CSE325 Computers and Sculpture

1. CSE325 Computersand Sculpture Prof. George Hart

2. 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

3. Intuitive uses of “symmetry” • left side = right side • Human body or face • n-fold rotation • Flower petals • Other ways?

4. 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, …

5. 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.

7. 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.

8. Wallpaper Groups o 2222 xx ** *2222 22*

9. Wallpaper Groups *442 x* 22x 2*22 442 4*2

10. Wallpaper Groups 333 *333 3*3 Images by Xah Lee 632 *632

11. 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

12. Roll up a Frieze into a Cylinder

13. 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

14. Symmetries of cube = Symmetries of octahedron In “dual position” symmetry axes line up

15. 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

16. 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

17. 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.

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

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

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

21. People

22. Candy BoxM.C. Escher

23. Sphere with FishM.C. Escher, 1940

24. Carlo Sequin, after Escher

25. Polyhedron with FlowersM.C. Escher, 1958

26. Carlo Sequin, after Escher

27. Sphere with Angels and DevilsM.C. Escher, 1942

28. Carlo Sequin, after Escher

29. M.C. Escher

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

31. The following slides are borrowed from Carlo Sequin

32. Escher Sphere Construction Kit Jane YenCarlo SéquinUC BerkeleyI3D 2001 [1] M.C. Escher, His Life and Complete Graphic Work

33. Introduction • M.C. Escher • graphic artist & print maker • myriad of famous planar tilings • why so few 3D designs? [2] M.C. Escher: Visions of Symmetry

34. Spherical Tilings • Spherical Symmetry is difficult • Hard to understand • Hard to visualize • Hard to make the final object [1]

35. 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

36. [1] Interface Design • How can we make the system intuitive and easy to use? • What is the best way to communicate how spherical symmetry works?

37. tetrahedron octahedron cube dodecahedron icosahedron R3 R5 R3 R3 R5 R2 Spherical Symmetry • The Platonic Solids

38. 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

39. 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 2 Tile 1 Tile 2

40. Modifying the Tile • Insert and move boundary points • system automatically updates the tile based on symmetry • Add interior detail points

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

42. 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

43. Video

44. 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

45. FDM Fabrication moving head Inside the FDM machine support material

46. Z-Corp Fabrication infiltration de-powdering

47. Results FDM

48. Results FDM | Z-Corp

49. Results FDM | Z-Corp

50. Results Z-Corp