Symmetrical Hamiltonian ManifoldsSymmetrical Hamiltonian Manifoldson Regular 3D and 4D Polytopeson Regular 3D and 4D Polytopes

Carlo H. Séquin

EECS Computer Science Division

University of California, Berkeley

Coxeter Day, Banff, Canada, August 3, 2005

IntroductionIntroduction

Eulerian Path: Uses all edges of a graph.

Eulerian Cycle: A closed Eulerian Paththat returns to the start. END

START

Hamiltonian Path: Visits all vertices once.

Hamiltonian Cycle: A closed Ham. Path.

Map of KönigsbergMap of Königsberg

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.

The Platonic Solids in 3DThe Platonic Solids in 3D

Hamiltonian Cycles ? Eulerian Cycles ?

The OctahedronThe Octahedron 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.

How many different such Hamiltonian cycles are there ?

Can we do the same for all the other Platonic solids ?

Hamiltonian DissectionsHamiltonian 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 ...

Dissection of the TetrahedronDissection of the Tetrahedron

Two congruent parts

Dissection of the Hexahedron (Cube)Dissection of the Hexahedron (Cube)

Two congruent parts

Dissection of the OctahedronDissection of the Octahedron

Two congruent parts

The Other Octahedron DissectionThe Other Octahedron Dissection

3-fold symmetry

complement edges are not a Ham. cycle

Dissection of the DodecahedronDissection of the Dodecahedron

¼ + ½ + ¼

Dissection of the IcosahedronDissection of the Icosahedron

based on cycle with S6 - Symmetry

Hamiltonian Cycles on the IcosahedronHamiltonian 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

*

Another Dissection of the IcosahedronAnother Dissection of the Icosahedron

Not just a conical extrusion from the centroid;

Extra edges in the slide-apart direction.

Multiple Uniform Coverage 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!

Double Edge Coverage of TetrahedronDouble Edge Coverage of Tetrahedron

3 congruent Hamiltonian cycles

Double Edge Coverage, DodecahedronDouble Edge Coverage, Dodecahedron

3 congruent Hamiltonian cycles

Double Edge Coverage on IcosahedronDouble Edge Coverage on Icosahedron

5 congruent Hamiltonian cycles

Double Edge Coverage on CubeDouble Edge Coverage on Cube

Using 3 Hamiltonian paths – not cycles !

The Different Hamiltonian CyclesThe Different Hamiltonian Cycles

Edges # of H.C. # Dissect. Uniform edge cover

Tetrahedron 4 1 1 yes

Cube 12 1 1 (yes)

Octahedron 12 2 2 yes

Dodecahedron 30 1 0 yes

Icosahedron 30 11 2 yes

Talk OutlineTalk 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)

Constructivist SculpturesConstructivist Sculptures

Use Hamiltonian Paths to make constructivist sculptures.

Inspiration by: Peter Verhoeff, Popke Bakker, Rinus Roelofs

Peter VerhoeffPeter Verhoeff

truncatedicosahedron

Hamiltonian CycleHamiltonian Cycle

on the edges of a dodecahedron

CS 184, Fall 2004CS 184, Fall 2004

Student homework

HamCycle_2HamCycle_2

on two stacked dodecahedra

CS 184, F’04CS 184, F’04

““Hamiltonian Path” by Rinus RoelofsHamiltonian Path” by Rinus Roelofs

Space diagonals in a dodecahedron

Dodecahedron with Face DiagonalsDodecahedron with Face Diagonals

Only non-crossing diagonals may be used !

Ham. Cycle with 5-fold SymmetryHam. Cycle with 5-fold Symmetry

on the face diagonals of the dodecahedron

Hamiltonian Cycle with CHamiltonian Cycle with C22-Symmetry-Symmetry

on the face diagonals of the dodecahedron

Sculpture Model of CSculpture Model of C22 Ham. Cycle Ham. Cycle

made on FDM machine

With Prismatic Beams ...With Prismatic Beams ...

... mitring might be tricky !

Sculpture Model of CSculpture Model of C22 Ham. Cycle Ham. Cycle

made on Zcorporation 3D-Printer

““CC22-Symmetrical Hamiltonian Cycle”-Symmetrical Hamiltonian Cycle”

... on face diagonals of the dodecahedron

Count of Different Hamiltonian CyclesCount of Different Hamiltonian Cycles

Edges Face Diag. Space Diag. Diam. Axes

Tetra 4 1HC 0 --- 0 --- 0 ---

Octa 12 2HC 0 --- 0 --- 3 0HC

(three pairs)

Cube 12 1HC 12 0HC

(two tetras)

0 --- 4 0HC

(four pairs)

Icosa 30 11HC

0 --- 30 0HC

( 10 diagonals)

6 0HC

(six pairs)

Dodeca 30 1HC 60 2 ? 60 2 ??

30 0HC

10 0HC

(ten pairs)

Disjoint setsCrossing constraintInteresting !

