CHSCHSUCBUCB ISIS Symmetry Congress 2001ISIS Symmetry Congress 2001

Symmetries on the Sphere

Carlo H. Séquin

University of California, Berkeley

CHSCHSUCBUCB OutlineOutline

2 Tools to design / construct artistic artefacts:

“Escher Balls”: Spherical Escher Tilings

“Viae Globi”: Closed Curves on a Sphere

Discuss the use of Symmetry

Discuss Symmetry-Breaking in order to obtain artistically more interesting results.

CHSCHSUCBUCB

Jane YenJane YenCarlo SCarlo Sééquinquin

UC BerkeleyUC Berkeley

[1] M.C. Escher, His Life and Complete Graphic Work

Spherical Escher TilingsSpherical Escher Tilings

CHSCHSUCBUCB IntroductionIntroduction

M.C. Escher

graphic artist & print maker

myriad of famous planar tilings

why so few 3D designs?

[2] M.C. Escher: Visions of Symmetry

CHSCHSUCBUCB Spherical TilingsSpherical Tilings

Spherical Symmetry is difficult

Hard to understand

Hard to visualize

Hard to make the final object

[1]

CHSCHSUCBUCB Our GoalOur Goal

Develop a system to easily design and manufacture “Escher spheres” = spherical balls composed of identical tiles.

Provide visual feedback

Guarantee that the tiles join properly

Allow for bas-relief decorations

Output for manufacturing of physical models

CHSCHSUCBUCB Interface DesignInterface Design

How can we make the system intuitive and easy to use?

What is the best way to communicate how spherical symmetry works?

[1]

CHSCHSUCBUCB Spherical SymmetrySpherical Symmetry

The Platonic Solids

tetrahedron octahedron cube dodecahedron icosahedron

R3 R5 R5R3 R3 R2

CHSCHSUCBUCB Introduction to TilingIntroduction to Tiling

Spherical Symmetry - defined by 7 groups

1) oriented tetrahedron 12 elem: E, 8C3, 3C2

2) straight tetrahedron 24 elem: E, 8C3, 3C2, 6S4, 6sd

3) double tetrahedron 24 elem: E, 8C3, 3C2, i, 8S4, 3sd

4) oriented octahedron/cube 24 elem: E, 8C3, 6C2, 6C4, 3C42

5) straight octahedron/cube 48 elem: E, 8C3, 6C2, 6C4, 3C42, i, 8S6, 6S4, 6sd, 6sd

6) oriented icosa/dodecah. 60 elem: E, 20C3, 15C2, 12C5, 12C52

7) straight icosa/dodecah. 120 elem: E, 20C3, 15C2, 12C5, 12C52, i, 20S6, 12S10,

12S103, 15s

Platonic Solids:

With duals:

1,2)

3)

6,7)4,5)

CHSCHSUCBUCB

Escher Sphere EditorEscher Sphere Editor

CHSCHSUCBUCB How the Program WorksHow the Program Works

Choose symmetry based on a Platonic solid

Choose an initial tiling pattern to edit

= starting place

Example:

Tetrahedron

R3

R2R3

R2

R3

R3

R3

R2

Tile 1 Tile 2

R3

R2

CHSCHSUCBUCB Using an Initial Tiling PatternUsing an Initial Tiling Pattern

• Easier to understand consequences of moving points• Guarantees proper tiling• Requires user to select the “right” initial tile

[2]

Tile 1 Tile 2 Tile 2

CHSCHSUCBUCB Modifying the TileModifying the Tile

Insert and move boundary points

system automatically updates the tile based on symmetry

Add interior detail points

CHSCHSUCBUCB

Adding Bas-ReliefAdding Bas-Relief

Stereographically project and triangulate:

Radial offsets can be given to points: individually or in groups separate mode from editing boundary points

CHSCHSUCBUCB

Creating a SolidCreating a Solid

The surface is extruded radially inward or outward extrusion, spherical or detailed base

