Soddy'sHJexlet::usage="The Soddy hexlet will be built, from the inversion of an annular hexlet.​References: Wikipedia; https://www.geogrbra.org: Sexteto de Soddy, Ignacio Larrosa Cañestro; ResourceFunction: InversionCurve; Personal ";
Theoretical considerations for the annular hexlet :
θ=
π
n
;
Let ρ the radius of each of the n spheres be:
Sin
π
n
=
ρ
r+ρ
;
The value of R will be R = r + 2 ρ.
The centers of the n spheres will be on the circumference:
c={(r+ρ)Cos[k1]+x1,(r+ρ)Sin[k1]+y1,z1};
For the animation, we will consider r = 2 and we wish n = 6 spheres. Base spheres have center in {4, 6, 4}.
Development:
In[]:=
SolveSin
π
6
==
ρ
r+ρ
,ρ
{{ρr}};
In[]:=
​​Manipulate​​Showr=2;n=6;ρ=r;x1=4;y1=6;z1=4;​​c={(r+ρ)Cos[t1+2πk1]+x1,(r+ρ)Sin[t1+2πk1]+y1,z1};​​ParametricPlot3D[{2,3,4+r}+u{1,-2,0}+v{3,-5,0},{u,-50,50},{v,-50,50},PlotStyle->Directive[Blue,Specularity[White,40],Opacity[0.4]],MeshFalse],​​ParametricPlot3D[{2,3,4-r}+u{1,-2,0}+v{3,-5,0},{u,-50,50},{v,-50,50},PlotStyle->Directive[Yellow,Specularity[White,40],Opacity[0.4]],MeshFalse],​​Graphics3D[{Blue,Specularity[White,40],Opacity[0.8],Sphere[{x1,y1,z1},r]}],​​TableGraphics3DRGBColor
π
k1+1
,
π
k1+5
,0,
π
k1+3
,Specularity[White,40],Opacity[0.8],Sphere[c,ρ],t1,0,2π,
2π
n
,​​ParametricPlot3D[{2rCos[θ]+x1,2rSin[θ]+y1,z1+r},{θ,0,2π},PlotStyle{Black,Thick}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,PlotRange{{-1,15},{-5,10},{0,8}},ViewPoint{1.3,-2.4,0.5},Style["Annular hexlet:",Bold,Medium],Delimiter,{{k1,0,"Value (θ)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Annular hexlet:
Value (θ)
Inversion:
It will be done with respect to the pole {2, 2, 8} with an inversion radius k = 12.
In[]:=
​​Manipulate​​Showr=2;n=6;ρ=r;x1=4;y1=6;z1=4;k=12;​​ParametricPlot3D{2,2,8}+
2
k
2
(u+3v)
+
2
(1-2u-5v)
+
2
(-4+r)
{u+3v,1-2u-5v,-4+r},{u,-35,35},{v,-10,10},PlotStyle->Directive[Yellow,Specularity[White,40],Opacity[0.01]],MeshFalse,MaxRecursion->4,​​ParametricPlot3D{2,2,8}+
2
k
2
(u+3v)
+
2
(1-2u-5v)
+
2
(-4-r)
{u+3v,1-2u-5v,-4-r},{u,-35,35},{v,-50,50},PlotStyle->Directive[Blue,Specularity[White,40],Opacity[0.8]],MeshFalse,MaxRecursion->4,​​ParametricPlot3D{2,2,8}+
2
k
2
(rCos[u]Sin[v]+x1-2)
+
2
(rSin[u]Sin[v]+y1-2)
+
2
(rCos[v]+z1-8)
{rCos[u]Sin[v]+x1-2,rSin[u]Sin[v]+y1-2,rCos[v]+z1-8},{u,0,2π},{v,0,π},PlotStyleDirective[Blue,Specularity[White,40],Opacity[0.8]],MeshFalse,PerformanceGoal->"Quality",​​TableGraphics3DRGBColor
π
k1+1
,
π
k1+5
,0,
π
k1+3
,Specularity[White,40],Opacity[0.8],Sphere{2,2,8}+
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-2)
+
2
((r+ρ)Sin[t1+2πk1]+y1-2)
+
2
(z1-8)
-
2
ρ
{(r+ρ)Cos[t1+2πk1]+x1-2,(r+ρ)Sin[t1+2πk1]+y1-2,z1-8},Abs
2
k
2
((r+ρ)Cos[t1+2πk1]+x1-2)
+
2
((r+ρ)Sin[t1+2πk1]+y1-2)
+
2
(z1-8)
-
2
ρ
ρ,t1,0,2π,
2π
n
,ParametricPlot3D2+
144(2+4Cos[θ])
4+
2
(2+4Cos[θ])
+
2
(4+4Sin[θ])
,2+
144(4+4Sin[θ])
4+
2
(2+4Cos[θ])
+
2
(4+4Sin[θ])
,8-
288
4+
2
(2+4Cos[θ])
+
2
(4+4Sin[θ])
,{θ,0,2π},PlotStyleBlack/.Line[pts_,rest___]:>Tube[pts,0.2,rest],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,PlotRangeFull,ViewPoint{24,30,-25},PerformanceGoal->"Quality",Style["Soddy's Hexlet:",Bold,Medium],Delimiter,{{k1,0,"Value (θ)"},0,6,0.2},ControlPlacementTop
Out[]=
​
Soddy's Hexlet:
Value (θ)