Escher Sphere Construction Kit

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

• Spherical Symmetry is difficult
• Hard to understand
• Hard to visualize
• Hard to make the final object

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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

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Interface Design

• How can we make the system intuitive and easy to
  use?
• What is the best way to communicate how spherical
  symmetry works?

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Spherical Symmetry

• The Platonic Solids

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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
R2
Tile 1
Tile 2
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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)
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Tile 2
Tile 1
Tile 2
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Modifying the Tile

• Insert and move boundary points (blue)
• system automatically updates all tiles based on
  symmetry
• Add interior detail points (pink)

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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

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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

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Video
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Fabrication Issues

• Many kinds of rapid prototyping technologies . .
  .
• we use two types of layered manufacturing

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FDM Fabrication
moving head
Inside the FDM machine
support material
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Z-Corp Fabrication
infiltration
de-powdering
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Results
FDM
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Results
FDM Z-Corp
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Results
FDM Z-Corp
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Results
Z-Corp
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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)

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Introduction to Tiling

• Planar Tiling
• Start with a shape that tiles the plane
• Modify the shape using translation, rotation,
  glides, or mirrors
• Example

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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
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Introduction to Tiling

• Tetrahedral Symmetry - a closer look
• 24 elements E, 8C3, 3C2, 6sd, 6S4

2-Fold Rotation
3-Fold Rotation
Identity
90 C2 Inversion (i)
Improper Rotation
Mirror
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Introduction to Tiling

• What do the tiles look like?

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Introduction to Tiling

• Rotational Symmetry Only
• 12 elements E, 8C3, 3C2

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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, 6sd 3) double
tetrahedron 24 elem E, 8C3, 3C2,
i, 8S4, 3sd 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, 6sd, 6sd 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, 15s
Platonic Solids
With Duals