Gosset Subscript[4, 21] polytope, 600-cells, compound of five 24-cells, and compounding of tetrahedral 4-Groups into a 20-Group
Author
Richard Clawson
Title
Gosset Subscript[4, 21] polytope, 600-cells, compound of five 24-cells, and compounding of tetrahedral 4-Groups into a 20-Group
Description
The vertices of a 600-cell partitioned into a compound of 5 24-cells implicitly induce the icosahedrally symmetric compound of 5 cuboctahedra.
Category
Working Material
Keywords
E8, Elser-Sloane quasicrystal, Gosset polytope, Gosset 421, 600-cell, 24-cell, compound of five cuboctahedra, icosahedral symmetry, H3 symmetry, H4 symmetry, isoclinic rotations, 20-Group of tetrahedra
URL
http://www.notebookarchive.org/2024-02-2v1915y/
DOI
https://notebookarchive.org/2024-02-2v1915y
Date Added
2024-02-06
Date Last Modified
2024-02-06
File Size
420.36 kilobytes
Supplements
Rights
CC BY-SA 4.0
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Gosset 421 polytope, 600-cells, compound of five 24-cells, and compounding of tetrahedral 4-Groups into a 20-Group
Gosset polytope, 600-cells, compound of five 24-cells, and compounding of tetrahedral 4-Groups into a 20-Group
4
21
Richard Clawson
Quantum Gravity Research
Los Angeles, CA, USA
2024-02-02
richard@quantumgravityresearch.org
Quantum Gravity Research
Los Angeles, CA, USA
2024-02-02
richard@quantumgravityresearch.org
The Gosset polytope in 8D projects by the Elser-Sloane projection to a pair of concentric 600-cells in 4D. The vertices of a 600-cell can be seen as the vertices of a compound of 5 24-cells. Indeed, one can generate all the vertices of the 600-cell by starting with one 24-cell, making 4 more copies of it, and applying a 2kπ/5 isoclinic rotation to the k-th one. These isoclinic rotations can be either left or right.
Each 24-cell has 12 distinct 3D cuboctahedral equators. We arbitrarily choose a single one of these cuboctahedra from a single 24-cell and call it the primary. Its 3D space we call V. Within V, we can apply to this cuboctahedron the standard procedure for making the icosahedrally symmetric compound of 5 cuboctahedra. It turns out that there is a natural unique pairwise correspondence between the 5 24-cells and the 5 cuboctahedra. For if one takes one of the cuboctahedra in 24-cell k and isoclinically rotates it to V, using the opposite-handed rotation as was used to construct the 24-cell 5-compound, then it coincides with one specific member of the cuboctahedron 5-compound -- and this is independent of the choice of cuboctahedron one makes in 24-cell k!
If one rotates any of the cuboctahedra in k to V using the same handed rotation as was used to make the 24-cell 5-compound, then it rotates it back to the primary cuboctahedron, regardless of which cuboctahedron one started with. If one uses, instead, a simple direct rotation, from the 3D space of the chosen cuboctahedron to V, then at lands in V with an incoherent, non-symmetric relationship with the primary cuboctahedron; this orientation varies widely based on which cuboctahedron was chosen in k. But if one uses the opposite isoclinic rotation, then it lands exactly in the position of one member of the cuboctahedron 5-compound.
The cuboctahedron can be made into a 4-Group (4G) of tetrahedra by selecting a disjoint set of triangular faces and and connecting their vertices to the centroid, thus forming regular tetrahedra. When these are compounded in the same icosahedrally symmetric manner, they form the 20-Group, having chiral icosahedral symmetry. On each cuboctahedron, there are two ways to choose a disjoint set of triangular faces, and a certain care must be taken in this choice to make a consistent selection that yields the chirally symmetric object.
This notebook generates the Gosset polytope and the Elser-Sloane projection to two 600-cells. It then decomposes one of them into a compound of 5 25-cells, identifies all their cuboctahedral equators, and constructs their rotations to the space V of a chosen primary cuboctahedron. The graphic at the end shows the results of these rotated into V, one cuboctahedron from each 24-cell. You can choose which of the 12 cuboctahedra to view from each 24-cell, and which of the 5 to view. You can slide the rotations from left- to right-isoclinic to see how at one extreme all cuboctahedra coincide, at the other extreme they all combine in the icosahedrally symmetric compound, and in the middle (simple direct rotation) they form an incoherent jumble. Yu can also select whether to view the cuboctahedra or the 4-Groups.
