正五角形(Regular pentagon、外角の大きさはpi/5=0.6283185ラジアンあるいは180/5=36度、内角の大きさは2*pi*(5-2)/5=3.769911ラジアンあるいは180*(5-2)/5=108度)の場合
対象円に内接する正五角形の対角線(1+sqrt(5))/2)をd(diagonal)とする。外接円に対する内接円の半径の比率は指数サイズで推移せず(0.809017倍)、空間充填性(同じ立体で空間を埋め尽くす能力)は備えるも平面充填性(同じ図面で平面を埋め尽くす能力)は備えない。立体としては正12面体(Rregular dodecahedron, 辺数30、頂点数20)を構成し、その表面積は3*sqrt(25+10*sqrt(5))*a^2、体積は(15+7*sqrt(5))/4*a^3で表される。
- 2*tan(pi/5)=1.453085
- 2*tan(pi/5)/cos(pi/5)=1.796112
- cos(pi/5)=0.809017
一辺の長さがaの正五角形に内接する円の半径r
- r=a*(2*tan(pi/5))=a*1.453085
- a=r/(2*tan(pi/5))=r/1.453085
一辺の長さがaの正五角形に外接する円の半径R
- R=a/(2*tan(pi/5))/cos(pi/5)=a/1.796112
- a=R*(2*tan(pi/5))/cos(pi/5)=R*1.796112
一辺の長さがaの正五角形の外接円の半径と内接円の関係
- r=R*cos(pi/5)=R*0.809017
- R=r/cos(pi/5)=r/0.809017
結果は「3.440955<=π<=8.98056」で正解範囲…
統計言語Rによる作表例
target_size<-c("5^-1","5^-1","5^-1","5^-1","5^-0.2722623","...","5^-0.131683","5^0","5^0","5^0","5^0","5^0","5^0","5^0.1316829","...","5^0.535628","5^1","5^1","5^1","5^1")
target_names<-c("5^-1","5^-1d","5^-1a1","5^-1a1*5","5^-0.2722623","...","5^-0.131683","5^0a0","5^0a0*5","5^0=1","5^0d","5^0a1","5^0a1*5","5^0.7277376","...","5^0.535628","5^0a1","5^0a1*5","5^1","5^1d")
target_values<-c("1/5=0.2","(1+sqrt(5))/2*0.2=0.3236068","0.2*1.796112=0.3592224","0.3592224*5=1.796112","0.3592224*1.796112=-0.2722623","...","0.8090168","1/1.453085=0.688191","0.5567581*5=3.440955","1.0","(1+sqrt(5))/2=1.618034","1.796112","1.796112*5=8.98056","1.796112*1.796112=3.226018","...","3.440955*1.453085=2.368034","5/1.453085=3.440955","3.440955*5=17.20478","5","*1/2)*5=8.09017")
Regula_falsi05<-data.frame(Target_size=target_size,Target_names=target_names,Target_values=target_values)
library(xtable)
print(xtable(Regula_falsi05), type = "html")
Target_size | Target_names | Target_values | |
---|---|---|---|
1 | 5^-1 | 5^-1 | 1/5=0.2 |
2 | 5^-1 | 5^-1d | (1+sqrt(5))/2*0.2=0.3236068 |
3 | 5^-1 | 5^-1a1 | 0.2*1.796112=0.3592224 |
4 | 5^-1 | 5^-1a1*5 | 0.3592224*5=1.796112 |
5 | 5^-0.2722623 | 5^-0.2722623 | 0.3592224*1.796112=-0.2722623 |
6 | ... | ... | ... |
7 | 5^-0.131683 | 5^-0.131683 | 0.8090168 |
8 | 5^0 | 5^0a0 | 1/1.453085=0.688191 |
9 | 5^0 | 5^0a0*5 | 0.5567581*5=3.440955 |
10 | 5^0 | 5^0=1 | 1.0 |
11 | 5^0 | 5^0d | (1+sqrt(5))/2=1.618034 |
12 | 5^0 | 5^0a1 | 1.796112 |
13 | 5^0 | 5^0a1*5 | 1.796112*5=8.98056 |
14 | 5^0.1316829 | 5^0.7277376 | 1.796112*1.796112=3.226018 |
15 | ... | ... | ... |
16 | 5^0.535628 | 5^0.535628 | 3.440955*1.453085=2.368034 |
17 | 5^1 | 5^0a1 | 5/1.453085=3.440955 |
18 | 5^1 | 5^0a1*5 | 3.440955*5=17.20478 |
19 | 5^1 | 5^1 | 5 |
20 | 5^1 | 5^1d | *2/2)*5=8.09017 |
「1の原始冪根級数(One Primitive Sone series)」の一環としてζ^5に従う以下の5点が基本円(Basic circle,半径1,円周2π)に内接する正五角形を描く。
- 1=(1+0i)
- (-1+sqrt(5)+(0+1i)*sqrt(10+2*sqrt(5)))/4
- (-1-sqrt(5)+(0+1i)*sqrt(10-2*sqrt(5)))/4
- (-1-sqrt(5)-(0+1i)*sqrt(10-2*sqrt(5)))/4
- (-1+sqrt(5)-(0+1i)*sqrt(10+2*sqrt(5)))/4
統計言語Rによる作図例
#正5角形
library(rgl)
Rtime<-seq(0,2,length=6)
