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

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

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

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The Platonic Solids in 3D u Hamiltonian Cycles ? u Eulerian Cycles ?

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

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

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Dissection of the Tetrahedron Two congruent parts

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Dissection of the Hexahedron (Cube) Two congruent parts

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Dissection of the Octahedron Two congruent parts

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

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Dissection of the Dodecahedron ¼ + ½ + ¼

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Dissection of the Icosahedron based on cycle with S 6 - Symmetry

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Hamiltonian Cycles on the Icosahedron... that split the surface into two congruent parts that transform into each other with a C 2 -rotation. Some have even higher symmetry, e.g., D 2 *

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

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

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Double Edge Coverage of Tetrahedron 3 congruent Hamiltonian cycles

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Double Edge Coverage, Dodecahedron 3 congruent Hamiltonian cycles

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Double Edge Coverage on Icosahedron 5 congruent Hamiltonian cycles

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

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The Different Hamiltonian Cycles Edges# of H.C.# Dissect.Uniform edge cover Tetrahedron 4 1 1yes Cube12 1 1(yes) Octahedron12 2 2yes Dodecahedron30 1 0yes Icosahedron30 11 2yes

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

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

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Peter Verhoeff truncated icosahedron

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Hamiltonian Cycle on the edges of a dodecahedron

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CS 184, Fall 2004 Student homework

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HamCycle_2 u on two stacked dodecahedra

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CS 184, F’04

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

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Dodecahedron with Face Diagonals u Only non-crossing diagonals may be used !

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

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Hamiltonian Cycle with C 2 -Symmetry on the face diagonals of the dodecahedron

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Sculpture Model of C 2 Ham. Cycle made on FDM machine

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

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Sculpture Model of C 2 Ham. Cycle made on Zcorporation 3D-Printer

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“C 2 -Symmetrical Hamiltonian Cycle”... on face diagonals of the dodecahedron

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Count of Different Hamiltonian Cycles EdgesFace Diag.Space Diag.Diam. Axes Tetra 4 1 HC 0 --- Octa12 2 HC 0 --- 3 0 HC (three pairs) Cube12 1 HC 12 0 HC (two tetras) 0 --- 4 0 HC (four pairs) Icosa 30 11 HC 0 ---30 0 HC ( 10 diagonals) 6 0 HC (six pairs) Dodeca30 1 HC 60 2 ?60 2 ?? 30 0 HC 10 0 HC (ten pairs) Disjoint sets Crossing constraint Interesting !

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

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

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

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Hamiltonian Cycles on the 4D Simplex Two identical paths, complementing each other C2C2 From BRIDGES’2004 Talk

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Ham. Cycles on the 4D Cross Polytope All vertices have valence 6 need 3 paths C3C3

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

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24-Cell: 4 Hamiltonian Cycles Aligned to show 4-fold symmetry

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

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Rapid Prototyping Model of the 24-Cell u Notice the 3-fold permutation of colors Made on the Z-corp machine.

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

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

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

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

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Shell-Based Search on 600-Cell u Shell Pre-Coloring: l For each (half-)shell of 12 edges assign two prototype edges of one color, so that five differently colored copies of this pair can be placed without causing any interferences. l We always find exactly 12 different such assignments. u Shell “Rotation”: l Add one of the 12 possible shell solutions l Check color condition: each node has 2 edges of all 6 colors l Check loop condition: no cycle shorter than 120 edges allowed. l If necessary, backtrack!

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One Ham. Cycle on the 600-Cell Thanks to Daniel Chen for programming this.

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Hamiltonian Cycles on the 600-Cell 1 cycle

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Hamiltonian Cycles on the 600-Cell 2 cycles

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Hamiltonian Cycles on the 600-Cell 4 cycles

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Hamiltonian Cycles on the 600-Cell 6 cycles

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The Uncolored 120-Cell u 600 vertices of valence 4, 1200 edges. u Find 2 congruent Hamiltonian cycles length 600.

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3D Color Printer (Z Corporation)

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2004 (Brute-force Approach) for 120-Cell u Build both cycles simultaneously: Edges mirrored at 3D centroid get different colors l Possible plane-mirror operations or C2 rotations are excluded, because they all map some edges of the dodecahedron back onto themselves. u Do (single) path search with backtracking: Extend path without closing loop before length 600. u Result: We came to a length of 550/600, but then painted ourselves in a corner ! (i.e., could not connect back to the start). Thanks to Mike Pao for his programming efforts !

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u Trying to reduce the depth of the search tree, look for symmetries in prototype path itself. u Neither 3-fold nor 5-fold symmetry is possible: u We can also rule out inside/outside (w) symmetry, because of contradiction on intra_shell vs7 (see paper). Legal coloring, but asymmetrical: C3-symmetrical, but illegal coloring: Legal coloring, but asymmetrical: C5-symmetrical, but illegal cycle: Symmetry Exploits for the 120-Cell

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Shells in the 120-Cell

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Shell-based Approach for 120-Cell ? In the meantime we had solved the 600-Cell. u Shell approach is not practical for 120-Cell u Up to 120 edges per shell, only 2 colors: too many possible shell colorings ! impractical to pre-compute !

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Edge-Based Coloring Approach u Grow multiple path segments, filling up shells in an orderly manner, avoiding any loop building: over-constrained impasses at the end. u Grow multiple path segments, extending segments in random order, but coloring constrained junctions first: very quick success ! AB

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One Ham. Cycle on the 120-Cell Thanks to Daniel Chen for programming this.

