LoginSignup
1
1

More than 3 years have passed since last update.

@ochimusha01(yasunori matuki)

【トーラス構造と古典数学】「単位円筒」から「トーラス構造」へ

Last updated at Posted at 2020-03-07

今回の投稿の出発点はこれ。
【初心者向け】物理学における「単位円筒(Unit Cylinder)」の概念について。
20191018074109.gif
XY軸（円弧）
20191018074133.png
ZX軸（Cos波）
20191101090256.png
ZY軸（Sin波）
20191101090408.png
【初心者向け】「単位円筒」から「単位球面」へ
20191101180804.gif
XY面にZ座標に応じた「曲率」を付与。
20190915101244.gif
実は単位円筒(Unit Cylinder)」から拡張可能な立体構造は単位球面(Unit Sphere)だけではありません。

「トーラス構造」の登場

①まずは平面トーラス（折り返し円柱）を準備する。
トーラス(単数系torus, 複数形tori)-Wikipedia
20190915192200.gif

libraly(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real,Imaginal,Rtime,type="l",xlim=c(-1.5,1.5),ylim=c(-1.5,1.5),zlim=c(-1.5,1.5))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test01")

②平面トーラス（折り返し円柱）の表裏両面に、それぞれ真逆の曲率を付与する。
20190915193504.gif
20190915193539.png

libraly(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
r0<-f0(radians)
f1<-function(x){sqrt(1-x^2)}
c0<-seq(-1,1,length=1500)
c1<-c(f1(c0)/2,-1*f1(c0)/2)
#c1<-c(f1(c0),seq(0,0,length=1500))/2)
r1<-r0*(1+c1)
Real<-Re(r1)
Imaginal<-Im(r1)
Rtime<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real,Imaginal,Rtime,type="l",xlim=c(-1.5,1.5),ylim=c(-1.5,1.5),zlim=c(-1.5,1.5))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test02")

例えば時間軸に沿って描画がなされたと想定すると、こう見えたりもします。盃？
20191020093523.gif

libraly(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
r0<-f0(radians)
f1<-function(x){sqrt(1-x^2)}
c0<-seq(-1,1,length=1500)
c1<-c(f1(c0)/2,-1*f1(c0)/2)
#c1<-c(f1(c0),seq(0,0,length=1500))/2)
r1<-r0*(1+c1)
Real<-Re(r1)
Imaginal<-Im(r1)
Rtime<-c(seq(2,1,length=1500),seq(1,0,length=1500))
plot3d(Real,Imaginal,Rtime,type="l",xlim=c(-1.5,1.5),ylim=c(-1.5,1.5),zlim=c(0,2))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test02")

ちなみに大半径(Major Radius)1,小半径(Minor Radius)1の場合。
image.gif
image.png

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}
r0<-f0(radians)
c0<-seq(-1,1,length=1500)
#c1<-c(f1(c0)/2,-1*f1(c0)/2)
c1<-c(f1(c0),-1*f1(c0))
r1<-r0*(1+c1)
Real<-Re(r1)
Imaginal<-Im(r1)
Rtime<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real,Imaginal,Rtime,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),xlab="",ylab="",zlab="")

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/unit07")

大円と小円を追加する。
image.gif
スクリーンショット 2021-02-20 21.57.22.png

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}
r0<-f0(radians)
c0<-seq(-1,1,length=1500)
#c1<-c(f1(c0)/2,-1*f1(c0)/2)
c1<-c(f1(c0),-1*f1(c0))
r1<-r0*(1+c1)
Real<-Re(r1)
Imaginal<-Im(r1)
Rtime<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real,Imaginal,Rtime,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),xlab="",ylab="",zlab="")

#大円描写
Real01<-Re(f0(radians))
Imaginal01<-Im(f0(radians))
Rtime01<-seq(1/2,1/2,length=3000)
points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))

