Visualizing Coxeter' s Regular Polytopes

In[]:=
addNullPoints=False;​​doCoxeterSections@in_:=Module[{groupVerts,sortVerts,out},​​groupVerts=GroupBy[in〚All,;;4〛,First];​​sortVerts=Sort[groupVerts,N[#1〚1,1〛/.φRep]<N[#2〚1,1〛/.φRep]&];​​out={#,octsym@sortVerts〚#,1,1〛,hulls3D@sortVerts〚#,All,2;;〛}&/@Range@Length@sortVerts;​​out//MatrixForm];
600-cell (I) generated using quaternion Weyl orbit construction with
D
4
24-cell T vertices in Coxeter’s Table V iii {3,3,5} vertex first (9=0...8) sections in 3D
In[]:=
doCoxeterSections@ILΦSymList
Out[]//MatrixForm=
1
-1
tallyList=
{1}
Combined Hulls=
Overall Hull=
2
-
φ
2
tallyList=
{12}
Hull # = 1
with 12 vertices
of 3D Norm
=
1
2
1+
1
2
φ
=
1
4
+
1
2
(1+
5
)
=
0.58779
Vertex #'s = {1,12}
Combined Hulls=
Overall Hull=
3
-
1
2
tallyList=
{20}
Hull # = 1
with 20 vertices
of 3D Norm
=
3
2
=
3
2
=
0.86603
Vertex #'s = {1,20}
Combined Hulls=
Overall Hull=
4
-
1
2φ
tallyList=
{12}
Hull # = 1
with 12 vertices
of 3D Norm
=
1+
2
φ
2
=
1
2
1
2
(5+
5
)
=
0.95106
Vertex #'s = {1,12}
Combined Hulls=
Overall Hull=
5
0.
tallyList=
{30}
Hull # = 1
with 30 vertices
of 3D Norm
=
1
=
1
=
1.
Vertex #'s = {1,30}
Combined Hulls=
Overall Hull=
6
1
2φ
tallyList=
{12}
Hull # = 1
with 12 vertices
of 3D Norm
=
1+
2
φ
2
=
1
2
1
2
(5+
5
)
=
0.95106
Vertex #'s = {1,12}
Combined Hulls=
Overall Hull=
7
1
2
tallyList=
{20}
Hull # = 1
with 20 vertices
of 3D Norm
=
3
2
=
3
2
=
0.86603
Vertex #'s = {1,20}
Combined Hulls=
Overall Hull=
8
φ
2
tallyList=
{12}
Hull # = 1
with 12 vertices
of 3D Norm
=
1
2
1+
1
2
φ
=
1
4
+
1
2
(1+
5
)
=
0.58779
Vertex #'s = {1,12}
Combined Hulls=
Overall Hull=
9
1
tallyList=
{1}
Combined Hulls=
Overall Hull=
120-cell (J) using generated quaternion Weyl orbit construction with
T’ (or against Ip) vertices {5,3,3} Coxeter’s Table V iv cell first (15=1...15) sections in 3D
120-cell (J’=Jp) using generated quaternion Weyl orbit construction with
T (or against I) vertices {5,3,3} Coxeter’s Table V v vertex first (31=0...30) sections in 3D