Output in a format for free-form fabrication individual tiles or entire ball

CHSCHSUCBUCB Several Fabrication TechnologiesSeveral Fabrication Technologies

Both are layered manufacturing technologies

Fused Deposition Modeling Z-Corp 3D Color Printer

- parts are made of plastic - starch powder glued together - each part is a solid color - parts can have multiple colors => assembly

CHSCHSUCBUCB Fused Deposition ModelingFused Deposition Modeling

supportmaterial

movinghead

inside the FDM machine

CHSCHSUCBUCB 3D-Printing (Z-Corporation)3D-Printing (Z-Corporation)

infiltrationde-powdering

CHSCHSUCBUCB 12 Lizard Tiles (FDM)12 Lizard Tiles (FDM)

Pattern 1R3

R2R3

R2

R3

R3

R3

R2

Pattern 2

CHSCHSUCBUCB 12 Fish Tiles (4 colors)12 Fish Tiles (4 colors)

Z-CorpSolid monolithic ball

FDM Hollow, hand-assembled

CHSCHSUCBUCB 24 Bird Tiles24 Bird Tiles

Z-Corp4-color tiling

FDM2-color tiling

CHSCHSUCBUCB Tiles Spanning Half the SphereTiles Spanning Half the Sphere

Z-Corp6-color tiling

FDM4-color tiling

CHSCHSUCBUCB Hollow StructuresHollow Structures

Z-CorpBlow loose powder

from eye holes

FDM Hard to remove the

support material

CHSCHSUCBUCB Frame StructuresFrame Structures

Z-CorpColorful but fragile

FDM Support removal tricky,but sturdy end-product

CHSCHSUCBUCB

3D PrinterZ-Corp.

60 Highly Interlocking Tiles60 Highly Interlocking Tiles

CHSCHSUCBUCB 60 Butterfly Tiles (FDM)60 Butterfly Tiles (FDM)

CHSCHSUCBUCB PART 2: “Viae Globi”PART 2: “Viae Globi” (Roads on a Sphere)(Roads on a Sphere)

• Symmetrical, Symmetrical, closed closed curvescurves on a sphereon a sphere

• Inspiration: Inspiration:

Brent Collins’ Brent Collins’ “Pax Mundi”“Pax Mundi”

CHSCHSUCBUCB Sculptures by Naum GaboSculptures by Naum Gabo

Pathway on a sphere:

Edge of surface is like seam of tennis ball;

==> 2-period Gabo curve.

CHSCHSUCBUCB 2-period Gabo curve2-period Gabo curve

Approximation with quartic B-spline

with 8 control points per period,

but only 3 DOF are used.

CHSCHSUCBUCB 3-period Gabo curve3-period Gabo curve

Same construction as for 2-period curve

CHSCHSUCBUCB ““Pax Mundi” RevisitedPax Mundi” Revisited

Can be seen as:

Amplitude modulated, 4-period Gabo curve

CHSCHSUCBUCB SLIDE-UI for “Pax Mundi” ShapesSLIDE-UI for “Pax Mundi” Shapes

Good combination of interactive 3D graphicsand parameterizable procedural constructs.

CHSCHSUCBUCB FDM Part with SupportFDM Part with Support

as it comes out of the machine

CHSCHSUCBUCB ““Viae Globi” Family Viae Globi” Family (Roads on a Sphere)(Roads on a Sphere)

2 3 4 5 periods

CHSCHSUCBUCB 2-period Gabo sculpture2-period Gabo sculpture

Looks more like a surface than a ribbon on a sphere.

CHSCHSUCBUCB Via Globi 3 (Stone)Via Globi 3 (Stone)

Wilmin Martono

CHSCHSUCBUCB Via Globi 5 (Wood)Via Globi 5 (Wood)

Wilmin Martono

CHSCHSUCBUCB Via Globi 5 (Gold)Via Globi 5 (Gold)

Wilmin Martono

CHSCHSUCBUCB More Complex PathwaysMore Complex Pathways

Tried to maintain high degree of symmetry,

but wanted higly convoluted paths …

Not as easy as I thought !

