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Symmetrical Hamiltonian Manifolds on Regular 3D and 4D Polytopes

Coxeter Day, Banff, Canada , August 3, 2005. Symmetrical Hamiltonian Manifolds on Regular 3D and 4D Polytopes. Carlo H. Séquin EECS Computer Science Division University of California, Berkeley. START. END. Hamiltonian Path: Visits all vertices once.

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Symmetrical Hamiltonian Manifolds on Regular 3D and 4D Polytopes

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  1. Coxeter Day, Banff, Canada, August 3, 2005 Symmetrical Hamiltonian Manifoldson Regular 3D and 4D Polytopes Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. START END • Hamiltonian Path:Visits all vertices once. • Hamiltonian Cycle:A closed Ham. Path. Introduction • Eulerian Path:Uses all edges of a graph. • Eulerian Cycle:A closed Eulerian Paththat returns to the start.

  3. Map of Königsberg Leonhard Euler (1707-83) says: NO ! (1735)– because there are vertices with odd valence. • Can you find a path that crosses all seven bridges exactly once – and then returns to the start ?

  4. The Platonic Solids in 3D • Hamiltonian Cycles ? • Eulerian Cycles ?

  5. The Octahedron • How many different such Hamiltonian cycles are there ? • Can we do the same for all the other Platonic solids ? • All vertices have valence 4. • They admit 2 paths passing through. • Pink edges form Hamiltonian cycle. • Yellow edges form Hamiltonian cycle. • The two paths are congruent ! • All edges are covered. • Together they form a Eulerian cycle.

  6. Hamiltonian Dissections • Hamiltonian Cycles clearly split genus zero surfaces into two domains. • Are these domains of equal size ? • Are these domains congruent ? • Can they be used to split the solid objectso that it can be taken apart ? • ... A nice way to visualize these cycles ...

  7. Dissection of the Tetrahedron Two congruent parts

  8. Dissection of the Hexahedron (Cube) Two congruent parts

  9. Dissection of the Octahedron Two congruent parts

  10. The Other Octahedron Dissection • 3-fold symmetry • complement edges are not a Ham. cycle

  11. Dissection of the Dodecahedron ¼+ ½ +¼

  12. Dissection of the Icosahedron based on cycle with S6 - Symmetry

  13. Hamiltonian Cycles on the Icosahedron * ... that split the surface into two congruent parts that transform into each other with a C2-rotation. Some have even higher symmetry, e.g., D2

  14. Another Dissection of the Icosahedron • Not just a conical extrusion from the centroid; • Extra edges in the slide-apart direction.

  15. Multiple Uniform Coverage • Can we do what we did for the octahedron also for the other Platonic solids ?. • The problem is:those have vertices with odd valences. • If we allow to pass every edge twice, this is no longer a problem. Example: valence_3 vertex: • Try to obtain uniform double edge coverage with multiple copies of one Hamiltonian cycle!

  16. Double Edge Coverage of Tetrahedron 3 congruent Hamiltonian cycles

  17. Double Edge Coverage, Dodecahedron 3 congruent Hamiltonian cycles

  18. Double Edge Coverage on Icosahedron 5 congruent Hamiltonian cycles

  19. Double Edge Coverage on Cube Using 3 Hamiltonian paths – not cycles !

  20. The Different Hamiltonian Cycles

  21. Talk Outline • Introduction of the Hamiltonian cycle • The various Ham. cycles on the Platonic solids • Hamiltonian dissections of the Platonic solids • Multiple uniform edge coverage with Ham. cycles • Ham. cycles as constructivist sculptures (art) • Ham. cycles on the 4D regular polytopes • Solutions of the 600-Cell and the 120-Cell • Hamiltonian 2-manifolds on 4D polytopes • Volution surfaces suspended in Ham. cycles (art)

  22. Constructivist Sculptures • Use Hamiltonian Paths to make constructivist sculptures. • Inspiration by: Peter Verhoeff, Popke Bakker, Rinus Roelofs

