CSE325 Computersand Sculpture Prof. George Hart
Symmetry • Intuitive notion – mirrors, rotations, … • Mathematical concept — set of transformations • Possible 2D and 3D symmetries • Sculpture examples: • M.C. Escher sculpture • Carlo Sequin’s EscherBall program • Constructions this week based on symmetry
Intuitive uses of “symmetry” • left side = right side • Human body or face • n-fold rotation • Flower petals • Other ways?
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, …
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.
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.
Wallpaper Groups o 2222 xx ** *2222 22*
Wallpaper Groups *442 x* 22x 2*22 442 4*2
Wallpaper Groups 333 *333 3*3 Images by Xah Lee 632 *632
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
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
Symmetries of cube = Symmetries of octahedron In “dual position” symmetry axes line up
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 • 9 + 8 + 6 + 1 = 24
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
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.
Tetrahedron Rotations Four 3-fold axes (vertex to opposite face center). Three 2-fold axes.
Tetrahedral Mirrors • Regular tetrahedron has 6 mirrors (1 per edge) • “Squiggled” tetrahedron has 0 mirrors. • “Pyrite symmetry” has tetrahedral rotations but 3 mirrors:
Symmetry in Sculpture • People Sculpture (G. Hart) • Sculpture by M.C. Escher • Replicas of Escher by Carlo Sequin • Original designs by Carlo Sequin
Construction this Week • Wormballs • Pipe-cleaner constructions • Based on one line in a 2D tessellation
The following slides are borrowed from Carlo Sequin
Escher Sphere Construction Kit Jane YenCarlo SéquinUC BerkeleyI3D 2001  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?  M.C. Escher: Visions of Symmetry
Spherical Tilings • Spherical Symmetry is difficult • Hard to understand • Hard to visualize • Hard to make the final object 
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
 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 • starting place • Example: Tetrahedron R3 R2 Tile 2 Tile 1
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  Tile 2 Tile 1 Tile 2
Modifying the Tile • Insert and move boundary points • system automatically updates the tile based on symmetry • Add interior detail points
Adding Bas-Relief • Stereographically projected and triangulated • Radial offsets can be given to points • individually or in groups • separate mode from editing boundary points
Creating a Solid • The surface is extruded radially • inward or outward extrusion, 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 - starch powder glued together • each part is a solid color - parts can have multiple colors • assembly Fabrication Issues • Many kinds of manufacturing technology • we use two types based on a layer-by-layer approach
FDM Fabrication moving head Inside the FDM machine support material
Z-Corp Fabrication infiltration de-powdering
Results FDM | Z-Corp
Results FDM | Z-Corp