PappusChain::usage="In the Pappus chainwe have in mind the concept of infinity by observing the continuos increase in the number of circumferences (spheres)and the decrease in their size because they occupy a finite space.Reference: Wolfram MathWorld: Pappus chain; Personal"
Animation in 2D:
In[]:=
Manipulater=(7+k1)10;A=-
-1-2r-
2
r
4r
;B=-3-5r;C1=-
3(1+2r+
2
r
)
r
;D1=-
-9-14r-4
2
r
r
;​​Show​​ParametricPlot[{5Cos[θ]+4,5Sin[θ]+6},{θ,0,2π},PlotStyle->{DarkYellow,Thick}],ParametricPlot
7+k1
2
Cos[θ]+
5
2
+
7-(7-k1)
2
,
7+k1
2
Sin[θ]+6,{θ,0,2π},PlotStyle->{Blue,Thick},​​​​ParametricPlot
-4AD1+A
2
B
+
2
C1
4A
Cos[θ]-
B
2
,
-4AD1+A
2
B
+
2
C1
4
2
A
Sin[θ]-
C1
2A
,{θ,0,2π},PlotStyle->{DarkYellow,Thick},TableGraphicsBlack,PointSize[0.005],Point
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,
10nr(1-r)
2
n
2
(1-r)
+r
+6,{n,0,20+50k1,1},​​TableGraphicsBlack,PointSize[0.005],Point
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,-
10nr(1-r)
2
n
2
(1-r)
+r
+6,{n,0,20+50k1,1},​​TableGraphicsPurple,Thick,Circle
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,
10nr(1-r)
2
n
2
(1-r)
+r
+6,
10(1-r)r
2(
2
n
2
(1-r)
+r)
,{n,0,20+50k1,1},TableGraphicsPurple,Thick,Circle
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,-
10nr(1-r)
2
n
2
(1-r)
+r
+6,
10(1-r)r
2(
2
n
2
(1-r)
+r)
,{n,0,20+50k1,1},AxesLabel->(Style[#,15,Blue]&/@{"X","Y"}),AxesOrigin{0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,PlotRange{{-1,10},{1,12}},Style["Pappus chain in 2D:",Bold,Medium],Delimiter,{{k1,0,"Value (k1)"},0,3,0.1},ControlPlacementTop
Out[]=
​
Pappus chain in 2D:
Value (k1)
Animation in 3D:
In[]:=
Manipulater=(7+k1)10;A=-
-1-2r-
2
r
4r
;B=-3-5r;C1=-
3(1+2r+
2
r
)
r
;D1=-
-9-14r-4
2
r
r
;​​Show​​ParametricPlot3D[{5Cos[u]Sin[v]+4,5Sin[u]Sin[v]+6,5Cos[v]+4},{u,0,2π},{v,0,π},PlotStyleDirective[Yellow,Specularity[White,40],Opacity[0.2]],Mesh->None,PerformanceGoal->"Quality",PlotPoints->60],ParametricPlot3D
7+k1
2
Cos[u]Sin[v]+
5
2
+
7-(7-k1)
2
,
7+k1
2
Sin[u]Sin[v]+6,
7+k1
2
Cos[v]+4,{u,0,2π},{v,0,π},PlotStyleDirective[Blue,Specularity[White,40],Opacity[0.8]],Mesh->None,PerformanceGoal->"Quality",PlotPoints->60,​​ParametricPlot3D
-4AD1+A
2
B
+
2
C1
4A
Cos[θ]-
B
2
,
-4AD1+A
2
B
+
2
C1
4
2
A
Sin[θ]-
C1
2A
,4,{θ,0,2π},PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.03,rest],​​TableGraphics3DBlack,Ball
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,
10nr(1-r)
2
n
2
(1-r)
+r
+6,4,0.07,{n,0,20+50k1,1},​​TableGraphics3DBlack,Ball
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,-
10nr(1-r)
2
n
2
(1-r)
+r
+6,4,0.07,{n,0,20+50k1,1},​​TableGraphics3DPurple,Specularity[White,40],Opacity[0.5],Sphere
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,
10nr(1-r)
2
n
2
(1-r)
+r
+6,4,
10(1-r)r
2(
2
n
2
(1-r)
+r)
,{n,0,20+50k1,1},TableGraphics3DPurple,Specularity[White,40],Opacity[0.5],Sphere
10r(1+r)
2(
2
n
2
(1-r)
+r)
-1,-
10nr(1-r)
2
n
2
(1-r)
+r
+6,4,
10(1-r)r
2(
2
n
2
(1-r)
+r)
,{n,0,20+50k1,1},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize900,PlotRange{{-2,9},{1,12},{-1,10}},ViewPoint{12,-2.4,10.},Style["Pappus chain in 3D:",Bold,Medium],Delimiter,{{k1,0,"Value (k1)"},0,3,0.1},ControlPlacementTop
Out[]=
​
Pappus chain in 3D:
Value (k1)
View of the all angles (below):