tr01<-c(1,(-1+sqrt(5)+(0+1i)*sqrt(10+2*sqrt(5)))/4,(-1-sqrt(5)+(0+1i)*sqrt(10-2*sqrt(5)))/4,(-1-sqrt(5)-(0+1i)*sqrt(10-2*sqrt(5)))/4,(-1+sqrt(5)-(0+1i)*sqrt(10+2*sqrt(5)))/4,1)
Real<-Re(tr01)
Imag<-Im(tr01)
#plot(Real,Imag,type="l")
plot3d(Real,Imag,Rtime,type="l",xlim=c(-1,1),ylim=c(-1,1),zlim=c(0,2))
movie3d(spin3d(axis=c(0,0,1),rpm=5),duration=10,fps=25,movie="~/Desktop/test13")
基本円(Basic circle)上に同心円を描く任意の正方形内で正方形を回すと、その片長の比はsqrt(2):sqrt(2)と1:2の間を反復し続ける。その比は多層化された内接円と外接円のサイズ比に等しい。
正五角形 統計言語Rによる実装例
Three_square_theorem05<-function(x){
c0<-seq(0,2*pi,length=6)
c0_cos<-cos(c0)
c0_sin<-sin(c0)
plot(c0_cos,c0_sin,type="l",main="Regular polygon rotation",xlab="Cos(θ)",ylab="Sin(θ)")
text(c0_cos,c0_sin, labels=c("a","b","c","d","e",""),col=c(rgb(1,0,0),rgb(1,0,0),rgb(1,0,0),rgb(1,0,0),rgb(1,0,0),rgb(1,0,0)),cex=c(2,2,2,2,2,2))
line_scale_01_cos<-seq(cos(c0[1]),cos(c0[2]),length=15)
line_scale_01_sin<-seq(sin(c0[1]),sin(c0[2]),length=15)
line_scale_02_cos<-seq(cos(c0[2]),cos(c0[3]),length=15)
line_scale_02_sin<-seq(sin(c0[2]),sin(c0[3]),length=15)
line_scale_03_cos<-seq(cos(c0[3]),cos(c0[4]),length=15)
line_scale_03_sin<-seq(sin(c0[3]),sin(c0[4]),length=15)
line_scale_04_cos<-seq(cos(c0[4]),cos(c0[5]),length=15)
line_scale_04_sin<-seq(sin(c0[4]),sin(c0[5]),length=15)
line_scale_05_cos<-seq(cos(c0[5]),cos(c0[1]),length=15)
line_scale_05_sin<-seq(sin(c0[5]),sin(c0[1]),length=15)
text(c(line_scale_01_cos[x],line_scale_02_cos[x],line_scale_03_cos[x],line_scale_04_cos[x],line_scale_05_cos[x]),c(line_scale_01_sin[x],line_scale_02_sin[x],line_scale_03_sin[x],line_scale_04_sin[x],line_scale_05_sin[x]), labels=c("ab","bc","cd","de","ea",""),col=c(rgb(0,0,1),rgb(0,0,1),rgb(0,0,1),rgb(0,0,1),rgb(0,0,1),rgb(0,0,1)),cex=c(2,2,2,2,2,2))#塗りつぶし
polygon(c(c0_cos[1],line_scale_05_cos[x],line_scale_01_cos[x],c0_cos[1]), #x
c(c0_sin[1],line_scale_05_sin[x],line_scale_01_sin[x],c0_sin[1]), #y
density=c(30), #塗りつぶす濃度
angle=c(45), #塗りつぶす斜線の角度
col=rgb(0,1,0))#塗りつぶし
polygon(c(c0_cos[2],line_scale_01_cos[x],line_scale_02_cos[x],c0_cos[2]), #x
c(c0_sin[2],line_scale_01_sin[x],line_scale_02_sin[x],c0_sin[2]), #y
density=c(30), #塗りつぶす濃度
angle=c(45), #塗りつぶす斜線の角度
col=rgb(0,1,0))#塗りつぶし
polygon(c(c0_cos[3],line_scale_02_cos[x],line_scale_03_cos[x],c0_cos[3]), #x
c(c0_sin[3],line_scale_02_sin[x],line_scale_03_sin[x],c0_sin[3]), #y
density=c(30), #塗りつぶす濃度
angle=c(45), #塗りつぶす斜線の角度
col=rgb(0,1,0))#塗りつぶし
polygon(c(c0_cos[4],line_scale_03_cos[x],line_scale_04_cos[x],c0_cos[4]), #x
c(c0_sin[4],line_scale_03_sin[x],line_scale_04_sin[x],c0_sin[4]), #y
density=c(30), #塗りつぶす濃度
angle=c(45), #塗りつぶす斜線の角度
col=rgb(0,1,0))#塗りつぶし
polygon(c(c0_cos[5],line_scale_04_cos[x],line_scale_05_cos[x],c0_cos[5]), #x
c(c0_sin[5],line_scale_04_sin[x],line_scale_05_sin[x],c0_sin[5]), #y
density=c(30), #塗りつぶす濃度
angle=c(45), #塗りつぶす斜線の角度
col=rgb(0,1,0))}
#アニメーション
library("animation")
Time_Code=c(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
saveGIF({
for (i in Time_Code){
Three_square_theorem05(i)
}
}, interval = 0.1, movie.name = "TEST005.gif")