「正多面体のデータを作る」を Mathematica のデータにしてみました。
データは
- 頂点の一つは (0,0,1)
- 半径１の球に内接
の条件で作られています。
以下、Mathematica 用のコードと頂点データです。
正四面体
正四面体
RegularTetrahedronVertexData = Module[
{
r2 = Sqrt[2],
r3 = Sqrt[3],
x1, x2, y1, z1
},
x1 = (2*r2)/3;
x2 = r2/3;
y1 = r2/r3;
z1 = 1/3;
{
{ 0, 0, 1 },
{ x1, 0, -z1 },
{ -x2, +y1, -z1 },
{ -x2, -y1, -z1 }
}
]
RegularTetrahedronVertexIndex = {
{ 1, 2, 3 },
{ 1, 3, 4 },
{ 1, 4, 2 },
{ 2, 4, 3 }
}
RegularTetrahedron = Polyhedron[RegularTetrahedronVertexData, RegularTetrahedronVertexIndex]
Graphics3D[%]
頂点データと三角形の対応
| x | y | z | ||||||
|---|---|---|---|---|---|---|---|---|
| 1 | $0$ | $0$ | $1$ | ● | ● | ● | ||
| 2 | $\frac{2\sqrt{2}}{3}$ | $0$ | $-\frac{1}{3}$ | ● | ● | ● | ||
| 3 | $-\frac{\sqrt{2}}{3}$ | $\sqrt{\frac{2}{3}}$ | $-\frac{1}{3}$ | ● | ● | ● | ||
| 4 | $-\frac{\sqrt{2}}{3}$ | $-\sqrt{\frac{2}{3}}$ | $-\frac{1}{3}$ | ● | ● | ● |
縦が頂点番号で、縦方向の「●」が三角形で、横方向の「●」は接する面になる。
正六面体
正六面体
RegularHexahedronVertexData = Module[
{
r2 = Sqrt[2],
r3 = Sqrt[3],
x1, x2, y1,
z1 = 1/3
},
x1 = (2*r2)/3;
x2 = r2/3;
y1 = r2/r3;
{
{ 0, 0, 1 },
{ x1, 0, z1 },
{ -x2, y1, z1 },
{ -x2, -y1, z1 },
{ x2, -y1, -z1 },
{ x2, y1, -z1 },
{ -x1, 0, -z1 },
{ 0, 0, -1 }
}
]
RegularHexahedronVertexIndex = {
{ 1, 2, 6, 3 },
{ 1, 3, 7, 4 },
{ 1, 4, 5, 2 },
{ 2, 5, 8, 6 },
{ 3, 6, 8, 7 },
{ 4, 7, 8, 5 }
}
RegularHexahedron = Polyhedron[RegularHexahedronVertexData, RegularHexahedronVertexIndex]
Graphics3D[%]
頂点データと四角形の対応
| x | y | z | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | $0$ | $0$ | $1$ | ● | ● | ● | ||||
| 2 | $\frac{2\sqrt{2}}{3}$ | $0$ | $\frac{1}{3}$ | ● | ● | |||||
| 3 | $-\frac{\sqrt{2}}{3}$ | $\sqrt{\frac{2}{3}}$ | $\frac{1}{3}$ | ● | ● | ● | ||||
| 4 | $-\frac{\sqrt{2}}{3}$ | $-\sqrt{\frac{2}{3}}$ | $\frac{1}{3}$ | ● | ● | ● | ||||
| 5 | $\frac{\sqrt{2}}{3}$ | $-\sqrt{\frac{2}{3}}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||
| 6 | $\frac{\sqrt{2}}{3}$ | $\sqrt{\frac{2}{3}}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||
| 7 | $-\frac{2\sqrt{2}}{3}$ | $0$ | $-\frac{1}{3}$ | ● | ● | ● | ||||
| 8 | $0$ | $0$ | $-1$ | ● | ● | ● |
縦が頂点番号で、縦方向の「●」が四角形で、横方向の「●」は接する面になる。
正八面体
正八面体
RegularOctahedronVertexData = {
{ 0, 0, 1 },
{ 1, 0, 0 },
{ 0, 1, 0 },
{ -1, 0, 0 },
{ 0, -1, 0 },
{ 0, 0, -1 }
}
RegularOctahedronVertexIndex = {
{ 1, 2, 3 },
{ 1, 3, 4 },
{ 1, 4, 5 },
{ 1, 5, 2 },
{ 2, 5, 6 },
{ 2, 6, 3 },
{ 3, 6, 4 },
{ 4, 6, 5 }
}
RegularOctahedron = Polyhedron[RegularOctahedronVertexData, RegularOctahedronVertexIndex]
Graphics3D[%]
頂点データ
| x | y | z | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | $0$ | $0$ | $1$ | ● | ● | ● | ● | |||||
| 2 | $1$ | $0$ | $0$ | ● | ● | ● | ● | |||||
| 3 | $0$ | $1$ | $0$ | ● | ● | ● | ● | |||||
| 4 | $-1$ | $0$ | $0$ | ● | ● | ● | ● | |||||
| 5 | $0$ | $-1$ | $0$ | ● | ● | ● | ● | |||||
| 6 | $0$ | $0$ | $-1$ | ● | ● | ● | ● |
縦が頂点番号で、縦方向の「●」が三角形で、横方向の「●」は接する面になる。
正十二面体
正十二面体
RegularDodecahedronVertexData = Module[
{
r2 = Sqrt[2],
