ISIS Symmetry Congress 2001 Florida 1999 ISIS Symmetry Congress 2001 Symmetries on the Sphere Carlo H. Séquin University of California, Berkeley

Outline 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.

Spherical Escher Tilings Jane Yen Carlo Séquin UC Berkeley [1] M.C. Escher, His Life and Complete Graphic Work

Introduction M.C. Escher graphic artist & print maker myriad of famous planar tilings why so few 3D designs? [2] M.C. Escher: Visions of Symmetry

Spherical Tilings Spherical Symmetry is difficult Hard to understand Hard to visualize Hard to make the final object [1]

Our 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

Interface Design How can we make the system intuitive and easy to use? What is the best way to communicate how spherical symmetry works? [1]

Spherical Symmetry The Platonic Solids tetrahedron octahedron cube dodecahedron icosahedron R3 R5 R3 R2

Introduction 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: 1,2) 4,5) 6,7) With duals: 3)

Escher Sphere Editor

How the Program Works Choose symmetry based on a Platonic solid Choose an initial tiling pattern to edit = starting place Example: Tetrahedron R3 R2 R3 R2 R3 R2 Tile 1 Tile 2

Using 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

Modifying the Tile Insert and move boundary points system automatically updates the tile based on symmetry Add interior detail points

Adding Bas-Relief Stereographically project and triangulate: Radial offsets can be given to points: individually or in groups separate mode from editing boundary points

Creating 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

Several Fabrication 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 Both are layered manufacturing technologies

Fused Deposition Modeling moving head inside the FDM machine support material

3D-Printing (Z-Corporation) de-powdering infiltration

12 Lizard Tiles (FDM) R3 R2 R3 R2 Pattern 1 Pattern 2

Hollow, hand-assembled 12 Fish Tiles (4 colors) FDM Hollow, hand-assembled Z-Corp Solid monolithic ball

24 Bird Tiles FDM 2-color tiling Z-Corp 4-color tiling

Tiles Spanning Half the Sphere FDM 4-color tiling Z-Corp 6-color tiling

Blow loose powder from eye holes Hollow Structures FDM Hard to remove the support material Z-Corp Blow loose powder from eye holes

Support removal tricky, but sturdy end-product Frame Structures FDM Support removal tricky, but sturdy end-product Z-Corp Colorful but fragile

60 Highly Interlocking Tiles 3D Printer Z-Corp.

60 Butterfly Tiles (FDM)

PART 2: “Viae Globi” (Roads on a Sphere) Symmetrical, closed curves on a sphere Inspiration: Brent Collins’ “Pax Mundi”

Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis ball; ==> 2-period Gabo curve.

2-period Gabo curve Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used.

3-period Gabo curve Same construction as for 2-period curve

“Pax Mundi” Revisited Can be seen as: Amplitude modulated, 4-period Gabo curve

SLIDE-UI for “Pax Mundi” Shapes Florida 1999 SLIDE-UI for “Pax Mundi” Shapes Good combination of interactive 3D graphics and parameterizable procedural constructs. Good combination of interactive 3D graphics and parameterizable procedural constructs.

FDM Part with Support as it comes out of the machine

“Viae Globi” Family (Roads on a Sphere) 2 3 4 5 periods

2-period Gabo sculpture Looks more like a surface than a ribbon on a sphere.

Via Globi 3 (Stone) Wilmin Martono

Via Globi 5 (Wood) Wilmin Martono

Via Globi 5 (Gold) Wilmin Martono

More 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 paths on the edges of a Platonic solid, but had only moderate success. Used free-hand sketching with C-splines, then edited control vertices coordinates to adhere to desired symmetry group.

“Viae Globi” Sometimes I started by sketching on a tennis ball !

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 sections along these spherical paths: ==> SLIDE. A way to fabricate the resulting designs: ==> Our FDM machine.

“Circle-Splines” (SIGGRAPH 2001) Carlo Séquin Jane Yen On the plane -- and on the sphere

Defining 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 …

Hamiltonian Paths Strictly realizable only on octahedron!  Gabo-2 path. Pseudo Hamiltonian path (multiple vertex visits)  Gabo-3 path.

Another Conceptual Approach Start from a closed curve, e.g., the equator And gradually deform it by introducing twisting vortex movements:

“Maloja” -- FDM part A rather winding Swiss mountain pass road in the upper Engadin.

“Stelvio” An even more convoluted alpine pass in Italy.

“Altamont” Celebrating American multi-lane highways.

“Lombard” A very famous crooked street in San Francisco Note that I switched to a flat ribbon.

Varying the Azimuth Parameter Florida 1999 Varying the Azimuth Parameter Setting the orientation of the cross section … The shape of the sweep curve on the sphere is just ONE aspect of the sculpture ! We can also play with the cross section And with the orientation of the cross section around the sweep curve. … by Frenet frame … using torsion-minimization with two different azimuth values

“Aurora” Path ~ Via Globi 2 Ribbon now lies perpendicular to sphere surface. Reminded me of the bands in an Aurora Borrealis.

“Aurora - T” Same sweep path ~ Via Globi 2 Ribbon now lies tangential to sphere surface.

“Aurora – F” (views from 3 sides) Still the same sweep path ~ Via Globi 2 Ribbon orientation now determined by Frenet frame.

“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.

Conclusions 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.

We have come full circle … QUESTIONS ?