Créer une présentation
Télécharger la présentation

Télécharger la présentation
## Bridges 2008, Leeuwarden

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Bridges 2008, Leeuwarden**Carlo H. Séquin EECS Computer Science Division University of California, Berkeley Intricate IsohedralTilingsof 3D Euclidean Space**My Fascination with Escher Tilings**in the plane on the sphere on the torus M.C. Escher Jane Yen, 1997 Young Shon, 2002**My Fascination with Escher Tilings**• on higher-genus surfaces: London Bridges 2006 • What next ?**A fascinating intellectual excursion !**Celebrating the Spirit of M.C. Escher Try to do Escher-tilings in 3D …**A very large domain**• keep it somewhat limited**Monohedral vs. Isohedral**monohedral tiling isohedral tiling In an isohedral tiling any tile can be transformed to any other tile location by mapping the whole tiling back onto itself.**Still a Large Domain! Outline**• Genus 0 • Modulated extrusions • Multi-layer tiles • Metamorphoses • 3D Shape Editing • Genus 1: “Toroids” • Tiles of Higher Genus • Interlinked Knot-Tiles**How to Make an “Escher Tiling”**• Start from a regular tiling • Distort all equivalent edges in the same way**Genus 0: Simple Extrusions**• Start from one of Escher’s 2D tilings … • Add 3rd dimension by extruding shape.**Extruded “2.5D” Fish-Tiles**Isohedral Fish-Tiles Go beyond 2.5D !**Modulated Extrusions**• Do something with top and bottom surfaces ! Tailor the surface height before extrusion.**Three tiles overlaid**Manufactured Tiles (FDM) Red part is viewed from the bottom**Offset (Shifted) Overlay**• Let thick and thin areas complement each other: • RED = Thick areas;BLUE = THIN areas;**Shift Fish Outline to Desired Position**• CAD tool calculates intersections with underlying height map of repeated fish tiles.**As QuickSlice sees the shape …**3D Shape is Saved in .STL Format**Fabricated Tiles …**Top and bottom view Snug fit in the plane …**Adding Tiles in a 2nd Layer**• Snug fit also in the third dimension !**Building Fish in Discrete Layers**• How would these tiles fit together ? need to fill 2D plane in each layer ! • How to turn these shapes into isohedral tiles ? selectively glue together pieces on individual layers.**M. Goerner’s Tile**• Glue together elements from two subsequent layers.**Escher Night and Day**• Inspiration: Escher’s wonderful shape transformations (more by C. Kaplan…)**M.C. Escher: Metamorphosis**• Do similar “morph”-transformation in the 3rd dim.**Bird Fish**• A sweep-morph from bird into fish … and back**“FishBird”-Tile Fills 3D Space**1 red + 1 yellow isohedral tile**True 3D Tiles**• No preferential (special) editing direction. • Need a new CAD tool ! • Do in 3D what Escher did in 2D:modify the fundamental domain of a chosen tiling lattice**A 3D Escher Tile Editor**• Start with truncated octahedron cell of the BCC lattice. • Each cell shares one face with 14 neighbors. • Allow arbitrary distortions and individual vertex moves.**BCC Cell: Editing Result**• A fish-like tile shape that tessellates 3D space**Another Fundamental Cell**• Based on densest sphere packing. • Each cell has 12 neighbors. • Symmetrical form is the rhombic dodecahedron. • Add edge- and face-mid-points to yield 3x3 array of face vertices,making them quadratic Bézier patches.**Cell 2: Editing Result**• Can yield fish-like shapes • Need more editing capabilities to add details …**Corresponding vertices will follow !**Can select and drag individual vertices Adam Megacz’ Compound Cell Editor “Hammerhead” starting configuration**Final Edited Shape**• “Butterfly-Stingray” by Adam Megacz**and between the planes!**The Fabricated Tiles … Snug fit in the plane …**Lessons Learned:**• To make such a 3D editing tool is hard. • To use it to make good 3D tile designsis tedious and difficult. • Some vertices are shared by 4 cells, and thus show up 4 times on the cell-boundary; editing the front messes up back (and some sides!). • Can we let aprogramdo the editing ?**Iterative Shape Approximation**A closest matching shape is found among the 93 possible marked isohedral tilings; That cell is then adaptively distorted to match the desired goal shape as close as possible. • Try simulated annealing to find isohedral shape:“Escherization,” Kaplan and Salesin, SIGGRAPH 2000).**“Escherization” Resultsby Kaplan and Salesin, 2000**• Two different isohedral tilings.**Towards 3D Escherization**• The basic cell, based on a rhombic dodecahedron • Each cell has 12 direct neighbors**The Goal Shape**• Designed in a separate CAD program**Subdivided and partially annealed 3D fish tile**Simulated Annealing in Action • Basic cell and goal shape (wire frame)**The Final Result**• made on a Fused Deposition Modeling Machine, • then hand painted.**More “Sim-Fish”**• At different resolutions**Part II: Tiles of Genus > 0**• In 3D you can interlink tiles topologically !**Genus 1: Toroids**• An assembly of 4-segment rings,based on the BCC lattice (Séquin, 1995)**Toroidal Tiles,Variations**12 F Based on cubic lattice 24 facets 16 F**Square Wire Frames in BCC Lattice**• Tiles are approx. Voronoi regions around wire loops**Diamond Lattice & “Triamond” Lattice**• We can do the same with two other lattices !**Diamond Lattice**SLS model by George Hart