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Hamiltonian Cycles on the 120-Cell path differentiation with profiles:

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120-Cell in De-powder Station

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120-Cell with Hamiltonian Cycles

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Hamiltonian Cycles on 120-Cell u two paths distinguished by cross sections of the beams (circular / triangular)

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Hamiltonian 2-Manifolds where:what:connects how:what: on edge graph: Ham. Path (1-manifold) touches allvertices (0-manifolds) on edge graph: Ham. Cycle (1-manifold) passes thru allvertices (0-manifolds) on polytope Ham. Surface (2-manifold) touches alledges (1-manifolds) on polytope Ham. Shell (2-manifold) passes thru alledges (1-manifolds)

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Three Levels of Challenges 1.) Find a Hamiltonian shell or surface for each 4D polytope. 2.) Find such a 2-manifold of proper geometry, so that multiple copies of it can lead to a uniform coverage of all polytope faces. 3.) Look for maximal symmetry and for other nice properties...

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Hamiltonian Surface on 4D Simplex u Moebius strip of 5 triangles: 5 open edges, 5 inner edges; u Inner/outer edges of same color form Hamiltonian cycles ! u Two of these will cover all 10 faces of the 4D simplex.

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Hamiltonian Closed Shell on Hypercube u Uses 16 out of 24 faces; all inner edges; u 3 copies of this 2-manifold yield double coverage.

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Hamiltonian Surface on Hypercube u Uses 12 out of 24 faces; 16 inner, 16 outer edges; u This surface is congruent to its complement in 4D ! u Two copies (in 4D, not in 3D) yield simple coverage.

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Ham. 2-Manifold on 4D Cross Polytope u 16 triangles form a closed Hamiltonian shell (torus); u 2 copies of those cover all faces of the Cross Polytope.

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What About the 3 Big Ones ?? Work in progress: u 24-Cell: almost there... ? u 120-Cell: first useful results u 600-Cell: have not seriously started yet

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2-Manifold Coverage of the 24-Cell Some basic arithmetic: u There are 96 edges of valence 3 u Possibility #1: Closed shell of 64 faces, passing through all 96 edges. Euler: 96{#E} – 24{#V} –64{#F} + 2 = 10 Genus 5; should partition 24-Cell into 2 sets of 12 octahedra. u Possibility #2: Open surface of 48 faces, with 48 border edges, and passing through 48 edges. GEP: 1 – 24{V} + 96{E} – 48{F} = 25 Ribbon Loops; might be a single closed band touching itself 24 times (with only 48 border edges, it’s a pretty tangled mess).

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Ham. 2-Manifold on 24-Cell u Found 2 loops of 24 triangles each, -- not yet the desired solution!

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2-Manifold Coverage of the 24-Cell u Symmetrical partial solution around z-axis

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2-Manifold Coverage of the 120-Cell Some basic arithmetic: u There are 1200 edges of valence 3. u Looking for: Open surface of 360 pentagons, with 600 border edges, and passing through 600 edges. u GEP: 1 –600{V} +1200{E} –360{F} = 241 Ribbon Loops. l Imagine a main loop with 240 side loops; l Needs 480 branch points. l On each pentagon on average 3.333 edges are used by faces of the same color; this is equivalent to 1.333 branches. l 360 pentagons * 1.333 branches 480 branch points !

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2-Manifold Coverage of 120-Cell Study of the emerging coloring patterns at the core.

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2-Manifold Coverage of the 120-Cell u We have found a 2-manifold coverage, with 1-2 pentagons on each edge, and exactly 3 pentagons around each vertex. u This is not congruent to its complement. u Probably does not have maximal possible symmetry. u Can we also have all the pass-thru edges of one color form a Hamiltonian cycle ?

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2-Manifold Coverage of the 600-Cell Some basic arithmetic: u There are 720 edges of valence 5, Valence 5 causes extra conceptual difficulties. 3600 edge uses. u There are several possibilities, e.g.: Open 2-manifold with 400 triangles, with 240 border edges, and passing through 480 edges (aim for coverage with 3 copies of this surface). GEP: 1 – 120{V} + 720{E} – 400{F} = 201 Ribbon Loops. Needs 400 branch points. Every triangle must serve as a branch points – but where do open edges come from ?? u Perhaps, try something else...

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2-Manifold Coverage of the 600-Cell Another attempt: u Open 2-manifold with 480 triangles, with 600 border edges, and passing through 120 edges (aim for double coverage with 5 copies of this surface). Not enough inner edges to hang everything together... u Need more thinking... u Stay tuned... !

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Conclusions u Wonderful abstract beauty ! u Symmetries, interactions between Ham. cycles and Ham. 2-manifolds. u Mind-bending, headache-creating... u End on an easier note... u Make surfaces of a different kind...

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“Volution” Surfaces Spanning Hamiltonian Cycles u Back to 3D-space and art...

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Volution Surfaces (Bridges 2003) “Volution’s Evolution” Minimal surfaces of different genus suspended in a wire frame composed of 12 quarter-circles on the surface of a cube.

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New Volution Surfaces u Use the Hamiltonian Cycles found on the Platonic solids to suspend Volution surfaces.

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On the Dodecahedron 2 holes

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On the Icosahedron + 4 tubes

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Many Different Models for Icosahedron

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How I Start Designing these Objects

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Or with Zome-Tool Models Paper cylinders mark positions of tunnels.

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Make a Crude Polyhedral Model refine with Brakke’s “Surface Evolver”

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Make a 3D Object u Import to SLIDE, apply some surface offset; u export as an STL file, and send to an RP machine.

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Icosa_Vol_J9 6 tubes

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Questions ?

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QUESTIONS ?

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