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

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.

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

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

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 ?

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...

Dissection of the Tetrahedron Two congruent parts

Dissection of the Hexahedron (Cube) Two congruent parts

Dissection of the Octahedron Two congruent parts

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

Dissection of the Dodecahedron ¼ + ½ + ¼

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

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 *

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

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!

Double Edge Coverage of Tetrahedron 3 congruent Hamiltonian cycles

Double Edge Coverage, Dodecahedron 3 congruent Hamiltonian cycles

Double Edge Coverage on Icosahedron 5 congruent Hamiltonian cycles

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

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

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)

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

Peter Verhoeff truncated icosahedron

Hamiltonian Cycle on the edges of a dodecahedron

CS 184, Fall 2004 Student homework

HamCycle_2 u on two stacked dodecahedra

CS 184, F’04

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

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

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

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

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

With Prismatic Beams mitring might be tricky !

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

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

Count of Different Hamiltonian Cycles EdgesFace Diag.Space Diag.Diam. Axes Tetra 4 1 HC Octa12 2 HC HC (three pairs) Cube12 1 HC 12 0 HC (two tetras) HC (four pairs) Icosa 30  11 HC 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 !

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.

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

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

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

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

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

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

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.

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

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...

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

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

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

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!

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

Hamiltonian Cycles on the 600-Cell 1 cycle

Hamiltonian Cycles on the 600-Cell 2 cycles

Hamiltonian Cycles on the 600-Cell 4 cycles

Hamiltonian Cycles on the 600-Cell 6 cycles

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

3D Color Printer (Z Corporation)

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 !

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

Shells in the 120-Cell

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 !

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

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

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

120-Cell in De-powder Station

120-Cell with Hamiltonian Cycles

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

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)

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...

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.

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.

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.

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.

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

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).

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

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

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 edges are used by faces of the same color; this is equivalent to branches. l 360 pentagons * branches  480 branch points !

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

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 ?

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...

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... !

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...

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

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.

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

On the Dodecahedron 2 holes

On the Icosahedron + 4 tubes

Many Different Models for Icosahedron

How I Start Designing these Objects

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

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

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

Icosa_Vol_J9 6 tubes

Questions ?

QUESTIONS ?