Jane YenCarlo Séquin

UC Berkeley

I3D 2001

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

Escher Sphere Construction Kit

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 tiles

– provide visual feedback– guarantee that the tiles join properly– allow for bas-relief– 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 R5R3 R3 R2

How the Program Works

Choose a symmetry based on a Platonic solid Choose an initial tiling pattern to edit

= good place to start . . .

Example: Tetrahedron:

2 different tiles:

R3

R2R3

R2

R3

R3

R3

R2

Tile 1 Tile 2

R3

R2

Initial Tiling Pattern+ Easy to understand consequences of moving points+ Guarantees proper overall tiling~ Requires user to select the “right” initial tile – This can only make monohedral tiles (one single type)

[2]

Tile 1 Tile 2 Tile 2

Modifying the Tile Insert and move boundary points (blue)

– system automatically updates all tiles based on symmetry

Add interior detail points (pink)

Adding Bas-Relief Stereographically project tile 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 radialy

– inward or outward extrusion; with a spherical or detailed base

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

Video

Fabrication Issues Many kinds of rapid prototyping technologies . . .

– we use two types of layered manufacturing:

Fused Deposition Modeling (FDM) Z-Corp 3D Color Printer

- parts made of plastic - plaster powder glued together - each part is a solid color - parts can have multiple colors assembly

FDM Fabrication

supportmaterial

movinghead

Inside the FDM machine

Z-Corp Fabrication

infiltrationde-powdering

ResultsFDM

ResultsFDM | Z-Corp

ResultsFDM | Z-Corp

ResultsZ-Corp

Conclusions Intuitive Conceptual Model

– symmetry groups have little meaning to user– need to give the user an easy to understand starting place

Editing in Context– need to see all the tiles together– need to edit (and see) the tile on the sphere

• editing in the plane is not good enough (distortions)

Part Fabrication– need limitations so that designs can be manufactured

• radial “height” manipulation of vertices

Future Work– predefined color symmetry– injection molded parts (puzzles)– tessellating over arbitrary shapes (any genus)

Introduction to Tiling

Planar Tiling– Start with a shape that tiles the plane

– Modify the shape using translation, rotation, glides, or mirrors

– Example:

Introduction to Tiling

Spherical Tiling - a first try– Start with a shape that tiles the sphere (platonic solid)

– Modify the face shape using rotation or mirrors

– Project the platonic solid onto the sphere

– Example:

• icosahedron• 3-fold symmetric triangle faces

tetrahedron octahedron cube dodecahedron icosahedron

Introduction to Tiling

Tetrahedral Symmetry - a closer look• 24 elements: {E, 8C3, 3C2, 6d, 6S4}

C2C3E

d

Identity3-Fold Rotation

2-Fold Rotation

MirrorImproper Rotation

S4

90° C2+

Inversion (i)

Introduction to Tiling

What do the tiles look like?

C2M

C3 C3

C3

C2

C2C2

C3

Introduction to Tiling

Rotational Symmetry Only• 12 elements: {E, 8C3, 3C2}

C3

C3

C3

C3

C2

C2C2

C2 C3

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, d

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

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, d,

d

6) oriented icosa/dodeca-hedron 60 elem: E, 20C3, 15C2, 12C5, 12C52

7) straight icosa/dodeca-hedron 120 elem: E, 20C3, 15C2, 12C5, 12C52, i, 20S6,

12S10, 12S103,

Platonic Solids

With Duals