Jane Yen Carlo Séquin UC Berkeley I3D 2001 [1] M.C. Escher, His Life and Complete Graphic Work Escher Sphere Construction Kit

Introduction n 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 n Spherical Symmetry is difficult –Hard to understand –Hard to visualize –Hard to make the final object [1]

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

Spherical Symmetry n The Platonic Solids tetrahedronoctahedroncubedodecahedronicosahedron R3 R5 R3 R2

How the Program Works n Choose a symmetry based on a Platonic solid n Choose an initial tiling pattern to edit = good place to start... n Example: Tetrahedron: 2 different tiles: R3 R2 R3 R2 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

Modifying the Tile n Insert and move boundary points (blue) –system automatically updates all tiles based on symmetry n Add interior detail points (pink)

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

Creating a Solid n The surface is extruded radialy –inward or outward extrusion; with a spherical or detailed base n Output in a format for free-form fabrication –individual tiles, or entire ball

Video

Fabrication Issues n 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 support material moving head Inside the FDM machine

Z-Corp Fabrication infiltration de-powdering

Results FDM

Results FDM | Z-Corp

Results FDM | Z-Corp

Results Z-Corp

Conclusions n Intuitive Conceptual Model –symmetry groups have little meaning to user –need to give the user an easy to understand starting place n 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) n Part Fabrication –need limitations so that designs can be manufactured radial “height” manipulation of vertices n Future Work –predefined color symmetry –injection molded parts (puzzles) –tessellating over arbitrary shapes (any genus)

Introduction to Tiling n Planar Tiling –Start with a shape that tiles the plane –Modify the shape using translation, rotation, glides, or mirrors –Example:

Introduction to Tiling n 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 tetrahedronoctahedroncubedodecahedronicosahedron

Introduction to Tiling n Tetrahedral Symmetry - a closer look 24 elements: { E, 8C 3, 3C 2, 6  d, 6S 4 } C2C2 C3C3 E dd Identity 3-Fold Rotation 2-Fold Rotation Mirror Improper Rotation S4S4 90° C2 + Inversion (i)

Introduction to Tiling n What do the tiles look like? C2C2 M C3C3 C3C3 C3C3 C2C2 C2C2 C2C2 C3C3

Introduction to Tiling n Rotational Symmetry Only 12 elements: { E, 8C 3, 3C 2 } C3C3 C3C3 C3C3 C3C3 C2C2 C2C2 C2C2 C2C2 C3C3

Introduction to Tiling n Spherical Symmetry - defined by 7 groups 1) oriented tetrahedron 12 elem: E, 8C 3, 3C 2 2) straight tetrahedron 24 elem: E, 8C 3, 3C 2, 6S 4,  d 3) double tetrahedron 24 elem: E, 8C 3, 3C 2, i, 8S 4,  d 4) oriented octahedron/cube 24 elem: E, 8C 3, 6C 2, 6C 4, 3C 4 2 5) straight octahedron/cube 48 elem: E, 8C 3, 6C 2, 6C 4, 3C 4 2, i, 8S 6, 6S 4,  d,  d 6) oriented icosa/dodeca-hedron 60 elem: E, 20C 3, 15C 2, 12C 5, 12C 5 2 7) straight icosa/dodeca-hedron 120 elem: E, 20C 3, 15C 2, 12C 5, 12C 5 2, i, 20S 6, 12S 10, 12S 10 3,  Platonic Solids With Duals