QuadratrixOfHipias::usage="The BMN quadratrix (see figure) is defined as the geometric locus of the points obtained by the intersection of the segments AE and FG. The circumference with radius a and the square is side a. The Quadratix of Hipias serves to solve the problem of the trisection of an angle and for the quadrature of the circle, two problems that together with the duplication of the cube, were the classic problems of the greeks.
​ References: Ayerbe Toledano, José María. La cuadratriz de Hipias y los problemas clásicos griegos. Epistemología e Historia de la Ciencia. Vol. 9, No. 1 (2025). Universidad de Sevilla; Personal.";
Development:
Be M = {x, y}:
AE
segment’s equation:
{aCos[θ],aSin[θ]};{0,0};​​y==Tan[θ]x;​​θ==ArcTan
y
x
;
θ is the angle DAE.
The following proportion between the angles and the sides met (because to the definition of the quadratrix):
DABangle
DAEangle
==
AB
AF
;
Cartesian equation of the quadratrix:
π
2
ArcTan
y
x

==
a
y
;​​πy==2aArcTan
y
x
;​​x=yCot
πy
2a
;
For the animation:
In[]:=
LimityCot
πy
2a
,y0
2a
π
;
Animation:
In[]:=
Manipulatea=4;A={0,0};D1={a,0};C1={a,a};B={0,a};N1=
2a
π
,0;F[y_]:={0,y};G[y_]:={a,y};​​ζ[θ_]:={aCos[θ],aSin[θ]};Q[y_]:=yCot
πy
2a
,y;​​Show​​ParametricPlotζ[θ],θ,0,
π
2
,PlotStyle->{Blue,Thick},​​ParametricPlot[Q[y],{y,4,a-t},PlotStyle->{Red,Thick}],​​GraphicsBlack,PointSize[0.01],​​Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],​​Point[D1],Text[Style["D",14,Bold],D1,{0,-1.2}],​​Point[C1],Text[Style["C",14,Bold],C1,{0,-1.2}],​​Point[B],Text[Style["B",14,Bold],B,{0,-1.2}],​​Point[N1],Text[Style["N",14,Bold],N1,{0,-1.2}],​​Point[Q[a-t]],Text[Style["M",14,Bold],Q[a-t],{0,-1.2}],​​Point[F[a-t]],Text[Style["F",14,Bold],F[a-t],{0,-1.2}],​​Point[G[a-t]],Text[Style["G",14,Bold],G[a-t],{0,-1.2}],​​Pointζ
π
8
(a-t),TextStyle["E",14,Bold],ζ
π
8
(a-t),{0,-1.2},​​{EdgeForm[Black],FaceForm[None],Rectangle[A,C1]},​​Thick,Line[{F[a-t],G[a-t]}],LineA,ζ
π
8
(a-t),AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-0.1,a},{-0.1,a}},ImageSize700,Style["Quadratix of Hipias",Bold,Large],{{t,0.00002,"Value (t)"},0,a,0.00001},ControlPlacementTop
Out[]=
​
Quadratix of Hipias
Value (t)