Let \(X\) be a translation surface. Then a systole of \(X\) is a shortest, simple closed, not null-homotopic geodesic of \(\overline{X}\). We denote by \(\mathrm{sys}(X)\) the length of the systole. The systolic ratio of \(X\) is the ratio \(\mathrm{SR}(X) := \mathrm{sys}(X)^2/\mathrm{area}(X)\).
‣ SystoleLength( O[, equilateral] ) | ( function ) |
Returns: a record containing the length of the systole and its combinatorial length
Computes the length of a systole \(\gamma\) of \(\texttt{O}\) and the combinatorial length of \(\gamma\). If the optional parameter \(\texttt{equilateral}\) is true, then it is assumed that \(\texttt{O}\) consists of equilateral triangles.
gap> O := Origami((1,8,4,6,2,3,5), (2,5,7,6,8), 8); Origami((1,8,4,6,2,3,5), (2,5,7,6,8), 8) gap> SystoleLength(O); rec( combinatorial_length := 1, systole := 1. )
‣ SystolicRatio( O[, equilateral] ) | ( function ) |
Returns: a record containing the systolic ratio and the combinatorial length of the systole used in the computation
Computes the systolic ratio \(\mathrm{SR}(\texttt{O})\) of \(\texttt{O}\). If the optional parameter \(\texttt{equilateral}\) is true, then it is assumed that \(\texttt{O}\) consists of equilateral triangles.
gap> O := Origami((1,8,4,6,2,3,5), (2,5,7,6,8), 8); Origami((1,8,4,6,2,3,5), (2,5,7,6,8), 8) gap> SystolicRatio(O); rec( combinatorial_length := 1, systolic_ratio := 0.125 )
‣ MaximalSystolicRatioInStratum( from, to, stratum[, equilateral] ) | ( function ) |
Returns: a record containing the maximal systolic ratio in \(\texttt{stratum}\), an origami representing the maximum and a bool value indicating if a combinatorial length of three occured during the computation
Computes the maximal systolic ratio of all origamis from degree \(\texttt{from}\) to degree \(\texttt{to}\) in the stratum \(\texttt{stratum}\). If the optional parameter \(\texttt{equilateral}\) is true, then it is assumed that all origamis consist of equilateral triangles.
gap> MaximalSystolicRatioInStratum(4,7,[1,1]); rec( origami := Origami((1,2)(3,4), (1,2,3,4), 4), systolic_ratio := 0.5, three_occured := false )
‣ MaximalSystolicRatioByDegree( d[, equilateral] ) | ( function ) |
Returns: a record containing the maximal systolic ratio of all origamis with degree \(\texttt{d}\), an origami representing the maximum and a bool value indicating if a combinatorial length of three occured during the computation
Computes the maximal systolic ratio of all origamis with degree \(\texttt{d}\). If the optional parameter \(\texttt{equilateral}\) is true, then it is assumed that all origamis consist of equilateral triangles.
gap> MaximalSystolicRatioByDegree(6); rec( origami := Origami((1,2)(3,4)(5,6), (1,2,3,4,5,6), 6), systolic_ratio := 0.333333, three_occured := false )
‣ MaximalSystolicRatioOfList( origamis[, equilateral] ) | ( function ) |
Returns: a record containing the maximal systolic ratio of all origamis in the list \(\texttt{origamis}\), an origami representing the maximum and a bool value indicating if a combinatorial length of three occured during the computation
Computes the maximal systolic ratio of all origamis in the list \(\texttt{origamis}\). If the optional parameter \(\texttt{equilateral}\) is true, then it is assumed that all origamis consist of equilateral triangles.
gap> o1 := Origami((1,3,5,2), (2,4,3,5), 5); Origami((1,3,5,2), (2,4,3,5), 5) gap> o2 := Origami((1,4,3,2,6), (1,4,5)(2,6), 6); Origami((1,4,3,2,6), (1,4,5)(2,6), 6) gap> origamis := [o1, o2]; [ Origami((1,3,5,2), (2,4,3,5), 5), Origami((1,4,3,2,6), (1,4,5)(2,6), 6) ] gap> MaximalSystolicRatioOfList(origamis); rec( origami := Origami((1,3,5,2), (2,4,3,5), 5), systolic_ratio := 0.2, three_occured := false )
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