  23. Peter Verhoeff truncatedicosahedron

  24. Hamiltonian Cycle on the edges of a dodecahedron

  25. CS 184, Fall 2004 Student homework

  26. HamCycle_2 • on two stacked dodecahedra

  27. CS 184, F’04

  28. “Hamiltonian Path” by Rinus Roelofs Space diagonals in a dodecahedron

  29. Dodecahedron with Face Diagonals • Only non-crossing diagonals may be used !

  30. Ham. Cycle with 5-fold Symmetry on the face diagonals of the dodecahedron

  31. Hamiltonian Cycle with C2-Symmetry on the face diagonals of the dodecahedron

  32. Sculpture Model of C2 Ham. Cycle made on FDM machine

  33. With Prismatic Beams ... ... mitring might be tricky !

  34. Sculpture Model of C2 Ham. Cycle made on Zcorporation 3D-Printer

  35. “C2-Symmetrical Hamiltonian Cycle” ... on face diagonals of the dodecahedron

  36. Count of Different Hamiltonian Cycles Interesting ! Crossing constraint Disjoint sets

  37. Paths on the 4D Edge Graphs • The 4D regular polytopes offer several very interesting graphs on which we can study Hamiltonian Eulerian coverage. • Start by finding Hamiltonian cycles. • Then try to obtain uniform edge coverage.

  38. The 6 Regular Polytopes in 4D From BRIDGES’2002 Talk

  39. Which 4D-to-3D Projection ?? Cell-first Face-first Edge-first Vertex-first • There are many possible ways to project the edge frame of the 4D polytopes to 3D.Example: Tesseract (Hypercube, 8-Cell) Use Cell-first: High symmetry; no coinciding vertices/edges

  40. C2 Hamiltonian Cycles on the 4D Simplex Two identical paths, complementing each other From BRIDGES’2004 Talk

  41. Ham. Cycles on the 4D Cross Polytope C3 All vertices have valence 6  need 3 paths

  42. Valence-4 vertices  requires 2 paths. There are many different solutions. Hamiltonian Cycles on the Hypercube

  43. 24-Cell: 4 Hamiltonian Cycles Alignedto show 4-fold symmetry

  44. Why Shells Make Task Easier • Decompose problem into smaller ones: • Find a suitable shell schedule; • Prepare components on shells compatible with schedule; • Find a coloring that fits the schedule and glues components together,by “rotating” the shells and connector edges within the chosen symmetry group. • Fewer combinations to deal with. • Easier to maintain desired symmetry.

  45. Rapid Prototyping Model of the 24-Cell • Noticethe 3-foldpermutationof colorsMade on the Z-corp machine.

  46. That is how far I got last year ... Solutions of the 600- and C120-Cell • 600-Cell solution found first: • Paths are “only” 120 edges long. • The 6 congruent copies add many constraints. • Shell-based approach worked well for this. • 120-Cell was tougher: • Only 2 colors:  Too many possibilities in each shell to enumerate all legal colorings. • Also a daunting challenge for backtracking, because each cycle is 600 edges long.

  47. The 600-Cell • 120 vertices,valence 12; • 720 edges;  Find 6 cycles, length 120.

  48. Shells in the 600-Cell OUTERMOST TETRAHEDRON INSIDE / OUTSIDE SYMMETRY INNERMOST TETRAHEDRON CONNECTORS SPANNING THE CENTRAL SHELL Number of segments of each type in each Hamiltonian cycle

  49. Shells in the 600-Cell • 15 shells of vertices • 49 different types of edges: • 4 intra shells with 6 (tetrahedral) edges, • 4 intra shells with 12 edges, • 28 connector shells with 12 edges, • 13 connector shells with 24 edges (= two 12-edge shells). • Inside/outside symmetry • Overall tetrahedral symmetry

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