4
21
Each 24-cell has 12 distinct 3D cuboctahedral equators. We arbitrarily choose a single one of these cuboctahedra from a single 24-cell and call it the primary. Its 3D space we call V. Within V, we can apply to this cuboctahedron the standard procedure for making the icosahedrally symmetric compound of 5 cuboctahedra. It turns out that there is a natural unique pairwise correspondence between the 5 24-cells and the 5 cuboctahedra. For if one takes one of the cuboctahedra in 24-cell k and isoclinically rotates it to V, using the opposite-handed rotation as was used to construct the 24-cell 5-compound, then it coincides with one specific member of the cuboctahedron 5-compound -- and this is independent of the choice of cuboctahedron one makes in 24-cell k!
If one rotates any of the cuboctahedra in k to V using the same handed rotation as was used to make the 24-cell 5-compound, then it rotates it back to the primary cuboctahedron, regardless of which cuboctahedron one started with. If one uses, instead, a simple direct rotation, from the 3D space of the chosen cuboctahedron to V, then at lands in V with an incoherent, non-symmetric relationship with the primary cuboctahedron; this orientation varies widely based on which cuboctahedron was chosen in k. But if one uses the opposite isoclinic rotation, then it lands exactly in the position of one member of the cuboctahedron 5-compound.
The cuboctahedron can be made into a 4-Group (4G) of tetrahedra by selecting a disjoint set of triangular faces and and connecting their vertices to the centroid, thus forming regular tetrahedra. When these are compounded in the same icosahedrally symmetric manner, they form the 20-Group, having chiral icosahedral symmetry. On each cuboctahedron, there are two ways to choose a disjoint set of triangular faces, and a certain care must be taken in this choice to make a consistent selection that yields the chirally symmetric object.
This notebook generates the Gosset polytope and the Elser-Sloane projection to two 600-cells. It then decomposes one of them into a compound of 5 25-cells, identifies all their cuboctahedral equators, and constructs their rotations to the space V of a chosen primary cuboctahedron. The graphic at the end shows the results of these rotated into V, one cuboctahedron from each 24-cell. You can choose which of the 12 cuboctahedra to view from each 24-cell, and which of the 5 to view. You can slide the rotations from left- to right-isoclinic to see how at one extreme all cuboctahedra coincide, at the other extreme they all combine in the icosahedrally symmetric compound, and in the middle (simple direct rotation) they form an incoherent jumble. Yu can also select whether to view the cuboctahedra or the 4-Groups.
Init
Packages and constants
Packages and constants
In[]:=
Needs["ConvexPolytope`"]Needs["Vector`"]
In[]:=
ctr3={0,0,0};ctr4={0,0,0,0};φ=GoldenRatio;
In[]:=
SetOptions[Graphics3D,Boxed->False,SphericalRegion->True];SetOptions[Manipulate,LabelStyle->{10}];
Functions
Functions
Find the equatorial point set on a polytope with respect to some pole.
Find the equatorial point set on a polytope with respect to some pole.
Find the dual of a simple rotation R in 4D
Find the dual of a simple rotation R in 4D
Making a 4G from a cuboctahedron.
Making a 4G from a cuboctahedron.
Find the index of a point in a point list.
Find the index of a point in a point list.
Facet normals of a 4D polytope
Facet normals of a 4D polytope
Main
Gosset polytope
Gosset polytope
Projection to 4D Elser-Sloane space (ES)
Projection to 4D Elser-Sloane space (ES)
Orient Gosset for ES projection
Orient Gosset for ES projection
Vertices projected to 4D ES, to two 600-cells
Vertices projected to 4D ES, to two 600-cells
From C600 vertices, define 5-fold isoclinic rotation in ES
From C600 vertices, define 5-fold isoclinic rotation in ES
600-cell structures
600-cell structures
◼
Just for reference, what are the angles between tetrahedron cell normals in the 600-cell?
◼
. . . and what are the angles between vertices?
Select a vertex cap and a single tetrahedron.
Select a vertex cap and a single tetrahedron.
Select one 24-cell, then generate a set of 5 24-cells that partitions the vertices of C600
Select one 24-cell, then generate a set of 5 24-cells that partitions the vertices of C600
In C600, compare all cuboctahedra from the 5 24-cells, each rotated into the 3D space V of one “primary” cuboctahedron in 24-cell 1.
In C600, compare all cuboctahedra from the 5 24-cells, each rotated into the 3D space V of one “primary” cuboctahedron in 24-cell 1.
