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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