jane yen carlo séquin uc berkeley i3d 2001 [1] m.c. escher, his life and complete graphic work...
TRANSCRIPT
Jane YenCarlo Séquin
UC Berkeley
I3D 2001
[1] M.C. Escher, His Life and Complete Graphic Work
Escher Sphere Construction Kit
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
Spherical Tilings
Spherical Symmetry is difficult– Hard to understand– Hard to visualize– Hard to make the final object
[1]
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?
[1]
Spherical Symmetry
The Platonic Solids
tetrahedron octahedron cube dodecahedron icosahedron
R3 R5 R5R3 R3 R2
How the Program Works
Choose a symmetry based on a Platonic solid Choose an initial tiling pattern to edit
= good place to start . . .
Example: Tetrahedron:
2 different tiles:
R3
R2R3
R2
R3
R3
R3
R2
Tile 1 Tile 2
R3
R2
Initial Tiling Pattern+ Easy to understand consequences of moving points+ Guarantees proper overall tiling~ Requires user to select the “right” initial tile – This can only make monohedral tiles (one single type)
[2]
Tile 1 Tile 2 Tile 2
Modifying the Tile Insert and move boundary points (blue)
– system automatically updates all tiles based on symmetry
Add interior detail points (pink)
Adding Bas-Relief Stereographically project tile and triangulate
Radial offsets can be given to points– individually or in groups– separate mode from editing boundary points
Creating a Solid The surface is extruded radialy
– inward or outward extrusion; with a spherical or detailed base
Output in a format for free-form fabrication– individual tiles, or entire ball
Fabrication Issues Many kinds of rapid prototyping technologies . . .
– we use two types of layered manufacturing:
Fused Deposition Modeling (FDM) Z-Corp 3D Color Printer
- parts made of plastic - plaster powder glued together - each part is a solid color - parts can have multiple colors assembly
Conclusions Intuitive Conceptual Model
– 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 (and see) 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 “height” manipulation of vertices
Future Work– predefined color symmetry– injection molded parts (puzzles)– tessellating over arbitrary shapes (any genus)
Introduction to Tiling
Planar Tiling– Start with a shape that tiles the plane
– Modify the shape using translation, rotation, glides, or mirrors
– Example:
Introduction to Tiling
Spherical Tiling - a first try– Start with a shape that tiles the sphere (platonic solid)
– Modify the face shape using rotation or mirrors
– Project the platonic solid onto the sphere
– Example:
• icosahedron• 3-fold symmetric triangle faces
tetrahedron octahedron cube dodecahedron icosahedron
Introduction to Tiling
Tetrahedral Symmetry - a closer look• 24 elements: {E, 8C3, 3C2, 6d, 6S4}
C2C3E
d
Identity3-Fold Rotation
2-Fold Rotation
MirrorImproper Rotation
S4
90° C2+
Inversion (i)
Introduction to Tiling
Rotational Symmetry Only• 12 elements: {E, 8C3, 3C2}
C3
C3
C3
C3
C2
C2C2
C2 C3
Introduction to Tiling
Spherical Symmetry - defined by 7 groups
1) oriented tetrahedron 12 elem: E, 8C3, 3C2
2) straight tetrahedron 24 elem: E, 8C3, 3C2, 6S4, d
3) double tetrahedron 24 elem: E, 8C3, 3C2, i, 8S4, d
4) oriented octahedron/cube 24 elem: E, 8C3, 6C2, 6C4, 3C42
5) straight octahedron/cube 48 elem: E, 8C3, 6C2, 6C4, 3C42, i, 8S6, 6S4, d,
d
6) oriented icosa/dodeca-hedron 60 elem: E, 20C3, 15C2, 12C5, 12C52
7) straight icosa/dodeca-hedron 120 elem: E, 20C3, 15C2, 12C5, 12C52, i, 20S6,
12S10, 12S103,
Platonic Solids
With Duals