2Symmetry Intuitive notion – mirrors, rotations, … Mathematical concept — set of transformationsPossible 2D and 3D symmetriesSculpture examples:M.C. Escher sculptureCarlo Sequin’s EscherBall programConstructions this week based on symmetry
3Intuitive uses of “symmetry” left side = right sideHuman body or facen-fold rotationFlower petalsOther ways?
4Mathematical Definition Define geometric transformations:reflection, rotation, translation (“slide”),glide reflection (“slide and reflect”), identity, …A symmetry is a transformationThe symmetries of an object are the set of transformations which leave object looking unchangedThink of symmetries as axes, mirror lines, …
5Frieze 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.
7Wallpaper Groups Include 2 directions of translation Might have 2-fold, 3-fold, 6-fold rotations, mirrors, and glide-reflections17 possibilitiesSeveral standard notations. The following slides show the “orbifold” notation of John Conway.
10Wallpaper Groups333*3333*3Images by Xah Lee632*632
113D SymmetryThree 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 friezeSymmetry of pyramids and prismsEach of the seven can be 2-fold, 3-fold, 4-fold,…7 correspond to regular polyhedra
13Seven Polyhedra Groups Octahedral, with 0 or 9 mirrorsIcosahedral, with 0 or 15 mirrorsTetrahedral, with 0, 3, or 6 mirrorsCube and octahedron have same symmetryDodecahedron and icosahedron have same symmetry
14Symmetries of cube = Symmetries of octahedron In “dual position” symmetry axes line up
15Cube Rotational Symmetry Axes of rotation:Three 4-fold — through opposite face centersfour 3-fold — through opposite verticessix 2-fold — through opposite edge midpointsCount the Symmetry transformations:1, 2, or 3 times 90 degrees on each 4-fold axis1 or 2 times 120 degrees on each 3-fold axis180 degrees on each 2-fold axisIdentity transformation= 24
16Cube Rotations may or may not Come with Mirrors If any mirrors, then 9 mirror planes.If put “squiggles” on each face, then 0 mirrors
17Icosahedral = Dodecahedral Symmetry Six 5-fold axes. Ten 3-fold axes. Fifteen 2-fold axesThere are 15 mirror planes. Or squiggle each face for 0 mirrors.
18Tetrahedron Rotations Four 3-fold axes (vertex to opposite face center). Three 2-fold axes.
19Tetrahedral Mirrors Regular tetrahedron has 6 mirrors (1 per edge) “Squiggled” tetrahedron has 0 mirrors.“Pyrite symmetry” has tetrahedral rotations but 3 mirrors:
20Symmetry in Sculpture People Sculpture (G. Hart) Sculpture by M.C. EscherReplicas of Escher by Carlo SequinOriginal designs by Carlo Sequin
30Construction this Week WormballsPipe-cleaner constructionsBased on one line in a 2D tessellation
31The following slides are borrowed from Carlo Sequin
32Escher Sphere Construction Kit Jane Yen Carlo Séquin UC Berkeley I3D 2001 M.C. Escher, His Life and Complete Graphic Work
33Introduction M.C. Escher graphic artist & print maker myriad of famous planar tilingswhy so few 3D designs? M.C. Escher: Visions of Symmetry
34Spherical Tilings Spherical Symmetry is difficult Hard to understand Hard to visualizeHard to make the final object
35Our GoalDevelop a system to easily design and manufacture “Escher spheres” - spherical balls composed of tilesprovide visual feedbackguarantee that the tiles join properlyallow for bas-reliefoutput for manufacturing of physical models
36Interface Design How can we make the system intuitive and easy to use? What is the best way to communicate how spherical symmetry works?
37Spherical Symmetry The Platonic Solids R3 R5 tetrahedron octahedron cubedodecahedronicosahedronR3R2R5R3
38How the Program Works Choose a symmetry based on a Platonic solid Choose an initial tiling pattern to editstarting placeExample:TetrahedronR3R2R3R2R3R2Tile 1Tile 2
39Initial Tiling Pattern + easier to understand consequences of moving points+ guarantees proper tiling~ requires user to select the “right” initial tile- can only make monohedral tilesTile 1Tile 2Tile 2
40Modifying the Tile Insert and move boundary points system automatically updates the tile based on symmetryAdd interior detail points
41Adding Bas-Relief Stereographically projected and triangulated Radial offsets can be given to pointsindividually or in groupsseparate mode from editing boundary points
42Creating a Solid The surface is extruded radially inward or outward extrusion, spherical or detailed baseOutput in a format for free-form fabricationindividual tiles or entire ball
44Fabrication Issues Many kinds of manufacturing technology we use two types based on a layer-by-layer approachFused Deposition Modeling (FDM) Z-Corp 3D Color Printer- parts made of plastic starch powder glued togethereach part is a solid color parts can have multiple colorsassembly
45FDM FabricationmovingheadInside the FDM machinesupportmaterial
51Conclusions Intuitive Conceptual Model Editing in Context symmetry groups have little meaning to userneed to give the user an easy to understand starting placeEditing in Contextneed to see all the tiles togetherneed to edit the tile on the sphereediting in the plane is not good enough (distortions)Part Fabricationneed limitations so that designs can be manufacturedradial manipulationFuture Workpredefined color symmetryinjection molded parts (puzzles)tessellating over arbitrary shapes (any genus)