Display
Display
In[]:=
Manipulate[cbctSelections={1,Cbct2,Cbct3,Cbct4,Cbct5};Which[shapeChoice=="4Gs",{shapes=n4GsInV;primaryShape=n4GPrimary3D;mapDepth=2;plottingFxn=Tetrahedron},shapeChoice=="Cuboctahdera",{shapes=cbctsInV;primaryShape=CbctPrimary3D;mapDepth=1;plottingFxn=ConvexHullMesh}];C24cbctDualRots=Map[rotMatrixToDual4D[#,α]&,Table[C24cbctRotsToV[[k,cbctSelections[[k]]]],{k,1,5}]];shapesInVtwisted=MapThread[Function[{rotMat,ptSets},Map[rotMat.#&,ptSets,{mapDepth}]][#1,#2]&,{C24cbctDualRots,Table[shapes[[k,cbctSelections[[k]]]],{k,1,5}]}];shapesInV3D=Map[VectorProject[#,NullSpace[{normalV}]]&,shapesInVtwisted,{mapDepth}];Column[{Graphics3D[{(*Magenta*)Yellow,If[MemberQ[n4GList,1],plottingFxn@(1.01*primaryShape)],Yellow,plottingFxn/@shapesInV3D[[DeleteCases[n4GList,1]]],If[showAxes,{Blue,Table[Tube[{ctr3,.8*compoundingAxes3D[[k,cbctSelections[[k]]]]},.015],{k,DeleteCases[n4GList,1]}]}]},ImageSize->Medium,PlotRange->{{-.75,.75},{-.75,.75},{-.75,.75}},SphericalRegion->True(*,ViewProjection->"Orthographic"*),ViewPoint{3.277,0.756,-0.378},ViewVertical{0.410,0.905,0.113},ViewAngle->.4]}],{C24cbctDualRots,None},{shapes,None},{primaryShape,None},{mapDepth,None},{plottingFxn,None},{shapesInVtwisted,None},{shapesInV3D,None}(*,{cmpdAxes3D,None}*),{{shapeChoice,"4Gs","Shapes to show"},{"4Gs","Cuboctahdera"}},{{showAxes,False,"Show Compounding Axes"},{True,False}},{{n4GList,{1,2,3,4,5},"Show Cuboctahedron/4G from Selected 24-cells"},{1,2,3,4,5},TogglerBar},Text["Select which Cuboctahedron/4G from each 24-cell"],{{Cbct2,1,"4G/Cuboct from 24-cell 2"},1,12,1},{{Cbct3,1,"4G/Cuboct from 24-cell 3"},1,12,1},{{Cbct4,1,"4G/Cuboct from 24-cell 4"},1,12,1},{{Cbct5,1,"4G/Cuboct from 24-cell 5"},1,12,1},Delimiter,{{α,0,"Twist fraction"},-1,1},LabelStyle->{11},SaveDefinitions->True]
Out[]=
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This graphic shows the results of 5 cuboctahedra (or 4-Groups, selectable) from the 5 respective 24-cells all rotated into V. You can choose which of the 12 cuboctahedra to view from each 24-cell, and which of those 5 to view (except for the primary, which is fixed, and always shown). You can slide the rotation α from left- to right-isoclinic, to see how at one extreme all cuboctahedra coincide, at the other extreme they all combine in the icosahedrally symmetric compound, and in the middle (simple direct rotation) they form an incoherent jumble.
The default values in the graphic select the first cuboctahedron from each 24-cell. This is a special case: even with no twist (α = 0), the compound is a 20-Group or C5C (depending on whether one chooses to show 4Gs or cuboctahedra) .
For all other cases, the shapes form a chaotic jumble when rotated directly into V with no twist . But with the left twist (α = 1) they all converge to a single 4G/cuboctahedron; with the right twist (α = -1), they form a right-handed 20-Group. If the original compound of 24-cells had been constructed using a right-isoclinic rotation, and then the 4Gs/cuboctahedra were compounded using a left-isoclinic rotation, we would get a left-handed 20-Group .
The default values in the graphic select the first cuboctahedron from each 24-cell. This is a special case: even with no twist (α = 0), the compound is a 20-Group or C5C (depending on whether one chooses to show 4Gs or cuboctahedra) .
For all other cases, the shapes form a chaotic jumble when rotated directly into V with no twist . But with the left twist (α = 1) they all converge to a single 4G/cuboctahedron; with the right twist (α = -1), they form a right-handed 20-Group. If the original compound of 24-cells had been constructed using a right-isoclinic rotation, and then the 4Gs/cuboctahedra were compounded using a left-isoclinic rotation, we would get a left-handed 20-Group .
Cite this as: Richard Clawson, "Gosset Subscript[4, 21] polytope, 600-cells, compound of five 24-cells, and compounding of tetrahedral 4-Groups into a 20-Group" from the Notebook Archive (2024), https://notebookarchive.org/2024-02-2v1915y
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