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CSE325 Computers and Sculpture Prof. George Hart

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Symmetry Intuitive notion – mirrors, rotations, … Mathematical concept set of transformations Possible 2D and 3D symmetries Sculpture examples: –M.C. Escher sculpture –Carlo Sequins EscherBall program Constructions this week based on symmetry

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Intuitive uses of symmetry left side = right side –Human body or face n-fold rotation –Flower petals Other ways?

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Mathematical Definition Define geometric transformations: –reflection, rotation, translation (slide), –glide reflection (slide and reflect), identity, … A symmetry is a transformation The symmetries of an object are the set of transformations which leave object looking unchanged Think of symmetries as axes, mirror lines, …

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Frieze Patterns Imagine as infinitely long. Each frieze has translations. A smallest translation generates all translations by repetition and inverse. Some have vertical mirror lines. Some have a horizontal mirror. Some have 2-fold rotations. Analysis shows there are exactly seven possibilities for the symmetry.

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Wallpaper Groups Include 2 directions of translation Might have 2-fold, 3-fold, 6-fold rotations, mirrors, and glide-reflections 17 possibilities Several standard notations. The following slides show the orbifold notation of John Conway.

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Wallpaper Groups o2222 ***222222* xx

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Wallpaper Groups 22x x* *442 4*2 4422*22

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Wallpaper Groups 333 * *3*333 Images by Xah Lee

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3D Symmetry Three translation directions give the 230 crystallographic space groups of infinite lattices. If no translations, center is fixed, giving the 14 types of polyhedral groups: 7 families correspond to a rolled-up frieze –Symmetry of pyramids and prisms –Each of the seven can be 2-fold, 3-fold, 4-fold,… 7 correspond to regular polyhedra

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Roll up a Frieze into a Cylinder

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Seven Polyhedra Groups Octahedral, with 0 or 9 mirrors Icosahedral, with 0 or 15 mirrors Tetrahedral, with 0, 3, or 6 mirrors Cube and octahedron have same symmetry Dodecahedron and icosahedron have same symmetry

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Symmetries of cube = Symmetries of octahedron In dual position symmetry axes line up

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Cube Rotational Symmetry Axes of rotation: –Three 4-fold through opposite face centers –four 3-fold through opposite vertices –six 2-fold through opposite edge midpoints Count the Symmetry transformations: –1, 2, or 3 times 90 degrees on each 4-fold axis –1 or 2 times 120 degrees on each 3-fold axis –180 degrees on each 2-fold axis –Identity transformation – = 24

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Cube Rotations may or may not Come with Mirrors If any mirrors, then 9 mirror planes. If put squiggles on each face, then 0 mirrors

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Icosahedral = Dodecahedral Symmetry Six 5-fold axes. Ten 3-fold axes. Fifteen 2-fold axes There are 15 mirror planes. Or squiggle each face for 0 mirrors.

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Tetrahedron Rotations Four 3-fold axes (vertex to opposite face center). Three 2-fold axes.

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Tetrahedral Mirrors Regular tetrahedron has 6 mirrors (1 per edge) Squiggled tetrahedron has 0 mirrors. Pyrite symmetry has tetrahedral rotations but 3 mirrors:

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Symmetry in Sculpture People Sculpture (G. Hart) Sculpture by M.C. Escher Replicas of Escher by Carlo Sequin Original designs by Carlo Sequin

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People

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Candy Box M.C. Escher

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Sphere with Fish M.C. Escher, 1940

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Carlo Sequin, after Escher

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Polyhedron with Flowers M.C. Escher, 1958

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Carlo Sequin, after Escher

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Sphere with Angels and Devils M.C. Escher, 1942

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Carlo Sequin, after Escher

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M.C. Escher

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Construction this Week Wormballs –Pipe-cleaner constructions –Based on one line in a 2D tessellation

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The following slides are borrowed from Carlo Sequin

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

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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

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

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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

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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]

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Spherical Symmetry n The Platonic Solids tetrahedronoctahedroncubedodecahedronicosahedron R3 R5 R3 R2

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How the Program Works n Choose a symmetry based on a Platonic solid n Choose an initial tiling pattern to edit –starting place n Example: Tetrahedron R3 R2 R3 R2 R3 R2 Tile 1 Tile 2 R3 R2

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Initial Tiling Pattern + easier to understand consequences of moving points + guarantees proper tiling ~ requires user to select the right initial tile - can only make monohedral tiles [2] Tile 1 Tile 2

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Modifying the Tile n Insert and move boundary points –system automatically updates the tile based on symmetry n Add interior detail points

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

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

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Video

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Fabrication Issues n Many kinds of manufacturing technology –we use two types based on a layer-by-layer approach Fused Deposition Modeling (FDM) Z-Corp 3D Color Printer - parts made of plastic - starch powder glued together - each part is a solid color - parts can have multiple colors assembly

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FDM Fabrication support material moving head Inside the FDM machine

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Z-Corp Fabrication infiltration de-powdering

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Results FDM

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Results FDM | Z-Corp

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Results FDM | Z-Corp

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Results Z-Corp

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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 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 manipulation n Future Work –predefined color symmetry –injection molded parts (puzzles) –tessellating over arbitrary shapes (any genus)

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