Tried to work with Hamiltonian pathson the edges of a Platonic solid,but had only moderate success.

Used free-hand sketching with C-splines,

then edited control vertices coordinatesto adhere to desired symmetry group.

CHSCHSUCBUCB ““Viae Globi”Viae Globi”

Sometimes I started by sketching on a tennis ball !

CHSCHSUCBUCB A Better CAD Tool is Needed !A Better CAD Tool is Needed !

A way to make nice curvy paths on the surface of a sphere:==> C-splines.

A way to sweep interesting cross sectionsalong these spherical paths:==> SLIDE.

A way to fabricate the resulting designs:==> Our FDM machine.

CHSCHSUCBUCB ““Circle-Splines” (SIGGRAPH 2001)Circle-Splines” (SIGGRAPH 2001)

Carlo SéquinJane Yen

On the plane -- and on the sphere

CHSCHSUCBUCB Defining the Basic Path ShapesDefining the Basic Path Shapes

Use Platonic or Archimedean solids as “guides”:

Place control points of an approximating spline at the vertices,

or place control points of an interpolating spline at edge-midpoints.

Spline formalism will do the smoothing.

Maintain some desirable degree of symmetry,

and make sure that curve closes – difficult !

Often leads to the same basic shapes again …

CHSCHSUCBUCB Hamiltonian PathsHamiltonian Paths

Strictly realizable only on octahedron! Gabo-2 path.

Pseudo Hamiltonian path (multiple vertex visits) Gabo-3 path.

CHSCHSUCBUCB Another Conceptual ApproachAnother Conceptual Approach

Start from a closed curve, e.g., the equator

And gradually deform it by introducing twisting vortex movements:

CHSCHSUCBUCB ““Maloja” -- FDM partMaloja” -- FDM part

A rather winding Swiss mountain pass road in the upper Engadin.

CHSCHSUCBUCB ““Stelvio”Stelvio”

An even more convoluted alpine pass in Italy.

CHSCHSUCBUCB ““Altamont”Altamont”

Celebrating American multi-lane highways.

CHSCHSUCBUCB ““Lombard”Lombard”

A very famous crooked street in San Francisco

Note that I switched to a flat ribbon.

CHSCHSUCBUCB Varying the Azimuth ParameterVarying the Azimuth Parameter

Setting the orientation of the cross section …

… by Frenet frame … using torsion-minimization withtwo different azimuth values

CHSCHSUCBUCB ““Aurora”Aurora”

Path ~ Via Globi 2

Ribbon now lies perpendicular to sphere surface.

Reminded me ofthe bands in anAurora Borrealis.

CHSCHSUCBUCB ““Aurora - T”Aurora - T”

Same sweep path ~ Via Globi 2

Ribbon now lies tangential to sphere surface.

CHSCHSUCBUCB ““Aurora – F” (views from 3 sides)Aurora – F” (views from 3 sides)

Still the same sweep path ~ Via Globi 2

Ribbon orientation now determined by Frenet frame.

CHSCHSUCBUCB ““Aurora-M”Aurora-M”

Same path on sphere,

but more play with the swept cross section.

This is a Moebius band.

It is morphed from a concave shape at the bottom to a flat ribbon at the top of the flower.

CHSCHSUCBUCB ConclusionsConclusions

Focus on spherical symmetries to make artistic artefacts.

Undecorated Platonic solids are artistically not too interesting (too much symmetry).

Breaking the mirror symmetries leads to more interesting shapes (snubcube)

use tiles with rotational symmetries, or asymmetrical wiggles on Gabo curves.

Can also break symmetry with a varying orientation of the swept cross section.

CHSCHSUCBUCB We have come full circle …We have come full circle …