Escher Sphere Construction Kit

1. Escher Sphere Construction Kit Jane YenCarlo SéquinUC BerkeleyI3D 2001 [1] M.C. Escher, His Life and Complete Graphic Work

2. 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

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

4. 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

5. [1] Interface Design • How can we make the system intuitive and easy to use? • What is the best way to communicate how spherical symmetry works?

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

7. R3 R3 R2 R2 R2 R3 R3 R3 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 R2 Tile 2 Tile 1

8. 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 2 Tile 1 Tile 2

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

10. 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

11. 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

12. Video

13. 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 Fabrication Issues • Many kinds of rapid prototyping technologies . . . • we use two types of layered manufacturing:

14. FDM Fabrication moving head Inside the FDM machine support material

15. Z-Corp Fabrication infiltration de-powdering

16. Results FDM

17. Results FDM | Z-Corp

18. Results FDM | Z-Corp

19. Results Z-Corp

20. 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)

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

23. 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

24. C2 S4 C3 E sd Introduction to Tiling • Tetrahedral Symmetry - a closer look • 24 elements: {E, 8C3, 3C2, 6sd, 6S4} 2-Fold Rotation 3-Fold Rotation Identity 90° C2 + Inversion (i) Improper Rotation Mirror

25. M C2 C3 C3 C2 C2 C3 C3 C2 Introduction to Tiling • What do the tiles look like?

26. C3 C2 C3 C2 C2 C3 C3 C3 C2 Introduction to Tiling • Rotational Symmetry Only • 12 elements: {E,8C3, 3C2}

27. 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/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, 15s Platonic Solids With Duals