1. CSE325 Computers and 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
= 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 Box M.C. Escher

23. Sphere with Fish M.C. Escher, 1940

24. Carlo Sequin, after Escher

25. Polyhedron with Flowers M.C. Escher, 1958

26. Carlo Sequin, after Escher

27. Sphere with Angels and Devils M.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 Yen Carlo Séquin UC Berkeley I3D 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. Interface Design How can we make the system intuitive and easy to use?
What is the best way to communicate how spherical symmetry works?
[1]

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

38. 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 R3 R2 R3 R2
Tile 1
Tile 2

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 1
Tile 2
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. Fabrication Issues 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

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

46. Z-Corp Fabrication
de-powdering
infiltration

47. Results
FDM

48. Results
FDM | Z-Corp

49. Results
FDM | Z-Corp

50. Results
Z-Corp

51. Conclusions Intuitive Conceptual Model Editing in Context
symmetry groups have little meaning to user
need to give the user an easy to understand starting place
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)
Part Fabrication
need limitations so that designs can be manufactured
radial manipulation
Future Work
predefined color symmetry
injection molded parts (puzzles)
tessellating over arbitrary shapes (any genus)