_TikzArrayOfIntegers(table,rec(highlights:=arr));;