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Explore the world of Escher spheres and create mesmerizing designs with this innovative system. Easily visualize, edit, and manufacture spherical balls made of tiles. This tool ensures proper tile joining and bas-relief output for physical models. Interface design focuses on intuitive user experience for editing and creating solids. Drive your creativity with the possibilities of free-form fabrication and multiple color selections. Discover the future potential of injecting molded parts and tessellating over arbitrary shapes.
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Escher Sphere Construction Kit Jane YenCarlo SéquinUC BerkeleyI3D 2001 [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 tiles • provide visual feedback • guarantee that the tiles join properly • allow for bas-relief • output for manufacturing of physical models
[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?
tetrahedron octahedron cube dodecahedron icosahedron R3 R5 R3 R3 R5 R2 Spherical Symmetry • The Platonic Solids
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
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
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
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:
FDM Fabrication moving head Inside the FDM machine support material
Z-Corp Fabrication infiltration de-powdering
Results FDM
Results FDM | Z-Corp
Results FDM | Z-Corp
Results Z-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
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
M C2 C3 C3 C2 C2 C3 C3 C2 Introduction to Tiling • What do the tiles look like?
C3 C2 C3 C2 C2 C3 C3 C3 C2 Introduction to Tiling • Rotational Symmetry Only • 12 elements: {E,8C3, 3C2}
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