#小円描写
Real01<-Re(f0(radians))+1
Imaginal01<-Im(f0(radians))/2+1/2
Rtime01<-seq(0,0,length=3000)
points3d(Rtime01,Real01,Imaginal01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/unit16")

座標系(Coordinate System)としては媒介変数t,p(0≦t≦2π,0≦p≦2π)を用いて以下の様に表します。
image.png

  • x=大半径R×cos(t)+小半径r×cos(p)×cos(t)
  • y=大半径R×cos(t)+小半径r×cos(p)×sin(t)
  • z=小半径r×sin(p)

ここで重要なのが以下。

  • 大半径R=0,小半径R=1の時、単位球面(Unit Sphere)を二重に描く。従って経緯度法(経度-180度~180度に対して緯度-90度~90度)や3次元極座標系(水平角φ=-π~πに対して垂直角0~π)は適用範囲を半分に減らしてなお単位球面(Unit Sphere)上の全座標を示せる訳である。
  • 逆に大半径R=1,小半径R=0の時、Z=0となってただの単位円(Unit Circle)になる。

そしてこの変遷はオイラーの公式（Eulerian Formula）Cos(θ)+Sin(θ)iの一般形Cos(θ)+Cos(θ-π/NoS)i(NoS=Number of Sides)上における二辺形(Bilateral,NoS=2)すなわち「1辺がπの円弧2分割図形/1辺が2の往復線分」から円そのもの(NoS=Inf(inity))への変遷に合致するのです。
【Rで球面幾何学】二辺形と一辺形？ - Qiita
image.gif

  • この概念上における一辺形(One Side, NoS=1)は「1辺が2πの円弧=円そのもの」であり、ここにある種の循環性を見てとる事が出来そうである。
    一辺形(One Side)Cos(θ)-Cos(θ)i
    image.png
    二辺形(Bilateral)Cos(θ)+Sin(θ)i
    image.png
    円そのもの(Circle Itself)Cos(θ)+Cos(θ)i
    image.png

これは偶奇性(Evenness)とも連動してくる概念です。
【初心者向け】偶奇性概念と共役関係概念の連続性について。
image.gif
半径(Radius)1の二重球面たる単位球面(Unit Sphere)から出発します。偶奇性(Eveness)という時、重要なのはそこに現れてる数が偶数/奇数の関係にある事そのものより「ピッチが半周期ズレている」事なんですね。例えば同心円集合(Concentric Circle Set)半径R={0,1,2,3,4,…,Inf(inity)}から出発すると、これを偶数層(Even Layer)として奇数層(Odd Layer)が半径R={0.5,1.5,2.5,3.5,4.5,…,Inf(inity)}という形で現れるのです。
偶数層(Even Layer)0.0-1.0
image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#外球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime02<-seq(0,2,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real02<-Real*c01
Imaginal02<-Imaginal*c01
plot3d(Real02*2,Imaginal02*2,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),xlab="",ylab="",zlab="",col=c(200,200,200))

#内部球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(1/2,3/2,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01
points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test39")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test16")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01,col=rgb(0,0,1))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test19")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01,col=rgb(0,0,1))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test24")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01-1,col=rgb(0,0,1))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test27")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01-1,col=rgb(0,0,1))
#Rtime01段
points3d(Real01+2,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01-2,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01-2,Rtime01,col=rgb(0,0,1))
#Rtime01+1段
points3d(Real01+2,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01+2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01-2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01+2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01-2,Rtime01+1,col=rgb(0,0,1))
#Rtime01-1段
points3d(Real01+2,Imaginal01,Rtime01-1,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01-1,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01+2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01-2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01+2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01-2,Rtime01-1,col=rgb(0,0,1))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test71")

奇数層(Odd Layer)0.5
image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#外球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime02<-seq(0,2,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real02<-Real*c01
Imaginal02<-Imaginal*c01
plot3d(Real02*2,Imaginal02*2,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),xlab="",ylab="",zlab="",col=c(200,200,200))