r3 = Sqrt[3],
r5 = Sqrt[5],
x1, x2, x3, x4, x5,
y1, y2, y3,
z1, z2
},
x1 = 2/3;
x2 = 1/3;
x3 = r5/3;
x4 = (3-r5)/6;
x5 = (3+r5)/6;
y1 = 1/r3;
y2 = Sqrt[(3+r5)/6];
y3 = Sqrt[(3-r5)/6];
z1 = r5/3;
z2 = 1/3;
{
{ 0, 0, 1 },
{ x1, 0, z1 },
{ -x2, y1, z1 },
{ -x2, -y1, z1 },
{ x3, y1, z2 },
{ x3, -y1, z2 },
{ x4, y2, z2 },
{ x4, -y2, z2 },
{ -x5, y3, z2 },
{ -x5, -y3, z2 },
{ x5, -y3, -z2 },
{ x5, y3, -z2 },
{ -x4, -y2, -z2 },
{ -x4, y2, -z2 },
{ -x3, -y1, -z2 },
{ -x3, y1, -z2 },
{ x2, -y1, -z1 },
{ x2, y1, -z1 },
{ -x1, 0, -z1 },
{ 0, 0, -1 }
}
]
RegularDodecahedronVertexIndex = {
{ 1, 2, 5, 7, 3 },
{ 1, 3, 9, 10, 4 },
{ 1, 4, 8, 6, 2 },
{ 2, 6, 11, 12, 5 },
{ 3, 7, 14, 16, 9 },
{ 4, 10, 15, 13, 8 },
{ 5, 12, 18, 14, 7 },
{ 6, 8, 13, 17, 11 },
{ 9, 16, 19, 15, 10 },
{ 11, 17, 20, 18, 12 },
{ 13, 15, 19, 20, 17 },
{ 14, 18, 20, 19, 16 }
}
RegularDodecahedron = Polyhedron[RegularDodecahedronVertexData, RegularDodecahedronVertexIndex]
Graphics3D[%]
頂点データ
| x | y | z | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | $0$ | $0$ | $1$ | ● | ● | ● | ||||||||||
| 2 | $\frac{2}{3}$ | $0$ | $\frac{\sqrt{5}}{3}$ | ● | ● | ● | ||||||||||
| 3 | $-\frac{1}{3}$ | $ \frac{1}{\sqrt{3}}$ | $\frac{\sqrt{5}}{3}$ | ● | ● | ● | ||||||||||
| 4 | $-\frac{1}{3}$ | $-\frac{1}{\sqrt{3}}$ | $\frac{\sqrt{5}}{3}$ | ● | ● | ● | ||||||||||
| 5 | $\frac{\sqrt{5}}{3}$ | $ \frac{1}{\sqrt{3}}$ | $\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 6 | $\frac{\sqrt{5}}{3}$ | $-\frac{1}{\sqrt{3}}$ | $\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 7 | $ \frac{1}{6}\left(3-\sqrt{5}\right)$ | $ \sqrt{\frac{1}{6}\left(3+\sqrt{5}\right)}$ | $\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 8 | $ \frac{1}{6}\left(3-\sqrt{5}\right)$ | $-\sqrt{\frac{1}{6}\left(3+\sqrt{5}\right)}$ | $\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 9 | $-\frac{1}{6}\left(3+\sqrt{5}\right)$ | $ \sqrt{\frac{1}{6}\left(3-\sqrt{5}\right)}$ | $\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 10 | $-\frac{1}{6}\left(3+\sqrt{5}\right)$ | $-\sqrt{\frac{1}{6}\left(3-\sqrt{5}\right)}$ | $\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 11 | $ \frac{1}{6}\left(3+\sqrt{5}\right)$ | $-\sqrt{\frac{1}{6}\left(3-\sqrt{5}\right)}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 12 | $ \frac{1}{6}\left(3+\sqrt{5}\right)$ | $ \sqrt{\frac{1}{6}\left(3-\sqrt{5}\right)}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 13 | $-\frac{1}{6}\left(3-\sqrt{5}\right)$ | $-\sqrt{\frac{1}{6}\left(3+\sqrt{5}\right)}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 14 | $-\frac{1}{6}\left(3-\sqrt{5}\right)$ | $ \sqrt{\frac{1}{6}\left(3+\sqrt{5}\right)}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 15 | $-\frac{\sqrt{5}}{3}$ | $-\frac{1}{\sqrt{3}}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 16 | $-\frac{\sqrt{5}}{3}$ | $ \frac{1}{\sqrt{3}}$ | $-\frac{1}{3}$ | ● | ● | ● | ||||||||||
