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