#内部球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(1/4,7/4,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01
points3d(Real01*1.5,Imaginal01*1.5,Rtime01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test41")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写

r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test15")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写

r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01,Rtime01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test20")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写

r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01+1,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01-1,Rtime01,col=rgb(1,0,0))
movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test23")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01+1,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01-1,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01,Imaginal01,Rtime01-1/2,col=rgb(1,0,0))
movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test28")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01+1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test42")

偶数層(Even Layer)0.0-2.0+奇数層(Odd Layer)0.5
image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写

r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test17")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写

r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01,Rtime01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test21")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写

r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01+1,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01-1,Rtime01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test22")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01+1,Rtime01,col=rgb(0,0,1))
points3d(Real01-1,Imaginal01+1,Rtime01,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01-1,Rtime01,col=rgb(0,0,1))
points3d(Real01-1,Imaginal01-1,Rtime01,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01+1,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01-1,Rtime01,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test74")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01-1,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01+1,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01-1,Rtime01,col=rgb(1,0,0))
points3d(Real01,Imaginal01,Rtime01+1,col=rgb(1,0,0))
points3d(Real01,Imaginal01,Rtime01-1,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test30")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test61")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01-1,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test32")

image.gif

library(rgl)
radians<-seq(0,60*pi,length=3000)
f0<-function(x){exp(x*(0+1i))}
f1<-function(x){sqrt(1-x^2)}

#トーラス描写
r1<-f0(radians)
c0B<-seq(-1,1,length=1500)
c02<-c(f1(c0B),-1*f1(c0B))
r2<-r1*(1+c02)
Real02<-Re(r2)
Imaginal02<-Im(r2)
Rtime02<-c(seq(0,1,length=1500),seq(1,0,length=1500))
plot3d(Real02,Imaginal02,Rtime02,type="l",xlim=c(-2,2),ylim=c(-2,2),zlim=c(0,2),col=c(200,200,200))

#球面描写
Real<-Re(f0(radians))
Imaginal<-Im(f0(radians))
Rtime01<-seq(0,1,length=3000)
c0A<-seq(-1,1,length=3000)
c01<-f1(c0A)
Real01<-Real*c01
Imaginal01<-Imaginal*c01

points3d(Real01,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01,Rtime01-1,col=rgb(0,0,1))
points3d(Real01+1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01+1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01+1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01+1,Rtime01-1/2,col=rgb(1,0,0))
points3d(Real01-1,Imaginal01-1,Rtime01-1/2,col=rgb(1,0,0))
#Rtime01段
points3d(Real01+2,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01-2,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01+2,Rtime01,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01-2,Rtime01,col=rgb(0,0,1))
#Rtime01+1段
points3d(Real01+2,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01+1,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01+2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01-2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01+2,Rtime01+1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01-2,Rtime01+1,col=rgb(0,0,1))
#Rtime01-1段
points3d(Real01+2,Imaginal01,Rtime01-1,col=rgb(0,0,1))
points3d(Real01,Imaginal01+2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01,Rtime01-1,col=rgb(0,0,1))
points3d(Real01,Imaginal01-2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01+2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01+2,Imaginal01-2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01+2,Rtime01-1,col=rgb(0,0,1))
points3d(Real01-2,Imaginal01-2,Rtime01-1,col=rgb(0,0,1))

movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test34")

それにつけても何かを連想させませんか？　そう、立体充填性(Space filling)における正四面体(Regular Tetrahedron)と正八面体(Regular Octahedron)の相補関係です。
【オイラーの多面体定理と正多面体】とある「球面幾何学」の出発点…
triangles004.gif

あれ、数論(Number Theory)の話をしてた筈なのに、いつの間にか分子構造(Molecular structure)みたいな世界に足を踏み入れている!!

1
1
0

Register as a new user and use Qiita more conveniently

  1. You get articles that match your needs
  2. You can efficiently read back useful information
  3. You can use dark theme
What you can do with signing up
Sign upLogin
1
1