| 17 | $ \frac{1}{3}$ | $-\frac{1}{\sqrt{3}}$ | $-\frac{\sqrt{5}}{3}$ | ● | ● | ● | ||||||||||
| 18 | $ \frac{1}{3}$ | $ \frac{1}{\sqrt{3}}$ | $-\frac{\sqrt{5}}{3}$ | ● | ● | ● | ||||||||||
| 19 | $-\frac{2}{3}$ | $0$ | $-\frac{\sqrt{5}}{3}$ | ● | ● | ● | ||||||||||
| 20 | $0$ | $0$ | $-1$ | ● | ● | ● |
縦が頂点番号で、縦方向の「●」が五角形で、横方向の「●」は接する面になる。
正二十面体
正二十面体
RegularIcosahedronVertexData = Module[
{
r2 = Sqrt[2],
r3 = Sqrt[3],
r5 = Sqrt[5],
x1, x2, x3,
y1, y2,
z1
},
x1 = 2/r5;
x2 = (5-r5)/10;
x3 = (5+r5)/10;
y1 = Sqrt[(5+r5)/10];
y2 = Sqrt[(5-r5)/10];
z1 = 1/r5;
{
{ 0, 0, 1 },
{ x1, 0, z1 },
{ x2, y1, z1 },
{ x2, -y1, z1 },
{ -x3, y2, z1 },
{ -x3, -y2, z1 },
{ x3, -y2, -z1 },
{ x3, y2, -z1 },
{ -x2, -y1, -z1 },
{ -x2, y1, -z1 },
{ -x1, 0, -z1 },
{ 0, 0, -1 }
}
]
RegularIcosahedronVertexIndex = {
{ 1, 2, 3 },
{ 1, 3, 5 },
{ 1, 5, 6 },
{ 1, 6, 4 },
{ 1, 4, 2 },
{ 2, 4, 7 },
{ 2, 7, 8 },
{ 2, 8, 3 },
{ 3, 8, 10 },
{ 3, 10, 5 },
{ 4, 6, 9 },
{ 4, 9, 7 },
{ 5, 10, 11 },
{ 5, 11, 6 },
{ 6, 11, 9 },
{ 7, 9, 12 },
{ 7, 12, 8 },
{ 8, 12, 10 },
{ 9, 11, 12 },
{ 10, 12, 11 }
}
RegularIcosahedron = Polyhedron[RegularIcosahedronVertexData, RegularIcosahedronVertexIndex]
Graphics3D[%]
頂点データ
| x | y | z | ||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | $0$ | $0$ | $1$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 2 | $\frac{2}{\sqrt{5}}$ | $0$ | $\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 3 | $ \frac{1}{10}\left(5-\sqrt{5}\right)$ | $ \sqrt{\frac{1}{10}\left(5+\sqrt{5}\right)}$ | $\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 4 | $ \frac{1}{10}\left(5-\sqrt{5}\right)$ | $-\sqrt{\frac{1}{10}\left(5+\sqrt{5}\right)}$ | $\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 5 | $-\frac{1}{10}\left(5+\sqrt{5}\right)$ | $ \sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}$ | $\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 6 | $-\frac{1}{10}\left(5+\sqrt{5}\right)$ | $-\sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}$ | $\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 7 | $ \frac{1}{10}\left(5+\sqrt{5}\right)$ | $-\sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}$ | $-\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 8 | $ \frac{1}{10}\left(5+\sqrt{5}\right)$ | $ \sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}$ | $-\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 9 | $-\frac{1}{10}\left(5-\sqrt{5}\right)$ | $-\sqrt{\frac{1}{10}\left(5+\sqrt{5}\right)}$ | $-\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 10 | $-\frac{1}{10}\left(5-\sqrt{5}\right)$ | $ \sqrt{\frac{1}{10}\left(5+\sqrt{5}\right)}$ | $-\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 11 | $-\frac{2}{\sqrt{5}}$ | $0$ | $-\frac{1}{\sqrt{5}}$ | ● | ● | ● | ● | ● | ||||||||||||||||
| 12 | $0$ | $0$ | $-1$ | ● | ● | ● | ● | ● |
縦が頂点番号で、縦方向の「●」が三角形で、横方向の「●」は接する面になる。
Mathematica での生成プログラム
正多面体のデータを作る[改] (HTML+JavaScript版) の方法で計算します。
正十二面体と正二十面体の計算では時間がかかります。
RegularPolyhedron
