Let X be a translation surface. Then a systole of X is a shortest, simple closed, not null-homotopic geodesic of the completion of X. We denote by sys(X) the length of the systole. The systolic ratio of X is the ratio SR(X) := sys(X)^2/ 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 γ of O and the combinatorial length of γ. If the optional parameter equilateral is true, then it is assumed that 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 SR(O) of O. If the optional parameter equilateral is true, then it is assumed that 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 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 from to degree to in the stratum stratum. If the optional parameter 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 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 d. If the optional parameter 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 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 origamis. If the optional parameter 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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