1 / 24

G4G9

G4G9. A 10 -dimensional Jewel. Carlo H. Séquin. EECS Computer Science Division University of California, Berkeley. What Is a Regular Polytope ?. “Polytope” is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), “Polychoron” (4D), … to arbitrary dimensions.

kairos
Download Presentation

G4G9

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. G4G9 A 10-dimensional Jewel Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. What Is a Regular Polytope ? • “Polytope”is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), “Polychoron” (4D), …to arbitrary dimensions. • “Regular”means: All the vertices, edges, faces, cells…are indistinguishable from each another. • Examples in 2D: Regular n-gons:There are infinitely many of them!

  3. In 3 Dimensions . . . There are only 5 Platonic solids: They are composed from the regular 2D polygons. Only triangles, squares, and pentagons are useful: other n-gons are too ”round”;they cannot form nice 3D corners.

  4. In 4D Space . . . The same constructive approach continues: We can use the Platonic solids as building blocks to form the “crust” of regular 4D polychora. Only 6 constructions are successful. Only 4 of the 5 Platonic solids can be used; the icosahedron is too round (dihedral angle > 120°). This is the result . . . 

  5. The 6 Regular Polychorain 4-D . . .

  6. 120-Cell ( 600V, 1200E, 720F ) • Cell-first,extremeperspectiveprojection • Z-Corp. model

  7. 600-Cell ( 120V, 720E, 1200F ) (parallel proj.) • David Richter

  8. In Higher-Dimensional Spaces . . . We can recursively construct new regular polytopes Using the ones from one dimension lower spceas the boundary (“surface”) element. But from dimension 5 onwards, there are just 3 each: N-Simplices (like tetrahedron) N-Cubes (hypercubes, measure-polytopes) N-Orthoplexes (cross-polytopes = duals of n-cube)

  9. Thinking Outside the Box . . . Allow polyhedron faces to intersect . . . or even to be self-intersecting:

  10. In 3D:  Kepler-PoinsotSolids • Mutually intersecting faces: (all) • Faces in the form of pentagrams: (3,4) • + 10 such objects in 4D space ! 1 2 3 4 Gr. Dodeca, Gr. Icosa, Gr. Stell. Dodeca, Sm. Stell. Dodeca

  11. Single-Sided Polychora in 4D But we can do even wilder things ... Let’s allow single-sided polytope constructions like a Möbius band or a Klein bottle. In 4D we can make objects that close on themselves;they have the topology of the projective plane. The simplest one is the hemi-cube . . .

  12. Hemi-Cube • Single-sided;not a solid any more! • Has the connectivity of the projective plane! 3 faces only vertex graph K4 3 saddle faces

  13. Physical Model of a Hemi-cube Made on a Fused-Deposition Modeling Machine

  14. Hemi-Dodecahedron • A self-intersecting, single-sided 3D cell • Is only geometrically regular in 9D space connect oppositeperimeter points connectivity: Petersen graph six warped pentagons

  15. Hemi-Icosahedron • A self-intersecting, single-sided 3D cell • Is only geometrically regular in 5D  This is the BUILDING BLOCK for the 10D JEWEL ! connect oppositeperimeter points connectivity: graph K6 5-D simplex;warped octahedron

  16. The Complete Connectivity Diagram • From: Coxeter [2], colored by Tom Ruen

  17. Combining Two Cells Starter cell with4 tetra faces Add new cells on the inside ! All the edges of the first 5 cells.  A highly confusing, intersecting mess!

  18. Six More Cells !

  19. Regular Hendecachoron (11-Cell) Solid faces Transparency 11 vertices, 55 edges, 55 faces, 11 cells  self dual

  20. The Full 11-Cell

  21. The 10D Jewel 660 automorphisms – a building block of our universe ?

  22. Hands-on Construction Project This afternoon we will build card-board models of the hemi-icosahedron. Thanks to Chris Palmer (now at U.C. Berkeley):for designing the parameterized template and for laser cutting the 30 colored parts.

  23. What Is the 11-Cell Good For ? • A neat mathematical object ! • A piece of “absolute truth”:(Does not change with style, new experiments) • A 10-dimensional building block …(Physicists believe Universe may be 10-D)

  24. Are there More Polychora Like This ? • Yes – one more: the 57-Cell • Built from 57 Hemi-dodecahedra • 5 such single-sided cells join around edges • It is also self-dual: 57 V, 171 E, 171 F, 57 C. • I may talk about it at G4G57 . . .

More Related