RegularPolyhedron[nNormal_] := Module[{$Assumptions = Reals,
nVertex = nNormal + (2 + 6 Floor[Mod[nNormal + 3, 9]/4]) (1 - Mod[nNormal, 3]),
nVertPerPoly = 3 + Floor[Mod[nNormal^4, 202]/64],
nEdgePerVert = 3 + Floor[Mod[nNormal + 6, 9]/4],
fSimplify, fIsNew, fAddNew, fPickup, fAxis, fUpdate, fMake, fPVLink, fPolygon,
pCos, pSin, qCos, qSin, rCos, rSin, vertex, mRot, n1, n2, n3,
normal, vCos, vMap},
fSimplify = FullSimplify;
fIsNew = Module[{vectors = #1, target = #2, fTest},
fTest = fSimplify[# == target] &; (Length[Select[vectors, fTest]] == 0)] &;
fAddNew = Module[{pList = #1, nList = #2, data},
Do[data = nList[[i]]; If[fIsNew[pList, data], pList = Append[pList, data]], {i, Length[nList]}]; pList] &;
fPickup = Module[{center = #2, value = #3, fTest},
fTest = fSimplify[# . center == value] &; Select[#1, fTest]] &;
fAxis = Module[{x, y, z}, x = #1; y = fSimplify[Cross[x, #2]/#3];
z = fSimplify[Cross[x, y]]; {x, y, z}] &;
fUpdate = Module[{center = #1[[#2]], vcos = #1[[1]] . #1[[2]], vnew = #3, aSin = #4},
fAddNew[#1, Map[fSimplify[vnew . fAxis[center, #, aSin]] &, fPickup[#1, center, vcos]]]] &;
fMake = Module[{elen = #1, aSin = #2, vectors, plen, nlen, vnew},
vectors = #3;
vnew = fSimplify[fAxis[vectors[[1]], vectors[[2]], aSin] . vectors[[3]]];
plen = 1; nlen = Length[vectors];
While[plen < nlen,
Do[Print[{"Progress", i, Length[vectors], elen}];
If[Length[vectors] >= elen, Break[]];
vectors = fUpdate[vectors, i, vnew, aSin], {i, nlen}];
plen = nlen + 1; nlen = Length[vectors]];
FullSimplify[vectors]] &;
fPVLink = Module[{vectors = fPickup[#1, #2, #3], v1, v2 = #2, v3},
v1 = vectors[[1]]; v3 = vectors[[2]];
If[fSimplify[#4 == Normalize[Cross[v2 - v1, v3 - v2]]], v2 -> v3, v2 -> v1]] &;
fPolygon = Module[{vvecs = fPickup[#1, #2, #3], center = #2, vcos = #4, vmap, vp},
vmap = Association[];
Do[AssociateTo[vmap, fPVLink[vvecs, vvecs[[i]], vcos, center]], {i, Length[vvecs]}];
vp = vvecs[[1]]; Table[#5[vp = vmap[vp]], {i, Length[vvecs]}]] &;
pCos = fSimplify[2 Cos[Pi/nVertPerPoly]^2 Csc[Pi/nEdgePerVert]^2 - 1];
pSin = fSimplify[Sqrt[1 - pCos^2]];
rCos = Cos[2 Pi/nEdgePerVert]; rSin = Sin[2 Pi/nEdgePerVert];
vertex = fMake[nVertex, pSin, {{0, 0, 1}, {pSin, 0, pCos}, {fSimplify[pSin rCos], fSimplify[pSin rSin], pCos}}];
mRot = {{rCos, -rSin, 0}, {rSin, rCos, 0}, {0, 0, 1}};
n1 = fSimplify[Normalize[Cross[vertex[[1]] - vertex[[3]], vertex[[2]] - vertex[[1]]]]];
n2 = fSimplify[mRot . n1]; n3 = fSimplify[mRot . n2];
qCos = fSimplify[n1 . n2]; qSin = fSimplify[Sqrt[1 - qCos^2]];
normal = fMake[nNormal, qSin, {n1, n2, n3}];
vCos = fSimplify[n1[[3]]];
vMap = Association[]; Do[AssociateTo[vMap, vertex[[i]] -> i], {i, Length[vertex]}];
Polyhedron[vertex, Table[fPolygon[vertex, normal[[i]], vCos, pCos, vMap], {i, Length[normal]}]]]
(*正四面体*)Graphics3D[RegularPolyhedron[4]]
(*正六面体*)Graphics3D[RegularPolyhedron[6]]
(*正八面体*)Graphics3D[RegularPolyhedron[8]]
(*正十二面体*)Graphics3D[RegularPolyhedron[12]]
(*正二十面体*)Graphics3D[RegularPolyhedron[20]]