graphics lunch, nov. 15, 2007 the regular 4-dimensional 11-cell & 57-cell carlo h. séquin...
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![Page 1: Graphics Lunch, Nov. 15, 2007 The Regular 4-Dimensional 11-Cell & 57-Cell Carlo H. Séquin University of California, Berkeley](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649d545503460f94a31ab6/html5/thumbnails/1.jpg)
Graphics Lunch, Nov. 15, 2007Graphics Lunch, Nov. 15, 2007
The Regular 4-Dimensional
11-Cell & 57-Cell
Carlo H. Séquin
University of California, Berkeley
![Page 2: Graphics Lunch, Nov. 15, 2007 The Regular 4-Dimensional 11-Cell & 57-Cell Carlo H. Séquin University of California, Berkeley](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649d545503460f94a31ab6/html5/thumbnails/2.jpg)
4 Dimensions ??4 Dimensions ??
The 4th dimension exists !and it is NOT “time” !
The 11-Cell and 57-Cell are complex, self-intersecting, 4-dimensional geometrical objects.
They cannot be visualized / explained with a single image / model.
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San FranciscoSan Francisco
Cannot be understood from one single shot !
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To Get to Know San FranciscoTo Get to Know San Francisco
need a rich assembly of impressions,
then form an “image” in your mind...
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“Regular”means: All the vertices and edgesare indistinguishable from each another.
There are infinitely many regular n-gons !
Use them to build regular 3D objects
Regular Polygons in 2 DimensionsRegular Polygons in 2 Dimensions
. . .
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Regular Polyhedra in 3-DRegular Polyhedra in 3-D(made from regular 2-D n-gons)(made from regular 2-D n-gons)
The Platonic Solids:
There are only 5. Why ? …
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Why Only 5 Platonic Solids ?Why Only 5 Platonic Solids ?
Ways to build a regular convex corner: from triangles:
3, 4, or 5 around a corner; 3
from squares: only 3 around a corner; 1 . . .
from pentagons: only 3 around a corner; 1
from hexagons: planar tiling, does not close. 0
higher N-gons: do not fit around vertex without undulations (forming saddles).
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Let’s Build Some 4-D Polychora Let’s Build Some 4-D Polychora “multi-cell”“multi-cell”
By analogy with 3-D polyhedra:
Each will be bounded by 3-D cellsin the shape of some Platonic solid.
Around every edge the same small numberof Platonic cells will join together.(That number has to be small enough,so that some wedge of free 3D space is left.)
This gap then gets forcibly closed,thereby producing bending into 4-D.
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AllAll Regular “Platonic” Polychora in 4-D Regular “Platonic” Polychora in 4-D
Using Tetrahedra (Dihedral angle = 70.5°):
3 around an edge (211.5°) (5 cells) Simplex
4 around an edge (282.0°) (16 cells) Cross polytope
5 around an edge (352.5°) (600 cells) “600-Cell”
Using Cubes (90°):
3 around an edge (270.0°) (8 cells) Hypercube
Using Octahedra (109.5°):
3 around an edge (328.5°) (24 cells) Hyper-octahedron
Using Dodecahedra (116.5°):
3 around an edge (349.5°) (120 cells) “120-Cell”
Using Icosahedra (138.2° > 120° ):
NONE: angle too large (414.6°).
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How to View a Higher-D Polytope ?How to View a Higher-D Polytope ?
For a 3-D object on a 2-D screen:
Shadow of a solid object is mostly a blob.
Better to use wire frame, so we can also see what is going on on the back side.
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Oblique ProjectionsOblique Projections
Cavalier Projection
3-D Cube 2-D 4-D Cube 3-D ( 2-D )
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Projections of a Hypercube to 3-DProjections of a Hypercube to 3-D
Cell-first Face-first Edge-first Vertex-first
Use Cell-first: High symmetry; no coinciding vertices/edges
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The 6 Regular Polychora in 4-DThe 6 Regular Polychora in 4-D
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120-Cell 120-Cell ( 600V, 1200E, 720F )( 600V, 1200E, 720F )
Cell-first,extremeperspectiveprojection
Z-Corp. model
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600-Cell 600-Cell ( 120V, 720E, 1200F )( 120V, 720E, 1200F )
Cell-first,extremeperspectiveprojection
Z-Corp. model
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600-Cell 600-Cell (Parallel Projection)(Parallel Projection)
David Richter
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An 11-Cell ???An 11-Cell ???
Another Regular 4-D Polychoron ?Another Regular 4-D Polychoron ?
I have just shown that there are only 6.
“11” feels like a weird number;typical numbers are: 8, 16, 24, 120, 600.
The notion of a 4-D 11-Cell seems bizarre!
![Page 18: Graphics Lunch, Nov. 15, 2007 The Regular 4-Dimensional 11-Cell & 57-Cell Carlo H. Séquin University of California, Berkeley](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649d545503460f94a31ab6/html5/thumbnails/18.jpg)
Kepler-Poinsot Polyhedra in 3-DKepler-Poinsot Polyhedra in 3-D
Mutually intersecting faces (all above)
Faces in the form of pentagrams (#3,4)
Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca
1 2 3 4
But in 4-D we can do even “crazier” things ...