Paths on the 4D Edge GraphsPaths 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.

The 6 Regular Polytopes in 4DThe 6 Regular Polytopes in 4D

From BRIDGES’2002 Talk

Which 4D-to-3D Projection ??Which 4D-to-3D Projection ??

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 SimplexHamiltonian Cycles on the 4D Simplex

Two identical paths, complementing each other

C2

From BRIDGES’2004 Talk

Ham. Cycles on the 4D Cross PolytopeHam. Cycles on the 4D Cross Polytope

All vertices have valence 6 need 3 paths

C3

Hamiltonian Cycles on the HypercubeHamiltonian Cycles on the Hypercube

Valence-4 vertices requires 2 paths.

There are many different solutions.

24-Cell: 4 Hamiltonian Cycles24-Cell: 4 Hamiltonian Cycles

Aligned to show 4-fold symmetry

Why Shells Make Task EasierWhy 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.

Rapid Prototyping Model of the 24-CellRapid Prototyping Model of the 24-Cell

Noticethe 3-foldpermutationof colors

Made on the Z-corp machine.

Solutions of the 600- and C120-CellSolutions 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.

That is how far I got last year ...

The 600-CellThe 600-Cell

120 vertices,valence 12;

720 edges;

Find 6 cycles, length 120.

Shells in the 600-CellShells in the 600-Cell

Number of segments of each type in each Hamiltonian cycle

OUTERMOST TETRAHEDRON

INN

ER

MO

ST

TE

TR

AH

ED

RO

N

CONNECTORS SPANNING THE CENTRAL SHELL

INSIDE / OUTSIDE SYMMETRY

Shells in the 600-CellShells 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

Shell-Based Search on 600-CellShell-Based Search on 600-Cell Shell Pre-Coloring:

For each (half-)shell of 12 edgesassign two prototype edges of one color, so that five differently colored copies of this pair can be placed without causing any interferences.

We always find exactly 12 different such assignments.

Shell “Rotation”: Add one of the 12 possible shell solutions Check color condition:

each node has 2 edges of all 6 colors Check loop condition:

no cycle shorter than 120 edges allowed. If necessary, backtrack!

One Ham. Cycle on the 600-CellOne Ham. Cycle on the 600-Cell

Thanks to Daniel Chen for programming this.

Hamiltonian Cycles on the 600-CellHamiltonian Cycles on the 600-Cell

1 cycle

Hamiltonian Cycles on the 600-CellHamiltonian Cycles on the 600-Cell

2 cycles

Hamiltonian Cycles on the 600-CellHamiltonian Cycles on the 600-Cell

4 cycles

Hamiltonian Cycles on the 600-CellHamiltonian Cycles on the 600-Cell

6 cycles

The Uncolored 120-CellThe Uncolored 120-Cell

600 vertices of valence 4, 1200 edges.

Find 2 congruent Hamiltonian cycles length 600.

3D Color Printer3D Color Printer (Z Corporation)(Z Corporation)

2004 (Brute-force Approach) for 120-Cell2004 (Brute-force Approach) for 120-Cell

Build both cycles simultaneously: Edges mirrored at 3D centroid get different colors Possible plane-mirror operations or C2 rotations are excluded,

because they all map some edges of the dodecahedron back onto themselves.

Do (single) path search with backtracking:Extend path without closing loop before length 600.

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 !

Trying to reduce the depth of the search tree, look for symmetries in prototype path itself.

Neither 3-fold nor 5-fold symmetry is possible:

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-CellSymmetry Exploits for the 120-Cell

Shells in the 120-CellShells in the 120-Cell

Shell-based Approach for 120-Cell ?Shell-based Approach for 120-Cell ?

In the meantime we had solved the 600-Cell.

Shell approach is not practical for 120-Cell

Up to 120 edges per shell, only 2 colors: too many possible shell colorings ! impractical to pre-compute !

Edge-Based Coloring ApproachEdge-Based Coloring Approach

Grow multiple path segments, filling up shells in an orderly manner,avoiding any loop building: over-constrained impasses at the end.

Grow multiple path segments, extending segments in random order,but coloring constrained junctions first: very quick success !

A B

One Ham. Cycle on the 120-CellOne Ham. Cycle on the 120-Cell

Thanks to Daniel Chen for programming this.

Hamiltonian Cycles on the 120-CellHamiltonian Cycles on the 120-Cell

path differentiation with profiles:

120-Cell in De-powder Station120-Cell in De-powder Station

120-Cell with Hamiltonian Cycles120-Cell with Hamiltonian Cycles

Hamiltonian Cycles on 120-CellHamiltonian Cycles on 120-Cell

two paths distinguished by cross sections of the beams (circular / triangular)

Hamiltonian 2-ManifoldsHamiltonian 2-Manifoldswhere: what: connects how: what:

on edge graph:

Ham. Path(1-manifold)

touches all vertices(0-manifolds)

on edge graph:

Ham. Cycle(1-manifold)

passes thru all vertices(0-manifolds)

on polytope

Ham. Surface(2-manifold)

touches all edges(1-manifolds)

on polytope

Ham. Shell(2-manifold)

passes thru all edges(1-manifolds)

Three Levels of ChallengesThree 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 SimplexHamiltonian Surface on 4D Simplex

Moebius strip of 5 triangles: 5 open edges, 5 inner edges;

Inner/outer edges of same color form Hamiltonian cycles !