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Even “Weirder” Building Blocks:Even “Weirder” Building Blocks:
Non-orientable, self-intersecting 2D manifolds
Cross-cap Steiner’s Roman Surface
Klein bottle
Models of the 2D Projective Plane
Construct 2 regular 4D objects:the 11-Cell & the 57-Cell
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Hemi-cube Hemi-cube ((single-sidedsingle-sided, , not a solidnot a solid any more!) any more!)
Simplest object with the connectivity of the projective plane,
(But too simple to form 4-D polychora)
3 faces only vertex graph K4 3 saddle faces
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Physical Model of a Hemi-cubePhysical Model of a Hemi-cube
Made on a Fused-Deposition Modeling Machine
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Hemi-icosahedronHemi-icosahedron
A self-intersecting, single-sided 3D cell Is only geometrically regular in 5D
BUILDING BLOCK FOR THE 11-CELL
connect oppositeperimeter points
connectivity: graph K6
5-D Simplex;warped octahedron
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The Hemi-icosahedral Building BlockThe Hemi-icosahedral Building Block
Steiner’sRoman Surface
Polyhedral model with 10 triangles
with cut-out face centers
10 triangles – 15 edges – 6 vertices
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Gluing Two Steiner-Cells TogetherGluing Two Steiner-Cells Together
Two cells share one triangle face
Together they use 9 vertices
Hemi-icosahedron
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Adding Cells SequentiallyAdding Cells Sequentially
1 cell 2 cells inner faces 3rd cell 4th cell 5th cell
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A More Symmetrical ConstructionA More Symmetrical Construction Exploit the symmetry of the Steiner cell !
One Steiner cell 2nd cell added on “inside”Two cells with cut-out faces
4th white vertex used by next 3 cells
(central) 11th vertex used by last 6 cells
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How Much Further to Go ?How Much Further to Go ?
We have seen at most 5 of 11 cellsand it already looked busy (messy)!
This object cannot be “seen” in one model.It must be “assembled” in your head.
Use different ways to understand it:
Now try a “top-down” approach.
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Start With the Overall Plan ...Start With the Overall Plan ...
We know from:H.S.M. Coxeter: A Symmetrical Arrangement of Eleven Hemi-Icosahedra.Annals of Discrete Mathematics 20 (1984), pp 103-114.
The regular 4-D 11-Cellhas 11 vertices, 55 edges, 55 faces, 11 cells.
Its edges form the complete graph K11 .
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Start: Highly Symmetrical Vertex-SetStart: Highly Symmetrical Vertex-SetCenter Vertex + Tetrahedron + Octahedron
1 + 4 + 6 vertices all 55 edges shown
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The Complete Connectivity DiagramThe Complete Connectivity Diagram
Based on [ Coxeter 1984, Ann. Disc. Math 20 ]
7 6 2
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Views of the 11-CellViews of the 11-Cell
Solid faces Transparency
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The Full 11-Cell The Full 11-Cell
– a building block of our universe ?
660 automorphisms
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On to the 57-Cell . . .On to the 57-Cell . . .
It has a much more complex connectivity!
It is also self-dual: 57 V, 171 E, 171 F, 57 C.
Built from 57 Hemi-dodecahedra
5 such single-sided cells join around edges
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Hemi-dodecahedronHemi-dodecahedron
A self-intersecting, single-sided 3D cell
BUILDING BLOCK FOR THE 57-CELL
connect oppositeperimeter points
connectivity: Petersen graph
six warped pentagons
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Bottom-up Assembly of the 57-Cell (1)Bottom-up Assembly of the 57-Cell (1)
5 cells around a common edge (black)
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Bottom-up Assembly of the 57-Cell (2)Bottom-up Assembly of the 57-Cell (2)
10 cells around a common (central) vertex
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Vertex Vertex ClusterCluster
(v0)(v0)
10 cells with one corner at v0
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Edge ClusterEdge Clusteraround v1-v0around v1-v0
+ vertex clusters at both ends.
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Connectivity Graph of the 57-CellConnectivity Graph of the 57-Cell 57-Cell is self-dual. Thus the graph of all its edges
also represents the adjacency diagram of its cells.
Six edges joinat each vertex
Each cell has six neighbors
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Connectivity Graph of the 57-Cell (2)Connectivity Graph of the 57-Cell (2)
Thirty 2nd-nearest neighbors
No loops yet (graph girth is 5)
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Connectivity Graph of the 57-Cell (3)Connectivity Graph of the 57-Cell (3)
Every possible combination of 2 primary edges is used in a pentagonal face
Graphprojected into plane
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Connectivity in shell 2 : truncated hemi-icosahedron
Connectivity Graph of the 57-Cell (4)Connectivity Graph of the 57-Cell (4)
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Connectivity Graph of the 57-Cell (5)Connectivity Graph of the 57-Cell (5)
The 3 “shells” around a vertex
Diameter of graph is 3
20 vertices
30 vertices
6 vertices
1 vertex
57 vertices total
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Connectivity Graph of the 57-Cell (6)Connectivity Graph of the 57-Cell (6)
The 20 vertices in the outermost shellare connected as in a dodecahedron.
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An “Aerial Shot” of the 57-CellAn “Aerial Shot” of the 57-Cell
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A “Deconstruction” of the 57-CellA “Deconstruction” of the 57-Cell
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Questions ?Questions ?