Two of these will cover all 10 faces of the 4D simplex.

Hamiltonian Closed Shell on HypercubeHamiltonian Closed Shell on Hypercube

Uses 16 out of 24 faces; all inner edges;

3 copies of this 2-manifold yield double coverage.

Hamiltonian Surface on HypercubeHamiltonian Surface on Hypercube

Uses 12 out of 24 faces; 16 inner, 16 outer edges;

This surface is congruent to its complement in 4D !

Two copies (in 4D, not in 3D) yield simple coverage.

Ham. 2-Manifold on 4D Cross PolytopeHam. 2-Manifold on 4D Cross Polytope

16 triangles form a closed Hamiltonian shell (torus);

2 copies of those cover all faces of the Cross Polytope.

What About the 3 Big Ones ??What About the 3 Big Ones ??

Work in progress:

24-Cell: almost there ... ?

120-Cell: first useful results

600-Cell: have not seriously started yet

2-Manifold Coverage of the 24-Cell2-Manifold Coverage of the 24-Cell

Some basic arithmetic:

There are 96 edges of valence 3

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.

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-CellHam. 2-Manifold on 24-Cell

Found 2 loops of 24 triangles each,-- not yet the desired solution!

2-Manifold Coverage of the 24-Cell2-Manifold Coverage of the 24-Cell

Symmetrical partial solution around z-axis

2-Manifold Coverage of the 120-Cell2-Manifold Coverage of the 120-Cell

Some basic arithmetic:

There are 1200 edges of valence 3.

Looking for: Open surface of 360 pentagons, with 600 border edges, and passing through 600 edges.

GEP: 1 –600{V} +1200{E} –360{F} = 241 Ribbon Loops. Imagine a main loop with 240 side loops;

Needs 480 branch points.

On each pentagon on average 3.333 edges are used by faces of the same color; this is equivalent to 1.333 branches.

360 pentagons * 1.333 branches 480 branch points !

2-Manifold Coverage of 120-Cell2-Manifold Coverage of 120-Cell

Study of the emerging coloring patterns at the core.

2-Manifold Coverage of the 120-Cell2-Manifold Coverage of the 120-Cell

We have found a 2-manifold coverage,with 1-2 pentagons on each edge, and exactly 3 pentagons around each vertex.

This is not congruent to its complement.

Probably does not have maximal possible symmetry.

Can we also have all the pass-thru edges of one color form a Hamiltonian cycle ?

2-Manifold Coverage of the 600-Cell2-Manifold Coverage of the 600-Cell

Some basic arithmetic: There are 720 edges of valence 5,

Valence 5 causes extra conceptual difficulties. 3600 edge uses.

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 ??

Perhaps, try something else ...

2-Manifold Coverage of the 600-Cell2-Manifold Coverage of the 600-Cell

Another attempt:

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 ...

Need more thinking ...

Stay tuned ... !

ConclusionsConclusions

Wonderful abstract beauty !

Symmetries, interactions between Ham. cycles and Ham. 2-manifolds.

Mind-bending, headache-creating ...

End on an easier note ...

Make surfaces of a different kind ...

““Volution”Volution” Surfaces Spanning Surfaces Spanning Hamiltonian CyclesHamiltonian Cycles

Back to 3D-space and art ...

VolutionVolution Surfaces (Bridges 2003) Surfaces (Bridges 2003)

“Volution’s Evolution”

Minimal surfaces of different genussuspended in a wire frame composed of

12 quarter-circles on the surface of a cube.

New New VolutionVolution Surfaces Surfaces

Use the Hamiltonian Cycles

found on the Platonic solids

to suspend Volution surfaces.

On the DodecahedronOn the Dodecahedron

2 holes

On the IcosahedronOn the Icosahedron

+ 4 tubes

Many Different Models for IcosahedronMany Different Models for Icosahedron

How I Start Designing these ObjectsHow I Start Designing these Objects

Or with Zome-Tool ModelsOr with Zome-Tool Models

Paper cylinders mark positions of tunnels.

Make a Crude Polyhedral ModelMake a Crude Polyhedral Model

refine with Brakke’s “Surface Evolver”

Make a 3D ObjectMake a 3D Object

Import to SLIDE, apply some surface offset;

export as an STL file, and send to an RP machine.

Icosa_Vol_J9Icosa_Vol_J9

6 tubes

Questions ?Questions ?

QUESTIONS ?QUESTIONS ?